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Research ArticleDirected Bee Colony Optimization Algorithm toSolve the Nurse Rostering Problem
M Rajeswari1 J Amudhavel2 Sujatha Pothula1 and P Dhavachelvan1
1Department of CSE Pondicherry University Puducherry India2Department of CSE KL University Andhra Pradesh India
Correspondence should be addressed to M Rajeswari rajirajeswari18gmailcom
Received 26 October 2016 Revised 6 January 2017 Accepted 1 March 2017 Published 4 April 2017
Academic Editor Reinoud Maex
Copyright copy 2017 M Rajeswari et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The Nurse Rostering Problem is an NP-hard combinatorial optimization scheduling problem for assigning a set of nurses toshifts per day by considering both hard and soft constraints A novel metaheuristic technique is required for solving NurseRostering Problem (NRP) This work proposes a metaheuristic technique called Directed Bee Colony Optimization Algorithmusing the Modified Nelder-Mead Method for solving the NRP To solve the NRP the authors used a multiobjective mathematicalprogramming model and proposed a methodology for the adaptation of a Multiobjective Directed Bee Colony Optimization(MODBCO) MODBCO is used successfully for solving the multiobjective problem of optimizing the scheduling problems ThisMODBCO is an integration of deterministic local search multiagent particle system environment and honey bee decision-makingprocess The performance of the algorithm is assessed using the standard dataset INRC2010 and it reflects many real-world caseswhich vary in size and complexity The experimental analysis uses statistical tools to show the uniqueness of the algorithm onassessment criteria
1 Introduction
Metaheuristic techniques especially the Bee Colony Opti-mization Algorithm can be easily adapted to solve a largernumber of NP-hard combinatorial optimization problems bycombining other methodsThe metaheuristic method can bedivided into local search methods and global search meth-ods Local search methods such as tabu search simulatedannealing and the Nelder-Mead Methods are used to exploitsearch space of the problem while global search methodssuch as scatter search genetic algorithms and Bee ColonyOptimization focus on the exploration of the search spacearea [1] Exploitation is the process of intensifying the searchspace thismethod repeatedly restarts searching for each timefrom a different initial solution Exploration is the processof diversifying the search space to evade trapping in a localoptimum A hybrid method is used to obtain a balancebetween exploration and exploitation by introducing localsearch within global search to obtain a robust solution for the
NRP In a previous study the genetic algorithm was chosenfor global search and simulated annealing for a local searchto solve the NRP in [2]
In swarm intelligence the natural behavior of organismswill follow a simple basic rule to structure their environmentThe agents will not have any centralized structure to controlother individuals it uses the local interactions among theagents to determine the complex global behavior of the agents[3] Some of the inspired natural behavior of swarm intelli-gence comprises bird flocking ant colony fish schooling andanimal herding methods The various algorithms include theant colony optimization algorithm genetic algorithm andthe particle swarm optimization algorithm [4ndash6]The naturalforaging behavior of honey bees has inspired bee algorithmAll honey bees will start to collect nectar from various sitesaround their new hive and the process of finding out the bestnectar site is done by the group decision of honey bees Themode of communication among the honey bees is carried outby the process of the waggle dance to informhivemates about
HindawiComputational Intelligence and NeuroscienceVolume 2017 Article ID 6563498 26 pageshttpsdoiorg10115520176563498
2 Computational Intelligence and Neuroscience
the location of rich food sources Some of the algorithmswhich follow the waggle dance of communication performedby scout bees about the nectar site are bee system Bee ColonyOptimization [7] and Artificial Bee Colony [8]
TheDirected BeeColony (DBC)OptimizationAlgorithm[9] is inspired by the group decision-making process ofbee behavior for the selection of the nectar site The groupdecision process includes consensus and quorum methodsConsensus is the process of vote agreement and the votingpattern of the scouts is monitored The best nest site isselected once the quorum (threshold) value is reached Theexperimental result shows that the algorithm is robust andaccurate for generating the unique solutionThe contributionof this research article is the use of a hybrid Directed BeeColony Optimization with the Nelder-Mead Method foreffective local search The authors have adapted MODBCOfor solving multiobjective problems which integrate the fol-lowing processes At first a deterministic local searchmethodModifiedNelder-Mead is used to obtain the provisional opti-mal solutionThen a multiagent particle system environmentis used in the exploration and decision-making process forestablishing a new colony and nectar site selection Only fewhoney bees were active in the process of decision-making sothe energy conservation of the swarm is highly achievable
The Nurse Rostering Problem (NRP) is a staff schedulingproblem that intends to assign a set of nurses to workshifts to maximize hospital benefit by considering a setof hard and soft constraints like allotment of duty hourshospital regulations and so forth This nurse rostering is adelicate task of finding combinatorial solutions by satisfyingmultiple constraints [10] Satisfying the hard constraint ismandatory in any scheduling problem and a violation ofany soft constraints is allowable but penalized To achievean optimal global solution for the problem is impossible inmany cases [11] Many algorithmic techniques such as meta-heuristic method graph-based heuristics and mathematicalprogramming model have been proposed to solve automatedscheduling problems and timetabling problems over the lastdecades [12 13]
In this work the effectiveness of the hybrid algorithmis compared with different optimization algorithms usingperformancemetrics such as error rate convergence rate bestvalue and standard deviationThewell-known combinatorialscheduling problem NRP is chosen as the test bed toexperiment and analyze the effectiveness of the proposedalgorithm
This paper is organized as follows Section 2 presentsthe literature survey of existing algorithms to solve theNRP Section 3 highlights the mathematical model andthe formulation of hard and soft constraints of the NRPSection 4 explains the natural behavior of honey bees tohandle decision-making process and the Modified Nelder-Mead Method Section 5 describes the development of themetaheuristic approach and the effectiveness of the MOD-BCO algorithm to solve the NRP is demonstrated Section 6confers the computational experiments and the analysis ofresults for the formulated problem Finally Section 7 providesthe summary of the discussion and Section 8 will concludewith future directions of the research work
2 Literature Review
Berrada et al [19] considered multiple objectives to tacklethe nurse scheduling problem by considering various orderedsoft constraints The soft constraints are ordered based onpriority level and this determines the quality of the solutionBurke et al [20] proposed a multiobjective Pareto-basedsearch technique and used simulated annealing based on aweighted-sum evaluation function towards preferences and adominated-based evaluation function towards the Pareto setMany mathematical models are proposed to reduce the costand increase the performance of the taskThe performance ofthe problem greatly depends on the type of constraints used[21] Dowsland [22] proposed a technique of chain movesusing a multistate tabu search algorithm This algorithmexchanges the feasible and infeasible search space to increasethe transmission rate when the system gets disconnected Butthis algorithm fails to solve other problems in different searchspace instances
Burke et al [23] proposed a hybrid tabu search algorithmto solve the NRP in Belgian hospitals In their constraintsthe authors have added the previous roster along with hardand soft constraints To consider this they included heuris-tic search strategies in the general tabu search algorithmThis model provides flexibility and more user control Ahyperheuristic algorithm with tabu search is proposed forthe NRP by Burke et al [24] They developed a rule basedreinforcement learning which is domain specific but itchooses a little low-level heuristic to solve the NRP Theindirect genetic algorithm is problem dependent which usesencoding and decoding schemes with genetic operator tosolve NRP Burke et al [25] developed a memetic algorithmto solve the nurse scheduling problem and the authorshave compared memetic and tabu search algorithm Theexperimental result shows a memetic algorithm outperformswith better quality than the genetic algorithm and tabu searchalgorithm
Simulated annealing has been proposed to solve the NRPHadwan and Ayob [26] introduced a shift pattern approachwith simulated annealing The authors have proposed agreedy constructive heuristic algorithm to generate therequired shift patterns to solve the NRP at UKMMC (Univer-siti KebangsaanMalaysiaMedical Centre)Thismethodologywill reduce the complexity of the search space solutionto generate a roster by building two- or three-day shiftpatterns The efficiency of this algorithm was shown byexperimental results with respect to execution time per-formance considerations fairness and the quality of thesolution This approach was capable of handling all hard andsoft constraints and produces a quality roster pattern Sharifet al [27] proposed a hybridized heuristic approach withchanges in the neighborhood descent search algorithm tosolve the NRP at UKMMCThis heuristic is the hybridizationof cyclic schedule with noncyclic schedule They appliedrepairing mechanism which swaps the shifts between nursesto tackle the random shift arrangement in the solution Avariable neighborhood descent search algorithm (VNDS) isused to change the neighborhood structure using a localsearch and generate a quality duty roster In VNDS the first
Computational Intelligence and Neuroscience 3
neighborhood structure will reroster nurses to different shiftsand the second neighborhood structure will do repairingmechanism
Aickelin and Dowsland [28] proposed a technique forshift patterns they considered shift patterns with penaltypreferences and number of successive working days Theindirect genetic algorithm will generate various heuristicdecoders for shift patterns to reconstruct the shift roster forthe nurse A qualified roster is generated using decoderswith the help of the best permutations of nurses To generatebest search space solutions for the permutation of nursesthe authors used an adaptive iterative method to adjust theorder of nurses as scheduled one by one Asta et al [29] andAnwar et al [30] proposed a tensor-based hyperheuristic tosolve the NRP The authors tuned a specific group of datasetsand embedded a tensor-based machine learning algorithmA tensor-based hyperheuristic with memory managementis used to generate the best solution This approach isconsidered in life-long applications to extract knowledge anddesired behavior throughout the run time
Todorovic and Petrovic [31] proposed the Bee ColonyOptimization approach to solve the NRP all the unscheduledshifts are allocated to the available nurses in the constructivephase This algorithm combines the constructive move withlocal search to improve the quality of the solution For eachforward pass the predefined numbers of unscheduled shiftsare allocated to the nurses and discarded the solution withless improvement in the objective function The process ofintelligent reduction in neighborhood search had improvedthe current solution In construction phase unassigned shiftsare allotted to nurses and lead to violation of constraints tohigher penalties
Severalmethods have been proposed using the INRC2010dataset to solve the NRP the authors have consideredfive latest competitors to measure the effectiveness of theproposed algorithm Asaju et al [14] proposed Artificial BeeColony (ABC) algorithm to solve NRP This process is donein two phases at first heuristic based ordering of shift patternis used to generate the feasible solution In the second phaseto obtain the solution ABC algorithm is used In thismethodpremature convergence takes place and the solution getstrapped in local optima The lack of a local search algorithmof this process leads to yielding higher penalty Awadallah etal [15] developed a metaheuristic technique hybrid artificialbee colony (HABC) to solve the NRP In ABC algorithm theemployee bee phase was replaced by a hill climbing approachto increase exploitation process Use of hill climbing in ABCgenerates a higher value which leads to high computationaltime
The global best harmony search with pitch adjustmentdesign is used to tackle the NRP in [16] The author adaptedthe harmony search algorithm (HAS) in exploitation pro-cess and particle swarm optimization (PSO) in explorationprocess In HAS the solutions are generated based on threeoperator namely memory consideration random consider-ation and pitch adjustment for the improvisation processThey did two improvisations to solve the NRP multipitchadjustment to improve exploitation process and replaced ran-dom selectionwith global best to increase convergence speed
The hybrid harmony search algorithm with hill climbing isused to solve the NRP in [17] For local search metaheuristicharmony and hill climbing approach are used The memoryconsideration parameter in harmony is replaced by PSOalgorithm The derivative criteria will reduce the numberof iterations towards local minima This process considersmany parameters to construct the roster since improvisationprocess is to be at each iteration
Santos et al [18] used integer programming (IP) to solvetheNRP andproposedmonolith compact IPwith polynomialconstraints and variables The authors have used both upperand lower bounds for obtaining optimal cost They estimatedand improved lower bound values towards optimum and thismethod requires additional processing time
3 Mathematical Model
The NRP problem is a real-world problem at hospitals theproblem is to assign a predefined set of shifts (like S1-dayshift S2-noon shift S3-night shift and S4-Free-shift) of ascheduled period for a set of nurses of different preferencesand skills in each ward Figure 1 shows the illustrativeexample of the feasible nurse roster which consists of fourshifts namely day shift noon shift night shift and free shift(holiday) allocating five nurses over 11 days of scheduledperiod Each column in the scheduled table represents theday and the cell content represents the shift type allocatedto a nurse Each nurse is allocated one shift per day and thenumber of shifts is assigned based on the hospital contractsThis problem will have some variants on a number of shifttypes nurses nurse skills contracts and scheduling periodIn general both hard and soft constraints are considered forgenerating and assessing solutions
Hard constraints are the regulations which must besatisfied to achieve the feasible solution They cannot beviolated since hard constraints are demanded by hospitalregulations The hard constraints HC1 to HC5 must be filledto schedule the roster The soft constraints SC1 to SC14 aredesirable and the selection of soft constraints determines thequality of the roster Tables 1 and 2 list the set of hard andsoft constraints considered to solve the NRP This sectiondescribes the mathematical model required for hard and softconstraints extensively
The NRP consists of a set of nurses 119899 = 1 2 119873 whereeach row is specific to particular set of shifts 119904 = 1 2 119878for the given set day 119889 = 1 2 119863 The solution roster S forthe 01matrix dimension119873 lowast 119878119863 is as in
S119899119889119904 = 1 if nurse 119899 works 119904 shift for day 1198890 otherwise
(1)
HC1 In this constraint all demanded shifts are assigned to anurse
Figure 1 Illustrative example of Nurse Rostering Problem
Table 1
Hard constraintsHC1 All demanded shifts assigned to a nurseHC2 A nurse can work with only a single shift per dayHC3 The minimum number of nurses required for the shiftHC4 The total number of working days for the nurse should be between the maximum and minimum rangeHC5 A day shift followed by night shift is not allowed
Table 2
Soft constraintsSC1 The maximum number of shifts assigned to each nurseSC2 The minimum number of shifts assigned to each nurseSC3 The maximum number of consecutive working days assigned to each nurseSC4 The minimum number of consecutive working days assigned to each nurseSC5 The maximum number of consecutive working days assigned to each nurse on which no shift is allottedSC6 The minimum number of consecutive working days assigned to each nurse on which no shift is allottedSC7 The maximum number of consecutive working weekends with at least one shift assigned to each nurseSC8 The minimum number of consecutive working weekends with at least one shift assigned to each nurseSC9 The maximum number of weekends with at least one shift assigned to each nurseSC10 Specific working daySC11 Requested day offSC12 Specific shift onSC13 Specific shift offSC14 Nurse not working on the unwanted pattern
where 119864119889119904 is the number of nurses required for a day (119889) atshift (119904) and S119889119904 is the allocation of nurses in the feasiblesolution roster
HC2 In this constraint each nurse can work not more thanone shift per day
119878sum119904=1
S119904119899119889 le 1 forall119899 isin 119873 119889 isin 119863 (3)
where S119899119889 is the allocation of nurses (119899) in solution at shift (119904)for a day (119889)HC3This constraint deals with aminimumnumber of nursesrequired for each shift
119873sum119899=1
S119899119889119904 ge min119899119889119904 forall119889 isin 119863 119904 isin 119878 (4)
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where min119899119889119904 is the minimum number of nurses required fora shift (119904) on the day (119889)HC4 In this constraint the total number of working days foreach nurse should range between minimum and maximumrange for the given scheduled period
119882min le 119863sum119889=1
119878sum119904=1
S119889119904119899 le 119882max forall119899 isin 119873 (5)
The average working shift for nurse can be determined byusing
119882avg = 1119873 (119863sum119889=1
119878sum119904=1
S119889119904119899 forall119899 isin 119873) (6)
where 119882min and 119882max are the minimum and maximumnumber of days in scheduled period and119882avg is the averageworking shift of the nurse
HC5 In this constraint shift 1 followed by shift 3 is notallowed that is a day shift followed by a night shift is notallowed
119873sum119899=1
119863sum119889=1
S1198991198891199043 + S119899119889+11199041 le 1 forall119904 isin 119878 (7)
SC1 The maximum number of shifts assigned to each nursefor the given scheduled period is as follows
max(( 119863sum119889=1
119878sum119904=1
S119889119904119899 minus Φ119906119887119899 ) 0) forall119899 isin 119873 (8)
whereΦ119906119887119899 is themaximumnumber of shifts assigned to nurse(119899)SC2 The minimum number of shifts assigned to each nursefor the given scheduled period is as follows
whereΦ119897119887119899 is theminimumnumber of shifts assigned to nurse(119899)SC3 The maximum number of consecutive working daysassigned to each nurse on which a shift is allotted for thescheduled period is as follows
Ψ119899sum119896=1
max ((C119896119899 minus Θ119906119887119899 ) 0) forall119899 isin 119873 (10)
where Θ119906119887119899 is the maximum number of consecutive workingdays of nurse (119899) Ψ119899 is the total number of consecutive
working spans of nurse (119899) in the roster and C119896119899 is the countof the 119896th working spans of nurse (119899)SC4 The minimum number of consecutive working daysassigned to each nurse on which a shift is allotted for thescheduled period is as follows
Ψ119899sum119896=1
max ((Θ119897119887119899 minus C119896119899) 0) forall119899 isin 119873 (11)
where Θ119897119887119899 is the minimum number of consecutive workingdays of nurse (119899) Ψ119899 is the total number of consecutiveworking spans of nurse (119899) in the roster and C119896119899 is the countof the 119896th working span of the nurse (119899)SC5 The maximum number of consecutive working daysassigned to each nurse on which no shift is allotted for thegiven scheduled period is as follows
Γ119899sum119896=1
max ((eth119896119899 minus 120593119906119887119899 ) 0) forall119899 isin 119873 (12)
where120593119906119887119899 is themaximumnumber of consecutive free days ofnurse (119899) Γ119899 is the total number of consecutive free workingspans of nurse (119899) in the roster and eth119896119899 is the count of the 119896thworking span of the nurse (119899)SC6 The minimum number of consecutive working daysassigned to each nurse on which no shift is allotted for thegiven scheduled period is as follows
Γ119899sum119896=1
max ((120593119897119887119899 minus eth119896119899) 0) forall119899 isin 119873 (13)
where 120593119897119887119899 is theminimumnumber of consecutive free days ofnurse (119899) Γ119899 is the total number of consecutive free workingspans of nurse (119899) in the roster and eth119896119899 is the count of the 119896thworking span of the nurse (119899)SC7 The maximum number of consecutive working week-ends with at least one shift assigned to nurse for the givenscheduled period is as follows
Υ119899sum119896=1
max ((120577119896119899 minus Ω119906119887119899 ) 0) forall119899 isin 119873 (14)
where Ω119906119887119899 is the maximum number of consecutive workingweekends of nurse (119899) Υ119899 is the total number of consecutiveworking weekend spans of nurse (119899) in the roster and 120577119896119899 isthe count of the 119896th working weekend span of the nurse (119899)SC8 The minimum number of consecutive working week-ends with at least one shift assigned to nurse for the givenscheduled period is as follows
Υ119899sum119896=1
max ((Ω119897119887119899 minus 120577119896119899) 0) forall119899 isin 119873 (15)
6 Computational Intelligence and Neuroscience
where Ω119897119887119899 is the minimum number of consecutive workingweekends of nurse (119899) Υ119899 is the total number of consecutiveworking weekend spans of nurse (119899) in the roster and 120577119896119899 isthe count of the 119896th working weekend span of the nurse (119899)SC9 The maximum number of weekends with at least oneshift assigned to nurse in four weeks is as follows
119899sum119896=1
max ((119896119899 minus 120603119906119887119899 ) 0) forall119899 isin 119873 (16)
where 119896119899 is the number of working days at the 119896th weekendof nurse (119899) 120603119906119887119899 is the maximum number of working daysfor nurse (119899) and 119899 is the total count of the weekend in thescheduling period of nurse (119899)SC10 The nurse can request working on a particular day forthe given scheduled period
where 120582119889119899 is the day request from the nurse (119899) to work on anyshift on a particular day (119889)SC11 The nurse can request that they do not work on aparticular day for the given scheduled period
where 120582119889119899 is the request from the nurse (119899) not to work on anyshift on a particular day (119889)SC12 The nurse can request working on a particular shift ona particular day for the given scheduled period
where Υ119889119904119899 is the shift request from the nurse (119899) to work ona particular shift (119904) on particular day (119889)SC13 The nurse can request that they do not work on aparticular shift on a particular day for the given scheduledperiod
where Υ119889119904119899 is the shift request from the nurse (119899) not to workon a particular shift (119904) on particular day (119889)SC14 The nurse should not work on unwanted patternsuggested for the scheduled period
984858119899sum119906=1
120583119906119899 forall119899 isin 119873 (21)
where 120583119906119899 is the total count of occurring patterns for nurse (119899)of type 119906 984858119899 is the set of unwanted patterns suggested for thenurse (119899)
The objective function of the NRP is to maximize thenurse preferences and minimize the penalty cost from vio-lations of soft constraints in (22)
Here SC refers to the set of soft constraints indexed inTable 2 119875sc(119909) refers to the penalty weight violation of thesoft constraint and 119879sc(119909) refers to the total violations of thesoft constraints in roster solution It has to be noted that theusage of penalty function [32] in the NRP is to improve theperformance and provide the fair comparison with anotheroptimization algorithm
4 Bee Colony Optimization
41 Natural Behavior of Honey Bees Swarm intelligence isan emerging discipline for the study of problems whichrequires an optimal approach rather than the traditionalapproach The use of swarm intelligence is the part ofartificial intelligence based on the study of the behavior ofsocial insects The swarm intelligence is composed of manyindividual actions using decentralized and self-organizedsystem Swarm behavior is characterized by natural behaviorof many species such as fish schools herds of animals andflocks of birds formed for the biological requirements tostay together Swarm implies the aggregation of animalssuch as birds fishes ants and bees based on the collectivebehavior The individual agents in the swarm will have astochastic behavior which depends on the local perception ofthe neighborhood The communication between any insectscan be formed with the help of the colonies and it promotescollective intelligence among the colonies
The important features of swarms are proximity qualityresponse variability stability and adaptability The proximityof the swarm must be capable of providing simple spaceand time computations and it should respond to the qualityfactorsThe swarm should allow diverse activities and shouldnot be restricted among narrow channels The swarm shouldmaintain the stability nature and should not fluctuate basedon the behaviorThe adaptability of the swarmmust be able tochange the behavior mode when required Several hundredsof bees from the swarm work together to find nesting sitesand select the best nest site Bee Colony Optimization isinspired by the natural behavior of beesThe bee optimizationalgorithm is inspired by group decision-making processesof honey bees A honey bee searches the best nest site byconsidering speed and accuracy
In a bee colony there are three different types of beesa single queen bee thousands of male drone bees andthousands of worker bees
(1) The queen bee is responsible for creating new coloniesby laying eggs
Computational Intelligence and Neuroscience 7
(2) The male drone bees mated with the queen and werediscarded from the colonies
(3) The remaining female bees in the hive are calledworker bees and they are called the building block ofthe hiveThe responsibilities of the worker bees are tofeed guard and maintain the honey bee comb
Based on the responsibility worker bees are classifiedas scout bees and forager bees A scout bee flies in searchof food sources randomly and returns when the energygets exhausted After reaching a hive scout bees share theinformation and start to explore rich food source locationswith forager bees The scout beersquos information includesdirection quality quantity and distance of the food sourcethey found The way of communicating information about afood source to foragers is done using dance There are twotypes of dance round dance and waggle dance The rounddance will provide direction of the food source when thedistance is small The waggle dance indicates the positionand the direction of the food source the distance can bemeasured by the speed of the dance A greater speed indicatesa smaller distance and the quantity of the food depends onthe wriggling of the beeThe exchange of information amonghive mates is to acquire collective knowledge Forager beeswill silently observe the behavior of scout bee to acquireknowledge about the directions and information of the foodsource
The group decision process of honey bees is for searchingbest food source and nest siteThe decision-making process isbased on the swarming process of the honey bee Swarming isthe process inwhich the queen bee and half of theworker beeswill leave their hive to explore a new colony The remainingworker bees and daughter bee will remain in the old hiveto monitor the waggle dance After leaving their parentalhive swarm bees will form a cluster in search of the newnest site The waggle dance is used to communicate withquiescent bees which are inactive in the colonyThis providesprecise information about the direction of the flower patchbased on its quality and energy level The number of followerbees increases based on the quality of the food source andallows the colony to gather food quickly and efficiently Thedecision-making process can be done in two methods byswarm bees to find the best nest site They are consensusand quorum consensus is the group agreement taken intoaccount and quorum is the decision process taken when thebee vote reaches a threshold value
Bee Colony Optimization (BCO) algorithm is apopulation-based algorithm The bees in the populationare artificial bees and each bee finds its neighboring solutionfrom the current path This algorithm has a forward andbackward process In forwarding pass every bee starts toexplore the neighborhood of its current solution and enablesconstructive and improving moves In forward pass entirebees in the hive will start the constructive move and thenlocal search will start In backward pass bees share theobjective value obtained in the forward pass The bees withhigher priority are used to discard all nonimproving movesThe bees will continue to explore in next forward pass orcontinue the same process with neighborhoodThe flowchart
Forward pass
Initialization
Construction move
Backward pass
Update the bestsolution
Stopping criteriaFalse
True
Figure 2 Flowchart of BCO algorithm
for BCO is shown in Figure 2 The BCO is proficient insolving combinatorial optimization problems by creatingcolonies of the multiagent system The pseudocode for BCOis described in Algorithm 1 The bee colony system providesa standard well-organized and well-coordinated teamworkmultitasking performance [33]
42 Modified Nelder-Mead Method The Nelder-MeadMethod is a simplex method for finding a local minimumfunction of various variables and is a local search algorithmfor unconstrained optimization problems The whole searcharea is divided into different fragments and filled with beeagents To obtain the best solution each fragment can besearched by its bee agents through Modified Nelder-MeadMethod (MNMM) Each agent in the fragments passesinformation about the optimized point using MNMMBy using NMMM the best points are obtained and thebest solution is chosen by decision-making process ofhoney bees The algorithm is a simplex-based method119863-dimensional simplex is initialized with 119863 + 1 verticesthat is two dimensions and it forms a triangle if it has threedimensions it forms a tetrahedron To assign the best andworst point the vertices are evaluated and ordered based onthe objective function
The best point or vertex is considered to the minimumvalue of the objective function and the worst point is chosen
8 Computational Intelligence and Neuroscience
Bee Colony Optimization(1) Initialization Assign every bee to an empty solution(2) Forward Pass
For every bee(21) set 119894 = 1(22) Evaluate all possible construction moves(23) Based on the evaluation choose one move using Roulette Wheel(24) 119894 = 119894 + 1 if (119894 le 119873) Go to step (22)
where 119894 is the counter for construction move and119873 is the number of construction moves during one forwardpass
(3) Return to Hive(4) Backward Pass starts(5) Compute the objective function for each bee and sort accordingly(6) Calculate probability or logical reasoning to continue with the computed solution and become recruiter bee(7) For every follower choose the new solution from recruiters(8) If stopping criteria is not met Go to step (2)(9) Evaluate and find the best solution(10) Output the best solution
Algorithm 1 Pseudocode of BCO
with a maximum value of the computed objective functionTo form simplex new vertex function values are computedThismethod can be calculated using four procedures namelyreflection expansion contraction and shrinkage Figure 3shows the operators of the simplex triangle in MNMM
The simplex operations in each vertex are updated closerto its optimal solution the vertices are ordered based onfitness value and ordered The best vertex is 119860119887 the secondbest vertex is 119860 119904 and the worst vertex is 119860119908 calculated basedon the objective function Let 119860 = (119909 119910) be the vertex in atriangle as food source points 119860119887 = (119909119887 119910119887) 119860 119904 = (119909119904 119910119904)and119860119908 = (119909119908 119910119908) are the positions of the food source pointsthat is local optimal points The objective functions for 119860119887119860 119904 and 119860119908 are calculated based on (23) towards the foodsource points
The objective function to construct simplex to obtainlocal search using MNMM is formulated as
119891 (119909 119910) = 1199092 minus 4119909 + 1199102 minus 119910 minus 119909119910 (23)
Based on the objective function value the vertices foodpoints are ordered ascending with their corresponding honeybee agentsThe obtained values are ordered as119860119887 le 119860 119904 le 119860119908with their honey bee position and food points in the simplextriangle Figure 4 describes the search of best-minimizedcost value for the nurse based on objective function (22)The working principle of Modified Nelder-Mead Method(MNMM) for searching food particles is explained in detail
(1) In the simplex triangle the reflection coefficient 120572expansion coefficient 120574 contraction coefficient 120573 andshrinkage coefficient 120575 are initialized
(2) The objective function for the simplex triangle ver-tices is calculated and ordered The best vertex withlower objective value is 119860119887 the second best vertex is119860 119904 and the worst vertex is named as 119860119908 and thesevertices are ordered based on the objective functionas 119860119887 le 119860 119904 le 119860119908
(3) The first two best vertices are selected namely119860119887 and119860 119904 and the construction proceeds with calculatingthe midpoint of the line segment which joins the twobest vertices that is food positions The objectivefunction decreases as the honey agent associated withthe worst position vertex moves towards best andsecond best verticesThe value decreases as the honeyagent moves towards 119860119908 to 119860119887 and 119860119908 to 119860 119904 It isfeasible to calculate the midpoint vertex 119860119898 by theline joining best and second best vertices using
119860119898 = 119860119887 + 119860 1199042 (24)
(4) A reflecting vertex 119860119903 is generated by choosing thereflection of worst point 119860119908 The objective functionvalue for 119860119903 is 119891(119860119903) which is calculated and it iscompared with worst vertex 119860119908 objective functionvalue 119891(119860119908) If 119891(119860119903) lt 119891(119860119908) proceed with step(5) the reflection vertex can be calculated using
119860119903 = 119860119898 + 120572 (119860119898 minus 119860119908) where 120572 gt 0 (25)
(5) The expansion process starts when the objectivefunction value at reflection vertex 119860119903 is lesser thanworst vertex 119860119908 119891(119860119903) lt 119891(119860119908) and the linesegment is further extended to 119860119890 through 119860119903 and119860119908 The vertex point 119860119890 is calculated by (26) If theobjective function value at119860119890 is lesser than reflectionvertex 119860119903 119891(119860119890) lt 119891(119860119903) then the expansion isaccepted and the honey bee agent has found best foodposition compared with reflection point
119860119890 = 119860119903 + 120574 (119860119903 minus 119860119898) where 120574 gt 1 (26)
(6) The contraction process is carried out when 119891(119860119903) lt119891(119860 119904) and 119891(119860119903) le 119891(119860119887) for replacing 119860119887 with
Computational Intelligence and Neuroscience 9
AwAs
Ab
(a) Simplex triangle
Ar
As
Ab
Aw
(b) Reflection
Ae
Ar
As
Ab
Aw
(c) Expansion
Ac
As
Ab
Aw
(d) Contraction (119860ℎ lt 119860119903)
Ac
As
Ab
Aw
(e) Contraction (119860119903 lt 119860ℎ)
A㰀b
A㰀s
As
Ab
Aw
(f) Shrinkage
Figure 3 Nelder-Mead operations
119860119903 If 119891(119860119903) gt 119891(119860ℎ) then the direct contractionwithout the replacement of 119860119887 with 119860119903 is performedThe contraction vertex 119860119888 can be calculated using
119860119888 = 120573119860119903 + (1 minus 120573)119860119898 where 0 lt 120573 lt 1 (27)
If 119891(119860119903) le 119891(119860119887) the contraction can be done and119860119888 replaced with 119860ℎ go to step (8) or else proceed tostep (7)
(7) The shrinkage phase proceeds when the contractionprocess at step (6) fails and is done by shrinking allthe vertices of the simplex triangle except 119860ℎ using(28) The objective function value of reflection andcontraction phase is not lesser than the worst pointthen the vertices 119860 119904 and 119860119908 must be shrunk towards119860ℎThus the vertices of smaller value will form a newsimplex triangle with another two best vertices
119860 119894 = 120575119860 119894 + 1198601 (1 minus 120575) where 0 lt 120575 lt 1 (28)
(8) The calculations are stopped when the terminationcondition is met
Algorithm 2 describes the pseudocode for ModifiedNelder-Mead Method in detail It portraits the detailed pro-cess of MNMM to obtain the best solution for the NRP Theworkflow of the proposed MNMM is explained in Figure 5
5 MODBCO
Bee Colony Optimization is the metaheuristic algorithm tosolve various combinatorial optimization problems and itis inspired by the natural behavior of bee for their foodsources The algorithm consists of two steps forward andbackward pass During forwarding pass bees started toexplore the neighborhood of its current solution and findall possible ways In backward pass bees return to thehive and share the values of the objective function of theircurrent solution Calculate nectar amount using probability
10 Computational Intelligence and Neuroscience
Ab
Aw
Ar
As
Am
d
d
Ab
Aw
Ar
As
Am
d
d
Aed2
Ab
Aw
Ar
As
Am
Ac1
Ac2
Ab
Aw As
Am
Anew
Figure 4 Bees search movement based on MNMM
function and advertise the solution the bee which has thebetter solution is given higher priority The remaining beesbased on the probability value decide whether to explore thesolution or proceed with the advertised solution DirectedBee Colony Optimization is the computational system whereseveral bees work together in uniting and interact with eachother to achieve goals based on the group decision processThe whole search area of the bee is divided into multiplefragments different bees are sent to different fragments Thebest solution in each fragment is obtained by using a localsearch algorithmModified Nelder-Mead Method (MNMM)To obtain the best solution the total varieties of individualparameters are partitioned into individual volumes Eachvolume determines the starting point of the exploration offood particle by each bee The bees use developed MNMMalgorithm to find the best solution by remembering thelast two best food sites they obtained After obtaining thecurrent solution the bee starts to backward pass sharingof information obtained during forwarding pass The beesstarted to share information about optimized point by thenatural behavior of bees called waggle dance When all theinformation about the best food is shared the best among theoptimized point is chosen using a decision-making processcalled consensus and quorummethod in honey bees [34 35]
51 Multiagent System All agents live in an environmentwhich is well structured and organized Inmultiagent systemseveral agents work together and interact with each otherto obtain the goal According to Jiao and Shi [36] andZhong et al [37] all agents should possess the followingqualities agents should live and act in an environmenteach agent should sense its local environment each agent
should be capable of interacting with other agents in a localenvironment and agents attempt to perform their goal Allagents interact with each other and take the decision toachieve the desired goals The multiagent system is a com-putational system and provides an opportunity to optimizeand compute all complex problems In multiagent system allagents start to live and act in the same environment which iswell organized and structured Each agent in the environmentis fixed on a lattice point The size and dimension of thelattice point in the environment depend upon the variablesused The objective function can be calculated based on theparameters fixed
(1) Consider ldquo119890rdquo number of independent parameters tocalculate the objective function The range of the 119892thparameter can be calculated using [119876119892119894 119876119892119891] where119876119892119894 is the initial value of the 119892th parameter and 119876119892119891is the final value of the 119892th parameter chosen
(2) Thus the objective function can be formulated as 119890number of axes each axis will contain a total rangeof single parameter with different dimensions
(3) Each axis is divided into smaller parts each partis called a step So 119892th axis can be divided into 119899119892number of steps each with the length of 119871119892 where thevalue of 119892 depends upon parameters thus 119892 = 1 to 119890The relationship between 119899119892 and 119871119892 can be given as
119899119892 = 119876119892119894 minus 119876119892119891119871119892 (29)
(4) Then each axis is divided into branches foreach branch 119892 number of branches will form an
Computational Intelligence and Neuroscience 11
Modified Nelder-Mead Method for directed honey bee food search(1) Initialization119860119887 denotes the list of vertices in simplex where 119894 = 1 2 119899 + 1120572 120574 120573 and 120575 are the coefficients of reflection expansion contraction and shrinkage119891 is the objective function to be minimized(2)Ordering
Order the vertices in simplex from lowest objective function value 119891(1198601) to highest value 119891(119860119899+1) Ordered as 1198601le 1198602 le sdot sdot sdot le 119860119899+1(3)Midpoint
Calculate the midpoint for first two best vertices in simplex 119860119898 = sum(119860 119894119899) where 119894 = 1 2 119899(4) Reflection Process
Calculate reflection point 119860119903 by 119860119903 = 119860119898 + 120572(119860119898 minus 119860119899+1)if 119891(1198601) le 119891(1198602) le 119891(119860119899) then119860119899 larr 119860119903 and Go to to Step (8)end if
(5) Expansion Processif 119891(119860119903) le 119891(1198601) thenCalculate expansion point using 119860 119890 = 119860119903 + 120574(119860119903 minus 119860119898)end ifif 119891(119860 119890) lt 119891(119860119903) then119860119899 larr 119860 119890 and Go to to Step (8)else119860119899 larr 119860119903 and Go to to Step (8)end if
(6) Contraction Processif 119891(119860119899) le 119891(119860119903) le 119891(119860119899+1) thenCompute outside contraction by 119860 119888 = 120573119860119903 + (1 minus 120573)119860119898end ifif 119891(1198601) ge 119891(119860119899+1) thenCompute inside contraction by 119860 119888 = 120573119860119899+1 + (1 minus 120573)119860119898end ifif 119891(119860119903) ge 119891(119860119899) thenContraction is done between 119860119898 and the best vertex among 119860119903 and 119860119899+1end ifif 119891(119860 119888) lt 119891(119860119903) then119860119899 larr 119860 119888 and Go to to Step (8)else goes to Step (7)end ifif 119891(119860 119888) ge 119891(119860119899+1) then119860119899+1 larr 119860 119888 and Go to to Step (8)else Go to to Step (7)end if
(7) Shrinkage ProcessShrink towards the best solution with new vertices by 119860 119894 = 120575119860 119894 + 1198601(1 minus 120575) where 119894 = 2 119899 + 1
(8) Stopping CriteriaOrder and re-label new vertices of the simplex based on their objective function and go to step (4)
Algorithm 2 Pseudocode of Modified Nelder-Mead Method
119890-dimensional volume Total number of volumes 119873Vcan be formulated using
119873V = 119890prod119892=1
119899119892 (30)
(5) The starting point of the agent in the environmentwhich is one point inside volume is chosen bycalculating themidpoint of the volumeThemidpointof the lattice can be calculated as
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ] (31)
52 Decision-Making Process A key role of the honey beesis to select the best nest site and is done by the process ofdecision-making to produce a unified decisionThey follow adistributed decision-making process to find out the neighbornest site for their food particles The pseudocode for theproposed MODBCO algorithm is shown in Algorithm 3Figure 6 explains the workflow of the proposed algorithm forthe search of food particles by honey bees using MODBCO
521 Waggle Dance The scout bees after returning from thesearch of food particle report about the quality of the foodsite by communicationmode called waggle dance Scout beesperform thewaggle dance to other quiescent bees to advertise
12 Computational Intelligence and Neuroscience
Yes
Reflectionprocess
Order and label verticesbased on f(A)
Initialization
Coefficients 훼 훾 훽 훿
Objective function f(A)
f(Ab) lt f(Ar) lt f(Aw) Aw larr Ar
f(Ae) le f(Ar)
two best verticesAm forCalculate midpoint
Start
Terminationcriteria
Stop
Ar = Am + 훼(Am minus Aw)
ExpansionprocessNo
Yesf(Ar) le f(Aw) Aw larr Ae
No
b larr true Aw larr Ar
Contractionprocess
f(Ar) ge f(An)Yes
f(Ac) lt f(Ar)Aw larr Ac
b larr false
No
Shrinkageprocess
b larr true
Yes
Yes
No
Ae = Ar + 훾(Ar minus
Am)
Ac = 훽Ar + (1 minus 훽)Am
Ai = 훿Ai + A1(1 minus 훿)
Figure 5 Workflow of Modified Nelder-Mead Method
Computational Intelligence and Neuroscience 13
Multi-Objective Directed Bee Colony Optimization(1) Initialization119891(119909) is the objective function to be minimized
Initialize 119890 number of parameters and 119871119892 length of steps where 119892 = 0 to 119890Initialize initial value and the final value of the parameter as 119876119892119894 and 119876119892119891lowastlowast Solution Representation lowastlowastThe solutions are represented in the form of Binary values which can be generated as followsFor each solution 119894 = 1 119899119883119894 = 1199091198941 1199091198942 119909119894119889 | 119889 isin total days amp 119909119894119889 = rand ge 029 forall119889End for
(2) The number of steps in each step can be calculated using
119899119892 = 119876119892119894 minus 119876119892119891119871119892(3) The total number of volumes can be calculated using119873V = 119890prod
119892=1
119899119892(4) The midpoint of the volume to calculate starting point of the exploration can be calculated using
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ](5) Explore the search volume according to the Modified Nelder-Mead Method using Algorithm 2(6) The recorded value of the optimized point in vector table using[119891(1198811) 119891(1198812) 119891(119881119873V )](7) The globally optimized point is chosen based on Bee decision-making process using Consensus and Quorum
method approach 119891(119881119892) = min [119891(1198811) 119891(1198812) 119891(119881119873V )]Algorithm 3 Pseudocode of MODBCO
their best nest site for the exploration of food source Inthe multiagent system each agent after collecting individualsolution gives it to the centralized systems To select the bestoptimal solution forminimal optimal cases themathematicalformulation can be stated as
dance119894 = min (119891119894 (119881)) (32)
This mathematical formulation will find the minimaloptimal cases among the search solution where 119891119894(119881) is thesearch value calculated by the agent The search values arerecorded in the vector table 119881 119881 is the vector which consistsof 119890 number of elements The element 119890 contains the value ofthe parameter both optimal solution and parameter valuesare recorded in the vector table
522 Consensus Theconsensus is thewidespread agreementamong the group based on voting the voting pattern ofthe scout bees is monitored periodically to know whetherit reached an agreement and started acting on the decisionpattern Honey bees use the consensus method to select thebest search value the globally optimized point is chosen bycomparing the values in the vector table The globally opti-mized points are selected using themathematical formulation
523 Quorum In quorummethod the optimum solution iscalculated as the final solution based on the threshold levelobtained by the group decision-making process When thesolution reaches the optimal threshold level 120585119902 then the solu-tion is considered as a final solution based on unison decisionprocess The quorum threshold value describes the quality of
the food particle result When the threshold value is less thecomputation time decreases but it leads to inaccurate experi-mental resultsThe threshold value should be chosen to attainless computational timewith an accurate experimental result
6 Experimental Design and Analysis
61 Performance Metrics The performance of the proposedalgorithm MODBCO is assessed by comparing with fivedifferent competitor methods Here six performance metricsare considered to investigate the significance and evaluate theexperimental results The metrics are listed in this section
611 Least Error Rate Least Error Rate (LER) is the percent-age of the difference between known optimal value and thebest value obtained The LER can be calculated using
LER () = 119903sum119894=1
OptimalNRP-Instance minus fitness119894OptimalNRP-Instance
(34)
612 Average Convergence The Average Convergence is themeasure to evaluate the quality of the generated populationon average The Average Convergence (AC) is the percentageof the average of the convergence rate of solutions The per-formance of the convergence time is increased by the AverageConvergence to exploremore solutions in the populationTheAverage Convergence is calculated usingAC
= 119903sum119894=1
1 minus Avg_fitness119894 minusOptimalNRP-InstanceOptimalNRP-Instance
lowast 100 (35)
where (119903) is the number of instances in the given dataset
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
the location of rich food sources Some of the algorithmswhich follow the waggle dance of communication performedby scout bees about the nectar site are bee system Bee ColonyOptimization [7] and Artificial Bee Colony [8]
TheDirected BeeColony (DBC)OptimizationAlgorithm[9] is inspired by the group decision-making process ofbee behavior for the selection of the nectar site The groupdecision process includes consensus and quorum methodsConsensus is the process of vote agreement and the votingpattern of the scouts is monitored The best nest site isselected once the quorum (threshold) value is reached Theexperimental result shows that the algorithm is robust andaccurate for generating the unique solutionThe contributionof this research article is the use of a hybrid Directed BeeColony Optimization with the Nelder-Mead Method foreffective local search The authors have adapted MODBCOfor solving multiobjective problems which integrate the fol-lowing processes At first a deterministic local searchmethodModifiedNelder-Mead is used to obtain the provisional opti-mal solutionThen a multiagent particle system environmentis used in the exploration and decision-making process forestablishing a new colony and nectar site selection Only fewhoney bees were active in the process of decision-making sothe energy conservation of the swarm is highly achievable
The Nurse Rostering Problem (NRP) is a staff schedulingproblem that intends to assign a set of nurses to workshifts to maximize hospital benefit by considering a setof hard and soft constraints like allotment of duty hourshospital regulations and so forth This nurse rostering is adelicate task of finding combinatorial solutions by satisfyingmultiple constraints [10] Satisfying the hard constraint ismandatory in any scheduling problem and a violation ofany soft constraints is allowable but penalized To achievean optimal global solution for the problem is impossible inmany cases [11] Many algorithmic techniques such as meta-heuristic method graph-based heuristics and mathematicalprogramming model have been proposed to solve automatedscheduling problems and timetabling problems over the lastdecades [12 13]
In this work the effectiveness of the hybrid algorithmis compared with different optimization algorithms usingperformancemetrics such as error rate convergence rate bestvalue and standard deviationThewell-known combinatorialscheduling problem NRP is chosen as the test bed toexperiment and analyze the effectiveness of the proposedalgorithm
This paper is organized as follows Section 2 presentsthe literature survey of existing algorithms to solve theNRP Section 3 highlights the mathematical model andthe formulation of hard and soft constraints of the NRPSection 4 explains the natural behavior of honey bees tohandle decision-making process and the Modified Nelder-Mead Method Section 5 describes the development of themetaheuristic approach and the effectiveness of the MOD-BCO algorithm to solve the NRP is demonstrated Section 6confers the computational experiments and the analysis ofresults for the formulated problem Finally Section 7 providesthe summary of the discussion and Section 8 will concludewith future directions of the research work
2 Literature Review
Berrada et al [19] considered multiple objectives to tacklethe nurse scheduling problem by considering various orderedsoft constraints The soft constraints are ordered based onpriority level and this determines the quality of the solutionBurke et al [20] proposed a multiobjective Pareto-basedsearch technique and used simulated annealing based on aweighted-sum evaluation function towards preferences and adominated-based evaluation function towards the Pareto setMany mathematical models are proposed to reduce the costand increase the performance of the taskThe performance ofthe problem greatly depends on the type of constraints used[21] Dowsland [22] proposed a technique of chain movesusing a multistate tabu search algorithm This algorithmexchanges the feasible and infeasible search space to increasethe transmission rate when the system gets disconnected Butthis algorithm fails to solve other problems in different searchspace instances
Burke et al [23] proposed a hybrid tabu search algorithmto solve the NRP in Belgian hospitals In their constraintsthe authors have added the previous roster along with hardand soft constraints To consider this they included heuris-tic search strategies in the general tabu search algorithmThis model provides flexibility and more user control Ahyperheuristic algorithm with tabu search is proposed forthe NRP by Burke et al [24] They developed a rule basedreinforcement learning which is domain specific but itchooses a little low-level heuristic to solve the NRP Theindirect genetic algorithm is problem dependent which usesencoding and decoding schemes with genetic operator tosolve NRP Burke et al [25] developed a memetic algorithmto solve the nurse scheduling problem and the authorshave compared memetic and tabu search algorithm Theexperimental result shows a memetic algorithm outperformswith better quality than the genetic algorithm and tabu searchalgorithm
Simulated annealing has been proposed to solve the NRPHadwan and Ayob [26] introduced a shift pattern approachwith simulated annealing The authors have proposed agreedy constructive heuristic algorithm to generate therequired shift patterns to solve the NRP at UKMMC (Univer-siti KebangsaanMalaysiaMedical Centre)Thismethodologywill reduce the complexity of the search space solutionto generate a roster by building two- or three-day shiftpatterns The efficiency of this algorithm was shown byexperimental results with respect to execution time per-formance considerations fairness and the quality of thesolution This approach was capable of handling all hard andsoft constraints and produces a quality roster pattern Sharifet al [27] proposed a hybridized heuristic approach withchanges in the neighborhood descent search algorithm tosolve the NRP at UKMMCThis heuristic is the hybridizationof cyclic schedule with noncyclic schedule They appliedrepairing mechanism which swaps the shifts between nursesto tackle the random shift arrangement in the solution Avariable neighborhood descent search algorithm (VNDS) isused to change the neighborhood structure using a localsearch and generate a quality duty roster In VNDS the first
Computational Intelligence and Neuroscience 3
neighborhood structure will reroster nurses to different shiftsand the second neighborhood structure will do repairingmechanism
Aickelin and Dowsland [28] proposed a technique forshift patterns they considered shift patterns with penaltypreferences and number of successive working days Theindirect genetic algorithm will generate various heuristicdecoders for shift patterns to reconstruct the shift roster forthe nurse A qualified roster is generated using decoderswith the help of the best permutations of nurses To generatebest search space solutions for the permutation of nursesthe authors used an adaptive iterative method to adjust theorder of nurses as scheduled one by one Asta et al [29] andAnwar et al [30] proposed a tensor-based hyperheuristic tosolve the NRP The authors tuned a specific group of datasetsand embedded a tensor-based machine learning algorithmA tensor-based hyperheuristic with memory managementis used to generate the best solution This approach isconsidered in life-long applications to extract knowledge anddesired behavior throughout the run time
Todorovic and Petrovic [31] proposed the Bee ColonyOptimization approach to solve the NRP all the unscheduledshifts are allocated to the available nurses in the constructivephase This algorithm combines the constructive move withlocal search to improve the quality of the solution For eachforward pass the predefined numbers of unscheduled shiftsare allocated to the nurses and discarded the solution withless improvement in the objective function The process ofintelligent reduction in neighborhood search had improvedthe current solution In construction phase unassigned shiftsare allotted to nurses and lead to violation of constraints tohigher penalties
Severalmethods have been proposed using the INRC2010dataset to solve the NRP the authors have consideredfive latest competitors to measure the effectiveness of theproposed algorithm Asaju et al [14] proposed Artificial BeeColony (ABC) algorithm to solve NRP This process is donein two phases at first heuristic based ordering of shift patternis used to generate the feasible solution In the second phaseto obtain the solution ABC algorithm is used In thismethodpremature convergence takes place and the solution getstrapped in local optima The lack of a local search algorithmof this process leads to yielding higher penalty Awadallah etal [15] developed a metaheuristic technique hybrid artificialbee colony (HABC) to solve the NRP In ABC algorithm theemployee bee phase was replaced by a hill climbing approachto increase exploitation process Use of hill climbing in ABCgenerates a higher value which leads to high computationaltime
The global best harmony search with pitch adjustmentdesign is used to tackle the NRP in [16] The author adaptedthe harmony search algorithm (HAS) in exploitation pro-cess and particle swarm optimization (PSO) in explorationprocess In HAS the solutions are generated based on threeoperator namely memory consideration random consider-ation and pitch adjustment for the improvisation processThey did two improvisations to solve the NRP multipitchadjustment to improve exploitation process and replaced ran-dom selectionwith global best to increase convergence speed
The hybrid harmony search algorithm with hill climbing isused to solve the NRP in [17] For local search metaheuristicharmony and hill climbing approach are used The memoryconsideration parameter in harmony is replaced by PSOalgorithm The derivative criteria will reduce the numberof iterations towards local minima This process considersmany parameters to construct the roster since improvisationprocess is to be at each iteration
Santos et al [18] used integer programming (IP) to solvetheNRP andproposedmonolith compact IPwith polynomialconstraints and variables The authors have used both upperand lower bounds for obtaining optimal cost They estimatedand improved lower bound values towards optimum and thismethod requires additional processing time
3 Mathematical Model
The NRP problem is a real-world problem at hospitals theproblem is to assign a predefined set of shifts (like S1-dayshift S2-noon shift S3-night shift and S4-Free-shift) of ascheduled period for a set of nurses of different preferencesand skills in each ward Figure 1 shows the illustrativeexample of the feasible nurse roster which consists of fourshifts namely day shift noon shift night shift and free shift(holiday) allocating five nurses over 11 days of scheduledperiod Each column in the scheduled table represents theday and the cell content represents the shift type allocatedto a nurse Each nurse is allocated one shift per day and thenumber of shifts is assigned based on the hospital contractsThis problem will have some variants on a number of shifttypes nurses nurse skills contracts and scheduling periodIn general both hard and soft constraints are considered forgenerating and assessing solutions
Hard constraints are the regulations which must besatisfied to achieve the feasible solution They cannot beviolated since hard constraints are demanded by hospitalregulations The hard constraints HC1 to HC5 must be filledto schedule the roster The soft constraints SC1 to SC14 aredesirable and the selection of soft constraints determines thequality of the roster Tables 1 and 2 list the set of hard andsoft constraints considered to solve the NRP This sectiondescribes the mathematical model required for hard and softconstraints extensively
The NRP consists of a set of nurses 119899 = 1 2 119873 whereeach row is specific to particular set of shifts 119904 = 1 2 119878for the given set day 119889 = 1 2 119863 The solution roster S forthe 01matrix dimension119873 lowast 119878119863 is as in
S119899119889119904 = 1 if nurse 119899 works 119904 shift for day 1198890 otherwise
(1)
HC1 In this constraint all demanded shifts are assigned to anurse
Figure 1 Illustrative example of Nurse Rostering Problem
Table 1
Hard constraintsHC1 All demanded shifts assigned to a nurseHC2 A nurse can work with only a single shift per dayHC3 The minimum number of nurses required for the shiftHC4 The total number of working days for the nurse should be between the maximum and minimum rangeHC5 A day shift followed by night shift is not allowed
Table 2
Soft constraintsSC1 The maximum number of shifts assigned to each nurseSC2 The minimum number of shifts assigned to each nurseSC3 The maximum number of consecutive working days assigned to each nurseSC4 The minimum number of consecutive working days assigned to each nurseSC5 The maximum number of consecutive working days assigned to each nurse on which no shift is allottedSC6 The minimum number of consecutive working days assigned to each nurse on which no shift is allottedSC7 The maximum number of consecutive working weekends with at least one shift assigned to each nurseSC8 The minimum number of consecutive working weekends with at least one shift assigned to each nurseSC9 The maximum number of weekends with at least one shift assigned to each nurseSC10 Specific working daySC11 Requested day offSC12 Specific shift onSC13 Specific shift offSC14 Nurse not working on the unwanted pattern
where 119864119889119904 is the number of nurses required for a day (119889) atshift (119904) and S119889119904 is the allocation of nurses in the feasiblesolution roster
HC2 In this constraint each nurse can work not more thanone shift per day
119878sum119904=1
S119904119899119889 le 1 forall119899 isin 119873 119889 isin 119863 (3)
where S119899119889 is the allocation of nurses (119899) in solution at shift (119904)for a day (119889)HC3This constraint deals with aminimumnumber of nursesrequired for each shift
119873sum119899=1
S119899119889119904 ge min119899119889119904 forall119889 isin 119863 119904 isin 119878 (4)
Computational Intelligence and Neuroscience 5
where min119899119889119904 is the minimum number of nurses required fora shift (119904) on the day (119889)HC4 In this constraint the total number of working days foreach nurse should range between minimum and maximumrange for the given scheduled period
119882min le 119863sum119889=1
119878sum119904=1
S119889119904119899 le 119882max forall119899 isin 119873 (5)
The average working shift for nurse can be determined byusing
119882avg = 1119873 (119863sum119889=1
119878sum119904=1
S119889119904119899 forall119899 isin 119873) (6)
where 119882min and 119882max are the minimum and maximumnumber of days in scheduled period and119882avg is the averageworking shift of the nurse
HC5 In this constraint shift 1 followed by shift 3 is notallowed that is a day shift followed by a night shift is notallowed
119873sum119899=1
119863sum119889=1
S1198991198891199043 + S119899119889+11199041 le 1 forall119904 isin 119878 (7)
SC1 The maximum number of shifts assigned to each nursefor the given scheduled period is as follows
max(( 119863sum119889=1
119878sum119904=1
S119889119904119899 minus Φ119906119887119899 ) 0) forall119899 isin 119873 (8)
whereΦ119906119887119899 is themaximumnumber of shifts assigned to nurse(119899)SC2 The minimum number of shifts assigned to each nursefor the given scheduled period is as follows
whereΦ119897119887119899 is theminimumnumber of shifts assigned to nurse(119899)SC3 The maximum number of consecutive working daysassigned to each nurse on which a shift is allotted for thescheduled period is as follows
Ψ119899sum119896=1
max ((C119896119899 minus Θ119906119887119899 ) 0) forall119899 isin 119873 (10)
where Θ119906119887119899 is the maximum number of consecutive workingdays of nurse (119899) Ψ119899 is the total number of consecutive
working spans of nurse (119899) in the roster and C119896119899 is the countof the 119896th working spans of nurse (119899)SC4 The minimum number of consecutive working daysassigned to each nurse on which a shift is allotted for thescheduled period is as follows
Ψ119899sum119896=1
max ((Θ119897119887119899 minus C119896119899) 0) forall119899 isin 119873 (11)
where Θ119897119887119899 is the minimum number of consecutive workingdays of nurse (119899) Ψ119899 is the total number of consecutiveworking spans of nurse (119899) in the roster and C119896119899 is the countof the 119896th working span of the nurse (119899)SC5 The maximum number of consecutive working daysassigned to each nurse on which no shift is allotted for thegiven scheduled period is as follows
Γ119899sum119896=1
max ((eth119896119899 minus 120593119906119887119899 ) 0) forall119899 isin 119873 (12)
where120593119906119887119899 is themaximumnumber of consecutive free days ofnurse (119899) Γ119899 is the total number of consecutive free workingspans of nurse (119899) in the roster and eth119896119899 is the count of the 119896thworking span of the nurse (119899)SC6 The minimum number of consecutive working daysassigned to each nurse on which no shift is allotted for thegiven scheduled period is as follows
Γ119899sum119896=1
max ((120593119897119887119899 minus eth119896119899) 0) forall119899 isin 119873 (13)
where 120593119897119887119899 is theminimumnumber of consecutive free days ofnurse (119899) Γ119899 is the total number of consecutive free workingspans of nurse (119899) in the roster and eth119896119899 is the count of the 119896thworking span of the nurse (119899)SC7 The maximum number of consecutive working week-ends with at least one shift assigned to nurse for the givenscheduled period is as follows
Υ119899sum119896=1
max ((120577119896119899 minus Ω119906119887119899 ) 0) forall119899 isin 119873 (14)
where Ω119906119887119899 is the maximum number of consecutive workingweekends of nurse (119899) Υ119899 is the total number of consecutiveworking weekend spans of nurse (119899) in the roster and 120577119896119899 isthe count of the 119896th working weekend span of the nurse (119899)SC8 The minimum number of consecutive working week-ends with at least one shift assigned to nurse for the givenscheduled period is as follows
Υ119899sum119896=1
max ((Ω119897119887119899 minus 120577119896119899) 0) forall119899 isin 119873 (15)
6 Computational Intelligence and Neuroscience
where Ω119897119887119899 is the minimum number of consecutive workingweekends of nurse (119899) Υ119899 is the total number of consecutiveworking weekend spans of nurse (119899) in the roster and 120577119896119899 isthe count of the 119896th working weekend span of the nurse (119899)SC9 The maximum number of weekends with at least oneshift assigned to nurse in four weeks is as follows
119899sum119896=1
max ((119896119899 minus 120603119906119887119899 ) 0) forall119899 isin 119873 (16)
where 119896119899 is the number of working days at the 119896th weekendof nurse (119899) 120603119906119887119899 is the maximum number of working daysfor nurse (119899) and 119899 is the total count of the weekend in thescheduling period of nurse (119899)SC10 The nurse can request working on a particular day forthe given scheduled period
where 120582119889119899 is the day request from the nurse (119899) to work on anyshift on a particular day (119889)SC11 The nurse can request that they do not work on aparticular day for the given scheduled period
where 120582119889119899 is the request from the nurse (119899) not to work on anyshift on a particular day (119889)SC12 The nurse can request working on a particular shift ona particular day for the given scheduled period
where Υ119889119904119899 is the shift request from the nurse (119899) to work ona particular shift (119904) on particular day (119889)SC13 The nurse can request that they do not work on aparticular shift on a particular day for the given scheduledperiod
where Υ119889119904119899 is the shift request from the nurse (119899) not to workon a particular shift (119904) on particular day (119889)SC14 The nurse should not work on unwanted patternsuggested for the scheduled period
984858119899sum119906=1
120583119906119899 forall119899 isin 119873 (21)
where 120583119906119899 is the total count of occurring patterns for nurse (119899)of type 119906 984858119899 is the set of unwanted patterns suggested for thenurse (119899)
The objective function of the NRP is to maximize thenurse preferences and minimize the penalty cost from vio-lations of soft constraints in (22)
Here SC refers to the set of soft constraints indexed inTable 2 119875sc(119909) refers to the penalty weight violation of thesoft constraint and 119879sc(119909) refers to the total violations of thesoft constraints in roster solution It has to be noted that theusage of penalty function [32] in the NRP is to improve theperformance and provide the fair comparison with anotheroptimization algorithm
4 Bee Colony Optimization
41 Natural Behavior of Honey Bees Swarm intelligence isan emerging discipline for the study of problems whichrequires an optimal approach rather than the traditionalapproach The use of swarm intelligence is the part ofartificial intelligence based on the study of the behavior ofsocial insects The swarm intelligence is composed of manyindividual actions using decentralized and self-organizedsystem Swarm behavior is characterized by natural behaviorof many species such as fish schools herds of animals andflocks of birds formed for the biological requirements tostay together Swarm implies the aggregation of animalssuch as birds fishes ants and bees based on the collectivebehavior The individual agents in the swarm will have astochastic behavior which depends on the local perception ofthe neighborhood The communication between any insectscan be formed with the help of the colonies and it promotescollective intelligence among the colonies
The important features of swarms are proximity qualityresponse variability stability and adaptability The proximityof the swarm must be capable of providing simple spaceand time computations and it should respond to the qualityfactorsThe swarm should allow diverse activities and shouldnot be restricted among narrow channels The swarm shouldmaintain the stability nature and should not fluctuate basedon the behaviorThe adaptability of the swarmmust be able tochange the behavior mode when required Several hundredsof bees from the swarm work together to find nesting sitesand select the best nest site Bee Colony Optimization isinspired by the natural behavior of beesThe bee optimizationalgorithm is inspired by group decision-making processesof honey bees A honey bee searches the best nest site byconsidering speed and accuracy
In a bee colony there are three different types of beesa single queen bee thousands of male drone bees andthousands of worker bees
(1) The queen bee is responsible for creating new coloniesby laying eggs
Computational Intelligence and Neuroscience 7
(2) The male drone bees mated with the queen and werediscarded from the colonies
(3) The remaining female bees in the hive are calledworker bees and they are called the building block ofthe hiveThe responsibilities of the worker bees are tofeed guard and maintain the honey bee comb
Based on the responsibility worker bees are classifiedas scout bees and forager bees A scout bee flies in searchof food sources randomly and returns when the energygets exhausted After reaching a hive scout bees share theinformation and start to explore rich food source locationswith forager bees The scout beersquos information includesdirection quality quantity and distance of the food sourcethey found The way of communicating information about afood source to foragers is done using dance There are twotypes of dance round dance and waggle dance The rounddance will provide direction of the food source when thedistance is small The waggle dance indicates the positionand the direction of the food source the distance can bemeasured by the speed of the dance A greater speed indicatesa smaller distance and the quantity of the food depends onthe wriggling of the beeThe exchange of information amonghive mates is to acquire collective knowledge Forager beeswill silently observe the behavior of scout bee to acquireknowledge about the directions and information of the foodsource
The group decision process of honey bees is for searchingbest food source and nest siteThe decision-making process isbased on the swarming process of the honey bee Swarming isthe process inwhich the queen bee and half of theworker beeswill leave their hive to explore a new colony The remainingworker bees and daughter bee will remain in the old hiveto monitor the waggle dance After leaving their parentalhive swarm bees will form a cluster in search of the newnest site The waggle dance is used to communicate withquiescent bees which are inactive in the colonyThis providesprecise information about the direction of the flower patchbased on its quality and energy level The number of followerbees increases based on the quality of the food source andallows the colony to gather food quickly and efficiently Thedecision-making process can be done in two methods byswarm bees to find the best nest site They are consensusand quorum consensus is the group agreement taken intoaccount and quorum is the decision process taken when thebee vote reaches a threshold value
Bee Colony Optimization (BCO) algorithm is apopulation-based algorithm The bees in the populationare artificial bees and each bee finds its neighboring solutionfrom the current path This algorithm has a forward andbackward process In forwarding pass every bee starts toexplore the neighborhood of its current solution and enablesconstructive and improving moves In forward pass entirebees in the hive will start the constructive move and thenlocal search will start In backward pass bees share theobjective value obtained in the forward pass The bees withhigher priority are used to discard all nonimproving movesThe bees will continue to explore in next forward pass orcontinue the same process with neighborhoodThe flowchart
Forward pass
Initialization
Construction move
Backward pass
Update the bestsolution
Stopping criteriaFalse
True
Figure 2 Flowchart of BCO algorithm
for BCO is shown in Figure 2 The BCO is proficient insolving combinatorial optimization problems by creatingcolonies of the multiagent system The pseudocode for BCOis described in Algorithm 1 The bee colony system providesa standard well-organized and well-coordinated teamworkmultitasking performance [33]
42 Modified Nelder-Mead Method The Nelder-MeadMethod is a simplex method for finding a local minimumfunction of various variables and is a local search algorithmfor unconstrained optimization problems The whole searcharea is divided into different fragments and filled with beeagents To obtain the best solution each fragment can besearched by its bee agents through Modified Nelder-MeadMethod (MNMM) Each agent in the fragments passesinformation about the optimized point using MNMMBy using NMMM the best points are obtained and thebest solution is chosen by decision-making process ofhoney bees The algorithm is a simplex-based method119863-dimensional simplex is initialized with 119863 + 1 verticesthat is two dimensions and it forms a triangle if it has threedimensions it forms a tetrahedron To assign the best andworst point the vertices are evaluated and ordered based onthe objective function
The best point or vertex is considered to the minimumvalue of the objective function and the worst point is chosen
8 Computational Intelligence and Neuroscience
Bee Colony Optimization(1) Initialization Assign every bee to an empty solution(2) Forward Pass
For every bee(21) set 119894 = 1(22) Evaluate all possible construction moves(23) Based on the evaluation choose one move using Roulette Wheel(24) 119894 = 119894 + 1 if (119894 le 119873) Go to step (22)
where 119894 is the counter for construction move and119873 is the number of construction moves during one forwardpass
(3) Return to Hive(4) Backward Pass starts(5) Compute the objective function for each bee and sort accordingly(6) Calculate probability or logical reasoning to continue with the computed solution and become recruiter bee(7) For every follower choose the new solution from recruiters(8) If stopping criteria is not met Go to step (2)(9) Evaluate and find the best solution(10) Output the best solution
Algorithm 1 Pseudocode of BCO
with a maximum value of the computed objective functionTo form simplex new vertex function values are computedThismethod can be calculated using four procedures namelyreflection expansion contraction and shrinkage Figure 3shows the operators of the simplex triangle in MNMM
The simplex operations in each vertex are updated closerto its optimal solution the vertices are ordered based onfitness value and ordered The best vertex is 119860119887 the secondbest vertex is 119860 119904 and the worst vertex is 119860119908 calculated basedon the objective function Let 119860 = (119909 119910) be the vertex in atriangle as food source points 119860119887 = (119909119887 119910119887) 119860 119904 = (119909119904 119910119904)and119860119908 = (119909119908 119910119908) are the positions of the food source pointsthat is local optimal points The objective functions for 119860119887119860 119904 and 119860119908 are calculated based on (23) towards the foodsource points
The objective function to construct simplex to obtainlocal search using MNMM is formulated as
119891 (119909 119910) = 1199092 minus 4119909 + 1199102 minus 119910 minus 119909119910 (23)
Based on the objective function value the vertices foodpoints are ordered ascending with their corresponding honeybee agentsThe obtained values are ordered as119860119887 le 119860 119904 le 119860119908with their honey bee position and food points in the simplextriangle Figure 4 describes the search of best-minimizedcost value for the nurse based on objective function (22)The working principle of Modified Nelder-Mead Method(MNMM) for searching food particles is explained in detail
(1) In the simplex triangle the reflection coefficient 120572expansion coefficient 120574 contraction coefficient 120573 andshrinkage coefficient 120575 are initialized
(2) The objective function for the simplex triangle ver-tices is calculated and ordered The best vertex withlower objective value is 119860119887 the second best vertex is119860 119904 and the worst vertex is named as 119860119908 and thesevertices are ordered based on the objective functionas 119860119887 le 119860 119904 le 119860119908
(3) The first two best vertices are selected namely119860119887 and119860 119904 and the construction proceeds with calculatingthe midpoint of the line segment which joins the twobest vertices that is food positions The objectivefunction decreases as the honey agent associated withthe worst position vertex moves towards best andsecond best verticesThe value decreases as the honeyagent moves towards 119860119908 to 119860119887 and 119860119908 to 119860 119904 It isfeasible to calculate the midpoint vertex 119860119898 by theline joining best and second best vertices using
119860119898 = 119860119887 + 119860 1199042 (24)
(4) A reflecting vertex 119860119903 is generated by choosing thereflection of worst point 119860119908 The objective functionvalue for 119860119903 is 119891(119860119903) which is calculated and it iscompared with worst vertex 119860119908 objective functionvalue 119891(119860119908) If 119891(119860119903) lt 119891(119860119908) proceed with step(5) the reflection vertex can be calculated using
119860119903 = 119860119898 + 120572 (119860119898 minus 119860119908) where 120572 gt 0 (25)
(5) The expansion process starts when the objectivefunction value at reflection vertex 119860119903 is lesser thanworst vertex 119860119908 119891(119860119903) lt 119891(119860119908) and the linesegment is further extended to 119860119890 through 119860119903 and119860119908 The vertex point 119860119890 is calculated by (26) If theobjective function value at119860119890 is lesser than reflectionvertex 119860119903 119891(119860119890) lt 119891(119860119903) then the expansion isaccepted and the honey bee agent has found best foodposition compared with reflection point
119860119890 = 119860119903 + 120574 (119860119903 minus 119860119898) where 120574 gt 1 (26)
(6) The contraction process is carried out when 119891(119860119903) lt119891(119860 119904) and 119891(119860119903) le 119891(119860119887) for replacing 119860119887 with
Computational Intelligence and Neuroscience 9
AwAs
Ab
(a) Simplex triangle
Ar
As
Ab
Aw
(b) Reflection
Ae
Ar
As
Ab
Aw
(c) Expansion
Ac
As
Ab
Aw
(d) Contraction (119860ℎ lt 119860119903)
Ac
As
Ab
Aw
(e) Contraction (119860119903 lt 119860ℎ)
A㰀b
A㰀s
As
Ab
Aw
(f) Shrinkage
Figure 3 Nelder-Mead operations
119860119903 If 119891(119860119903) gt 119891(119860ℎ) then the direct contractionwithout the replacement of 119860119887 with 119860119903 is performedThe contraction vertex 119860119888 can be calculated using
119860119888 = 120573119860119903 + (1 minus 120573)119860119898 where 0 lt 120573 lt 1 (27)
If 119891(119860119903) le 119891(119860119887) the contraction can be done and119860119888 replaced with 119860ℎ go to step (8) or else proceed tostep (7)
(7) The shrinkage phase proceeds when the contractionprocess at step (6) fails and is done by shrinking allthe vertices of the simplex triangle except 119860ℎ using(28) The objective function value of reflection andcontraction phase is not lesser than the worst pointthen the vertices 119860 119904 and 119860119908 must be shrunk towards119860ℎThus the vertices of smaller value will form a newsimplex triangle with another two best vertices
119860 119894 = 120575119860 119894 + 1198601 (1 minus 120575) where 0 lt 120575 lt 1 (28)
(8) The calculations are stopped when the terminationcondition is met
Algorithm 2 describes the pseudocode for ModifiedNelder-Mead Method in detail It portraits the detailed pro-cess of MNMM to obtain the best solution for the NRP Theworkflow of the proposed MNMM is explained in Figure 5
5 MODBCO
Bee Colony Optimization is the metaheuristic algorithm tosolve various combinatorial optimization problems and itis inspired by the natural behavior of bee for their foodsources The algorithm consists of two steps forward andbackward pass During forwarding pass bees started toexplore the neighborhood of its current solution and findall possible ways In backward pass bees return to thehive and share the values of the objective function of theircurrent solution Calculate nectar amount using probability
10 Computational Intelligence and Neuroscience
Ab
Aw
Ar
As
Am
d
d
Ab
Aw
Ar
As
Am
d
d
Aed2
Ab
Aw
Ar
As
Am
Ac1
Ac2
Ab
Aw As
Am
Anew
Figure 4 Bees search movement based on MNMM
function and advertise the solution the bee which has thebetter solution is given higher priority The remaining beesbased on the probability value decide whether to explore thesolution or proceed with the advertised solution DirectedBee Colony Optimization is the computational system whereseveral bees work together in uniting and interact with eachother to achieve goals based on the group decision processThe whole search area of the bee is divided into multiplefragments different bees are sent to different fragments Thebest solution in each fragment is obtained by using a localsearch algorithmModified Nelder-Mead Method (MNMM)To obtain the best solution the total varieties of individualparameters are partitioned into individual volumes Eachvolume determines the starting point of the exploration offood particle by each bee The bees use developed MNMMalgorithm to find the best solution by remembering thelast two best food sites they obtained After obtaining thecurrent solution the bee starts to backward pass sharingof information obtained during forwarding pass The beesstarted to share information about optimized point by thenatural behavior of bees called waggle dance When all theinformation about the best food is shared the best among theoptimized point is chosen using a decision-making processcalled consensus and quorummethod in honey bees [34 35]
51 Multiagent System All agents live in an environmentwhich is well structured and organized Inmultiagent systemseveral agents work together and interact with each otherto obtain the goal According to Jiao and Shi [36] andZhong et al [37] all agents should possess the followingqualities agents should live and act in an environmenteach agent should sense its local environment each agent
should be capable of interacting with other agents in a localenvironment and agents attempt to perform their goal Allagents interact with each other and take the decision toachieve the desired goals The multiagent system is a com-putational system and provides an opportunity to optimizeand compute all complex problems In multiagent system allagents start to live and act in the same environment which iswell organized and structured Each agent in the environmentis fixed on a lattice point The size and dimension of thelattice point in the environment depend upon the variablesused The objective function can be calculated based on theparameters fixed
(1) Consider ldquo119890rdquo number of independent parameters tocalculate the objective function The range of the 119892thparameter can be calculated using [119876119892119894 119876119892119891] where119876119892119894 is the initial value of the 119892th parameter and 119876119892119891is the final value of the 119892th parameter chosen
(2) Thus the objective function can be formulated as 119890number of axes each axis will contain a total rangeof single parameter with different dimensions
(3) Each axis is divided into smaller parts each partis called a step So 119892th axis can be divided into 119899119892number of steps each with the length of 119871119892 where thevalue of 119892 depends upon parameters thus 119892 = 1 to 119890The relationship between 119899119892 and 119871119892 can be given as
119899119892 = 119876119892119894 minus 119876119892119891119871119892 (29)
(4) Then each axis is divided into branches foreach branch 119892 number of branches will form an
Computational Intelligence and Neuroscience 11
Modified Nelder-Mead Method for directed honey bee food search(1) Initialization119860119887 denotes the list of vertices in simplex where 119894 = 1 2 119899 + 1120572 120574 120573 and 120575 are the coefficients of reflection expansion contraction and shrinkage119891 is the objective function to be minimized(2)Ordering
Order the vertices in simplex from lowest objective function value 119891(1198601) to highest value 119891(119860119899+1) Ordered as 1198601le 1198602 le sdot sdot sdot le 119860119899+1(3)Midpoint
Calculate the midpoint for first two best vertices in simplex 119860119898 = sum(119860 119894119899) where 119894 = 1 2 119899(4) Reflection Process
Calculate reflection point 119860119903 by 119860119903 = 119860119898 + 120572(119860119898 minus 119860119899+1)if 119891(1198601) le 119891(1198602) le 119891(119860119899) then119860119899 larr 119860119903 and Go to to Step (8)end if
(5) Expansion Processif 119891(119860119903) le 119891(1198601) thenCalculate expansion point using 119860 119890 = 119860119903 + 120574(119860119903 minus 119860119898)end ifif 119891(119860 119890) lt 119891(119860119903) then119860119899 larr 119860 119890 and Go to to Step (8)else119860119899 larr 119860119903 and Go to to Step (8)end if
(6) Contraction Processif 119891(119860119899) le 119891(119860119903) le 119891(119860119899+1) thenCompute outside contraction by 119860 119888 = 120573119860119903 + (1 minus 120573)119860119898end ifif 119891(1198601) ge 119891(119860119899+1) thenCompute inside contraction by 119860 119888 = 120573119860119899+1 + (1 minus 120573)119860119898end ifif 119891(119860119903) ge 119891(119860119899) thenContraction is done between 119860119898 and the best vertex among 119860119903 and 119860119899+1end ifif 119891(119860 119888) lt 119891(119860119903) then119860119899 larr 119860 119888 and Go to to Step (8)else goes to Step (7)end ifif 119891(119860 119888) ge 119891(119860119899+1) then119860119899+1 larr 119860 119888 and Go to to Step (8)else Go to to Step (7)end if
(7) Shrinkage ProcessShrink towards the best solution with new vertices by 119860 119894 = 120575119860 119894 + 1198601(1 minus 120575) where 119894 = 2 119899 + 1
(8) Stopping CriteriaOrder and re-label new vertices of the simplex based on their objective function and go to step (4)
Algorithm 2 Pseudocode of Modified Nelder-Mead Method
119890-dimensional volume Total number of volumes 119873Vcan be formulated using
119873V = 119890prod119892=1
119899119892 (30)
(5) The starting point of the agent in the environmentwhich is one point inside volume is chosen bycalculating themidpoint of the volumeThemidpointof the lattice can be calculated as
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ] (31)
52 Decision-Making Process A key role of the honey beesis to select the best nest site and is done by the process ofdecision-making to produce a unified decisionThey follow adistributed decision-making process to find out the neighbornest site for their food particles The pseudocode for theproposed MODBCO algorithm is shown in Algorithm 3Figure 6 explains the workflow of the proposed algorithm forthe search of food particles by honey bees using MODBCO
521 Waggle Dance The scout bees after returning from thesearch of food particle report about the quality of the foodsite by communicationmode called waggle dance Scout beesperform thewaggle dance to other quiescent bees to advertise
12 Computational Intelligence and Neuroscience
Yes
Reflectionprocess
Order and label verticesbased on f(A)
Initialization
Coefficients 훼 훾 훽 훿
Objective function f(A)
f(Ab) lt f(Ar) lt f(Aw) Aw larr Ar
f(Ae) le f(Ar)
two best verticesAm forCalculate midpoint
Start
Terminationcriteria
Stop
Ar = Am + 훼(Am minus Aw)
ExpansionprocessNo
Yesf(Ar) le f(Aw) Aw larr Ae
No
b larr true Aw larr Ar
Contractionprocess
f(Ar) ge f(An)Yes
f(Ac) lt f(Ar)Aw larr Ac
b larr false
No
Shrinkageprocess
b larr true
Yes
Yes
No
Ae = Ar + 훾(Ar minus
Am)
Ac = 훽Ar + (1 minus 훽)Am
Ai = 훿Ai + A1(1 minus 훿)
Figure 5 Workflow of Modified Nelder-Mead Method
Computational Intelligence and Neuroscience 13
Multi-Objective Directed Bee Colony Optimization(1) Initialization119891(119909) is the objective function to be minimized
Initialize 119890 number of parameters and 119871119892 length of steps where 119892 = 0 to 119890Initialize initial value and the final value of the parameter as 119876119892119894 and 119876119892119891lowastlowast Solution Representation lowastlowastThe solutions are represented in the form of Binary values which can be generated as followsFor each solution 119894 = 1 119899119883119894 = 1199091198941 1199091198942 119909119894119889 | 119889 isin total days amp 119909119894119889 = rand ge 029 forall119889End for
(2) The number of steps in each step can be calculated using
119899119892 = 119876119892119894 minus 119876119892119891119871119892(3) The total number of volumes can be calculated using119873V = 119890prod
119892=1
119899119892(4) The midpoint of the volume to calculate starting point of the exploration can be calculated using
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ](5) Explore the search volume according to the Modified Nelder-Mead Method using Algorithm 2(6) The recorded value of the optimized point in vector table using[119891(1198811) 119891(1198812) 119891(119881119873V )](7) The globally optimized point is chosen based on Bee decision-making process using Consensus and Quorum
method approach 119891(119881119892) = min [119891(1198811) 119891(1198812) 119891(119881119873V )]Algorithm 3 Pseudocode of MODBCO
their best nest site for the exploration of food source Inthe multiagent system each agent after collecting individualsolution gives it to the centralized systems To select the bestoptimal solution forminimal optimal cases themathematicalformulation can be stated as
dance119894 = min (119891119894 (119881)) (32)
This mathematical formulation will find the minimaloptimal cases among the search solution where 119891119894(119881) is thesearch value calculated by the agent The search values arerecorded in the vector table 119881 119881 is the vector which consistsof 119890 number of elements The element 119890 contains the value ofthe parameter both optimal solution and parameter valuesare recorded in the vector table
522 Consensus Theconsensus is thewidespread agreementamong the group based on voting the voting pattern ofthe scout bees is monitored periodically to know whetherit reached an agreement and started acting on the decisionpattern Honey bees use the consensus method to select thebest search value the globally optimized point is chosen bycomparing the values in the vector table The globally opti-mized points are selected using themathematical formulation
523 Quorum In quorummethod the optimum solution iscalculated as the final solution based on the threshold levelobtained by the group decision-making process When thesolution reaches the optimal threshold level 120585119902 then the solu-tion is considered as a final solution based on unison decisionprocess The quorum threshold value describes the quality of
the food particle result When the threshold value is less thecomputation time decreases but it leads to inaccurate experi-mental resultsThe threshold value should be chosen to attainless computational timewith an accurate experimental result
6 Experimental Design and Analysis
61 Performance Metrics The performance of the proposedalgorithm MODBCO is assessed by comparing with fivedifferent competitor methods Here six performance metricsare considered to investigate the significance and evaluate theexperimental results The metrics are listed in this section
611 Least Error Rate Least Error Rate (LER) is the percent-age of the difference between known optimal value and thebest value obtained The LER can be calculated using
LER () = 119903sum119894=1
OptimalNRP-Instance minus fitness119894OptimalNRP-Instance
(34)
612 Average Convergence The Average Convergence is themeasure to evaluate the quality of the generated populationon average The Average Convergence (AC) is the percentageof the average of the convergence rate of solutions The per-formance of the convergence time is increased by the AverageConvergence to exploremore solutions in the populationTheAverage Convergence is calculated usingAC
= 119903sum119894=1
1 minus Avg_fitness119894 minusOptimalNRP-InstanceOptimalNRP-Instance
lowast 100 (35)
where (119903) is the number of instances in the given dataset
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
neighborhood structure will reroster nurses to different shiftsand the second neighborhood structure will do repairingmechanism
Aickelin and Dowsland [28] proposed a technique forshift patterns they considered shift patterns with penaltypreferences and number of successive working days Theindirect genetic algorithm will generate various heuristicdecoders for shift patterns to reconstruct the shift roster forthe nurse A qualified roster is generated using decoderswith the help of the best permutations of nurses To generatebest search space solutions for the permutation of nursesthe authors used an adaptive iterative method to adjust theorder of nurses as scheduled one by one Asta et al [29] andAnwar et al [30] proposed a tensor-based hyperheuristic tosolve the NRP The authors tuned a specific group of datasetsand embedded a tensor-based machine learning algorithmA tensor-based hyperheuristic with memory managementis used to generate the best solution This approach isconsidered in life-long applications to extract knowledge anddesired behavior throughout the run time
Todorovic and Petrovic [31] proposed the Bee ColonyOptimization approach to solve the NRP all the unscheduledshifts are allocated to the available nurses in the constructivephase This algorithm combines the constructive move withlocal search to improve the quality of the solution For eachforward pass the predefined numbers of unscheduled shiftsare allocated to the nurses and discarded the solution withless improvement in the objective function The process ofintelligent reduction in neighborhood search had improvedthe current solution In construction phase unassigned shiftsare allotted to nurses and lead to violation of constraints tohigher penalties
Severalmethods have been proposed using the INRC2010dataset to solve the NRP the authors have consideredfive latest competitors to measure the effectiveness of theproposed algorithm Asaju et al [14] proposed Artificial BeeColony (ABC) algorithm to solve NRP This process is donein two phases at first heuristic based ordering of shift patternis used to generate the feasible solution In the second phaseto obtain the solution ABC algorithm is used In thismethodpremature convergence takes place and the solution getstrapped in local optima The lack of a local search algorithmof this process leads to yielding higher penalty Awadallah etal [15] developed a metaheuristic technique hybrid artificialbee colony (HABC) to solve the NRP In ABC algorithm theemployee bee phase was replaced by a hill climbing approachto increase exploitation process Use of hill climbing in ABCgenerates a higher value which leads to high computationaltime
The global best harmony search with pitch adjustmentdesign is used to tackle the NRP in [16] The author adaptedthe harmony search algorithm (HAS) in exploitation pro-cess and particle swarm optimization (PSO) in explorationprocess In HAS the solutions are generated based on threeoperator namely memory consideration random consider-ation and pitch adjustment for the improvisation processThey did two improvisations to solve the NRP multipitchadjustment to improve exploitation process and replaced ran-dom selectionwith global best to increase convergence speed
The hybrid harmony search algorithm with hill climbing isused to solve the NRP in [17] For local search metaheuristicharmony and hill climbing approach are used The memoryconsideration parameter in harmony is replaced by PSOalgorithm The derivative criteria will reduce the numberof iterations towards local minima This process considersmany parameters to construct the roster since improvisationprocess is to be at each iteration
Santos et al [18] used integer programming (IP) to solvetheNRP andproposedmonolith compact IPwith polynomialconstraints and variables The authors have used both upperand lower bounds for obtaining optimal cost They estimatedand improved lower bound values towards optimum and thismethod requires additional processing time
3 Mathematical Model
The NRP problem is a real-world problem at hospitals theproblem is to assign a predefined set of shifts (like S1-dayshift S2-noon shift S3-night shift and S4-Free-shift) of ascheduled period for a set of nurses of different preferencesand skills in each ward Figure 1 shows the illustrativeexample of the feasible nurse roster which consists of fourshifts namely day shift noon shift night shift and free shift(holiday) allocating five nurses over 11 days of scheduledperiod Each column in the scheduled table represents theday and the cell content represents the shift type allocatedto a nurse Each nurse is allocated one shift per day and thenumber of shifts is assigned based on the hospital contractsThis problem will have some variants on a number of shifttypes nurses nurse skills contracts and scheduling periodIn general both hard and soft constraints are considered forgenerating and assessing solutions
Hard constraints are the regulations which must besatisfied to achieve the feasible solution They cannot beviolated since hard constraints are demanded by hospitalregulations The hard constraints HC1 to HC5 must be filledto schedule the roster The soft constraints SC1 to SC14 aredesirable and the selection of soft constraints determines thequality of the roster Tables 1 and 2 list the set of hard andsoft constraints considered to solve the NRP This sectiondescribes the mathematical model required for hard and softconstraints extensively
The NRP consists of a set of nurses 119899 = 1 2 119873 whereeach row is specific to particular set of shifts 119904 = 1 2 119878for the given set day 119889 = 1 2 119863 The solution roster S forthe 01matrix dimension119873 lowast 119878119863 is as in
S119899119889119904 = 1 if nurse 119899 works 119904 shift for day 1198890 otherwise
(1)
HC1 In this constraint all demanded shifts are assigned to anurse
Figure 1 Illustrative example of Nurse Rostering Problem
Table 1
Hard constraintsHC1 All demanded shifts assigned to a nurseHC2 A nurse can work with only a single shift per dayHC3 The minimum number of nurses required for the shiftHC4 The total number of working days for the nurse should be between the maximum and minimum rangeHC5 A day shift followed by night shift is not allowed
Table 2
Soft constraintsSC1 The maximum number of shifts assigned to each nurseSC2 The minimum number of shifts assigned to each nurseSC3 The maximum number of consecutive working days assigned to each nurseSC4 The minimum number of consecutive working days assigned to each nurseSC5 The maximum number of consecutive working days assigned to each nurse on which no shift is allottedSC6 The minimum number of consecutive working days assigned to each nurse on which no shift is allottedSC7 The maximum number of consecutive working weekends with at least one shift assigned to each nurseSC8 The minimum number of consecutive working weekends with at least one shift assigned to each nurseSC9 The maximum number of weekends with at least one shift assigned to each nurseSC10 Specific working daySC11 Requested day offSC12 Specific shift onSC13 Specific shift offSC14 Nurse not working on the unwanted pattern
where 119864119889119904 is the number of nurses required for a day (119889) atshift (119904) and S119889119904 is the allocation of nurses in the feasiblesolution roster
HC2 In this constraint each nurse can work not more thanone shift per day
119878sum119904=1
S119904119899119889 le 1 forall119899 isin 119873 119889 isin 119863 (3)
where S119899119889 is the allocation of nurses (119899) in solution at shift (119904)for a day (119889)HC3This constraint deals with aminimumnumber of nursesrequired for each shift
119873sum119899=1
S119899119889119904 ge min119899119889119904 forall119889 isin 119863 119904 isin 119878 (4)
Computational Intelligence and Neuroscience 5
where min119899119889119904 is the minimum number of nurses required fora shift (119904) on the day (119889)HC4 In this constraint the total number of working days foreach nurse should range between minimum and maximumrange for the given scheduled period
119882min le 119863sum119889=1
119878sum119904=1
S119889119904119899 le 119882max forall119899 isin 119873 (5)
The average working shift for nurse can be determined byusing
119882avg = 1119873 (119863sum119889=1
119878sum119904=1
S119889119904119899 forall119899 isin 119873) (6)
where 119882min and 119882max are the minimum and maximumnumber of days in scheduled period and119882avg is the averageworking shift of the nurse
HC5 In this constraint shift 1 followed by shift 3 is notallowed that is a day shift followed by a night shift is notallowed
119873sum119899=1
119863sum119889=1
S1198991198891199043 + S119899119889+11199041 le 1 forall119904 isin 119878 (7)
SC1 The maximum number of shifts assigned to each nursefor the given scheduled period is as follows
max(( 119863sum119889=1
119878sum119904=1
S119889119904119899 minus Φ119906119887119899 ) 0) forall119899 isin 119873 (8)
whereΦ119906119887119899 is themaximumnumber of shifts assigned to nurse(119899)SC2 The minimum number of shifts assigned to each nursefor the given scheduled period is as follows
whereΦ119897119887119899 is theminimumnumber of shifts assigned to nurse(119899)SC3 The maximum number of consecutive working daysassigned to each nurse on which a shift is allotted for thescheduled period is as follows
Ψ119899sum119896=1
max ((C119896119899 minus Θ119906119887119899 ) 0) forall119899 isin 119873 (10)
where Θ119906119887119899 is the maximum number of consecutive workingdays of nurse (119899) Ψ119899 is the total number of consecutive
working spans of nurse (119899) in the roster and C119896119899 is the countof the 119896th working spans of nurse (119899)SC4 The minimum number of consecutive working daysassigned to each nurse on which a shift is allotted for thescheduled period is as follows
Ψ119899sum119896=1
max ((Θ119897119887119899 minus C119896119899) 0) forall119899 isin 119873 (11)
where Θ119897119887119899 is the minimum number of consecutive workingdays of nurse (119899) Ψ119899 is the total number of consecutiveworking spans of nurse (119899) in the roster and C119896119899 is the countof the 119896th working span of the nurse (119899)SC5 The maximum number of consecutive working daysassigned to each nurse on which no shift is allotted for thegiven scheduled period is as follows
Γ119899sum119896=1
max ((eth119896119899 minus 120593119906119887119899 ) 0) forall119899 isin 119873 (12)
where120593119906119887119899 is themaximumnumber of consecutive free days ofnurse (119899) Γ119899 is the total number of consecutive free workingspans of nurse (119899) in the roster and eth119896119899 is the count of the 119896thworking span of the nurse (119899)SC6 The minimum number of consecutive working daysassigned to each nurse on which no shift is allotted for thegiven scheduled period is as follows
Γ119899sum119896=1
max ((120593119897119887119899 minus eth119896119899) 0) forall119899 isin 119873 (13)
where 120593119897119887119899 is theminimumnumber of consecutive free days ofnurse (119899) Γ119899 is the total number of consecutive free workingspans of nurse (119899) in the roster and eth119896119899 is the count of the 119896thworking span of the nurse (119899)SC7 The maximum number of consecutive working week-ends with at least one shift assigned to nurse for the givenscheduled period is as follows
Υ119899sum119896=1
max ((120577119896119899 minus Ω119906119887119899 ) 0) forall119899 isin 119873 (14)
where Ω119906119887119899 is the maximum number of consecutive workingweekends of nurse (119899) Υ119899 is the total number of consecutiveworking weekend spans of nurse (119899) in the roster and 120577119896119899 isthe count of the 119896th working weekend span of the nurse (119899)SC8 The minimum number of consecutive working week-ends with at least one shift assigned to nurse for the givenscheduled period is as follows
Υ119899sum119896=1
max ((Ω119897119887119899 minus 120577119896119899) 0) forall119899 isin 119873 (15)
6 Computational Intelligence and Neuroscience
where Ω119897119887119899 is the minimum number of consecutive workingweekends of nurse (119899) Υ119899 is the total number of consecutiveworking weekend spans of nurse (119899) in the roster and 120577119896119899 isthe count of the 119896th working weekend span of the nurse (119899)SC9 The maximum number of weekends with at least oneshift assigned to nurse in four weeks is as follows
119899sum119896=1
max ((119896119899 minus 120603119906119887119899 ) 0) forall119899 isin 119873 (16)
where 119896119899 is the number of working days at the 119896th weekendof nurse (119899) 120603119906119887119899 is the maximum number of working daysfor nurse (119899) and 119899 is the total count of the weekend in thescheduling period of nurse (119899)SC10 The nurse can request working on a particular day forthe given scheduled period
where 120582119889119899 is the day request from the nurse (119899) to work on anyshift on a particular day (119889)SC11 The nurse can request that they do not work on aparticular day for the given scheduled period
where 120582119889119899 is the request from the nurse (119899) not to work on anyshift on a particular day (119889)SC12 The nurse can request working on a particular shift ona particular day for the given scheduled period
where Υ119889119904119899 is the shift request from the nurse (119899) to work ona particular shift (119904) on particular day (119889)SC13 The nurse can request that they do not work on aparticular shift on a particular day for the given scheduledperiod
where Υ119889119904119899 is the shift request from the nurse (119899) not to workon a particular shift (119904) on particular day (119889)SC14 The nurse should not work on unwanted patternsuggested for the scheduled period
984858119899sum119906=1
120583119906119899 forall119899 isin 119873 (21)
where 120583119906119899 is the total count of occurring patterns for nurse (119899)of type 119906 984858119899 is the set of unwanted patterns suggested for thenurse (119899)
The objective function of the NRP is to maximize thenurse preferences and minimize the penalty cost from vio-lations of soft constraints in (22)
Here SC refers to the set of soft constraints indexed inTable 2 119875sc(119909) refers to the penalty weight violation of thesoft constraint and 119879sc(119909) refers to the total violations of thesoft constraints in roster solution It has to be noted that theusage of penalty function [32] in the NRP is to improve theperformance and provide the fair comparison with anotheroptimization algorithm
4 Bee Colony Optimization
41 Natural Behavior of Honey Bees Swarm intelligence isan emerging discipline for the study of problems whichrequires an optimal approach rather than the traditionalapproach The use of swarm intelligence is the part ofartificial intelligence based on the study of the behavior ofsocial insects The swarm intelligence is composed of manyindividual actions using decentralized and self-organizedsystem Swarm behavior is characterized by natural behaviorof many species such as fish schools herds of animals andflocks of birds formed for the biological requirements tostay together Swarm implies the aggregation of animalssuch as birds fishes ants and bees based on the collectivebehavior The individual agents in the swarm will have astochastic behavior which depends on the local perception ofthe neighborhood The communication between any insectscan be formed with the help of the colonies and it promotescollective intelligence among the colonies
The important features of swarms are proximity qualityresponse variability stability and adaptability The proximityof the swarm must be capable of providing simple spaceand time computations and it should respond to the qualityfactorsThe swarm should allow diverse activities and shouldnot be restricted among narrow channels The swarm shouldmaintain the stability nature and should not fluctuate basedon the behaviorThe adaptability of the swarmmust be able tochange the behavior mode when required Several hundredsof bees from the swarm work together to find nesting sitesand select the best nest site Bee Colony Optimization isinspired by the natural behavior of beesThe bee optimizationalgorithm is inspired by group decision-making processesof honey bees A honey bee searches the best nest site byconsidering speed and accuracy
In a bee colony there are three different types of beesa single queen bee thousands of male drone bees andthousands of worker bees
(1) The queen bee is responsible for creating new coloniesby laying eggs
Computational Intelligence and Neuroscience 7
(2) The male drone bees mated with the queen and werediscarded from the colonies
(3) The remaining female bees in the hive are calledworker bees and they are called the building block ofthe hiveThe responsibilities of the worker bees are tofeed guard and maintain the honey bee comb
Based on the responsibility worker bees are classifiedas scout bees and forager bees A scout bee flies in searchof food sources randomly and returns when the energygets exhausted After reaching a hive scout bees share theinformation and start to explore rich food source locationswith forager bees The scout beersquos information includesdirection quality quantity and distance of the food sourcethey found The way of communicating information about afood source to foragers is done using dance There are twotypes of dance round dance and waggle dance The rounddance will provide direction of the food source when thedistance is small The waggle dance indicates the positionand the direction of the food source the distance can bemeasured by the speed of the dance A greater speed indicatesa smaller distance and the quantity of the food depends onthe wriggling of the beeThe exchange of information amonghive mates is to acquire collective knowledge Forager beeswill silently observe the behavior of scout bee to acquireknowledge about the directions and information of the foodsource
The group decision process of honey bees is for searchingbest food source and nest siteThe decision-making process isbased on the swarming process of the honey bee Swarming isthe process inwhich the queen bee and half of theworker beeswill leave their hive to explore a new colony The remainingworker bees and daughter bee will remain in the old hiveto monitor the waggle dance After leaving their parentalhive swarm bees will form a cluster in search of the newnest site The waggle dance is used to communicate withquiescent bees which are inactive in the colonyThis providesprecise information about the direction of the flower patchbased on its quality and energy level The number of followerbees increases based on the quality of the food source andallows the colony to gather food quickly and efficiently Thedecision-making process can be done in two methods byswarm bees to find the best nest site They are consensusand quorum consensus is the group agreement taken intoaccount and quorum is the decision process taken when thebee vote reaches a threshold value
Bee Colony Optimization (BCO) algorithm is apopulation-based algorithm The bees in the populationare artificial bees and each bee finds its neighboring solutionfrom the current path This algorithm has a forward andbackward process In forwarding pass every bee starts toexplore the neighborhood of its current solution and enablesconstructive and improving moves In forward pass entirebees in the hive will start the constructive move and thenlocal search will start In backward pass bees share theobjective value obtained in the forward pass The bees withhigher priority are used to discard all nonimproving movesThe bees will continue to explore in next forward pass orcontinue the same process with neighborhoodThe flowchart
Forward pass
Initialization
Construction move
Backward pass
Update the bestsolution
Stopping criteriaFalse
True
Figure 2 Flowchart of BCO algorithm
for BCO is shown in Figure 2 The BCO is proficient insolving combinatorial optimization problems by creatingcolonies of the multiagent system The pseudocode for BCOis described in Algorithm 1 The bee colony system providesa standard well-organized and well-coordinated teamworkmultitasking performance [33]
42 Modified Nelder-Mead Method The Nelder-MeadMethod is a simplex method for finding a local minimumfunction of various variables and is a local search algorithmfor unconstrained optimization problems The whole searcharea is divided into different fragments and filled with beeagents To obtain the best solution each fragment can besearched by its bee agents through Modified Nelder-MeadMethod (MNMM) Each agent in the fragments passesinformation about the optimized point using MNMMBy using NMMM the best points are obtained and thebest solution is chosen by decision-making process ofhoney bees The algorithm is a simplex-based method119863-dimensional simplex is initialized with 119863 + 1 verticesthat is two dimensions and it forms a triangle if it has threedimensions it forms a tetrahedron To assign the best andworst point the vertices are evaluated and ordered based onthe objective function
The best point or vertex is considered to the minimumvalue of the objective function and the worst point is chosen
8 Computational Intelligence and Neuroscience
Bee Colony Optimization(1) Initialization Assign every bee to an empty solution(2) Forward Pass
For every bee(21) set 119894 = 1(22) Evaluate all possible construction moves(23) Based on the evaluation choose one move using Roulette Wheel(24) 119894 = 119894 + 1 if (119894 le 119873) Go to step (22)
where 119894 is the counter for construction move and119873 is the number of construction moves during one forwardpass
(3) Return to Hive(4) Backward Pass starts(5) Compute the objective function for each bee and sort accordingly(6) Calculate probability or logical reasoning to continue with the computed solution and become recruiter bee(7) For every follower choose the new solution from recruiters(8) If stopping criteria is not met Go to step (2)(9) Evaluate and find the best solution(10) Output the best solution
Algorithm 1 Pseudocode of BCO
with a maximum value of the computed objective functionTo form simplex new vertex function values are computedThismethod can be calculated using four procedures namelyreflection expansion contraction and shrinkage Figure 3shows the operators of the simplex triangle in MNMM
The simplex operations in each vertex are updated closerto its optimal solution the vertices are ordered based onfitness value and ordered The best vertex is 119860119887 the secondbest vertex is 119860 119904 and the worst vertex is 119860119908 calculated basedon the objective function Let 119860 = (119909 119910) be the vertex in atriangle as food source points 119860119887 = (119909119887 119910119887) 119860 119904 = (119909119904 119910119904)and119860119908 = (119909119908 119910119908) are the positions of the food source pointsthat is local optimal points The objective functions for 119860119887119860 119904 and 119860119908 are calculated based on (23) towards the foodsource points
The objective function to construct simplex to obtainlocal search using MNMM is formulated as
119891 (119909 119910) = 1199092 minus 4119909 + 1199102 minus 119910 minus 119909119910 (23)
Based on the objective function value the vertices foodpoints are ordered ascending with their corresponding honeybee agentsThe obtained values are ordered as119860119887 le 119860 119904 le 119860119908with their honey bee position and food points in the simplextriangle Figure 4 describes the search of best-minimizedcost value for the nurse based on objective function (22)The working principle of Modified Nelder-Mead Method(MNMM) for searching food particles is explained in detail
(1) In the simplex triangle the reflection coefficient 120572expansion coefficient 120574 contraction coefficient 120573 andshrinkage coefficient 120575 are initialized
(2) The objective function for the simplex triangle ver-tices is calculated and ordered The best vertex withlower objective value is 119860119887 the second best vertex is119860 119904 and the worst vertex is named as 119860119908 and thesevertices are ordered based on the objective functionas 119860119887 le 119860 119904 le 119860119908
(3) The first two best vertices are selected namely119860119887 and119860 119904 and the construction proceeds with calculatingthe midpoint of the line segment which joins the twobest vertices that is food positions The objectivefunction decreases as the honey agent associated withthe worst position vertex moves towards best andsecond best verticesThe value decreases as the honeyagent moves towards 119860119908 to 119860119887 and 119860119908 to 119860 119904 It isfeasible to calculate the midpoint vertex 119860119898 by theline joining best and second best vertices using
119860119898 = 119860119887 + 119860 1199042 (24)
(4) A reflecting vertex 119860119903 is generated by choosing thereflection of worst point 119860119908 The objective functionvalue for 119860119903 is 119891(119860119903) which is calculated and it iscompared with worst vertex 119860119908 objective functionvalue 119891(119860119908) If 119891(119860119903) lt 119891(119860119908) proceed with step(5) the reflection vertex can be calculated using
119860119903 = 119860119898 + 120572 (119860119898 minus 119860119908) where 120572 gt 0 (25)
(5) The expansion process starts when the objectivefunction value at reflection vertex 119860119903 is lesser thanworst vertex 119860119908 119891(119860119903) lt 119891(119860119908) and the linesegment is further extended to 119860119890 through 119860119903 and119860119908 The vertex point 119860119890 is calculated by (26) If theobjective function value at119860119890 is lesser than reflectionvertex 119860119903 119891(119860119890) lt 119891(119860119903) then the expansion isaccepted and the honey bee agent has found best foodposition compared with reflection point
119860119890 = 119860119903 + 120574 (119860119903 minus 119860119898) where 120574 gt 1 (26)
(6) The contraction process is carried out when 119891(119860119903) lt119891(119860 119904) and 119891(119860119903) le 119891(119860119887) for replacing 119860119887 with
Computational Intelligence and Neuroscience 9
AwAs
Ab
(a) Simplex triangle
Ar
As
Ab
Aw
(b) Reflection
Ae
Ar
As
Ab
Aw
(c) Expansion
Ac
As
Ab
Aw
(d) Contraction (119860ℎ lt 119860119903)
Ac
As
Ab
Aw
(e) Contraction (119860119903 lt 119860ℎ)
A㰀b
A㰀s
As
Ab
Aw
(f) Shrinkage
Figure 3 Nelder-Mead operations
119860119903 If 119891(119860119903) gt 119891(119860ℎ) then the direct contractionwithout the replacement of 119860119887 with 119860119903 is performedThe contraction vertex 119860119888 can be calculated using
119860119888 = 120573119860119903 + (1 minus 120573)119860119898 where 0 lt 120573 lt 1 (27)
If 119891(119860119903) le 119891(119860119887) the contraction can be done and119860119888 replaced with 119860ℎ go to step (8) or else proceed tostep (7)
(7) The shrinkage phase proceeds when the contractionprocess at step (6) fails and is done by shrinking allthe vertices of the simplex triangle except 119860ℎ using(28) The objective function value of reflection andcontraction phase is not lesser than the worst pointthen the vertices 119860 119904 and 119860119908 must be shrunk towards119860ℎThus the vertices of smaller value will form a newsimplex triangle with another two best vertices
119860 119894 = 120575119860 119894 + 1198601 (1 minus 120575) where 0 lt 120575 lt 1 (28)
(8) The calculations are stopped when the terminationcondition is met
Algorithm 2 describes the pseudocode for ModifiedNelder-Mead Method in detail It portraits the detailed pro-cess of MNMM to obtain the best solution for the NRP Theworkflow of the proposed MNMM is explained in Figure 5
5 MODBCO
Bee Colony Optimization is the metaheuristic algorithm tosolve various combinatorial optimization problems and itis inspired by the natural behavior of bee for their foodsources The algorithm consists of two steps forward andbackward pass During forwarding pass bees started toexplore the neighborhood of its current solution and findall possible ways In backward pass bees return to thehive and share the values of the objective function of theircurrent solution Calculate nectar amount using probability
10 Computational Intelligence and Neuroscience
Ab
Aw
Ar
As
Am
d
d
Ab
Aw
Ar
As
Am
d
d
Aed2
Ab
Aw
Ar
As
Am
Ac1
Ac2
Ab
Aw As
Am
Anew
Figure 4 Bees search movement based on MNMM
function and advertise the solution the bee which has thebetter solution is given higher priority The remaining beesbased on the probability value decide whether to explore thesolution or proceed with the advertised solution DirectedBee Colony Optimization is the computational system whereseveral bees work together in uniting and interact with eachother to achieve goals based on the group decision processThe whole search area of the bee is divided into multiplefragments different bees are sent to different fragments Thebest solution in each fragment is obtained by using a localsearch algorithmModified Nelder-Mead Method (MNMM)To obtain the best solution the total varieties of individualparameters are partitioned into individual volumes Eachvolume determines the starting point of the exploration offood particle by each bee The bees use developed MNMMalgorithm to find the best solution by remembering thelast two best food sites they obtained After obtaining thecurrent solution the bee starts to backward pass sharingof information obtained during forwarding pass The beesstarted to share information about optimized point by thenatural behavior of bees called waggle dance When all theinformation about the best food is shared the best among theoptimized point is chosen using a decision-making processcalled consensus and quorummethod in honey bees [34 35]
51 Multiagent System All agents live in an environmentwhich is well structured and organized Inmultiagent systemseveral agents work together and interact with each otherto obtain the goal According to Jiao and Shi [36] andZhong et al [37] all agents should possess the followingqualities agents should live and act in an environmenteach agent should sense its local environment each agent
should be capable of interacting with other agents in a localenvironment and agents attempt to perform their goal Allagents interact with each other and take the decision toachieve the desired goals The multiagent system is a com-putational system and provides an opportunity to optimizeand compute all complex problems In multiagent system allagents start to live and act in the same environment which iswell organized and structured Each agent in the environmentis fixed on a lattice point The size and dimension of thelattice point in the environment depend upon the variablesused The objective function can be calculated based on theparameters fixed
(1) Consider ldquo119890rdquo number of independent parameters tocalculate the objective function The range of the 119892thparameter can be calculated using [119876119892119894 119876119892119891] where119876119892119894 is the initial value of the 119892th parameter and 119876119892119891is the final value of the 119892th parameter chosen
(2) Thus the objective function can be formulated as 119890number of axes each axis will contain a total rangeof single parameter with different dimensions
(3) Each axis is divided into smaller parts each partis called a step So 119892th axis can be divided into 119899119892number of steps each with the length of 119871119892 where thevalue of 119892 depends upon parameters thus 119892 = 1 to 119890The relationship between 119899119892 and 119871119892 can be given as
119899119892 = 119876119892119894 minus 119876119892119891119871119892 (29)
(4) Then each axis is divided into branches foreach branch 119892 number of branches will form an
Computational Intelligence and Neuroscience 11
Modified Nelder-Mead Method for directed honey bee food search(1) Initialization119860119887 denotes the list of vertices in simplex where 119894 = 1 2 119899 + 1120572 120574 120573 and 120575 are the coefficients of reflection expansion contraction and shrinkage119891 is the objective function to be minimized(2)Ordering
Order the vertices in simplex from lowest objective function value 119891(1198601) to highest value 119891(119860119899+1) Ordered as 1198601le 1198602 le sdot sdot sdot le 119860119899+1(3)Midpoint
Calculate the midpoint for first two best vertices in simplex 119860119898 = sum(119860 119894119899) where 119894 = 1 2 119899(4) Reflection Process
Calculate reflection point 119860119903 by 119860119903 = 119860119898 + 120572(119860119898 minus 119860119899+1)if 119891(1198601) le 119891(1198602) le 119891(119860119899) then119860119899 larr 119860119903 and Go to to Step (8)end if
(5) Expansion Processif 119891(119860119903) le 119891(1198601) thenCalculate expansion point using 119860 119890 = 119860119903 + 120574(119860119903 minus 119860119898)end ifif 119891(119860 119890) lt 119891(119860119903) then119860119899 larr 119860 119890 and Go to to Step (8)else119860119899 larr 119860119903 and Go to to Step (8)end if
(6) Contraction Processif 119891(119860119899) le 119891(119860119903) le 119891(119860119899+1) thenCompute outside contraction by 119860 119888 = 120573119860119903 + (1 minus 120573)119860119898end ifif 119891(1198601) ge 119891(119860119899+1) thenCompute inside contraction by 119860 119888 = 120573119860119899+1 + (1 minus 120573)119860119898end ifif 119891(119860119903) ge 119891(119860119899) thenContraction is done between 119860119898 and the best vertex among 119860119903 and 119860119899+1end ifif 119891(119860 119888) lt 119891(119860119903) then119860119899 larr 119860 119888 and Go to to Step (8)else goes to Step (7)end ifif 119891(119860 119888) ge 119891(119860119899+1) then119860119899+1 larr 119860 119888 and Go to to Step (8)else Go to to Step (7)end if
(7) Shrinkage ProcessShrink towards the best solution with new vertices by 119860 119894 = 120575119860 119894 + 1198601(1 minus 120575) where 119894 = 2 119899 + 1
(8) Stopping CriteriaOrder and re-label new vertices of the simplex based on their objective function and go to step (4)
Algorithm 2 Pseudocode of Modified Nelder-Mead Method
119890-dimensional volume Total number of volumes 119873Vcan be formulated using
119873V = 119890prod119892=1
119899119892 (30)
(5) The starting point of the agent in the environmentwhich is one point inside volume is chosen bycalculating themidpoint of the volumeThemidpointof the lattice can be calculated as
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ] (31)
52 Decision-Making Process A key role of the honey beesis to select the best nest site and is done by the process ofdecision-making to produce a unified decisionThey follow adistributed decision-making process to find out the neighbornest site for their food particles The pseudocode for theproposed MODBCO algorithm is shown in Algorithm 3Figure 6 explains the workflow of the proposed algorithm forthe search of food particles by honey bees using MODBCO
521 Waggle Dance The scout bees after returning from thesearch of food particle report about the quality of the foodsite by communicationmode called waggle dance Scout beesperform thewaggle dance to other quiescent bees to advertise
12 Computational Intelligence and Neuroscience
Yes
Reflectionprocess
Order and label verticesbased on f(A)
Initialization
Coefficients 훼 훾 훽 훿
Objective function f(A)
f(Ab) lt f(Ar) lt f(Aw) Aw larr Ar
f(Ae) le f(Ar)
two best verticesAm forCalculate midpoint
Start
Terminationcriteria
Stop
Ar = Am + 훼(Am minus Aw)
ExpansionprocessNo
Yesf(Ar) le f(Aw) Aw larr Ae
No
b larr true Aw larr Ar
Contractionprocess
f(Ar) ge f(An)Yes
f(Ac) lt f(Ar)Aw larr Ac
b larr false
No
Shrinkageprocess
b larr true
Yes
Yes
No
Ae = Ar + 훾(Ar minus
Am)
Ac = 훽Ar + (1 minus 훽)Am
Ai = 훿Ai + A1(1 minus 훿)
Figure 5 Workflow of Modified Nelder-Mead Method
Computational Intelligence and Neuroscience 13
Multi-Objective Directed Bee Colony Optimization(1) Initialization119891(119909) is the objective function to be minimized
Initialize 119890 number of parameters and 119871119892 length of steps where 119892 = 0 to 119890Initialize initial value and the final value of the parameter as 119876119892119894 and 119876119892119891lowastlowast Solution Representation lowastlowastThe solutions are represented in the form of Binary values which can be generated as followsFor each solution 119894 = 1 119899119883119894 = 1199091198941 1199091198942 119909119894119889 | 119889 isin total days amp 119909119894119889 = rand ge 029 forall119889End for
(2) The number of steps in each step can be calculated using
119899119892 = 119876119892119894 minus 119876119892119891119871119892(3) The total number of volumes can be calculated using119873V = 119890prod
119892=1
119899119892(4) The midpoint of the volume to calculate starting point of the exploration can be calculated using
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ](5) Explore the search volume according to the Modified Nelder-Mead Method using Algorithm 2(6) The recorded value of the optimized point in vector table using[119891(1198811) 119891(1198812) 119891(119881119873V )](7) The globally optimized point is chosen based on Bee decision-making process using Consensus and Quorum
method approach 119891(119881119892) = min [119891(1198811) 119891(1198812) 119891(119881119873V )]Algorithm 3 Pseudocode of MODBCO
their best nest site for the exploration of food source Inthe multiagent system each agent after collecting individualsolution gives it to the centralized systems To select the bestoptimal solution forminimal optimal cases themathematicalformulation can be stated as
dance119894 = min (119891119894 (119881)) (32)
This mathematical formulation will find the minimaloptimal cases among the search solution where 119891119894(119881) is thesearch value calculated by the agent The search values arerecorded in the vector table 119881 119881 is the vector which consistsof 119890 number of elements The element 119890 contains the value ofthe parameter both optimal solution and parameter valuesare recorded in the vector table
522 Consensus Theconsensus is thewidespread agreementamong the group based on voting the voting pattern ofthe scout bees is monitored periodically to know whetherit reached an agreement and started acting on the decisionpattern Honey bees use the consensus method to select thebest search value the globally optimized point is chosen bycomparing the values in the vector table The globally opti-mized points are selected using themathematical formulation
523 Quorum In quorummethod the optimum solution iscalculated as the final solution based on the threshold levelobtained by the group decision-making process When thesolution reaches the optimal threshold level 120585119902 then the solu-tion is considered as a final solution based on unison decisionprocess The quorum threshold value describes the quality of
the food particle result When the threshold value is less thecomputation time decreases but it leads to inaccurate experi-mental resultsThe threshold value should be chosen to attainless computational timewith an accurate experimental result
6 Experimental Design and Analysis
61 Performance Metrics The performance of the proposedalgorithm MODBCO is assessed by comparing with fivedifferent competitor methods Here six performance metricsare considered to investigate the significance and evaluate theexperimental results The metrics are listed in this section
611 Least Error Rate Least Error Rate (LER) is the percent-age of the difference between known optimal value and thebest value obtained The LER can be calculated using
LER () = 119903sum119894=1
OptimalNRP-Instance minus fitness119894OptimalNRP-Instance
(34)
612 Average Convergence The Average Convergence is themeasure to evaluate the quality of the generated populationon average The Average Convergence (AC) is the percentageof the average of the convergence rate of solutions The per-formance of the convergence time is increased by the AverageConvergence to exploremore solutions in the populationTheAverage Convergence is calculated usingAC
= 119903sum119894=1
1 minus Avg_fitness119894 minusOptimalNRP-InstanceOptimalNRP-Instance
lowast 100 (35)
where (119903) is the number of instances in the given dataset
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
Figure 1 Illustrative example of Nurse Rostering Problem
Table 1
Hard constraintsHC1 All demanded shifts assigned to a nurseHC2 A nurse can work with only a single shift per dayHC3 The minimum number of nurses required for the shiftHC4 The total number of working days for the nurse should be between the maximum and minimum rangeHC5 A day shift followed by night shift is not allowed
Table 2
Soft constraintsSC1 The maximum number of shifts assigned to each nurseSC2 The minimum number of shifts assigned to each nurseSC3 The maximum number of consecutive working days assigned to each nurseSC4 The minimum number of consecutive working days assigned to each nurseSC5 The maximum number of consecutive working days assigned to each nurse on which no shift is allottedSC6 The minimum number of consecutive working days assigned to each nurse on which no shift is allottedSC7 The maximum number of consecutive working weekends with at least one shift assigned to each nurseSC8 The minimum number of consecutive working weekends with at least one shift assigned to each nurseSC9 The maximum number of weekends with at least one shift assigned to each nurseSC10 Specific working daySC11 Requested day offSC12 Specific shift onSC13 Specific shift offSC14 Nurse not working on the unwanted pattern
where 119864119889119904 is the number of nurses required for a day (119889) atshift (119904) and S119889119904 is the allocation of nurses in the feasiblesolution roster
HC2 In this constraint each nurse can work not more thanone shift per day
119878sum119904=1
S119904119899119889 le 1 forall119899 isin 119873 119889 isin 119863 (3)
where S119899119889 is the allocation of nurses (119899) in solution at shift (119904)for a day (119889)HC3This constraint deals with aminimumnumber of nursesrequired for each shift
119873sum119899=1
S119899119889119904 ge min119899119889119904 forall119889 isin 119863 119904 isin 119878 (4)
Computational Intelligence and Neuroscience 5
where min119899119889119904 is the minimum number of nurses required fora shift (119904) on the day (119889)HC4 In this constraint the total number of working days foreach nurse should range between minimum and maximumrange for the given scheduled period
119882min le 119863sum119889=1
119878sum119904=1
S119889119904119899 le 119882max forall119899 isin 119873 (5)
The average working shift for nurse can be determined byusing
119882avg = 1119873 (119863sum119889=1
119878sum119904=1
S119889119904119899 forall119899 isin 119873) (6)
where 119882min and 119882max are the minimum and maximumnumber of days in scheduled period and119882avg is the averageworking shift of the nurse
HC5 In this constraint shift 1 followed by shift 3 is notallowed that is a day shift followed by a night shift is notallowed
119873sum119899=1
119863sum119889=1
S1198991198891199043 + S119899119889+11199041 le 1 forall119904 isin 119878 (7)
SC1 The maximum number of shifts assigned to each nursefor the given scheduled period is as follows
max(( 119863sum119889=1
119878sum119904=1
S119889119904119899 minus Φ119906119887119899 ) 0) forall119899 isin 119873 (8)
whereΦ119906119887119899 is themaximumnumber of shifts assigned to nurse(119899)SC2 The minimum number of shifts assigned to each nursefor the given scheduled period is as follows
whereΦ119897119887119899 is theminimumnumber of shifts assigned to nurse(119899)SC3 The maximum number of consecutive working daysassigned to each nurse on which a shift is allotted for thescheduled period is as follows
Ψ119899sum119896=1
max ((C119896119899 minus Θ119906119887119899 ) 0) forall119899 isin 119873 (10)
where Θ119906119887119899 is the maximum number of consecutive workingdays of nurse (119899) Ψ119899 is the total number of consecutive
working spans of nurse (119899) in the roster and C119896119899 is the countof the 119896th working spans of nurse (119899)SC4 The minimum number of consecutive working daysassigned to each nurse on which a shift is allotted for thescheduled period is as follows
Ψ119899sum119896=1
max ((Θ119897119887119899 minus C119896119899) 0) forall119899 isin 119873 (11)
where Θ119897119887119899 is the minimum number of consecutive workingdays of nurse (119899) Ψ119899 is the total number of consecutiveworking spans of nurse (119899) in the roster and C119896119899 is the countof the 119896th working span of the nurse (119899)SC5 The maximum number of consecutive working daysassigned to each nurse on which no shift is allotted for thegiven scheduled period is as follows
Γ119899sum119896=1
max ((eth119896119899 minus 120593119906119887119899 ) 0) forall119899 isin 119873 (12)
where120593119906119887119899 is themaximumnumber of consecutive free days ofnurse (119899) Γ119899 is the total number of consecutive free workingspans of nurse (119899) in the roster and eth119896119899 is the count of the 119896thworking span of the nurse (119899)SC6 The minimum number of consecutive working daysassigned to each nurse on which no shift is allotted for thegiven scheduled period is as follows
Γ119899sum119896=1
max ((120593119897119887119899 minus eth119896119899) 0) forall119899 isin 119873 (13)
where 120593119897119887119899 is theminimumnumber of consecutive free days ofnurse (119899) Γ119899 is the total number of consecutive free workingspans of nurse (119899) in the roster and eth119896119899 is the count of the 119896thworking span of the nurse (119899)SC7 The maximum number of consecutive working week-ends with at least one shift assigned to nurse for the givenscheduled period is as follows
Υ119899sum119896=1
max ((120577119896119899 minus Ω119906119887119899 ) 0) forall119899 isin 119873 (14)
where Ω119906119887119899 is the maximum number of consecutive workingweekends of nurse (119899) Υ119899 is the total number of consecutiveworking weekend spans of nurse (119899) in the roster and 120577119896119899 isthe count of the 119896th working weekend span of the nurse (119899)SC8 The minimum number of consecutive working week-ends with at least one shift assigned to nurse for the givenscheduled period is as follows
Υ119899sum119896=1
max ((Ω119897119887119899 minus 120577119896119899) 0) forall119899 isin 119873 (15)
6 Computational Intelligence and Neuroscience
where Ω119897119887119899 is the minimum number of consecutive workingweekends of nurse (119899) Υ119899 is the total number of consecutiveworking weekend spans of nurse (119899) in the roster and 120577119896119899 isthe count of the 119896th working weekend span of the nurse (119899)SC9 The maximum number of weekends with at least oneshift assigned to nurse in four weeks is as follows
119899sum119896=1
max ((119896119899 minus 120603119906119887119899 ) 0) forall119899 isin 119873 (16)
where 119896119899 is the number of working days at the 119896th weekendof nurse (119899) 120603119906119887119899 is the maximum number of working daysfor nurse (119899) and 119899 is the total count of the weekend in thescheduling period of nurse (119899)SC10 The nurse can request working on a particular day forthe given scheduled period
where 120582119889119899 is the day request from the nurse (119899) to work on anyshift on a particular day (119889)SC11 The nurse can request that they do not work on aparticular day for the given scheduled period
where 120582119889119899 is the request from the nurse (119899) not to work on anyshift on a particular day (119889)SC12 The nurse can request working on a particular shift ona particular day for the given scheduled period
where Υ119889119904119899 is the shift request from the nurse (119899) to work ona particular shift (119904) on particular day (119889)SC13 The nurse can request that they do not work on aparticular shift on a particular day for the given scheduledperiod
where Υ119889119904119899 is the shift request from the nurse (119899) not to workon a particular shift (119904) on particular day (119889)SC14 The nurse should not work on unwanted patternsuggested for the scheduled period
984858119899sum119906=1
120583119906119899 forall119899 isin 119873 (21)
where 120583119906119899 is the total count of occurring patterns for nurse (119899)of type 119906 984858119899 is the set of unwanted patterns suggested for thenurse (119899)
The objective function of the NRP is to maximize thenurse preferences and minimize the penalty cost from vio-lations of soft constraints in (22)
Here SC refers to the set of soft constraints indexed inTable 2 119875sc(119909) refers to the penalty weight violation of thesoft constraint and 119879sc(119909) refers to the total violations of thesoft constraints in roster solution It has to be noted that theusage of penalty function [32] in the NRP is to improve theperformance and provide the fair comparison with anotheroptimization algorithm
4 Bee Colony Optimization
41 Natural Behavior of Honey Bees Swarm intelligence isan emerging discipline for the study of problems whichrequires an optimal approach rather than the traditionalapproach The use of swarm intelligence is the part ofartificial intelligence based on the study of the behavior ofsocial insects The swarm intelligence is composed of manyindividual actions using decentralized and self-organizedsystem Swarm behavior is characterized by natural behaviorof many species such as fish schools herds of animals andflocks of birds formed for the biological requirements tostay together Swarm implies the aggregation of animalssuch as birds fishes ants and bees based on the collectivebehavior The individual agents in the swarm will have astochastic behavior which depends on the local perception ofthe neighborhood The communication between any insectscan be formed with the help of the colonies and it promotescollective intelligence among the colonies
The important features of swarms are proximity qualityresponse variability stability and adaptability The proximityof the swarm must be capable of providing simple spaceand time computations and it should respond to the qualityfactorsThe swarm should allow diverse activities and shouldnot be restricted among narrow channels The swarm shouldmaintain the stability nature and should not fluctuate basedon the behaviorThe adaptability of the swarmmust be able tochange the behavior mode when required Several hundredsof bees from the swarm work together to find nesting sitesand select the best nest site Bee Colony Optimization isinspired by the natural behavior of beesThe bee optimizationalgorithm is inspired by group decision-making processesof honey bees A honey bee searches the best nest site byconsidering speed and accuracy
In a bee colony there are three different types of beesa single queen bee thousands of male drone bees andthousands of worker bees
(1) The queen bee is responsible for creating new coloniesby laying eggs
Computational Intelligence and Neuroscience 7
(2) The male drone bees mated with the queen and werediscarded from the colonies
(3) The remaining female bees in the hive are calledworker bees and they are called the building block ofthe hiveThe responsibilities of the worker bees are tofeed guard and maintain the honey bee comb
Based on the responsibility worker bees are classifiedas scout bees and forager bees A scout bee flies in searchof food sources randomly and returns when the energygets exhausted After reaching a hive scout bees share theinformation and start to explore rich food source locationswith forager bees The scout beersquos information includesdirection quality quantity and distance of the food sourcethey found The way of communicating information about afood source to foragers is done using dance There are twotypes of dance round dance and waggle dance The rounddance will provide direction of the food source when thedistance is small The waggle dance indicates the positionand the direction of the food source the distance can bemeasured by the speed of the dance A greater speed indicatesa smaller distance and the quantity of the food depends onthe wriggling of the beeThe exchange of information amonghive mates is to acquire collective knowledge Forager beeswill silently observe the behavior of scout bee to acquireknowledge about the directions and information of the foodsource
The group decision process of honey bees is for searchingbest food source and nest siteThe decision-making process isbased on the swarming process of the honey bee Swarming isthe process inwhich the queen bee and half of theworker beeswill leave their hive to explore a new colony The remainingworker bees and daughter bee will remain in the old hiveto monitor the waggle dance After leaving their parentalhive swarm bees will form a cluster in search of the newnest site The waggle dance is used to communicate withquiescent bees which are inactive in the colonyThis providesprecise information about the direction of the flower patchbased on its quality and energy level The number of followerbees increases based on the quality of the food source andallows the colony to gather food quickly and efficiently Thedecision-making process can be done in two methods byswarm bees to find the best nest site They are consensusand quorum consensus is the group agreement taken intoaccount and quorum is the decision process taken when thebee vote reaches a threshold value
Bee Colony Optimization (BCO) algorithm is apopulation-based algorithm The bees in the populationare artificial bees and each bee finds its neighboring solutionfrom the current path This algorithm has a forward andbackward process In forwarding pass every bee starts toexplore the neighborhood of its current solution and enablesconstructive and improving moves In forward pass entirebees in the hive will start the constructive move and thenlocal search will start In backward pass bees share theobjective value obtained in the forward pass The bees withhigher priority are used to discard all nonimproving movesThe bees will continue to explore in next forward pass orcontinue the same process with neighborhoodThe flowchart
Forward pass
Initialization
Construction move
Backward pass
Update the bestsolution
Stopping criteriaFalse
True
Figure 2 Flowchart of BCO algorithm
for BCO is shown in Figure 2 The BCO is proficient insolving combinatorial optimization problems by creatingcolonies of the multiagent system The pseudocode for BCOis described in Algorithm 1 The bee colony system providesa standard well-organized and well-coordinated teamworkmultitasking performance [33]
42 Modified Nelder-Mead Method The Nelder-MeadMethod is a simplex method for finding a local minimumfunction of various variables and is a local search algorithmfor unconstrained optimization problems The whole searcharea is divided into different fragments and filled with beeagents To obtain the best solution each fragment can besearched by its bee agents through Modified Nelder-MeadMethod (MNMM) Each agent in the fragments passesinformation about the optimized point using MNMMBy using NMMM the best points are obtained and thebest solution is chosen by decision-making process ofhoney bees The algorithm is a simplex-based method119863-dimensional simplex is initialized with 119863 + 1 verticesthat is two dimensions and it forms a triangle if it has threedimensions it forms a tetrahedron To assign the best andworst point the vertices are evaluated and ordered based onthe objective function
The best point or vertex is considered to the minimumvalue of the objective function and the worst point is chosen
8 Computational Intelligence and Neuroscience
Bee Colony Optimization(1) Initialization Assign every bee to an empty solution(2) Forward Pass
For every bee(21) set 119894 = 1(22) Evaluate all possible construction moves(23) Based on the evaluation choose one move using Roulette Wheel(24) 119894 = 119894 + 1 if (119894 le 119873) Go to step (22)
where 119894 is the counter for construction move and119873 is the number of construction moves during one forwardpass
(3) Return to Hive(4) Backward Pass starts(5) Compute the objective function for each bee and sort accordingly(6) Calculate probability or logical reasoning to continue with the computed solution and become recruiter bee(7) For every follower choose the new solution from recruiters(8) If stopping criteria is not met Go to step (2)(9) Evaluate and find the best solution(10) Output the best solution
Algorithm 1 Pseudocode of BCO
with a maximum value of the computed objective functionTo form simplex new vertex function values are computedThismethod can be calculated using four procedures namelyreflection expansion contraction and shrinkage Figure 3shows the operators of the simplex triangle in MNMM
The simplex operations in each vertex are updated closerto its optimal solution the vertices are ordered based onfitness value and ordered The best vertex is 119860119887 the secondbest vertex is 119860 119904 and the worst vertex is 119860119908 calculated basedon the objective function Let 119860 = (119909 119910) be the vertex in atriangle as food source points 119860119887 = (119909119887 119910119887) 119860 119904 = (119909119904 119910119904)and119860119908 = (119909119908 119910119908) are the positions of the food source pointsthat is local optimal points The objective functions for 119860119887119860 119904 and 119860119908 are calculated based on (23) towards the foodsource points
The objective function to construct simplex to obtainlocal search using MNMM is formulated as
119891 (119909 119910) = 1199092 minus 4119909 + 1199102 minus 119910 minus 119909119910 (23)
Based on the objective function value the vertices foodpoints are ordered ascending with their corresponding honeybee agentsThe obtained values are ordered as119860119887 le 119860 119904 le 119860119908with their honey bee position and food points in the simplextriangle Figure 4 describes the search of best-minimizedcost value for the nurse based on objective function (22)The working principle of Modified Nelder-Mead Method(MNMM) for searching food particles is explained in detail
(1) In the simplex triangle the reflection coefficient 120572expansion coefficient 120574 contraction coefficient 120573 andshrinkage coefficient 120575 are initialized
(2) The objective function for the simplex triangle ver-tices is calculated and ordered The best vertex withlower objective value is 119860119887 the second best vertex is119860 119904 and the worst vertex is named as 119860119908 and thesevertices are ordered based on the objective functionas 119860119887 le 119860 119904 le 119860119908
(3) The first two best vertices are selected namely119860119887 and119860 119904 and the construction proceeds with calculatingthe midpoint of the line segment which joins the twobest vertices that is food positions The objectivefunction decreases as the honey agent associated withthe worst position vertex moves towards best andsecond best verticesThe value decreases as the honeyagent moves towards 119860119908 to 119860119887 and 119860119908 to 119860 119904 It isfeasible to calculate the midpoint vertex 119860119898 by theline joining best and second best vertices using
119860119898 = 119860119887 + 119860 1199042 (24)
(4) A reflecting vertex 119860119903 is generated by choosing thereflection of worst point 119860119908 The objective functionvalue for 119860119903 is 119891(119860119903) which is calculated and it iscompared with worst vertex 119860119908 objective functionvalue 119891(119860119908) If 119891(119860119903) lt 119891(119860119908) proceed with step(5) the reflection vertex can be calculated using
119860119903 = 119860119898 + 120572 (119860119898 minus 119860119908) where 120572 gt 0 (25)
(5) The expansion process starts when the objectivefunction value at reflection vertex 119860119903 is lesser thanworst vertex 119860119908 119891(119860119903) lt 119891(119860119908) and the linesegment is further extended to 119860119890 through 119860119903 and119860119908 The vertex point 119860119890 is calculated by (26) If theobjective function value at119860119890 is lesser than reflectionvertex 119860119903 119891(119860119890) lt 119891(119860119903) then the expansion isaccepted and the honey bee agent has found best foodposition compared with reflection point
119860119890 = 119860119903 + 120574 (119860119903 minus 119860119898) where 120574 gt 1 (26)
(6) The contraction process is carried out when 119891(119860119903) lt119891(119860 119904) and 119891(119860119903) le 119891(119860119887) for replacing 119860119887 with
Computational Intelligence and Neuroscience 9
AwAs
Ab
(a) Simplex triangle
Ar
As
Ab
Aw
(b) Reflection
Ae
Ar
As
Ab
Aw
(c) Expansion
Ac
As
Ab
Aw
(d) Contraction (119860ℎ lt 119860119903)
Ac
As
Ab
Aw
(e) Contraction (119860119903 lt 119860ℎ)
A㰀b
A㰀s
As
Ab
Aw
(f) Shrinkage
Figure 3 Nelder-Mead operations
119860119903 If 119891(119860119903) gt 119891(119860ℎ) then the direct contractionwithout the replacement of 119860119887 with 119860119903 is performedThe contraction vertex 119860119888 can be calculated using
119860119888 = 120573119860119903 + (1 minus 120573)119860119898 where 0 lt 120573 lt 1 (27)
If 119891(119860119903) le 119891(119860119887) the contraction can be done and119860119888 replaced with 119860ℎ go to step (8) or else proceed tostep (7)
(7) The shrinkage phase proceeds when the contractionprocess at step (6) fails and is done by shrinking allthe vertices of the simplex triangle except 119860ℎ using(28) The objective function value of reflection andcontraction phase is not lesser than the worst pointthen the vertices 119860 119904 and 119860119908 must be shrunk towards119860ℎThus the vertices of smaller value will form a newsimplex triangle with another two best vertices
119860 119894 = 120575119860 119894 + 1198601 (1 minus 120575) where 0 lt 120575 lt 1 (28)
(8) The calculations are stopped when the terminationcondition is met
Algorithm 2 describes the pseudocode for ModifiedNelder-Mead Method in detail It portraits the detailed pro-cess of MNMM to obtain the best solution for the NRP Theworkflow of the proposed MNMM is explained in Figure 5
5 MODBCO
Bee Colony Optimization is the metaheuristic algorithm tosolve various combinatorial optimization problems and itis inspired by the natural behavior of bee for their foodsources The algorithm consists of two steps forward andbackward pass During forwarding pass bees started toexplore the neighborhood of its current solution and findall possible ways In backward pass bees return to thehive and share the values of the objective function of theircurrent solution Calculate nectar amount using probability
10 Computational Intelligence and Neuroscience
Ab
Aw
Ar
As
Am
d
d
Ab
Aw
Ar
As
Am
d
d
Aed2
Ab
Aw
Ar
As
Am
Ac1
Ac2
Ab
Aw As
Am
Anew
Figure 4 Bees search movement based on MNMM
function and advertise the solution the bee which has thebetter solution is given higher priority The remaining beesbased on the probability value decide whether to explore thesolution or proceed with the advertised solution DirectedBee Colony Optimization is the computational system whereseveral bees work together in uniting and interact with eachother to achieve goals based on the group decision processThe whole search area of the bee is divided into multiplefragments different bees are sent to different fragments Thebest solution in each fragment is obtained by using a localsearch algorithmModified Nelder-Mead Method (MNMM)To obtain the best solution the total varieties of individualparameters are partitioned into individual volumes Eachvolume determines the starting point of the exploration offood particle by each bee The bees use developed MNMMalgorithm to find the best solution by remembering thelast two best food sites they obtained After obtaining thecurrent solution the bee starts to backward pass sharingof information obtained during forwarding pass The beesstarted to share information about optimized point by thenatural behavior of bees called waggle dance When all theinformation about the best food is shared the best among theoptimized point is chosen using a decision-making processcalled consensus and quorummethod in honey bees [34 35]
51 Multiagent System All agents live in an environmentwhich is well structured and organized Inmultiagent systemseveral agents work together and interact with each otherto obtain the goal According to Jiao and Shi [36] andZhong et al [37] all agents should possess the followingqualities agents should live and act in an environmenteach agent should sense its local environment each agent
should be capable of interacting with other agents in a localenvironment and agents attempt to perform their goal Allagents interact with each other and take the decision toachieve the desired goals The multiagent system is a com-putational system and provides an opportunity to optimizeand compute all complex problems In multiagent system allagents start to live and act in the same environment which iswell organized and structured Each agent in the environmentis fixed on a lattice point The size and dimension of thelattice point in the environment depend upon the variablesused The objective function can be calculated based on theparameters fixed
(1) Consider ldquo119890rdquo number of independent parameters tocalculate the objective function The range of the 119892thparameter can be calculated using [119876119892119894 119876119892119891] where119876119892119894 is the initial value of the 119892th parameter and 119876119892119891is the final value of the 119892th parameter chosen
(2) Thus the objective function can be formulated as 119890number of axes each axis will contain a total rangeof single parameter with different dimensions
(3) Each axis is divided into smaller parts each partis called a step So 119892th axis can be divided into 119899119892number of steps each with the length of 119871119892 where thevalue of 119892 depends upon parameters thus 119892 = 1 to 119890The relationship between 119899119892 and 119871119892 can be given as
119899119892 = 119876119892119894 minus 119876119892119891119871119892 (29)
(4) Then each axis is divided into branches foreach branch 119892 number of branches will form an
Computational Intelligence and Neuroscience 11
Modified Nelder-Mead Method for directed honey bee food search(1) Initialization119860119887 denotes the list of vertices in simplex where 119894 = 1 2 119899 + 1120572 120574 120573 and 120575 are the coefficients of reflection expansion contraction and shrinkage119891 is the objective function to be minimized(2)Ordering
Order the vertices in simplex from lowest objective function value 119891(1198601) to highest value 119891(119860119899+1) Ordered as 1198601le 1198602 le sdot sdot sdot le 119860119899+1(3)Midpoint
Calculate the midpoint for first two best vertices in simplex 119860119898 = sum(119860 119894119899) where 119894 = 1 2 119899(4) Reflection Process
Calculate reflection point 119860119903 by 119860119903 = 119860119898 + 120572(119860119898 minus 119860119899+1)if 119891(1198601) le 119891(1198602) le 119891(119860119899) then119860119899 larr 119860119903 and Go to to Step (8)end if
(5) Expansion Processif 119891(119860119903) le 119891(1198601) thenCalculate expansion point using 119860 119890 = 119860119903 + 120574(119860119903 minus 119860119898)end ifif 119891(119860 119890) lt 119891(119860119903) then119860119899 larr 119860 119890 and Go to to Step (8)else119860119899 larr 119860119903 and Go to to Step (8)end if
(6) Contraction Processif 119891(119860119899) le 119891(119860119903) le 119891(119860119899+1) thenCompute outside contraction by 119860 119888 = 120573119860119903 + (1 minus 120573)119860119898end ifif 119891(1198601) ge 119891(119860119899+1) thenCompute inside contraction by 119860 119888 = 120573119860119899+1 + (1 minus 120573)119860119898end ifif 119891(119860119903) ge 119891(119860119899) thenContraction is done between 119860119898 and the best vertex among 119860119903 and 119860119899+1end ifif 119891(119860 119888) lt 119891(119860119903) then119860119899 larr 119860 119888 and Go to to Step (8)else goes to Step (7)end ifif 119891(119860 119888) ge 119891(119860119899+1) then119860119899+1 larr 119860 119888 and Go to to Step (8)else Go to to Step (7)end if
(7) Shrinkage ProcessShrink towards the best solution with new vertices by 119860 119894 = 120575119860 119894 + 1198601(1 minus 120575) where 119894 = 2 119899 + 1
(8) Stopping CriteriaOrder and re-label new vertices of the simplex based on their objective function and go to step (4)
Algorithm 2 Pseudocode of Modified Nelder-Mead Method
119890-dimensional volume Total number of volumes 119873Vcan be formulated using
119873V = 119890prod119892=1
119899119892 (30)
(5) The starting point of the agent in the environmentwhich is one point inside volume is chosen bycalculating themidpoint of the volumeThemidpointof the lattice can be calculated as
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ] (31)
52 Decision-Making Process A key role of the honey beesis to select the best nest site and is done by the process ofdecision-making to produce a unified decisionThey follow adistributed decision-making process to find out the neighbornest site for their food particles The pseudocode for theproposed MODBCO algorithm is shown in Algorithm 3Figure 6 explains the workflow of the proposed algorithm forthe search of food particles by honey bees using MODBCO
521 Waggle Dance The scout bees after returning from thesearch of food particle report about the quality of the foodsite by communicationmode called waggle dance Scout beesperform thewaggle dance to other quiescent bees to advertise
12 Computational Intelligence and Neuroscience
Yes
Reflectionprocess
Order and label verticesbased on f(A)
Initialization
Coefficients 훼 훾 훽 훿
Objective function f(A)
f(Ab) lt f(Ar) lt f(Aw) Aw larr Ar
f(Ae) le f(Ar)
two best verticesAm forCalculate midpoint
Start
Terminationcriteria
Stop
Ar = Am + 훼(Am minus Aw)
ExpansionprocessNo
Yesf(Ar) le f(Aw) Aw larr Ae
No
b larr true Aw larr Ar
Contractionprocess
f(Ar) ge f(An)Yes
f(Ac) lt f(Ar)Aw larr Ac
b larr false
No
Shrinkageprocess
b larr true
Yes
Yes
No
Ae = Ar + 훾(Ar minus
Am)
Ac = 훽Ar + (1 minus 훽)Am
Ai = 훿Ai + A1(1 minus 훿)
Figure 5 Workflow of Modified Nelder-Mead Method
Computational Intelligence and Neuroscience 13
Multi-Objective Directed Bee Colony Optimization(1) Initialization119891(119909) is the objective function to be minimized
Initialize 119890 number of parameters and 119871119892 length of steps where 119892 = 0 to 119890Initialize initial value and the final value of the parameter as 119876119892119894 and 119876119892119891lowastlowast Solution Representation lowastlowastThe solutions are represented in the form of Binary values which can be generated as followsFor each solution 119894 = 1 119899119883119894 = 1199091198941 1199091198942 119909119894119889 | 119889 isin total days amp 119909119894119889 = rand ge 029 forall119889End for
(2) The number of steps in each step can be calculated using
119899119892 = 119876119892119894 minus 119876119892119891119871119892(3) The total number of volumes can be calculated using119873V = 119890prod
119892=1
119899119892(4) The midpoint of the volume to calculate starting point of the exploration can be calculated using
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ](5) Explore the search volume according to the Modified Nelder-Mead Method using Algorithm 2(6) The recorded value of the optimized point in vector table using[119891(1198811) 119891(1198812) 119891(119881119873V )](7) The globally optimized point is chosen based on Bee decision-making process using Consensus and Quorum
method approach 119891(119881119892) = min [119891(1198811) 119891(1198812) 119891(119881119873V )]Algorithm 3 Pseudocode of MODBCO
their best nest site for the exploration of food source Inthe multiagent system each agent after collecting individualsolution gives it to the centralized systems To select the bestoptimal solution forminimal optimal cases themathematicalformulation can be stated as
dance119894 = min (119891119894 (119881)) (32)
This mathematical formulation will find the minimaloptimal cases among the search solution where 119891119894(119881) is thesearch value calculated by the agent The search values arerecorded in the vector table 119881 119881 is the vector which consistsof 119890 number of elements The element 119890 contains the value ofthe parameter both optimal solution and parameter valuesare recorded in the vector table
522 Consensus Theconsensus is thewidespread agreementamong the group based on voting the voting pattern ofthe scout bees is monitored periodically to know whetherit reached an agreement and started acting on the decisionpattern Honey bees use the consensus method to select thebest search value the globally optimized point is chosen bycomparing the values in the vector table The globally opti-mized points are selected using themathematical formulation
523 Quorum In quorummethod the optimum solution iscalculated as the final solution based on the threshold levelobtained by the group decision-making process When thesolution reaches the optimal threshold level 120585119902 then the solu-tion is considered as a final solution based on unison decisionprocess The quorum threshold value describes the quality of
the food particle result When the threshold value is less thecomputation time decreases but it leads to inaccurate experi-mental resultsThe threshold value should be chosen to attainless computational timewith an accurate experimental result
6 Experimental Design and Analysis
61 Performance Metrics The performance of the proposedalgorithm MODBCO is assessed by comparing with fivedifferent competitor methods Here six performance metricsare considered to investigate the significance and evaluate theexperimental results The metrics are listed in this section
611 Least Error Rate Least Error Rate (LER) is the percent-age of the difference between known optimal value and thebest value obtained The LER can be calculated using
LER () = 119903sum119894=1
OptimalNRP-Instance minus fitness119894OptimalNRP-Instance
(34)
612 Average Convergence The Average Convergence is themeasure to evaluate the quality of the generated populationon average The Average Convergence (AC) is the percentageof the average of the convergence rate of solutions The per-formance of the convergence time is increased by the AverageConvergence to exploremore solutions in the populationTheAverage Convergence is calculated usingAC
= 119903sum119894=1
1 minus Avg_fitness119894 minusOptimalNRP-InstanceOptimalNRP-Instance
lowast 100 (35)
where (119903) is the number of instances in the given dataset
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
where min119899119889119904 is the minimum number of nurses required fora shift (119904) on the day (119889)HC4 In this constraint the total number of working days foreach nurse should range between minimum and maximumrange for the given scheduled period
119882min le 119863sum119889=1
119878sum119904=1
S119889119904119899 le 119882max forall119899 isin 119873 (5)
The average working shift for nurse can be determined byusing
119882avg = 1119873 (119863sum119889=1
119878sum119904=1
S119889119904119899 forall119899 isin 119873) (6)
where 119882min and 119882max are the minimum and maximumnumber of days in scheduled period and119882avg is the averageworking shift of the nurse
HC5 In this constraint shift 1 followed by shift 3 is notallowed that is a day shift followed by a night shift is notallowed
119873sum119899=1
119863sum119889=1
S1198991198891199043 + S119899119889+11199041 le 1 forall119904 isin 119878 (7)
SC1 The maximum number of shifts assigned to each nursefor the given scheduled period is as follows
max(( 119863sum119889=1
119878sum119904=1
S119889119904119899 minus Φ119906119887119899 ) 0) forall119899 isin 119873 (8)
whereΦ119906119887119899 is themaximumnumber of shifts assigned to nurse(119899)SC2 The minimum number of shifts assigned to each nursefor the given scheduled period is as follows
whereΦ119897119887119899 is theminimumnumber of shifts assigned to nurse(119899)SC3 The maximum number of consecutive working daysassigned to each nurse on which a shift is allotted for thescheduled period is as follows
Ψ119899sum119896=1
max ((C119896119899 minus Θ119906119887119899 ) 0) forall119899 isin 119873 (10)
where Θ119906119887119899 is the maximum number of consecutive workingdays of nurse (119899) Ψ119899 is the total number of consecutive
working spans of nurse (119899) in the roster and C119896119899 is the countof the 119896th working spans of nurse (119899)SC4 The minimum number of consecutive working daysassigned to each nurse on which a shift is allotted for thescheduled period is as follows
Ψ119899sum119896=1
max ((Θ119897119887119899 minus C119896119899) 0) forall119899 isin 119873 (11)
where Θ119897119887119899 is the minimum number of consecutive workingdays of nurse (119899) Ψ119899 is the total number of consecutiveworking spans of nurse (119899) in the roster and C119896119899 is the countof the 119896th working span of the nurse (119899)SC5 The maximum number of consecutive working daysassigned to each nurse on which no shift is allotted for thegiven scheduled period is as follows
Γ119899sum119896=1
max ((eth119896119899 minus 120593119906119887119899 ) 0) forall119899 isin 119873 (12)
where120593119906119887119899 is themaximumnumber of consecutive free days ofnurse (119899) Γ119899 is the total number of consecutive free workingspans of nurse (119899) in the roster and eth119896119899 is the count of the 119896thworking span of the nurse (119899)SC6 The minimum number of consecutive working daysassigned to each nurse on which no shift is allotted for thegiven scheduled period is as follows
Γ119899sum119896=1
max ((120593119897119887119899 minus eth119896119899) 0) forall119899 isin 119873 (13)
where 120593119897119887119899 is theminimumnumber of consecutive free days ofnurse (119899) Γ119899 is the total number of consecutive free workingspans of nurse (119899) in the roster and eth119896119899 is the count of the 119896thworking span of the nurse (119899)SC7 The maximum number of consecutive working week-ends with at least one shift assigned to nurse for the givenscheduled period is as follows
Υ119899sum119896=1
max ((120577119896119899 minus Ω119906119887119899 ) 0) forall119899 isin 119873 (14)
where Ω119906119887119899 is the maximum number of consecutive workingweekends of nurse (119899) Υ119899 is the total number of consecutiveworking weekend spans of nurse (119899) in the roster and 120577119896119899 isthe count of the 119896th working weekend span of the nurse (119899)SC8 The minimum number of consecutive working week-ends with at least one shift assigned to nurse for the givenscheduled period is as follows
Υ119899sum119896=1
max ((Ω119897119887119899 minus 120577119896119899) 0) forall119899 isin 119873 (15)
6 Computational Intelligence and Neuroscience
where Ω119897119887119899 is the minimum number of consecutive workingweekends of nurse (119899) Υ119899 is the total number of consecutiveworking weekend spans of nurse (119899) in the roster and 120577119896119899 isthe count of the 119896th working weekend span of the nurse (119899)SC9 The maximum number of weekends with at least oneshift assigned to nurse in four weeks is as follows
119899sum119896=1
max ((119896119899 minus 120603119906119887119899 ) 0) forall119899 isin 119873 (16)
where 119896119899 is the number of working days at the 119896th weekendof nurse (119899) 120603119906119887119899 is the maximum number of working daysfor nurse (119899) and 119899 is the total count of the weekend in thescheduling period of nurse (119899)SC10 The nurse can request working on a particular day forthe given scheduled period
where 120582119889119899 is the day request from the nurse (119899) to work on anyshift on a particular day (119889)SC11 The nurse can request that they do not work on aparticular day for the given scheduled period
where 120582119889119899 is the request from the nurse (119899) not to work on anyshift on a particular day (119889)SC12 The nurse can request working on a particular shift ona particular day for the given scheduled period
where Υ119889119904119899 is the shift request from the nurse (119899) to work ona particular shift (119904) on particular day (119889)SC13 The nurse can request that they do not work on aparticular shift on a particular day for the given scheduledperiod
where Υ119889119904119899 is the shift request from the nurse (119899) not to workon a particular shift (119904) on particular day (119889)SC14 The nurse should not work on unwanted patternsuggested for the scheduled period
984858119899sum119906=1
120583119906119899 forall119899 isin 119873 (21)
where 120583119906119899 is the total count of occurring patterns for nurse (119899)of type 119906 984858119899 is the set of unwanted patterns suggested for thenurse (119899)
The objective function of the NRP is to maximize thenurse preferences and minimize the penalty cost from vio-lations of soft constraints in (22)
Here SC refers to the set of soft constraints indexed inTable 2 119875sc(119909) refers to the penalty weight violation of thesoft constraint and 119879sc(119909) refers to the total violations of thesoft constraints in roster solution It has to be noted that theusage of penalty function [32] in the NRP is to improve theperformance and provide the fair comparison with anotheroptimization algorithm
4 Bee Colony Optimization
41 Natural Behavior of Honey Bees Swarm intelligence isan emerging discipline for the study of problems whichrequires an optimal approach rather than the traditionalapproach The use of swarm intelligence is the part ofartificial intelligence based on the study of the behavior ofsocial insects The swarm intelligence is composed of manyindividual actions using decentralized and self-organizedsystem Swarm behavior is characterized by natural behaviorof many species such as fish schools herds of animals andflocks of birds formed for the biological requirements tostay together Swarm implies the aggregation of animalssuch as birds fishes ants and bees based on the collectivebehavior The individual agents in the swarm will have astochastic behavior which depends on the local perception ofthe neighborhood The communication between any insectscan be formed with the help of the colonies and it promotescollective intelligence among the colonies
The important features of swarms are proximity qualityresponse variability stability and adaptability The proximityof the swarm must be capable of providing simple spaceand time computations and it should respond to the qualityfactorsThe swarm should allow diverse activities and shouldnot be restricted among narrow channels The swarm shouldmaintain the stability nature and should not fluctuate basedon the behaviorThe adaptability of the swarmmust be able tochange the behavior mode when required Several hundredsof bees from the swarm work together to find nesting sitesand select the best nest site Bee Colony Optimization isinspired by the natural behavior of beesThe bee optimizationalgorithm is inspired by group decision-making processesof honey bees A honey bee searches the best nest site byconsidering speed and accuracy
In a bee colony there are three different types of beesa single queen bee thousands of male drone bees andthousands of worker bees
(1) The queen bee is responsible for creating new coloniesby laying eggs
Computational Intelligence and Neuroscience 7
(2) The male drone bees mated with the queen and werediscarded from the colonies
(3) The remaining female bees in the hive are calledworker bees and they are called the building block ofthe hiveThe responsibilities of the worker bees are tofeed guard and maintain the honey bee comb
Based on the responsibility worker bees are classifiedas scout bees and forager bees A scout bee flies in searchof food sources randomly and returns when the energygets exhausted After reaching a hive scout bees share theinformation and start to explore rich food source locationswith forager bees The scout beersquos information includesdirection quality quantity and distance of the food sourcethey found The way of communicating information about afood source to foragers is done using dance There are twotypes of dance round dance and waggle dance The rounddance will provide direction of the food source when thedistance is small The waggle dance indicates the positionand the direction of the food source the distance can bemeasured by the speed of the dance A greater speed indicatesa smaller distance and the quantity of the food depends onthe wriggling of the beeThe exchange of information amonghive mates is to acquire collective knowledge Forager beeswill silently observe the behavior of scout bee to acquireknowledge about the directions and information of the foodsource
The group decision process of honey bees is for searchingbest food source and nest siteThe decision-making process isbased on the swarming process of the honey bee Swarming isthe process inwhich the queen bee and half of theworker beeswill leave their hive to explore a new colony The remainingworker bees and daughter bee will remain in the old hiveto monitor the waggle dance After leaving their parentalhive swarm bees will form a cluster in search of the newnest site The waggle dance is used to communicate withquiescent bees which are inactive in the colonyThis providesprecise information about the direction of the flower patchbased on its quality and energy level The number of followerbees increases based on the quality of the food source andallows the colony to gather food quickly and efficiently Thedecision-making process can be done in two methods byswarm bees to find the best nest site They are consensusand quorum consensus is the group agreement taken intoaccount and quorum is the decision process taken when thebee vote reaches a threshold value
Bee Colony Optimization (BCO) algorithm is apopulation-based algorithm The bees in the populationare artificial bees and each bee finds its neighboring solutionfrom the current path This algorithm has a forward andbackward process In forwarding pass every bee starts toexplore the neighborhood of its current solution and enablesconstructive and improving moves In forward pass entirebees in the hive will start the constructive move and thenlocal search will start In backward pass bees share theobjective value obtained in the forward pass The bees withhigher priority are used to discard all nonimproving movesThe bees will continue to explore in next forward pass orcontinue the same process with neighborhoodThe flowchart
Forward pass
Initialization
Construction move
Backward pass
Update the bestsolution
Stopping criteriaFalse
True
Figure 2 Flowchart of BCO algorithm
for BCO is shown in Figure 2 The BCO is proficient insolving combinatorial optimization problems by creatingcolonies of the multiagent system The pseudocode for BCOis described in Algorithm 1 The bee colony system providesa standard well-organized and well-coordinated teamworkmultitasking performance [33]
42 Modified Nelder-Mead Method The Nelder-MeadMethod is a simplex method for finding a local minimumfunction of various variables and is a local search algorithmfor unconstrained optimization problems The whole searcharea is divided into different fragments and filled with beeagents To obtain the best solution each fragment can besearched by its bee agents through Modified Nelder-MeadMethod (MNMM) Each agent in the fragments passesinformation about the optimized point using MNMMBy using NMMM the best points are obtained and thebest solution is chosen by decision-making process ofhoney bees The algorithm is a simplex-based method119863-dimensional simplex is initialized with 119863 + 1 verticesthat is two dimensions and it forms a triangle if it has threedimensions it forms a tetrahedron To assign the best andworst point the vertices are evaluated and ordered based onthe objective function
The best point or vertex is considered to the minimumvalue of the objective function and the worst point is chosen
8 Computational Intelligence and Neuroscience
Bee Colony Optimization(1) Initialization Assign every bee to an empty solution(2) Forward Pass
For every bee(21) set 119894 = 1(22) Evaluate all possible construction moves(23) Based on the evaluation choose one move using Roulette Wheel(24) 119894 = 119894 + 1 if (119894 le 119873) Go to step (22)
where 119894 is the counter for construction move and119873 is the number of construction moves during one forwardpass
(3) Return to Hive(4) Backward Pass starts(5) Compute the objective function for each bee and sort accordingly(6) Calculate probability or logical reasoning to continue with the computed solution and become recruiter bee(7) For every follower choose the new solution from recruiters(8) If stopping criteria is not met Go to step (2)(9) Evaluate and find the best solution(10) Output the best solution
Algorithm 1 Pseudocode of BCO
with a maximum value of the computed objective functionTo form simplex new vertex function values are computedThismethod can be calculated using four procedures namelyreflection expansion contraction and shrinkage Figure 3shows the operators of the simplex triangle in MNMM
The simplex operations in each vertex are updated closerto its optimal solution the vertices are ordered based onfitness value and ordered The best vertex is 119860119887 the secondbest vertex is 119860 119904 and the worst vertex is 119860119908 calculated basedon the objective function Let 119860 = (119909 119910) be the vertex in atriangle as food source points 119860119887 = (119909119887 119910119887) 119860 119904 = (119909119904 119910119904)and119860119908 = (119909119908 119910119908) are the positions of the food source pointsthat is local optimal points The objective functions for 119860119887119860 119904 and 119860119908 are calculated based on (23) towards the foodsource points
The objective function to construct simplex to obtainlocal search using MNMM is formulated as
119891 (119909 119910) = 1199092 minus 4119909 + 1199102 minus 119910 minus 119909119910 (23)
Based on the objective function value the vertices foodpoints are ordered ascending with their corresponding honeybee agentsThe obtained values are ordered as119860119887 le 119860 119904 le 119860119908with their honey bee position and food points in the simplextriangle Figure 4 describes the search of best-minimizedcost value for the nurse based on objective function (22)The working principle of Modified Nelder-Mead Method(MNMM) for searching food particles is explained in detail
(1) In the simplex triangle the reflection coefficient 120572expansion coefficient 120574 contraction coefficient 120573 andshrinkage coefficient 120575 are initialized
(2) The objective function for the simplex triangle ver-tices is calculated and ordered The best vertex withlower objective value is 119860119887 the second best vertex is119860 119904 and the worst vertex is named as 119860119908 and thesevertices are ordered based on the objective functionas 119860119887 le 119860 119904 le 119860119908
(3) The first two best vertices are selected namely119860119887 and119860 119904 and the construction proceeds with calculatingthe midpoint of the line segment which joins the twobest vertices that is food positions The objectivefunction decreases as the honey agent associated withthe worst position vertex moves towards best andsecond best verticesThe value decreases as the honeyagent moves towards 119860119908 to 119860119887 and 119860119908 to 119860 119904 It isfeasible to calculate the midpoint vertex 119860119898 by theline joining best and second best vertices using
119860119898 = 119860119887 + 119860 1199042 (24)
(4) A reflecting vertex 119860119903 is generated by choosing thereflection of worst point 119860119908 The objective functionvalue for 119860119903 is 119891(119860119903) which is calculated and it iscompared with worst vertex 119860119908 objective functionvalue 119891(119860119908) If 119891(119860119903) lt 119891(119860119908) proceed with step(5) the reflection vertex can be calculated using
119860119903 = 119860119898 + 120572 (119860119898 minus 119860119908) where 120572 gt 0 (25)
(5) The expansion process starts when the objectivefunction value at reflection vertex 119860119903 is lesser thanworst vertex 119860119908 119891(119860119903) lt 119891(119860119908) and the linesegment is further extended to 119860119890 through 119860119903 and119860119908 The vertex point 119860119890 is calculated by (26) If theobjective function value at119860119890 is lesser than reflectionvertex 119860119903 119891(119860119890) lt 119891(119860119903) then the expansion isaccepted and the honey bee agent has found best foodposition compared with reflection point
119860119890 = 119860119903 + 120574 (119860119903 minus 119860119898) where 120574 gt 1 (26)
(6) The contraction process is carried out when 119891(119860119903) lt119891(119860 119904) and 119891(119860119903) le 119891(119860119887) for replacing 119860119887 with
Computational Intelligence and Neuroscience 9
AwAs
Ab
(a) Simplex triangle
Ar
As
Ab
Aw
(b) Reflection
Ae
Ar
As
Ab
Aw
(c) Expansion
Ac
As
Ab
Aw
(d) Contraction (119860ℎ lt 119860119903)
Ac
As
Ab
Aw
(e) Contraction (119860119903 lt 119860ℎ)
A㰀b
A㰀s
As
Ab
Aw
(f) Shrinkage
Figure 3 Nelder-Mead operations
119860119903 If 119891(119860119903) gt 119891(119860ℎ) then the direct contractionwithout the replacement of 119860119887 with 119860119903 is performedThe contraction vertex 119860119888 can be calculated using
119860119888 = 120573119860119903 + (1 minus 120573)119860119898 where 0 lt 120573 lt 1 (27)
If 119891(119860119903) le 119891(119860119887) the contraction can be done and119860119888 replaced with 119860ℎ go to step (8) or else proceed tostep (7)
(7) The shrinkage phase proceeds when the contractionprocess at step (6) fails and is done by shrinking allthe vertices of the simplex triangle except 119860ℎ using(28) The objective function value of reflection andcontraction phase is not lesser than the worst pointthen the vertices 119860 119904 and 119860119908 must be shrunk towards119860ℎThus the vertices of smaller value will form a newsimplex triangle with another two best vertices
119860 119894 = 120575119860 119894 + 1198601 (1 minus 120575) where 0 lt 120575 lt 1 (28)
(8) The calculations are stopped when the terminationcondition is met
Algorithm 2 describes the pseudocode for ModifiedNelder-Mead Method in detail It portraits the detailed pro-cess of MNMM to obtain the best solution for the NRP Theworkflow of the proposed MNMM is explained in Figure 5
5 MODBCO
Bee Colony Optimization is the metaheuristic algorithm tosolve various combinatorial optimization problems and itis inspired by the natural behavior of bee for their foodsources The algorithm consists of two steps forward andbackward pass During forwarding pass bees started toexplore the neighborhood of its current solution and findall possible ways In backward pass bees return to thehive and share the values of the objective function of theircurrent solution Calculate nectar amount using probability
10 Computational Intelligence and Neuroscience
Ab
Aw
Ar
As
Am
d
d
Ab
Aw
Ar
As
Am
d
d
Aed2
Ab
Aw
Ar
As
Am
Ac1
Ac2
Ab
Aw As
Am
Anew
Figure 4 Bees search movement based on MNMM
function and advertise the solution the bee which has thebetter solution is given higher priority The remaining beesbased on the probability value decide whether to explore thesolution or proceed with the advertised solution DirectedBee Colony Optimization is the computational system whereseveral bees work together in uniting and interact with eachother to achieve goals based on the group decision processThe whole search area of the bee is divided into multiplefragments different bees are sent to different fragments Thebest solution in each fragment is obtained by using a localsearch algorithmModified Nelder-Mead Method (MNMM)To obtain the best solution the total varieties of individualparameters are partitioned into individual volumes Eachvolume determines the starting point of the exploration offood particle by each bee The bees use developed MNMMalgorithm to find the best solution by remembering thelast two best food sites they obtained After obtaining thecurrent solution the bee starts to backward pass sharingof information obtained during forwarding pass The beesstarted to share information about optimized point by thenatural behavior of bees called waggle dance When all theinformation about the best food is shared the best among theoptimized point is chosen using a decision-making processcalled consensus and quorummethod in honey bees [34 35]
51 Multiagent System All agents live in an environmentwhich is well structured and organized Inmultiagent systemseveral agents work together and interact with each otherto obtain the goal According to Jiao and Shi [36] andZhong et al [37] all agents should possess the followingqualities agents should live and act in an environmenteach agent should sense its local environment each agent
should be capable of interacting with other agents in a localenvironment and agents attempt to perform their goal Allagents interact with each other and take the decision toachieve the desired goals The multiagent system is a com-putational system and provides an opportunity to optimizeand compute all complex problems In multiagent system allagents start to live and act in the same environment which iswell organized and structured Each agent in the environmentis fixed on a lattice point The size and dimension of thelattice point in the environment depend upon the variablesused The objective function can be calculated based on theparameters fixed
(1) Consider ldquo119890rdquo number of independent parameters tocalculate the objective function The range of the 119892thparameter can be calculated using [119876119892119894 119876119892119891] where119876119892119894 is the initial value of the 119892th parameter and 119876119892119891is the final value of the 119892th parameter chosen
(2) Thus the objective function can be formulated as 119890number of axes each axis will contain a total rangeof single parameter with different dimensions
(3) Each axis is divided into smaller parts each partis called a step So 119892th axis can be divided into 119899119892number of steps each with the length of 119871119892 where thevalue of 119892 depends upon parameters thus 119892 = 1 to 119890The relationship between 119899119892 and 119871119892 can be given as
119899119892 = 119876119892119894 minus 119876119892119891119871119892 (29)
(4) Then each axis is divided into branches foreach branch 119892 number of branches will form an
Computational Intelligence and Neuroscience 11
Modified Nelder-Mead Method for directed honey bee food search(1) Initialization119860119887 denotes the list of vertices in simplex where 119894 = 1 2 119899 + 1120572 120574 120573 and 120575 are the coefficients of reflection expansion contraction and shrinkage119891 is the objective function to be minimized(2)Ordering
Order the vertices in simplex from lowest objective function value 119891(1198601) to highest value 119891(119860119899+1) Ordered as 1198601le 1198602 le sdot sdot sdot le 119860119899+1(3)Midpoint
Calculate the midpoint for first two best vertices in simplex 119860119898 = sum(119860 119894119899) where 119894 = 1 2 119899(4) Reflection Process
Calculate reflection point 119860119903 by 119860119903 = 119860119898 + 120572(119860119898 minus 119860119899+1)if 119891(1198601) le 119891(1198602) le 119891(119860119899) then119860119899 larr 119860119903 and Go to to Step (8)end if
(5) Expansion Processif 119891(119860119903) le 119891(1198601) thenCalculate expansion point using 119860 119890 = 119860119903 + 120574(119860119903 minus 119860119898)end ifif 119891(119860 119890) lt 119891(119860119903) then119860119899 larr 119860 119890 and Go to to Step (8)else119860119899 larr 119860119903 and Go to to Step (8)end if
(6) Contraction Processif 119891(119860119899) le 119891(119860119903) le 119891(119860119899+1) thenCompute outside contraction by 119860 119888 = 120573119860119903 + (1 minus 120573)119860119898end ifif 119891(1198601) ge 119891(119860119899+1) thenCompute inside contraction by 119860 119888 = 120573119860119899+1 + (1 minus 120573)119860119898end ifif 119891(119860119903) ge 119891(119860119899) thenContraction is done between 119860119898 and the best vertex among 119860119903 and 119860119899+1end ifif 119891(119860 119888) lt 119891(119860119903) then119860119899 larr 119860 119888 and Go to to Step (8)else goes to Step (7)end ifif 119891(119860 119888) ge 119891(119860119899+1) then119860119899+1 larr 119860 119888 and Go to to Step (8)else Go to to Step (7)end if
(7) Shrinkage ProcessShrink towards the best solution with new vertices by 119860 119894 = 120575119860 119894 + 1198601(1 minus 120575) where 119894 = 2 119899 + 1
(8) Stopping CriteriaOrder and re-label new vertices of the simplex based on their objective function and go to step (4)
Algorithm 2 Pseudocode of Modified Nelder-Mead Method
119890-dimensional volume Total number of volumes 119873Vcan be formulated using
119873V = 119890prod119892=1
119899119892 (30)
(5) The starting point of the agent in the environmentwhich is one point inside volume is chosen bycalculating themidpoint of the volumeThemidpointof the lattice can be calculated as
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ] (31)
52 Decision-Making Process A key role of the honey beesis to select the best nest site and is done by the process ofdecision-making to produce a unified decisionThey follow adistributed decision-making process to find out the neighbornest site for their food particles The pseudocode for theproposed MODBCO algorithm is shown in Algorithm 3Figure 6 explains the workflow of the proposed algorithm forthe search of food particles by honey bees using MODBCO
521 Waggle Dance The scout bees after returning from thesearch of food particle report about the quality of the foodsite by communicationmode called waggle dance Scout beesperform thewaggle dance to other quiescent bees to advertise
12 Computational Intelligence and Neuroscience
Yes
Reflectionprocess
Order and label verticesbased on f(A)
Initialization
Coefficients 훼 훾 훽 훿
Objective function f(A)
f(Ab) lt f(Ar) lt f(Aw) Aw larr Ar
f(Ae) le f(Ar)
two best verticesAm forCalculate midpoint
Start
Terminationcriteria
Stop
Ar = Am + 훼(Am minus Aw)
ExpansionprocessNo
Yesf(Ar) le f(Aw) Aw larr Ae
No
b larr true Aw larr Ar
Contractionprocess
f(Ar) ge f(An)Yes
f(Ac) lt f(Ar)Aw larr Ac
b larr false
No
Shrinkageprocess
b larr true
Yes
Yes
No
Ae = Ar + 훾(Ar minus
Am)
Ac = 훽Ar + (1 minus 훽)Am
Ai = 훿Ai + A1(1 minus 훿)
Figure 5 Workflow of Modified Nelder-Mead Method
Computational Intelligence and Neuroscience 13
Multi-Objective Directed Bee Colony Optimization(1) Initialization119891(119909) is the objective function to be minimized
Initialize 119890 number of parameters and 119871119892 length of steps where 119892 = 0 to 119890Initialize initial value and the final value of the parameter as 119876119892119894 and 119876119892119891lowastlowast Solution Representation lowastlowastThe solutions are represented in the form of Binary values which can be generated as followsFor each solution 119894 = 1 119899119883119894 = 1199091198941 1199091198942 119909119894119889 | 119889 isin total days amp 119909119894119889 = rand ge 029 forall119889End for
(2) The number of steps in each step can be calculated using
119899119892 = 119876119892119894 minus 119876119892119891119871119892(3) The total number of volumes can be calculated using119873V = 119890prod
119892=1
119899119892(4) The midpoint of the volume to calculate starting point of the exploration can be calculated using
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ](5) Explore the search volume according to the Modified Nelder-Mead Method using Algorithm 2(6) The recorded value of the optimized point in vector table using[119891(1198811) 119891(1198812) 119891(119881119873V )](7) The globally optimized point is chosen based on Bee decision-making process using Consensus and Quorum
method approach 119891(119881119892) = min [119891(1198811) 119891(1198812) 119891(119881119873V )]Algorithm 3 Pseudocode of MODBCO
their best nest site for the exploration of food source Inthe multiagent system each agent after collecting individualsolution gives it to the centralized systems To select the bestoptimal solution forminimal optimal cases themathematicalformulation can be stated as
dance119894 = min (119891119894 (119881)) (32)
This mathematical formulation will find the minimaloptimal cases among the search solution where 119891119894(119881) is thesearch value calculated by the agent The search values arerecorded in the vector table 119881 119881 is the vector which consistsof 119890 number of elements The element 119890 contains the value ofthe parameter both optimal solution and parameter valuesare recorded in the vector table
522 Consensus Theconsensus is thewidespread agreementamong the group based on voting the voting pattern ofthe scout bees is monitored periodically to know whetherit reached an agreement and started acting on the decisionpattern Honey bees use the consensus method to select thebest search value the globally optimized point is chosen bycomparing the values in the vector table The globally opti-mized points are selected using themathematical formulation
523 Quorum In quorummethod the optimum solution iscalculated as the final solution based on the threshold levelobtained by the group decision-making process When thesolution reaches the optimal threshold level 120585119902 then the solu-tion is considered as a final solution based on unison decisionprocess The quorum threshold value describes the quality of
the food particle result When the threshold value is less thecomputation time decreases but it leads to inaccurate experi-mental resultsThe threshold value should be chosen to attainless computational timewith an accurate experimental result
6 Experimental Design and Analysis
61 Performance Metrics The performance of the proposedalgorithm MODBCO is assessed by comparing with fivedifferent competitor methods Here six performance metricsare considered to investigate the significance and evaluate theexperimental results The metrics are listed in this section
611 Least Error Rate Least Error Rate (LER) is the percent-age of the difference between known optimal value and thebest value obtained The LER can be calculated using
LER () = 119903sum119894=1
OptimalNRP-Instance minus fitness119894OptimalNRP-Instance
(34)
612 Average Convergence The Average Convergence is themeasure to evaluate the quality of the generated populationon average The Average Convergence (AC) is the percentageof the average of the convergence rate of solutions The per-formance of the convergence time is increased by the AverageConvergence to exploremore solutions in the populationTheAverage Convergence is calculated usingAC
= 119903sum119894=1
1 minus Avg_fitness119894 minusOptimalNRP-InstanceOptimalNRP-Instance
lowast 100 (35)
where (119903) is the number of instances in the given dataset
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
where Ω119897119887119899 is the minimum number of consecutive workingweekends of nurse (119899) Υ119899 is the total number of consecutiveworking weekend spans of nurse (119899) in the roster and 120577119896119899 isthe count of the 119896th working weekend span of the nurse (119899)SC9 The maximum number of weekends with at least oneshift assigned to nurse in four weeks is as follows
119899sum119896=1
max ((119896119899 minus 120603119906119887119899 ) 0) forall119899 isin 119873 (16)
where 119896119899 is the number of working days at the 119896th weekendof nurse (119899) 120603119906119887119899 is the maximum number of working daysfor nurse (119899) and 119899 is the total count of the weekend in thescheduling period of nurse (119899)SC10 The nurse can request working on a particular day forthe given scheduled period
where 120582119889119899 is the day request from the nurse (119899) to work on anyshift on a particular day (119889)SC11 The nurse can request that they do not work on aparticular day for the given scheduled period
where 120582119889119899 is the request from the nurse (119899) not to work on anyshift on a particular day (119889)SC12 The nurse can request working on a particular shift ona particular day for the given scheduled period
where Υ119889119904119899 is the shift request from the nurse (119899) to work ona particular shift (119904) on particular day (119889)SC13 The nurse can request that they do not work on aparticular shift on a particular day for the given scheduledperiod
where Υ119889119904119899 is the shift request from the nurse (119899) not to workon a particular shift (119904) on particular day (119889)SC14 The nurse should not work on unwanted patternsuggested for the scheduled period
984858119899sum119906=1
120583119906119899 forall119899 isin 119873 (21)
where 120583119906119899 is the total count of occurring patterns for nurse (119899)of type 119906 984858119899 is the set of unwanted patterns suggested for thenurse (119899)
The objective function of the NRP is to maximize thenurse preferences and minimize the penalty cost from vio-lations of soft constraints in (22)
Here SC refers to the set of soft constraints indexed inTable 2 119875sc(119909) refers to the penalty weight violation of thesoft constraint and 119879sc(119909) refers to the total violations of thesoft constraints in roster solution It has to be noted that theusage of penalty function [32] in the NRP is to improve theperformance and provide the fair comparison with anotheroptimization algorithm
4 Bee Colony Optimization
41 Natural Behavior of Honey Bees Swarm intelligence isan emerging discipline for the study of problems whichrequires an optimal approach rather than the traditionalapproach The use of swarm intelligence is the part ofartificial intelligence based on the study of the behavior ofsocial insects The swarm intelligence is composed of manyindividual actions using decentralized and self-organizedsystem Swarm behavior is characterized by natural behaviorof many species such as fish schools herds of animals andflocks of birds formed for the biological requirements tostay together Swarm implies the aggregation of animalssuch as birds fishes ants and bees based on the collectivebehavior The individual agents in the swarm will have astochastic behavior which depends on the local perception ofthe neighborhood The communication between any insectscan be formed with the help of the colonies and it promotescollective intelligence among the colonies
The important features of swarms are proximity qualityresponse variability stability and adaptability The proximityof the swarm must be capable of providing simple spaceand time computations and it should respond to the qualityfactorsThe swarm should allow diverse activities and shouldnot be restricted among narrow channels The swarm shouldmaintain the stability nature and should not fluctuate basedon the behaviorThe adaptability of the swarmmust be able tochange the behavior mode when required Several hundredsof bees from the swarm work together to find nesting sitesand select the best nest site Bee Colony Optimization isinspired by the natural behavior of beesThe bee optimizationalgorithm is inspired by group decision-making processesof honey bees A honey bee searches the best nest site byconsidering speed and accuracy
In a bee colony there are three different types of beesa single queen bee thousands of male drone bees andthousands of worker bees
(1) The queen bee is responsible for creating new coloniesby laying eggs
Computational Intelligence and Neuroscience 7
(2) The male drone bees mated with the queen and werediscarded from the colonies
(3) The remaining female bees in the hive are calledworker bees and they are called the building block ofthe hiveThe responsibilities of the worker bees are tofeed guard and maintain the honey bee comb
Based on the responsibility worker bees are classifiedas scout bees and forager bees A scout bee flies in searchof food sources randomly and returns when the energygets exhausted After reaching a hive scout bees share theinformation and start to explore rich food source locationswith forager bees The scout beersquos information includesdirection quality quantity and distance of the food sourcethey found The way of communicating information about afood source to foragers is done using dance There are twotypes of dance round dance and waggle dance The rounddance will provide direction of the food source when thedistance is small The waggle dance indicates the positionand the direction of the food source the distance can bemeasured by the speed of the dance A greater speed indicatesa smaller distance and the quantity of the food depends onthe wriggling of the beeThe exchange of information amonghive mates is to acquire collective knowledge Forager beeswill silently observe the behavior of scout bee to acquireknowledge about the directions and information of the foodsource
The group decision process of honey bees is for searchingbest food source and nest siteThe decision-making process isbased on the swarming process of the honey bee Swarming isthe process inwhich the queen bee and half of theworker beeswill leave their hive to explore a new colony The remainingworker bees and daughter bee will remain in the old hiveto monitor the waggle dance After leaving their parentalhive swarm bees will form a cluster in search of the newnest site The waggle dance is used to communicate withquiescent bees which are inactive in the colonyThis providesprecise information about the direction of the flower patchbased on its quality and energy level The number of followerbees increases based on the quality of the food source andallows the colony to gather food quickly and efficiently Thedecision-making process can be done in two methods byswarm bees to find the best nest site They are consensusand quorum consensus is the group agreement taken intoaccount and quorum is the decision process taken when thebee vote reaches a threshold value
Bee Colony Optimization (BCO) algorithm is apopulation-based algorithm The bees in the populationare artificial bees and each bee finds its neighboring solutionfrom the current path This algorithm has a forward andbackward process In forwarding pass every bee starts toexplore the neighborhood of its current solution and enablesconstructive and improving moves In forward pass entirebees in the hive will start the constructive move and thenlocal search will start In backward pass bees share theobjective value obtained in the forward pass The bees withhigher priority are used to discard all nonimproving movesThe bees will continue to explore in next forward pass orcontinue the same process with neighborhoodThe flowchart
Forward pass
Initialization
Construction move
Backward pass
Update the bestsolution
Stopping criteriaFalse
True
Figure 2 Flowchart of BCO algorithm
for BCO is shown in Figure 2 The BCO is proficient insolving combinatorial optimization problems by creatingcolonies of the multiagent system The pseudocode for BCOis described in Algorithm 1 The bee colony system providesa standard well-organized and well-coordinated teamworkmultitasking performance [33]
42 Modified Nelder-Mead Method The Nelder-MeadMethod is a simplex method for finding a local minimumfunction of various variables and is a local search algorithmfor unconstrained optimization problems The whole searcharea is divided into different fragments and filled with beeagents To obtain the best solution each fragment can besearched by its bee agents through Modified Nelder-MeadMethod (MNMM) Each agent in the fragments passesinformation about the optimized point using MNMMBy using NMMM the best points are obtained and thebest solution is chosen by decision-making process ofhoney bees The algorithm is a simplex-based method119863-dimensional simplex is initialized with 119863 + 1 verticesthat is two dimensions and it forms a triangle if it has threedimensions it forms a tetrahedron To assign the best andworst point the vertices are evaluated and ordered based onthe objective function
The best point or vertex is considered to the minimumvalue of the objective function and the worst point is chosen
8 Computational Intelligence and Neuroscience
Bee Colony Optimization(1) Initialization Assign every bee to an empty solution(2) Forward Pass
For every bee(21) set 119894 = 1(22) Evaluate all possible construction moves(23) Based on the evaluation choose one move using Roulette Wheel(24) 119894 = 119894 + 1 if (119894 le 119873) Go to step (22)
where 119894 is the counter for construction move and119873 is the number of construction moves during one forwardpass
(3) Return to Hive(4) Backward Pass starts(5) Compute the objective function for each bee and sort accordingly(6) Calculate probability or logical reasoning to continue with the computed solution and become recruiter bee(7) For every follower choose the new solution from recruiters(8) If stopping criteria is not met Go to step (2)(9) Evaluate and find the best solution(10) Output the best solution
Algorithm 1 Pseudocode of BCO
with a maximum value of the computed objective functionTo form simplex new vertex function values are computedThismethod can be calculated using four procedures namelyreflection expansion contraction and shrinkage Figure 3shows the operators of the simplex triangle in MNMM
The simplex operations in each vertex are updated closerto its optimal solution the vertices are ordered based onfitness value and ordered The best vertex is 119860119887 the secondbest vertex is 119860 119904 and the worst vertex is 119860119908 calculated basedon the objective function Let 119860 = (119909 119910) be the vertex in atriangle as food source points 119860119887 = (119909119887 119910119887) 119860 119904 = (119909119904 119910119904)and119860119908 = (119909119908 119910119908) are the positions of the food source pointsthat is local optimal points The objective functions for 119860119887119860 119904 and 119860119908 are calculated based on (23) towards the foodsource points
The objective function to construct simplex to obtainlocal search using MNMM is formulated as
119891 (119909 119910) = 1199092 minus 4119909 + 1199102 minus 119910 minus 119909119910 (23)
Based on the objective function value the vertices foodpoints are ordered ascending with their corresponding honeybee agentsThe obtained values are ordered as119860119887 le 119860 119904 le 119860119908with their honey bee position and food points in the simplextriangle Figure 4 describes the search of best-minimizedcost value for the nurse based on objective function (22)The working principle of Modified Nelder-Mead Method(MNMM) for searching food particles is explained in detail
(1) In the simplex triangle the reflection coefficient 120572expansion coefficient 120574 contraction coefficient 120573 andshrinkage coefficient 120575 are initialized
(2) The objective function for the simplex triangle ver-tices is calculated and ordered The best vertex withlower objective value is 119860119887 the second best vertex is119860 119904 and the worst vertex is named as 119860119908 and thesevertices are ordered based on the objective functionas 119860119887 le 119860 119904 le 119860119908
(3) The first two best vertices are selected namely119860119887 and119860 119904 and the construction proceeds with calculatingthe midpoint of the line segment which joins the twobest vertices that is food positions The objectivefunction decreases as the honey agent associated withthe worst position vertex moves towards best andsecond best verticesThe value decreases as the honeyagent moves towards 119860119908 to 119860119887 and 119860119908 to 119860 119904 It isfeasible to calculate the midpoint vertex 119860119898 by theline joining best and second best vertices using
119860119898 = 119860119887 + 119860 1199042 (24)
(4) A reflecting vertex 119860119903 is generated by choosing thereflection of worst point 119860119908 The objective functionvalue for 119860119903 is 119891(119860119903) which is calculated and it iscompared with worst vertex 119860119908 objective functionvalue 119891(119860119908) If 119891(119860119903) lt 119891(119860119908) proceed with step(5) the reflection vertex can be calculated using
119860119903 = 119860119898 + 120572 (119860119898 minus 119860119908) where 120572 gt 0 (25)
(5) The expansion process starts when the objectivefunction value at reflection vertex 119860119903 is lesser thanworst vertex 119860119908 119891(119860119903) lt 119891(119860119908) and the linesegment is further extended to 119860119890 through 119860119903 and119860119908 The vertex point 119860119890 is calculated by (26) If theobjective function value at119860119890 is lesser than reflectionvertex 119860119903 119891(119860119890) lt 119891(119860119903) then the expansion isaccepted and the honey bee agent has found best foodposition compared with reflection point
119860119890 = 119860119903 + 120574 (119860119903 minus 119860119898) where 120574 gt 1 (26)
(6) The contraction process is carried out when 119891(119860119903) lt119891(119860 119904) and 119891(119860119903) le 119891(119860119887) for replacing 119860119887 with
Computational Intelligence and Neuroscience 9
AwAs
Ab
(a) Simplex triangle
Ar
As
Ab
Aw
(b) Reflection
Ae
Ar
As
Ab
Aw
(c) Expansion
Ac
As
Ab
Aw
(d) Contraction (119860ℎ lt 119860119903)
Ac
As
Ab
Aw
(e) Contraction (119860119903 lt 119860ℎ)
A㰀b
A㰀s
As
Ab
Aw
(f) Shrinkage
Figure 3 Nelder-Mead operations
119860119903 If 119891(119860119903) gt 119891(119860ℎ) then the direct contractionwithout the replacement of 119860119887 with 119860119903 is performedThe contraction vertex 119860119888 can be calculated using
119860119888 = 120573119860119903 + (1 minus 120573)119860119898 where 0 lt 120573 lt 1 (27)
If 119891(119860119903) le 119891(119860119887) the contraction can be done and119860119888 replaced with 119860ℎ go to step (8) or else proceed tostep (7)
(7) The shrinkage phase proceeds when the contractionprocess at step (6) fails and is done by shrinking allthe vertices of the simplex triangle except 119860ℎ using(28) The objective function value of reflection andcontraction phase is not lesser than the worst pointthen the vertices 119860 119904 and 119860119908 must be shrunk towards119860ℎThus the vertices of smaller value will form a newsimplex triangle with another two best vertices
119860 119894 = 120575119860 119894 + 1198601 (1 minus 120575) where 0 lt 120575 lt 1 (28)
(8) The calculations are stopped when the terminationcondition is met
Algorithm 2 describes the pseudocode for ModifiedNelder-Mead Method in detail It portraits the detailed pro-cess of MNMM to obtain the best solution for the NRP Theworkflow of the proposed MNMM is explained in Figure 5
5 MODBCO
Bee Colony Optimization is the metaheuristic algorithm tosolve various combinatorial optimization problems and itis inspired by the natural behavior of bee for their foodsources The algorithm consists of two steps forward andbackward pass During forwarding pass bees started toexplore the neighborhood of its current solution and findall possible ways In backward pass bees return to thehive and share the values of the objective function of theircurrent solution Calculate nectar amount using probability
10 Computational Intelligence and Neuroscience
Ab
Aw
Ar
As
Am
d
d
Ab
Aw
Ar
As
Am
d
d
Aed2
Ab
Aw
Ar
As
Am
Ac1
Ac2
Ab
Aw As
Am
Anew
Figure 4 Bees search movement based on MNMM
function and advertise the solution the bee which has thebetter solution is given higher priority The remaining beesbased on the probability value decide whether to explore thesolution or proceed with the advertised solution DirectedBee Colony Optimization is the computational system whereseveral bees work together in uniting and interact with eachother to achieve goals based on the group decision processThe whole search area of the bee is divided into multiplefragments different bees are sent to different fragments Thebest solution in each fragment is obtained by using a localsearch algorithmModified Nelder-Mead Method (MNMM)To obtain the best solution the total varieties of individualparameters are partitioned into individual volumes Eachvolume determines the starting point of the exploration offood particle by each bee The bees use developed MNMMalgorithm to find the best solution by remembering thelast two best food sites they obtained After obtaining thecurrent solution the bee starts to backward pass sharingof information obtained during forwarding pass The beesstarted to share information about optimized point by thenatural behavior of bees called waggle dance When all theinformation about the best food is shared the best among theoptimized point is chosen using a decision-making processcalled consensus and quorummethod in honey bees [34 35]
51 Multiagent System All agents live in an environmentwhich is well structured and organized Inmultiagent systemseveral agents work together and interact with each otherto obtain the goal According to Jiao and Shi [36] andZhong et al [37] all agents should possess the followingqualities agents should live and act in an environmenteach agent should sense its local environment each agent
should be capable of interacting with other agents in a localenvironment and agents attempt to perform their goal Allagents interact with each other and take the decision toachieve the desired goals The multiagent system is a com-putational system and provides an opportunity to optimizeand compute all complex problems In multiagent system allagents start to live and act in the same environment which iswell organized and structured Each agent in the environmentis fixed on a lattice point The size and dimension of thelattice point in the environment depend upon the variablesused The objective function can be calculated based on theparameters fixed
(1) Consider ldquo119890rdquo number of independent parameters tocalculate the objective function The range of the 119892thparameter can be calculated using [119876119892119894 119876119892119891] where119876119892119894 is the initial value of the 119892th parameter and 119876119892119891is the final value of the 119892th parameter chosen
(2) Thus the objective function can be formulated as 119890number of axes each axis will contain a total rangeof single parameter with different dimensions
(3) Each axis is divided into smaller parts each partis called a step So 119892th axis can be divided into 119899119892number of steps each with the length of 119871119892 where thevalue of 119892 depends upon parameters thus 119892 = 1 to 119890The relationship between 119899119892 and 119871119892 can be given as
119899119892 = 119876119892119894 minus 119876119892119891119871119892 (29)
(4) Then each axis is divided into branches foreach branch 119892 number of branches will form an
Computational Intelligence and Neuroscience 11
Modified Nelder-Mead Method for directed honey bee food search(1) Initialization119860119887 denotes the list of vertices in simplex where 119894 = 1 2 119899 + 1120572 120574 120573 and 120575 are the coefficients of reflection expansion contraction and shrinkage119891 is the objective function to be minimized(2)Ordering
Order the vertices in simplex from lowest objective function value 119891(1198601) to highest value 119891(119860119899+1) Ordered as 1198601le 1198602 le sdot sdot sdot le 119860119899+1(3)Midpoint
Calculate the midpoint for first two best vertices in simplex 119860119898 = sum(119860 119894119899) where 119894 = 1 2 119899(4) Reflection Process
Calculate reflection point 119860119903 by 119860119903 = 119860119898 + 120572(119860119898 minus 119860119899+1)if 119891(1198601) le 119891(1198602) le 119891(119860119899) then119860119899 larr 119860119903 and Go to to Step (8)end if
(5) Expansion Processif 119891(119860119903) le 119891(1198601) thenCalculate expansion point using 119860 119890 = 119860119903 + 120574(119860119903 minus 119860119898)end ifif 119891(119860 119890) lt 119891(119860119903) then119860119899 larr 119860 119890 and Go to to Step (8)else119860119899 larr 119860119903 and Go to to Step (8)end if
(6) Contraction Processif 119891(119860119899) le 119891(119860119903) le 119891(119860119899+1) thenCompute outside contraction by 119860 119888 = 120573119860119903 + (1 minus 120573)119860119898end ifif 119891(1198601) ge 119891(119860119899+1) thenCompute inside contraction by 119860 119888 = 120573119860119899+1 + (1 minus 120573)119860119898end ifif 119891(119860119903) ge 119891(119860119899) thenContraction is done between 119860119898 and the best vertex among 119860119903 and 119860119899+1end ifif 119891(119860 119888) lt 119891(119860119903) then119860119899 larr 119860 119888 and Go to to Step (8)else goes to Step (7)end ifif 119891(119860 119888) ge 119891(119860119899+1) then119860119899+1 larr 119860 119888 and Go to to Step (8)else Go to to Step (7)end if
(7) Shrinkage ProcessShrink towards the best solution with new vertices by 119860 119894 = 120575119860 119894 + 1198601(1 minus 120575) where 119894 = 2 119899 + 1
(8) Stopping CriteriaOrder and re-label new vertices of the simplex based on their objective function and go to step (4)
Algorithm 2 Pseudocode of Modified Nelder-Mead Method
119890-dimensional volume Total number of volumes 119873Vcan be formulated using
119873V = 119890prod119892=1
119899119892 (30)
(5) The starting point of the agent in the environmentwhich is one point inside volume is chosen bycalculating themidpoint of the volumeThemidpointof the lattice can be calculated as
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ] (31)
52 Decision-Making Process A key role of the honey beesis to select the best nest site and is done by the process ofdecision-making to produce a unified decisionThey follow adistributed decision-making process to find out the neighbornest site for their food particles The pseudocode for theproposed MODBCO algorithm is shown in Algorithm 3Figure 6 explains the workflow of the proposed algorithm forthe search of food particles by honey bees using MODBCO
521 Waggle Dance The scout bees after returning from thesearch of food particle report about the quality of the foodsite by communicationmode called waggle dance Scout beesperform thewaggle dance to other quiescent bees to advertise
12 Computational Intelligence and Neuroscience
Yes
Reflectionprocess
Order and label verticesbased on f(A)
Initialization
Coefficients 훼 훾 훽 훿
Objective function f(A)
f(Ab) lt f(Ar) lt f(Aw) Aw larr Ar
f(Ae) le f(Ar)
two best verticesAm forCalculate midpoint
Start
Terminationcriteria
Stop
Ar = Am + 훼(Am minus Aw)
ExpansionprocessNo
Yesf(Ar) le f(Aw) Aw larr Ae
No
b larr true Aw larr Ar
Contractionprocess
f(Ar) ge f(An)Yes
f(Ac) lt f(Ar)Aw larr Ac
b larr false
No
Shrinkageprocess
b larr true
Yes
Yes
No
Ae = Ar + 훾(Ar minus
Am)
Ac = 훽Ar + (1 minus 훽)Am
Ai = 훿Ai + A1(1 minus 훿)
Figure 5 Workflow of Modified Nelder-Mead Method
Computational Intelligence and Neuroscience 13
Multi-Objective Directed Bee Colony Optimization(1) Initialization119891(119909) is the objective function to be minimized
Initialize 119890 number of parameters and 119871119892 length of steps where 119892 = 0 to 119890Initialize initial value and the final value of the parameter as 119876119892119894 and 119876119892119891lowastlowast Solution Representation lowastlowastThe solutions are represented in the form of Binary values which can be generated as followsFor each solution 119894 = 1 119899119883119894 = 1199091198941 1199091198942 119909119894119889 | 119889 isin total days amp 119909119894119889 = rand ge 029 forall119889End for
(2) The number of steps in each step can be calculated using
119899119892 = 119876119892119894 minus 119876119892119891119871119892(3) The total number of volumes can be calculated using119873V = 119890prod
119892=1
119899119892(4) The midpoint of the volume to calculate starting point of the exploration can be calculated using
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ](5) Explore the search volume according to the Modified Nelder-Mead Method using Algorithm 2(6) The recorded value of the optimized point in vector table using[119891(1198811) 119891(1198812) 119891(119881119873V )](7) The globally optimized point is chosen based on Bee decision-making process using Consensus and Quorum
method approach 119891(119881119892) = min [119891(1198811) 119891(1198812) 119891(119881119873V )]Algorithm 3 Pseudocode of MODBCO
their best nest site for the exploration of food source Inthe multiagent system each agent after collecting individualsolution gives it to the centralized systems To select the bestoptimal solution forminimal optimal cases themathematicalformulation can be stated as
dance119894 = min (119891119894 (119881)) (32)
This mathematical formulation will find the minimaloptimal cases among the search solution where 119891119894(119881) is thesearch value calculated by the agent The search values arerecorded in the vector table 119881 119881 is the vector which consistsof 119890 number of elements The element 119890 contains the value ofthe parameter both optimal solution and parameter valuesare recorded in the vector table
522 Consensus Theconsensus is thewidespread agreementamong the group based on voting the voting pattern ofthe scout bees is monitored periodically to know whetherit reached an agreement and started acting on the decisionpattern Honey bees use the consensus method to select thebest search value the globally optimized point is chosen bycomparing the values in the vector table The globally opti-mized points are selected using themathematical formulation
523 Quorum In quorummethod the optimum solution iscalculated as the final solution based on the threshold levelobtained by the group decision-making process When thesolution reaches the optimal threshold level 120585119902 then the solu-tion is considered as a final solution based on unison decisionprocess The quorum threshold value describes the quality of
the food particle result When the threshold value is less thecomputation time decreases but it leads to inaccurate experi-mental resultsThe threshold value should be chosen to attainless computational timewith an accurate experimental result
6 Experimental Design and Analysis
61 Performance Metrics The performance of the proposedalgorithm MODBCO is assessed by comparing with fivedifferent competitor methods Here six performance metricsare considered to investigate the significance and evaluate theexperimental results The metrics are listed in this section
611 Least Error Rate Least Error Rate (LER) is the percent-age of the difference between known optimal value and thebest value obtained The LER can be calculated using
LER () = 119903sum119894=1
OptimalNRP-Instance minus fitness119894OptimalNRP-Instance
(34)
612 Average Convergence The Average Convergence is themeasure to evaluate the quality of the generated populationon average The Average Convergence (AC) is the percentageof the average of the convergence rate of solutions The per-formance of the convergence time is increased by the AverageConvergence to exploremore solutions in the populationTheAverage Convergence is calculated usingAC
= 119903sum119894=1
1 minus Avg_fitness119894 minusOptimalNRP-InstanceOptimalNRP-Instance
lowast 100 (35)
where (119903) is the number of instances in the given dataset
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
(2) The male drone bees mated with the queen and werediscarded from the colonies
(3) The remaining female bees in the hive are calledworker bees and they are called the building block ofthe hiveThe responsibilities of the worker bees are tofeed guard and maintain the honey bee comb
Based on the responsibility worker bees are classifiedas scout bees and forager bees A scout bee flies in searchof food sources randomly and returns when the energygets exhausted After reaching a hive scout bees share theinformation and start to explore rich food source locationswith forager bees The scout beersquos information includesdirection quality quantity and distance of the food sourcethey found The way of communicating information about afood source to foragers is done using dance There are twotypes of dance round dance and waggle dance The rounddance will provide direction of the food source when thedistance is small The waggle dance indicates the positionand the direction of the food source the distance can bemeasured by the speed of the dance A greater speed indicatesa smaller distance and the quantity of the food depends onthe wriggling of the beeThe exchange of information amonghive mates is to acquire collective knowledge Forager beeswill silently observe the behavior of scout bee to acquireknowledge about the directions and information of the foodsource
The group decision process of honey bees is for searchingbest food source and nest siteThe decision-making process isbased on the swarming process of the honey bee Swarming isthe process inwhich the queen bee and half of theworker beeswill leave their hive to explore a new colony The remainingworker bees and daughter bee will remain in the old hiveto monitor the waggle dance After leaving their parentalhive swarm bees will form a cluster in search of the newnest site The waggle dance is used to communicate withquiescent bees which are inactive in the colonyThis providesprecise information about the direction of the flower patchbased on its quality and energy level The number of followerbees increases based on the quality of the food source andallows the colony to gather food quickly and efficiently Thedecision-making process can be done in two methods byswarm bees to find the best nest site They are consensusand quorum consensus is the group agreement taken intoaccount and quorum is the decision process taken when thebee vote reaches a threshold value
Bee Colony Optimization (BCO) algorithm is apopulation-based algorithm The bees in the populationare artificial bees and each bee finds its neighboring solutionfrom the current path This algorithm has a forward andbackward process In forwarding pass every bee starts toexplore the neighborhood of its current solution and enablesconstructive and improving moves In forward pass entirebees in the hive will start the constructive move and thenlocal search will start In backward pass bees share theobjective value obtained in the forward pass The bees withhigher priority are used to discard all nonimproving movesThe bees will continue to explore in next forward pass orcontinue the same process with neighborhoodThe flowchart
Forward pass
Initialization
Construction move
Backward pass
Update the bestsolution
Stopping criteriaFalse
True
Figure 2 Flowchart of BCO algorithm
for BCO is shown in Figure 2 The BCO is proficient insolving combinatorial optimization problems by creatingcolonies of the multiagent system The pseudocode for BCOis described in Algorithm 1 The bee colony system providesa standard well-organized and well-coordinated teamworkmultitasking performance [33]
42 Modified Nelder-Mead Method The Nelder-MeadMethod is a simplex method for finding a local minimumfunction of various variables and is a local search algorithmfor unconstrained optimization problems The whole searcharea is divided into different fragments and filled with beeagents To obtain the best solution each fragment can besearched by its bee agents through Modified Nelder-MeadMethod (MNMM) Each agent in the fragments passesinformation about the optimized point using MNMMBy using NMMM the best points are obtained and thebest solution is chosen by decision-making process ofhoney bees The algorithm is a simplex-based method119863-dimensional simplex is initialized with 119863 + 1 verticesthat is two dimensions and it forms a triangle if it has threedimensions it forms a tetrahedron To assign the best andworst point the vertices are evaluated and ordered based onthe objective function
The best point or vertex is considered to the minimumvalue of the objective function and the worst point is chosen
8 Computational Intelligence and Neuroscience
Bee Colony Optimization(1) Initialization Assign every bee to an empty solution(2) Forward Pass
For every bee(21) set 119894 = 1(22) Evaluate all possible construction moves(23) Based on the evaluation choose one move using Roulette Wheel(24) 119894 = 119894 + 1 if (119894 le 119873) Go to step (22)
where 119894 is the counter for construction move and119873 is the number of construction moves during one forwardpass
(3) Return to Hive(4) Backward Pass starts(5) Compute the objective function for each bee and sort accordingly(6) Calculate probability or logical reasoning to continue with the computed solution and become recruiter bee(7) For every follower choose the new solution from recruiters(8) If stopping criteria is not met Go to step (2)(9) Evaluate and find the best solution(10) Output the best solution
Algorithm 1 Pseudocode of BCO
with a maximum value of the computed objective functionTo form simplex new vertex function values are computedThismethod can be calculated using four procedures namelyreflection expansion contraction and shrinkage Figure 3shows the operators of the simplex triangle in MNMM
The simplex operations in each vertex are updated closerto its optimal solution the vertices are ordered based onfitness value and ordered The best vertex is 119860119887 the secondbest vertex is 119860 119904 and the worst vertex is 119860119908 calculated basedon the objective function Let 119860 = (119909 119910) be the vertex in atriangle as food source points 119860119887 = (119909119887 119910119887) 119860 119904 = (119909119904 119910119904)and119860119908 = (119909119908 119910119908) are the positions of the food source pointsthat is local optimal points The objective functions for 119860119887119860 119904 and 119860119908 are calculated based on (23) towards the foodsource points
The objective function to construct simplex to obtainlocal search using MNMM is formulated as
119891 (119909 119910) = 1199092 minus 4119909 + 1199102 minus 119910 minus 119909119910 (23)
Based on the objective function value the vertices foodpoints are ordered ascending with their corresponding honeybee agentsThe obtained values are ordered as119860119887 le 119860 119904 le 119860119908with their honey bee position and food points in the simplextriangle Figure 4 describes the search of best-minimizedcost value for the nurse based on objective function (22)The working principle of Modified Nelder-Mead Method(MNMM) for searching food particles is explained in detail
(1) In the simplex triangle the reflection coefficient 120572expansion coefficient 120574 contraction coefficient 120573 andshrinkage coefficient 120575 are initialized
(2) The objective function for the simplex triangle ver-tices is calculated and ordered The best vertex withlower objective value is 119860119887 the second best vertex is119860 119904 and the worst vertex is named as 119860119908 and thesevertices are ordered based on the objective functionas 119860119887 le 119860 119904 le 119860119908
(3) The first two best vertices are selected namely119860119887 and119860 119904 and the construction proceeds with calculatingthe midpoint of the line segment which joins the twobest vertices that is food positions The objectivefunction decreases as the honey agent associated withthe worst position vertex moves towards best andsecond best verticesThe value decreases as the honeyagent moves towards 119860119908 to 119860119887 and 119860119908 to 119860 119904 It isfeasible to calculate the midpoint vertex 119860119898 by theline joining best and second best vertices using
119860119898 = 119860119887 + 119860 1199042 (24)
(4) A reflecting vertex 119860119903 is generated by choosing thereflection of worst point 119860119908 The objective functionvalue for 119860119903 is 119891(119860119903) which is calculated and it iscompared with worst vertex 119860119908 objective functionvalue 119891(119860119908) If 119891(119860119903) lt 119891(119860119908) proceed with step(5) the reflection vertex can be calculated using
119860119903 = 119860119898 + 120572 (119860119898 minus 119860119908) where 120572 gt 0 (25)
(5) The expansion process starts when the objectivefunction value at reflection vertex 119860119903 is lesser thanworst vertex 119860119908 119891(119860119903) lt 119891(119860119908) and the linesegment is further extended to 119860119890 through 119860119903 and119860119908 The vertex point 119860119890 is calculated by (26) If theobjective function value at119860119890 is lesser than reflectionvertex 119860119903 119891(119860119890) lt 119891(119860119903) then the expansion isaccepted and the honey bee agent has found best foodposition compared with reflection point
119860119890 = 119860119903 + 120574 (119860119903 minus 119860119898) where 120574 gt 1 (26)
(6) The contraction process is carried out when 119891(119860119903) lt119891(119860 119904) and 119891(119860119903) le 119891(119860119887) for replacing 119860119887 with
Computational Intelligence and Neuroscience 9
AwAs
Ab
(a) Simplex triangle
Ar
As
Ab
Aw
(b) Reflection
Ae
Ar
As
Ab
Aw
(c) Expansion
Ac
As
Ab
Aw
(d) Contraction (119860ℎ lt 119860119903)
Ac
As
Ab
Aw
(e) Contraction (119860119903 lt 119860ℎ)
A㰀b
A㰀s
As
Ab
Aw
(f) Shrinkage
Figure 3 Nelder-Mead operations
119860119903 If 119891(119860119903) gt 119891(119860ℎ) then the direct contractionwithout the replacement of 119860119887 with 119860119903 is performedThe contraction vertex 119860119888 can be calculated using
119860119888 = 120573119860119903 + (1 minus 120573)119860119898 where 0 lt 120573 lt 1 (27)
If 119891(119860119903) le 119891(119860119887) the contraction can be done and119860119888 replaced with 119860ℎ go to step (8) or else proceed tostep (7)
(7) The shrinkage phase proceeds when the contractionprocess at step (6) fails and is done by shrinking allthe vertices of the simplex triangle except 119860ℎ using(28) The objective function value of reflection andcontraction phase is not lesser than the worst pointthen the vertices 119860 119904 and 119860119908 must be shrunk towards119860ℎThus the vertices of smaller value will form a newsimplex triangle with another two best vertices
119860 119894 = 120575119860 119894 + 1198601 (1 minus 120575) where 0 lt 120575 lt 1 (28)
(8) The calculations are stopped when the terminationcondition is met
Algorithm 2 describes the pseudocode for ModifiedNelder-Mead Method in detail It portraits the detailed pro-cess of MNMM to obtain the best solution for the NRP Theworkflow of the proposed MNMM is explained in Figure 5
5 MODBCO
Bee Colony Optimization is the metaheuristic algorithm tosolve various combinatorial optimization problems and itis inspired by the natural behavior of bee for their foodsources The algorithm consists of two steps forward andbackward pass During forwarding pass bees started toexplore the neighborhood of its current solution and findall possible ways In backward pass bees return to thehive and share the values of the objective function of theircurrent solution Calculate nectar amount using probability
10 Computational Intelligence and Neuroscience
Ab
Aw
Ar
As
Am
d
d
Ab
Aw
Ar
As
Am
d
d
Aed2
Ab
Aw
Ar
As
Am
Ac1
Ac2
Ab
Aw As
Am
Anew
Figure 4 Bees search movement based on MNMM
function and advertise the solution the bee which has thebetter solution is given higher priority The remaining beesbased on the probability value decide whether to explore thesolution or proceed with the advertised solution DirectedBee Colony Optimization is the computational system whereseveral bees work together in uniting and interact with eachother to achieve goals based on the group decision processThe whole search area of the bee is divided into multiplefragments different bees are sent to different fragments Thebest solution in each fragment is obtained by using a localsearch algorithmModified Nelder-Mead Method (MNMM)To obtain the best solution the total varieties of individualparameters are partitioned into individual volumes Eachvolume determines the starting point of the exploration offood particle by each bee The bees use developed MNMMalgorithm to find the best solution by remembering thelast two best food sites they obtained After obtaining thecurrent solution the bee starts to backward pass sharingof information obtained during forwarding pass The beesstarted to share information about optimized point by thenatural behavior of bees called waggle dance When all theinformation about the best food is shared the best among theoptimized point is chosen using a decision-making processcalled consensus and quorummethod in honey bees [34 35]
51 Multiagent System All agents live in an environmentwhich is well structured and organized Inmultiagent systemseveral agents work together and interact with each otherto obtain the goal According to Jiao and Shi [36] andZhong et al [37] all agents should possess the followingqualities agents should live and act in an environmenteach agent should sense its local environment each agent
should be capable of interacting with other agents in a localenvironment and agents attempt to perform their goal Allagents interact with each other and take the decision toachieve the desired goals The multiagent system is a com-putational system and provides an opportunity to optimizeand compute all complex problems In multiagent system allagents start to live and act in the same environment which iswell organized and structured Each agent in the environmentis fixed on a lattice point The size and dimension of thelattice point in the environment depend upon the variablesused The objective function can be calculated based on theparameters fixed
(1) Consider ldquo119890rdquo number of independent parameters tocalculate the objective function The range of the 119892thparameter can be calculated using [119876119892119894 119876119892119891] where119876119892119894 is the initial value of the 119892th parameter and 119876119892119891is the final value of the 119892th parameter chosen
(2) Thus the objective function can be formulated as 119890number of axes each axis will contain a total rangeof single parameter with different dimensions
(3) Each axis is divided into smaller parts each partis called a step So 119892th axis can be divided into 119899119892number of steps each with the length of 119871119892 where thevalue of 119892 depends upon parameters thus 119892 = 1 to 119890The relationship between 119899119892 and 119871119892 can be given as
119899119892 = 119876119892119894 minus 119876119892119891119871119892 (29)
(4) Then each axis is divided into branches foreach branch 119892 number of branches will form an
Computational Intelligence and Neuroscience 11
Modified Nelder-Mead Method for directed honey bee food search(1) Initialization119860119887 denotes the list of vertices in simplex where 119894 = 1 2 119899 + 1120572 120574 120573 and 120575 are the coefficients of reflection expansion contraction and shrinkage119891 is the objective function to be minimized(2)Ordering
Order the vertices in simplex from lowest objective function value 119891(1198601) to highest value 119891(119860119899+1) Ordered as 1198601le 1198602 le sdot sdot sdot le 119860119899+1(3)Midpoint
Calculate the midpoint for first two best vertices in simplex 119860119898 = sum(119860 119894119899) where 119894 = 1 2 119899(4) Reflection Process
Calculate reflection point 119860119903 by 119860119903 = 119860119898 + 120572(119860119898 minus 119860119899+1)if 119891(1198601) le 119891(1198602) le 119891(119860119899) then119860119899 larr 119860119903 and Go to to Step (8)end if
(5) Expansion Processif 119891(119860119903) le 119891(1198601) thenCalculate expansion point using 119860 119890 = 119860119903 + 120574(119860119903 minus 119860119898)end ifif 119891(119860 119890) lt 119891(119860119903) then119860119899 larr 119860 119890 and Go to to Step (8)else119860119899 larr 119860119903 and Go to to Step (8)end if
(6) Contraction Processif 119891(119860119899) le 119891(119860119903) le 119891(119860119899+1) thenCompute outside contraction by 119860 119888 = 120573119860119903 + (1 minus 120573)119860119898end ifif 119891(1198601) ge 119891(119860119899+1) thenCompute inside contraction by 119860 119888 = 120573119860119899+1 + (1 minus 120573)119860119898end ifif 119891(119860119903) ge 119891(119860119899) thenContraction is done between 119860119898 and the best vertex among 119860119903 and 119860119899+1end ifif 119891(119860 119888) lt 119891(119860119903) then119860119899 larr 119860 119888 and Go to to Step (8)else goes to Step (7)end ifif 119891(119860 119888) ge 119891(119860119899+1) then119860119899+1 larr 119860 119888 and Go to to Step (8)else Go to to Step (7)end if
(7) Shrinkage ProcessShrink towards the best solution with new vertices by 119860 119894 = 120575119860 119894 + 1198601(1 minus 120575) where 119894 = 2 119899 + 1
(8) Stopping CriteriaOrder and re-label new vertices of the simplex based on their objective function and go to step (4)
Algorithm 2 Pseudocode of Modified Nelder-Mead Method
119890-dimensional volume Total number of volumes 119873Vcan be formulated using
119873V = 119890prod119892=1
119899119892 (30)
(5) The starting point of the agent in the environmentwhich is one point inside volume is chosen bycalculating themidpoint of the volumeThemidpointof the lattice can be calculated as
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ] (31)
52 Decision-Making Process A key role of the honey beesis to select the best nest site and is done by the process ofdecision-making to produce a unified decisionThey follow adistributed decision-making process to find out the neighbornest site for their food particles The pseudocode for theproposed MODBCO algorithm is shown in Algorithm 3Figure 6 explains the workflow of the proposed algorithm forthe search of food particles by honey bees using MODBCO
521 Waggle Dance The scout bees after returning from thesearch of food particle report about the quality of the foodsite by communicationmode called waggle dance Scout beesperform thewaggle dance to other quiescent bees to advertise
12 Computational Intelligence and Neuroscience
Yes
Reflectionprocess
Order and label verticesbased on f(A)
Initialization
Coefficients 훼 훾 훽 훿
Objective function f(A)
f(Ab) lt f(Ar) lt f(Aw) Aw larr Ar
f(Ae) le f(Ar)
two best verticesAm forCalculate midpoint
Start
Terminationcriteria
Stop
Ar = Am + 훼(Am minus Aw)
ExpansionprocessNo
Yesf(Ar) le f(Aw) Aw larr Ae
No
b larr true Aw larr Ar
Contractionprocess
f(Ar) ge f(An)Yes
f(Ac) lt f(Ar)Aw larr Ac
b larr false
No
Shrinkageprocess
b larr true
Yes
Yes
No
Ae = Ar + 훾(Ar minus
Am)
Ac = 훽Ar + (1 minus 훽)Am
Ai = 훿Ai + A1(1 minus 훿)
Figure 5 Workflow of Modified Nelder-Mead Method
Computational Intelligence and Neuroscience 13
Multi-Objective Directed Bee Colony Optimization(1) Initialization119891(119909) is the objective function to be minimized
Initialize 119890 number of parameters and 119871119892 length of steps where 119892 = 0 to 119890Initialize initial value and the final value of the parameter as 119876119892119894 and 119876119892119891lowastlowast Solution Representation lowastlowastThe solutions are represented in the form of Binary values which can be generated as followsFor each solution 119894 = 1 119899119883119894 = 1199091198941 1199091198942 119909119894119889 | 119889 isin total days amp 119909119894119889 = rand ge 029 forall119889End for
(2) The number of steps in each step can be calculated using
119899119892 = 119876119892119894 minus 119876119892119891119871119892(3) The total number of volumes can be calculated using119873V = 119890prod
119892=1
119899119892(4) The midpoint of the volume to calculate starting point of the exploration can be calculated using
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ](5) Explore the search volume according to the Modified Nelder-Mead Method using Algorithm 2(6) The recorded value of the optimized point in vector table using[119891(1198811) 119891(1198812) 119891(119881119873V )](7) The globally optimized point is chosen based on Bee decision-making process using Consensus and Quorum
method approach 119891(119881119892) = min [119891(1198811) 119891(1198812) 119891(119881119873V )]Algorithm 3 Pseudocode of MODBCO
their best nest site for the exploration of food source Inthe multiagent system each agent after collecting individualsolution gives it to the centralized systems To select the bestoptimal solution forminimal optimal cases themathematicalformulation can be stated as
dance119894 = min (119891119894 (119881)) (32)
This mathematical formulation will find the minimaloptimal cases among the search solution where 119891119894(119881) is thesearch value calculated by the agent The search values arerecorded in the vector table 119881 119881 is the vector which consistsof 119890 number of elements The element 119890 contains the value ofthe parameter both optimal solution and parameter valuesare recorded in the vector table
522 Consensus Theconsensus is thewidespread agreementamong the group based on voting the voting pattern ofthe scout bees is monitored periodically to know whetherit reached an agreement and started acting on the decisionpattern Honey bees use the consensus method to select thebest search value the globally optimized point is chosen bycomparing the values in the vector table The globally opti-mized points are selected using themathematical formulation
523 Quorum In quorummethod the optimum solution iscalculated as the final solution based on the threshold levelobtained by the group decision-making process When thesolution reaches the optimal threshold level 120585119902 then the solu-tion is considered as a final solution based on unison decisionprocess The quorum threshold value describes the quality of
the food particle result When the threshold value is less thecomputation time decreases but it leads to inaccurate experi-mental resultsThe threshold value should be chosen to attainless computational timewith an accurate experimental result
6 Experimental Design and Analysis
61 Performance Metrics The performance of the proposedalgorithm MODBCO is assessed by comparing with fivedifferent competitor methods Here six performance metricsare considered to investigate the significance and evaluate theexperimental results The metrics are listed in this section
611 Least Error Rate Least Error Rate (LER) is the percent-age of the difference between known optimal value and thebest value obtained The LER can be calculated using
LER () = 119903sum119894=1
OptimalNRP-Instance minus fitness119894OptimalNRP-Instance
(34)
612 Average Convergence The Average Convergence is themeasure to evaluate the quality of the generated populationon average The Average Convergence (AC) is the percentageof the average of the convergence rate of solutions The per-formance of the convergence time is increased by the AverageConvergence to exploremore solutions in the populationTheAverage Convergence is calculated usingAC
= 119903sum119894=1
1 minus Avg_fitness119894 minusOptimalNRP-InstanceOptimalNRP-Instance
lowast 100 (35)
where (119903) is the number of instances in the given dataset
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
Bee Colony Optimization(1) Initialization Assign every bee to an empty solution(2) Forward Pass
For every bee(21) set 119894 = 1(22) Evaluate all possible construction moves(23) Based on the evaluation choose one move using Roulette Wheel(24) 119894 = 119894 + 1 if (119894 le 119873) Go to step (22)
where 119894 is the counter for construction move and119873 is the number of construction moves during one forwardpass
(3) Return to Hive(4) Backward Pass starts(5) Compute the objective function for each bee and sort accordingly(6) Calculate probability or logical reasoning to continue with the computed solution and become recruiter bee(7) For every follower choose the new solution from recruiters(8) If stopping criteria is not met Go to step (2)(9) Evaluate and find the best solution(10) Output the best solution
Algorithm 1 Pseudocode of BCO
with a maximum value of the computed objective functionTo form simplex new vertex function values are computedThismethod can be calculated using four procedures namelyreflection expansion contraction and shrinkage Figure 3shows the operators of the simplex triangle in MNMM
The simplex operations in each vertex are updated closerto its optimal solution the vertices are ordered based onfitness value and ordered The best vertex is 119860119887 the secondbest vertex is 119860 119904 and the worst vertex is 119860119908 calculated basedon the objective function Let 119860 = (119909 119910) be the vertex in atriangle as food source points 119860119887 = (119909119887 119910119887) 119860 119904 = (119909119904 119910119904)and119860119908 = (119909119908 119910119908) are the positions of the food source pointsthat is local optimal points The objective functions for 119860119887119860 119904 and 119860119908 are calculated based on (23) towards the foodsource points
The objective function to construct simplex to obtainlocal search using MNMM is formulated as
119891 (119909 119910) = 1199092 minus 4119909 + 1199102 minus 119910 minus 119909119910 (23)
Based on the objective function value the vertices foodpoints are ordered ascending with their corresponding honeybee agentsThe obtained values are ordered as119860119887 le 119860 119904 le 119860119908with their honey bee position and food points in the simplextriangle Figure 4 describes the search of best-minimizedcost value for the nurse based on objective function (22)The working principle of Modified Nelder-Mead Method(MNMM) for searching food particles is explained in detail
(1) In the simplex triangle the reflection coefficient 120572expansion coefficient 120574 contraction coefficient 120573 andshrinkage coefficient 120575 are initialized
(2) The objective function for the simplex triangle ver-tices is calculated and ordered The best vertex withlower objective value is 119860119887 the second best vertex is119860 119904 and the worst vertex is named as 119860119908 and thesevertices are ordered based on the objective functionas 119860119887 le 119860 119904 le 119860119908
(3) The first two best vertices are selected namely119860119887 and119860 119904 and the construction proceeds with calculatingthe midpoint of the line segment which joins the twobest vertices that is food positions The objectivefunction decreases as the honey agent associated withthe worst position vertex moves towards best andsecond best verticesThe value decreases as the honeyagent moves towards 119860119908 to 119860119887 and 119860119908 to 119860 119904 It isfeasible to calculate the midpoint vertex 119860119898 by theline joining best and second best vertices using
119860119898 = 119860119887 + 119860 1199042 (24)
(4) A reflecting vertex 119860119903 is generated by choosing thereflection of worst point 119860119908 The objective functionvalue for 119860119903 is 119891(119860119903) which is calculated and it iscompared with worst vertex 119860119908 objective functionvalue 119891(119860119908) If 119891(119860119903) lt 119891(119860119908) proceed with step(5) the reflection vertex can be calculated using
119860119903 = 119860119898 + 120572 (119860119898 minus 119860119908) where 120572 gt 0 (25)
(5) The expansion process starts when the objectivefunction value at reflection vertex 119860119903 is lesser thanworst vertex 119860119908 119891(119860119903) lt 119891(119860119908) and the linesegment is further extended to 119860119890 through 119860119903 and119860119908 The vertex point 119860119890 is calculated by (26) If theobjective function value at119860119890 is lesser than reflectionvertex 119860119903 119891(119860119890) lt 119891(119860119903) then the expansion isaccepted and the honey bee agent has found best foodposition compared with reflection point
119860119890 = 119860119903 + 120574 (119860119903 minus 119860119898) where 120574 gt 1 (26)
(6) The contraction process is carried out when 119891(119860119903) lt119891(119860 119904) and 119891(119860119903) le 119891(119860119887) for replacing 119860119887 with
Computational Intelligence and Neuroscience 9
AwAs
Ab
(a) Simplex triangle
Ar
As
Ab
Aw
(b) Reflection
Ae
Ar
As
Ab
Aw
(c) Expansion
Ac
As
Ab
Aw
(d) Contraction (119860ℎ lt 119860119903)
Ac
As
Ab
Aw
(e) Contraction (119860119903 lt 119860ℎ)
A㰀b
A㰀s
As
Ab
Aw
(f) Shrinkage
Figure 3 Nelder-Mead operations
119860119903 If 119891(119860119903) gt 119891(119860ℎ) then the direct contractionwithout the replacement of 119860119887 with 119860119903 is performedThe contraction vertex 119860119888 can be calculated using
119860119888 = 120573119860119903 + (1 minus 120573)119860119898 where 0 lt 120573 lt 1 (27)
If 119891(119860119903) le 119891(119860119887) the contraction can be done and119860119888 replaced with 119860ℎ go to step (8) or else proceed tostep (7)
(7) The shrinkage phase proceeds when the contractionprocess at step (6) fails and is done by shrinking allthe vertices of the simplex triangle except 119860ℎ using(28) The objective function value of reflection andcontraction phase is not lesser than the worst pointthen the vertices 119860 119904 and 119860119908 must be shrunk towards119860ℎThus the vertices of smaller value will form a newsimplex triangle with another two best vertices
119860 119894 = 120575119860 119894 + 1198601 (1 minus 120575) where 0 lt 120575 lt 1 (28)
(8) The calculations are stopped when the terminationcondition is met
Algorithm 2 describes the pseudocode for ModifiedNelder-Mead Method in detail It portraits the detailed pro-cess of MNMM to obtain the best solution for the NRP Theworkflow of the proposed MNMM is explained in Figure 5
5 MODBCO
Bee Colony Optimization is the metaheuristic algorithm tosolve various combinatorial optimization problems and itis inspired by the natural behavior of bee for their foodsources The algorithm consists of two steps forward andbackward pass During forwarding pass bees started toexplore the neighborhood of its current solution and findall possible ways In backward pass bees return to thehive and share the values of the objective function of theircurrent solution Calculate nectar amount using probability
10 Computational Intelligence and Neuroscience
Ab
Aw
Ar
As
Am
d
d
Ab
Aw
Ar
As
Am
d
d
Aed2
Ab
Aw
Ar
As
Am
Ac1
Ac2
Ab
Aw As
Am
Anew
Figure 4 Bees search movement based on MNMM
function and advertise the solution the bee which has thebetter solution is given higher priority The remaining beesbased on the probability value decide whether to explore thesolution or proceed with the advertised solution DirectedBee Colony Optimization is the computational system whereseveral bees work together in uniting and interact with eachother to achieve goals based on the group decision processThe whole search area of the bee is divided into multiplefragments different bees are sent to different fragments Thebest solution in each fragment is obtained by using a localsearch algorithmModified Nelder-Mead Method (MNMM)To obtain the best solution the total varieties of individualparameters are partitioned into individual volumes Eachvolume determines the starting point of the exploration offood particle by each bee The bees use developed MNMMalgorithm to find the best solution by remembering thelast two best food sites they obtained After obtaining thecurrent solution the bee starts to backward pass sharingof information obtained during forwarding pass The beesstarted to share information about optimized point by thenatural behavior of bees called waggle dance When all theinformation about the best food is shared the best among theoptimized point is chosen using a decision-making processcalled consensus and quorummethod in honey bees [34 35]
51 Multiagent System All agents live in an environmentwhich is well structured and organized Inmultiagent systemseveral agents work together and interact with each otherto obtain the goal According to Jiao and Shi [36] andZhong et al [37] all agents should possess the followingqualities agents should live and act in an environmenteach agent should sense its local environment each agent
should be capable of interacting with other agents in a localenvironment and agents attempt to perform their goal Allagents interact with each other and take the decision toachieve the desired goals The multiagent system is a com-putational system and provides an opportunity to optimizeand compute all complex problems In multiagent system allagents start to live and act in the same environment which iswell organized and structured Each agent in the environmentis fixed on a lattice point The size and dimension of thelattice point in the environment depend upon the variablesused The objective function can be calculated based on theparameters fixed
(1) Consider ldquo119890rdquo number of independent parameters tocalculate the objective function The range of the 119892thparameter can be calculated using [119876119892119894 119876119892119891] where119876119892119894 is the initial value of the 119892th parameter and 119876119892119891is the final value of the 119892th parameter chosen
(2) Thus the objective function can be formulated as 119890number of axes each axis will contain a total rangeof single parameter with different dimensions
(3) Each axis is divided into smaller parts each partis called a step So 119892th axis can be divided into 119899119892number of steps each with the length of 119871119892 where thevalue of 119892 depends upon parameters thus 119892 = 1 to 119890The relationship between 119899119892 and 119871119892 can be given as
119899119892 = 119876119892119894 minus 119876119892119891119871119892 (29)
(4) Then each axis is divided into branches foreach branch 119892 number of branches will form an
Computational Intelligence and Neuroscience 11
Modified Nelder-Mead Method for directed honey bee food search(1) Initialization119860119887 denotes the list of vertices in simplex where 119894 = 1 2 119899 + 1120572 120574 120573 and 120575 are the coefficients of reflection expansion contraction and shrinkage119891 is the objective function to be minimized(2)Ordering
Order the vertices in simplex from lowest objective function value 119891(1198601) to highest value 119891(119860119899+1) Ordered as 1198601le 1198602 le sdot sdot sdot le 119860119899+1(3)Midpoint
Calculate the midpoint for first two best vertices in simplex 119860119898 = sum(119860 119894119899) where 119894 = 1 2 119899(4) Reflection Process
Calculate reflection point 119860119903 by 119860119903 = 119860119898 + 120572(119860119898 minus 119860119899+1)if 119891(1198601) le 119891(1198602) le 119891(119860119899) then119860119899 larr 119860119903 and Go to to Step (8)end if
(5) Expansion Processif 119891(119860119903) le 119891(1198601) thenCalculate expansion point using 119860 119890 = 119860119903 + 120574(119860119903 minus 119860119898)end ifif 119891(119860 119890) lt 119891(119860119903) then119860119899 larr 119860 119890 and Go to to Step (8)else119860119899 larr 119860119903 and Go to to Step (8)end if
(6) Contraction Processif 119891(119860119899) le 119891(119860119903) le 119891(119860119899+1) thenCompute outside contraction by 119860 119888 = 120573119860119903 + (1 minus 120573)119860119898end ifif 119891(1198601) ge 119891(119860119899+1) thenCompute inside contraction by 119860 119888 = 120573119860119899+1 + (1 minus 120573)119860119898end ifif 119891(119860119903) ge 119891(119860119899) thenContraction is done between 119860119898 and the best vertex among 119860119903 and 119860119899+1end ifif 119891(119860 119888) lt 119891(119860119903) then119860119899 larr 119860 119888 and Go to to Step (8)else goes to Step (7)end ifif 119891(119860 119888) ge 119891(119860119899+1) then119860119899+1 larr 119860 119888 and Go to to Step (8)else Go to to Step (7)end if
(7) Shrinkage ProcessShrink towards the best solution with new vertices by 119860 119894 = 120575119860 119894 + 1198601(1 minus 120575) where 119894 = 2 119899 + 1
(8) Stopping CriteriaOrder and re-label new vertices of the simplex based on their objective function and go to step (4)
Algorithm 2 Pseudocode of Modified Nelder-Mead Method
119890-dimensional volume Total number of volumes 119873Vcan be formulated using
119873V = 119890prod119892=1
119899119892 (30)
(5) The starting point of the agent in the environmentwhich is one point inside volume is chosen bycalculating themidpoint of the volumeThemidpointof the lattice can be calculated as
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ] (31)
52 Decision-Making Process A key role of the honey beesis to select the best nest site and is done by the process ofdecision-making to produce a unified decisionThey follow adistributed decision-making process to find out the neighbornest site for their food particles The pseudocode for theproposed MODBCO algorithm is shown in Algorithm 3Figure 6 explains the workflow of the proposed algorithm forthe search of food particles by honey bees using MODBCO
521 Waggle Dance The scout bees after returning from thesearch of food particle report about the quality of the foodsite by communicationmode called waggle dance Scout beesperform thewaggle dance to other quiescent bees to advertise
12 Computational Intelligence and Neuroscience
Yes
Reflectionprocess
Order and label verticesbased on f(A)
Initialization
Coefficients 훼 훾 훽 훿
Objective function f(A)
f(Ab) lt f(Ar) lt f(Aw) Aw larr Ar
f(Ae) le f(Ar)
two best verticesAm forCalculate midpoint
Start
Terminationcriteria
Stop
Ar = Am + 훼(Am minus Aw)
ExpansionprocessNo
Yesf(Ar) le f(Aw) Aw larr Ae
No
b larr true Aw larr Ar
Contractionprocess
f(Ar) ge f(An)Yes
f(Ac) lt f(Ar)Aw larr Ac
b larr false
No
Shrinkageprocess
b larr true
Yes
Yes
No
Ae = Ar + 훾(Ar minus
Am)
Ac = 훽Ar + (1 minus 훽)Am
Ai = 훿Ai + A1(1 minus 훿)
Figure 5 Workflow of Modified Nelder-Mead Method
Computational Intelligence and Neuroscience 13
Multi-Objective Directed Bee Colony Optimization(1) Initialization119891(119909) is the objective function to be minimized
Initialize 119890 number of parameters and 119871119892 length of steps where 119892 = 0 to 119890Initialize initial value and the final value of the parameter as 119876119892119894 and 119876119892119891lowastlowast Solution Representation lowastlowastThe solutions are represented in the form of Binary values which can be generated as followsFor each solution 119894 = 1 119899119883119894 = 1199091198941 1199091198942 119909119894119889 | 119889 isin total days amp 119909119894119889 = rand ge 029 forall119889End for
(2) The number of steps in each step can be calculated using
119899119892 = 119876119892119894 minus 119876119892119891119871119892(3) The total number of volumes can be calculated using119873V = 119890prod
119892=1
119899119892(4) The midpoint of the volume to calculate starting point of the exploration can be calculated using
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ](5) Explore the search volume according to the Modified Nelder-Mead Method using Algorithm 2(6) The recorded value of the optimized point in vector table using[119891(1198811) 119891(1198812) 119891(119881119873V )](7) The globally optimized point is chosen based on Bee decision-making process using Consensus and Quorum
method approach 119891(119881119892) = min [119891(1198811) 119891(1198812) 119891(119881119873V )]Algorithm 3 Pseudocode of MODBCO
their best nest site for the exploration of food source Inthe multiagent system each agent after collecting individualsolution gives it to the centralized systems To select the bestoptimal solution forminimal optimal cases themathematicalformulation can be stated as
dance119894 = min (119891119894 (119881)) (32)
This mathematical formulation will find the minimaloptimal cases among the search solution where 119891119894(119881) is thesearch value calculated by the agent The search values arerecorded in the vector table 119881 119881 is the vector which consistsof 119890 number of elements The element 119890 contains the value ofthe parameter both optimal solution and parameter valuesare recorded in the vector table
522 Consensus Theconsensus is thewidespread agreementamong the group based on voting the voting pattern ofthe scout bees is monitored periodically to know whetherit reached an agreement and started acting on the decisionpattern Honey bees use the consensus method to select thebest search value the globally optimized point is chosen bycomparing the values in the vector table The globally opti-mized points are selected using themathematical formulation
523 Quorum In quorummethod the optimum solution iscalculated as the final solution based on the threshold levelobtained by the group decision-making process When thesolution reaches the optimal threshold level 120585119902 then the solu-tion is considered as a final solution based on unison decisionprocess The quorum threshold value describes the quality of
the food particle result When the threshold value is less thecomputation time decreases but it leads to inaccurate experi-mental resultsThe threshold value should be chosen to attainless computational timewith an accurate experimental result
6 Experimental Design and Analysis
61 Performance Metrics The performance of the proposedalgorithm MODBCO is assessed by comparing with fivedifferent competitor methods Here six performance metricsare considered to investigate the significance and evaluate theexperimental results The metrics are listed in this section
611 Least Error Rate Least Error Rate (LER) is the percent-age of the difference between known optimal value and thebest value obtained The LER can be calculated using
LER () = 119903sum119894=1
OptimalNRP-Instance minus fitness119894OptimalNRP-Instance
(34)
612 Average Convergence The Average Convergence is themeasure to evaluate the quality of the generated populationon average The Average Convergence (AC) is the percentageof the average of the convergence rate of solutions The per-formance of the convergence time is increased by the AverageConvergence to exploremore solutions in the populationTheAverage Convergence is calculated usingAC
= 119903sum119894=1
1 minus Avg_fitness119894 minusOptimalNRP-InstanceOptimalNRP-Instance
lowast 100 (35)
where (119903) is the number of instances in the given dataset
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
119860119903 If 119891(119860119903) gt 119891(119860ℎ) then the direct contractionwithout the replacement of 119860119887 with 119860119903 is performedThe contraction vertex 119860119888 can be calculated using
119860119888 = 120573119860119903 + (1 minus 120573)119860119898 where 0 lt 120573 lt 1 (27)
If 119891(119860119903) le 119891(119860119887) the contraction can be done and119860119888 replaced with 119860ℎ go to step (8) or else proceed tostep (7)
(7) The shrinkage phase proceeds when the contractionprocess at step (6) fails and is done by shrinking allthe vertices of the simplex triangle except 119860ℎ using(28) The objective function value of reflection andcontraction phase is not lesser than the worst pointthen the vertices 119860 119904 and 119860119908 must be shrunk towards119860ℎThus the vertices of smaller value will form a newsimplex triangle with another two best vertices
119860 119894 = 120575119860 119894 + 1198601 (1 minus 120575) where 0 lt 120575 lt 1 (28)
(8) The calculations are stopped when the terminationcondition is met
Algorithm 2 describes the pseudocode for ModifiedNelder-Mead Method in detail It portraits the detailed pro-cess of MNMM to obtain the best solution for the NRP Theworkflow of the proposed MNMM is explained in Figure 5
5 MODBCO
Bee Colony Optimization is the metaheuristic algorithm tosolve various combinatorial optimization problems and itis inspired by the natural behavior of bee for their foodsources The algorithm consists of two steps forward andbackward pass During forwarding pass bees started toexplore the neighborhood of its current solution and findall possible ways In backward pass bees return to thehive and share the values of the objective function of theircurrent solution Calculate nectar amount using probability
10 Computational Intelligence and Neuroscience
Ab
Aw
Ar
As
Am
d
d
Ab
Aw
Ar
As
Am
d
d
Aed2
Ab
Aw
Ar
As
Am
Ac1
Ac2
Ab
Aw As
Am
Anew
Figure 4 Bees search movement based on MNMM
function and advertise the solution the bee which has thebetter solution is given higher priority The remaining beesbased on the probability value decide whether to explore thesolution or proceed with the advertised solution DirectedBee Colony Optimization is the computational system whereseveral bees work together in uniting and interact with eachother to achieve goals based on the group decision processThe whole search area of the bee is divided into multiplefragments different bees are sent to different fragments Thebest solution in each fragment is obtained by using a localsearch algorithmModified Nelder-Mead Method (MNMM)To obtain the best solution the total varieties of individualparameters are partitioned into individual volumes Eachvolume determines the starting point of the exploration offood particle by each bee The bees use developed MNMMalgorithm to find the best solution by remembering thelast two best food sites they obtained After obtaining thecurrent solution the bee starts to backward pass sharingof information obtained during forwarding pass The beesstarted to share information about optimized point by thenatural behavior of bees called waggle dance When all theinformation about the best food is shared the best among theoptimized point is chosen using a decision-making processcalled consensus and quorummethod in honey bees [34 35]
51 Multiagent System All agents live in an environmentwhich is well structured and organized Inmultiagent systemseveral agents work together and interact with each otherto obtain the goal According to Jiao and Shi [36] andZhong et al [37] all agents should possess the followingqualities agents should live and act in an environmenteach agent should sense its local environment each agent
should be capable of interacting with other agents in a localenvironment and agents attempt to perform their goal Allagents interact with each other and take the decision toachieve the desired goals The multiagent system is a com-putational system and provides an opportunity to optimizeand compute all complex problems In multiagent system allagents start to live and act in the same environment which iswell organized and structured Each agent in the environmentis fixed on a lattice point The size and dimension of thelattice point in the environment depend upon the variablesused The objective function can be calculated based on theparameters fixed
(1) Consider ldquo119890rdquo number of independent parameters tocalculate the objective function The range of the 119892thparameter can be calculated using [119876119892119894 119876119892119891] where119876119892119894 is the initial value of the 119892th parameter and 119876119892119891is the final value of the 119892th parameter chosen
(2) Thus the objective function can be formulated as 119890number of axes each axis will contain a total rangeof single parameter with different dimensions
(3) Each axis is divided into smaller parts each partis called a step So 119892th axis can be divided into 119899119892number of steps each with the length of 119871119892 where thevalue of 119892 depends upon parameters thus 119892 = 1 to 119890The relationship between 119899119892 and 119871119892 can be given as
119899119892 = 119876119892119894 minus 119876119892119891119871119892 (29)
(4) Then each axis is divided into branches foreach branch 119892 number of branches will form an
Computational Intelligence and Neuroscience 11
Modified Nelder-Mead Method for directed honey bee food search(1) Initialization119860119887 denotes the list of vertices in simplex where 119894 = 1 2 119899 + 1120572 120574 120573 and 120575 are the coefficients of reflection expansion contraction and shrinkage119891 is the objective function to be minimized(2)Ordering
Order the vertices in simplex from lowest objective function value 119891(1198601) to highest value 119891(119860119899+1) Ordered as 1198601le 1198602 le sdot sdot sdot le 119860119899+1(3)Midpoint
Calculate the midpoint for first two best vertices in simplex 119860119898 = sum(119860 119894119899) where 119894 = 1 2 119899(4) Reflection Process
Calculate reflection point 119860119903 by 119860119903 = 119860119898 + 120572(119860119898 minus 119860119899+1)if 119891(1198601) le 119891(1198602) le 119891(119860119899) then119860119899 larr 119860119903 and Go to to Step (8)end if
(5) Expansion Processif 119891(119860119903) le 119891(1198601) thenCalculate expansion point using 119860 119890 = 119860119903 + 120574(119860119903 minus 119860119898)end ifif 119891(119860 119890) lt 119891(119860119903) then119860119899 larr 119860 119890 and Go to to Step (8)else119860119899 larr 119860119903 and Go to to Step (8)end if
(6) Contraction Processif 119891(119860119899) le 119891(119860119903) le 119891(119860119899+1) thenCompute outside contraction by 119860 119888 = 120573119860119903 + (1 minus 120573)119860119898end ifif 119891(1198601) ge 119891(119860119899+1) thenCompute inside contraction by 119860 119888 = 120573119860119899+1 + (1 minus 120573)119860119898end ifif 119891(119860119903) ge 119891(119860119899) thenContraction is done between 119860119898 and the best vertex among 119860119903 and 119860119899+1end ifif 119891(119860 119888) lt 119891(119860119903) then119860119899 larr 119860 119888 and Go to to Step (8)else goes to Step (7)end ifif 119891(119860 119888) ge 119891(119860119899+1) then119860119899+1 larr 119860 119888 and Go to to Step (8)else Go to to Step (7)end if
(7) Shrinkage ProcessShrink towards the best solution with new vertices by 119860 119894 = 120575119860 119894 + 1198601(1 minus 120575) where 119894 = 2 119899 + 1
(8) Stopping CriteriaOrder and re-label new vertices of the simplex based on their objective function and go to step (4)
Algorithm 2 Pseudocode of Modified Nelder-Mead Method
119890-dimensional volume Total number of volumes 119873Vcan be formulated using
119873V = 119890prod119892=1
119899119892 (30)
(5) The starting point of the agent in the environmentwhich is one point inside volume is chosen bycalculating themidpoint of the volumeThemidpointof the lattice can be calculated as
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ] (31)
52 Decision-Making Process A key role of the honey beesis to select the best nest site and is done by the process ofdecision-making to produce a unified decisionThey follow adistributed decision-making process to find out the neighbornest site for their food particles The pseudocode for theproposed MODBCO algorithm is shown in Algorithm 3Figure 6 explains the workflow of the proposed algorithm forthe search of food particles by honey bees using MODBCO
521 Waggle Dance The scout bees after returning from thesearch of food particle report about the quality of the foodsite by communicationmode called waggle dance Scout beesperform thewaggle dance to other quiescent bees to advertise
12 Computational Intelligence and Neuroscience
Yes
Reflectionprocess
Order and label verticesbased on f(A)
Initialization
Coefficients 훼 훾 훽 훿
Objective function f(A)
f(Ab) lt f(Ar) lt f(Aw) Aw larr Ar
f(Ae) le f(Ar)
two best verticesAm forCalculate midpoint
Start
Terminationcriteria
Stop
Ar = Am + 훼(Am minus Aw)
ExpansionprocessNo
Yesf(Ar) le f(Aw) Aw larr Ae
No
b larr true Aw larr Ar
Contractionprocess
f(Ar) ge f(An)Yes
f(Ac) lt f(Ar)Aw larr Ac
b larr false
No
Shrinkageprocess
b larr true
Yes
Yes
No
Ae = Ar + 훾(Ar minus
Am)
Ac = 훽Ar + (1 minus 훽)Am
Ai = 훿Ai + A1(1 minus 훿)
Figure 5 Workflow of Modified Nelder-Mead Method
Computational Intelligence and Neuroscience 13
Multi-Objective Directed Bee Colony Optimization(1) Initialization119891(119909) is the objective function to be minimized
Initialize 119890 number of parameters and 119871119892 length of steps where 119892 = 0 to 119890Initialize initial value and the final value of the parameter as 119876119892119894 and 119876119892119891lowastlowast Solution Representation lowastlowastThe solutions are represented in the form of Binary values which can be generated as followsFor each solution 119894 = 1 119899119883119894 = 1199091198941 1199091198942 119909119894119889 | 119889 isin total days amp 119909119894119889 = rand ge 029 forall119889End for
(2) The number of steps in each step can be calculated using
119899119892 = 119876119892119894 minus 119876119892119891119871119892(3) The total number of volumes can be calculated using119873V = 119890prod
119892=1
119899119892(4) The midpoint of the volume to calculate starting point of the exploration can be calculated using
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ](5) Explore the search volume according to the Modified Nelder-Mead Method using Algorithm 2(6) The recorded value of the optimized point in vector table using[119891(1198811) 119891(1198812) 119891(119881119873V )](7) The globally optimized point is chosen based on Bee decision-making process using Consensus and Quorum
method approach 119891(119881119892) = min [119891(1198811) 119891(1198812) 119891(119881119873V )]Algorithm 3 Pseudocode of MODBCO
their best nest site for the exploration of food source Inthe multiagent system each agent after collecting individualsolution gives it to the centralized systems To select the bestoptimal solution forminimal optimal cases themathematicalformulation can be stated as
dance119894 = min (119891119894 (119881)) (32)
This mathematical formulation will find the minimaloptimal cases among the search solution where 119891119894(119881) is thesearch value calculated by the agent The search values arerecorded in the vector table 119881 119881 is the vector which consistsof 119890 number of elements The element 119890 contains the value ofthe parameter both optimal solution and parameter valuesare recorded in the vector table
522 Consensus Theconsensus is thewidespread agreementamong the group based on voting the voting pattern ofthe scout bees is monitored periodically to know whetherit reached an agreement and started acting on the decisionpattern Honey bees use the consensus method to select thebest search value the globally optimized point is chosen bycomparing the values in the vector table The globally opti-mized points are selected using themathematical formulation
523 Quorum In quorummethod the optimum solution iscalculated as the final solution based on the threshold levelobtained by the group decision-making process When thesolution reaches the optimal threshold level 120585119902 then the solu-tion is considered as a final solution based on unison decisionprocess The quorum threshold value describes the quality of
the food particle result When the threshold value is less thecomputation time decreases but it leads to inaccurate experi-mental resultsThe threshold value should be chosen to attainless computational timewith an accurate experimental result
6 Experimental Design and Analysis
61 Performance Metrics The performance of the proposedalgorithm MODBCO is assessed by comparing with fivedifferent competitor methods Here six performance metricsare considered to investigate the significance and evaluate theexperimental results The metrics are listed in this section
611 Least Error Rate Least Error Rate (LER) is the percent-age of the difference between known optimal value and thebest value obtained The LER can be calculated using
LER () = 119903sum119894=1
OptimalNRP-Instance minus fitness119894OptimalNRP-Instance
(34)
612 Average Convergence The Average Convergence is themeasure to evaluate the quality of the generated populationon average The Average Convergence (AC) is the percentageof the average of the convergence rate of solutions The per-formance of the convergence time is increased by the AverageConvergence to exploremore solutions in the populationTheAverage Convergence is calculated usingAC
= 119903sum119894=1
1 minus Avg_fitness119894 minusOptimalNRP-InstanceOptimalNRP-Instance
lowast 100 (35)
where (119903) is the number of instances in the given dataset
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
function and advertise the solution the bee which has thebetter solution is given higher priority The remaining beesbased on the probability value decide whether to explore thesolution or proceed with the advertised solution DirectedBee Colony Optimization is the computational system whereseveral bees work together in uniting and interact with eachother to achieve goals based on the group decision processThe whole search area of the bee is divided into multiplefragments different bees are sent to different fragments Thebest solution in each fragment is obtained by using a localsearch algorithmModified Nelder-Mead Method (MNMM)To obtain the best solution the total varieties of individualparameters are partitioned into individual volumes Eachvolume determines the starting point of the exploration offood particle by each bee The bees use developed MNMMalgorithm to find the best solution by remembering thelast two best food sites they obtained After obtaining thecurrent solution the bee starts to backward pass sharingof information obtained during forwarding pass The beesstarted to share information about optimized point by thenatural behavior of bees called waggle dance When all theinformation about the best food is shared the best among theoptimized point is chosen using a decision-making processcalled consensus and quorummethod in honey bees [34 35]
51 Multiagent System All agents live in an environmentwhich is well structured and organized Inmultiagent systemseveral agents work together and interact with each otherto obtain the goal According to Jiao and Shi [36] andZhong et al [37] all agents should possess the followingqualities agents should live and act in an environmenteach agent should sense its local environment each agent
should be capable of interacting with other agents in a localenvironment and agents attempt to perform their goal Allagents interact with each other and take the decision toachieve the desired goals The multiagent system is a com-putational system and provides an opportunity to optimizeand compute all complex problems In multiagent system allagents start to live and act in the same environment which iswell organized and structured Each agent in the environmentis fixed on a lattice point The size and dimension of thelattice point in the environment depend upon the variablesused The objective function can be calculated based on theparameters fixed
(1) Consider ldquo119890rdquo number of independent parameters tocalculate the objective function The range of the 119892thparameter can be calculated using [119876119892119894 119876119892119891] where119876119892119894 is the initial value of the 119892th parameter and 119876119892119891is the final value of the 119892th parameter chosen
(2) Thus the objective function can be formulated as 119890number of axes each axis will contain a total rangeof single parameter with different dimensions
(3) Each axis is divided into smaller parts each partis called a step So 119892th axis can be divided into 119899119892number of steps each with the length of 119871119892 where thevalue of 119892 depends upon parameters thus 119892 = 1 to 119890The relationship between 119899119892 and 119871119892 can be given as
119899119892 = 119876119892119894 minus 119876119892119891119871119892 (29)
(4) Then each axis is divided into branches foreach branch 119892 number of branches will form an
Computational Intelligence and Neuroscience 11
Modified Nelder-Mead Method for directed honey bee food search(1) Initialization119860119887 denotes the list of vertices in simplex where 119894 = 1 2 119899 + 1120572 120574 120573 and 120575 are the coefficients of reflection expansion contraction and shrinkage119891 is the objective function to be minimized(2)Ordering
Order the vertices in simplex from lowest objective function value 119891(1198601) to highest value 119891(119860119899+1) Ordered as 1198601le 1198602 le sdot sdot sdot le 119860119899+1(3)Midpoint
Calculate the midpoint for first two best vertices in simplex 119860119898 = sum(119860 119894119899) where 119894 = 1 2 119899(4) Reflection Process
Calculate reflection point 119860119903 by 119860119903 = 119860119898 + 120572(119860119898 minus 119860119899+1)if 119891(1198601) le 119891(1198602) le 119891(119860119899) then119860119899 larr 119860119903 and Go to to Step (8)end if
(5) Expansion Processif 119891(119860119903) le 119891(1198601) thenCalculate expansion point using 119860 119890 = 119860119903 + 120574(119860119903 minus 119860119898)end ifif 119891(119860 119890) lt 119891(119860119903) then119860119899 larr 119860 119890 and Go to to Step (8)else119860119899 larr 119860119903 and Go to to Step (8)end if
(6) Contraction Processif 119891(119860119899) le 119891(119860119903) le 119891(119860119899+1) thenCompute outside contraction by 119860 119888 = 120573119860119903 + (1 minus 120573)119860119898end ifif 119891(1198601) ge 119891(119860119899+1) thenCompute inside contraction by 119860 119888 = 120573119860119899+1 + (1 minus 120573)119860119898end ifif 119891(119860119903) ge 119891(119860119899) thenContraction is done between 119860119898 and the best vertex among 119860119903 and 119860119899+1end ifif 119891(119860 119888) lt 119891(119860119903) then119860119899 larr 119860 119888 and Go to to Step (8)else goes to Step (7)end ifif 119891(119860 119888) ge 119891(119860119899+1) then119860119899+1 larr 119860 119888 and Go to to Step (8)else Go to to Step (7)end if
(7) Shrinkage ProcessShrink towards the best solution with new vertices by 119860 119894 = 120575119860 119894 + 1198601(1 minus 120575) where 119894 = 2 119899 + 1
(8) Stopping CriteriaOrder and re-label new vertices of the simplex based on their objective function and go to step (4)
Algorithm 2 Pseudocode of Modified Nelder-Mead Method
119890-dimensional volume Total number of volumes 119873Vcan be formulated using
119873V = 119890prod119892=1
119899119892 (30)
(5) The starting point of the agent in the environmentwhich is one point inside volume is chosen bycalculating themidpoint of the volumeThemidpointof the lattice can be calculated as
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ] (31)
52 Decision-Making Process A key role of the honey beesis to select the best nest site and is done by the process ofdecision-making to produce a unified decisionThey follow adistributed decision-making process to find out the neighbornest site for their food particles The pseudocode for theproposed MODBCO algorithm is shown in Algorithm 3Figure 6 explains the workflow of the proposed algorithm forthe search of food particles by honey bees using MODBCO
521 Waggle Dance The scout bees after returning from thesearch of food particle report about the quality of the foodsite by communicationmode called waggle dance Scout beesperform thewaggle dance to other quiescent bees to advertise
12 Computational Intelligence and Neuroscience
Yes
Reflectionprocess
Order and label verticesbased on f(A)
Initialization
Coefficients 훼 훾 훽 훿
Objective function f(A)
f(Ab) lt f(Ar) lt f(Aw) Aw larr Ar
f(Ae) le f(Ar)
two best verticesAm forCalculate midpoint
Start
Terminationcriteria
Stop
Ar = Am + 훼(Am minus Aw)
ExpansionprocessNo
Yesf(Ar) le f(Aw) Aw larr Ae
No
b larr true Aw larr Ar
Contractionprocess
f(Ar) ge f(An)Yes
f(Ac) lt f(Ar)Aw larr Ac
b larr false
No
Shrinkageprocess
b larr true
Yes
Yes
No
Ae = Ar + 훾(Ar minus
Am)
Ac = 훽Ar + (1 minus 훽)Am
Ai = 훿Ai + A1(1 minus 훿)
Figure 5 Workflow of Modified Nelder-Mead Method
Computational Intelligence and Neuroscience 13
Multi-Objective Directed Bee Colony Optimization(1) Initialization119891(119909) is the objective function to be minimized
Initialize 119890 number of parameters and 119871119892 length of steps where 119892 = 0 to 119890Initialize initial value and the final value of the parameter as 119876119892119894 and 119876119892119891lowastlowast Solution Representation lowastlowastThe solutions are represented in the form of Binary values which can be generated as followsFor each solution 119894 = 1 119899119883119894 = 1199091198941 1199091198942 119909119894119889 | 119889 isin total days amp 119909119894119889 = rand ge 029 forall119889End for
(2) The number of steps in each step can be calculated using
119899119892 = 119876119892119894 minus 119876119892119891119871119892(3) The total number of volumes can be calculated using119873V = 119890prod
119892=1
119899119892(4) The midpoint of the volume to calculate starting point of the exploration can be calculated using
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ](5) Explore the search volume according to the Modified Nelder-Mead Method using Algorithm 2(6) The recorded value of the optimized point in vector table using[119891(1198811) 119891(1198812) 119891(119881119873V )](7) The globally optimized point is chosen based on Bee decision-making process using Consensus and Quorum
method approach 119891(119881119892) = min [119891(1198811) 119891(1198812) 119891(119881119873V )]Algorithm 3 Pseudocode of MODBCO
their best nest site for the exploration of food source Inthe multiagent system each agent after collecting individualsolution gives it to the centralized systems To select the bestoptimal solution forminimal optimal cases themathematicalformulation can be stated as
dance119894 = min (119891119894 (119881)) (32)
This mathematical formulation will find the minimaloptimal cases among the search solution where 119891119894(119881) is thesearch value calculated by the agent The search values arerecorded in the vector table 119881 119881 is the vector which consistsof 119890 number of elements The element 119890 contains the value ofthe parameter both optimal solution and parameter valuesare recorded in the vector table
522 Consensus Theconsensus is thewidespread agreementamong the group based on voting the voting pattern ofthe scout bees is monitored periodically to know whetherit reached an agreement and started acting on the decisionpattern Honey bees use the consensus method to select thebest search value the globally optimized point is chosen bycomparing the values in the vector table The globally opti-mized points are selected using themathematical formulation
523 Quorum In quorummethod the optimum solution iscalculated as the final solution based on the threshold levelobtained by the group decision-making process When thesolution reaches the optimal threshold level 120585119902 then the solu-tion is considered as a final solution based on unison decisionprocess The quorum threshold value describes the quality of
the food particle result When the threshold value is less thecomputation time decreases but it leads to inaccurate experi-mental resultsThe threshold value should be chosen to attainless computational timewith an accurate experimental result
6 Experimental Design and Analysis
61 Performance Metrics The performance of the proposedalgorithm MODBCO is assessed by comparing with fivedifferent competitor methods Here six performance metricsare considered to investigate the significance and evaluate theexperimental results The metrics are listed in this section
611 Least Error Rate Least Error Rate (LER) is the percent-age of the difference between known optimal value and thebest value obtained The LER can be calculated using
LER () = 119903sum119894=1
OptimalNRP-Instance minus fitness119894OptimalNRP-Instance
(34)
612 Average Convergence The Average Convergence is themeasure to evaluate the quality of the generated populationon average The Average Convergence (AC) is the percentageof the average of the convergence rate of solutions The per-formance of the convergence time is increased by the AverageConvergence to exploremore solutions in the populationTheAverage Convergence is calculated usingAC
= 119903sum119894=1
1 minus Avg_fitness119894 minusOptimalNRP-InstanceOptimalNRP-Instance
lowast 100 (35)
where (119903) is the number of instances in the given dataset
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
Modified Nelder-Mead Method for directed honey bee food search(1) Initialization119860119887 denotes the list of vertices in simplex where 119894 = 1 2 119899 + 1120572 120574 120573 and 120575 are the coefficients of reflection expansion contraction and shrinkage119891 is the objective function to be minimized(2)Ordering
Order the vertices in simplex from lowest objective function value 119891(1198601) to highest value 119891(119860119899+1) Ordered as 1198601le 1198602 le sdot sdot sdot le 119860119899+1(3)Midpoint
Calculate the midpoint for first two best vertices in simplex 119860119898 = sum(119860 119894119899) where 119894 = 1 2 119899(4) Reflection Process
Calculate reflection point 119860119903 by 119860119903 = 119860119898 + 120572(119860119898 minus 119860119899+1)if 119891(1198601) le 119891(1198602) le 119891(119860119899) then119860119899 larr 119860119903 and Go to to Step (8)end if
(5) Expansion Processif 119891(119860119903) le 119891(1198601) thenCalculate expansion point using 119860 119890 = 119860119903 + 120574(119860119903 minus 119860119898)end ifif 119891(119860 119890) lt 119891(119860119903) then119860119899 larr 119860 119890 and Go to to Step (8)else119860119899 larr 119860119903 and Go to to Step (8)end if
(6) Contraction Processif 119891(119860119899) le 119891(119860119903) le 119891(119860119899+1) thenCompute outside contraction by 119860 119888 = 120573119860119903 + (1 minus 120573)119860119898end ifif 119891(1198601) ge 119891(119860119899+1) thenCompute inside contraction by 119860 119888 = 120573119860119899+1 + (1 minus 120573)119860119898end ifif 119891(119860119903) ge 119891(119860119899) thenContraction is done between 119860119898 and the best vertex among 119860119903 and 119860119899+1end ifif 119891(119860 119888) lt 119891(119860119903) then119860119899 larr 119860 119888 and Go to to Step (8)else goes to Step (7)end ifif 119891(119860 119888) ge 119891(119860119899+1) then119860119899+1 larr 119860 119888 and Go to to Step (8)else Go to to Step (7)end if
(7) Shrinkage ProcessShrink towards the best solution with new vertices by 119860 119894 = 120575119860 119894 + 1198601(1 minus 120575) where 119894 = 2 119899 + 1
(8) Stopping CriteriaOrder and re-label new vertices of the simplex based on their objective function and go to step (4)
Algorithm 2 Pseudocode of Modified Nelder-Mead Method
119890-dimensional volume Total number of volumes 119873Vcan be formulated using
119873V = 119890prod119892=1
119899119892 (30)
(5) The starting point of the agent in the environmentwhich is one point inside volume is chosen bycalculating themidpoint of the volumeThemidpointof the lattice can be calculated as
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ] (31)
52 Decision-Making Process A key role of the honey beesis to select the best nest site and is done by the process ofdecision-making to produce a unified decisionThey follow adistributed decision-making process to find out the neighbornest site for their food particles The pseudocode for theproposed MODBCO algorithm is shown in Algorithm 3Figure 6 explains the workflow of the proposed algorithm forthe search of food particles by honey bees using MODBCO
521 Waggle Dance The scout bees after returning from thesearch of food particle report about the quality of the foodsite by communicationmode called waggle dance Scout beesperform thewaggle dance to other quiescent bees to advertise
12 Computational Intelligence and Neuroscience
Yes
Reflectionprocess
Order and label verticesbased on f(A)
Initialization
Coefficients 훼 훾 훽 훿
Objective function f(A)
f(Ab) lt f(Ar) lt f(Aw) Aw larr Ar
f(Ae) le f(Ar)
two best verticesAm forCalculate midpoint
Start
Terminationcriteria
Stop
Ar = Am + 훼(Am minus Aw)
ExpansionprocessNo
Yesf(Ar) le f(Aw) Aw larr Ae
No
b larr true Aw larr Ar
Contractionprocess
f(Ar) ge f(An)Yes
f(Ac) lt f(Ar)Aw larr Ac
b larr false
No
Shrinkageprocess
b larr true
Yes
Yes
No
Ae = Ar + 훾(Ar minus
Am)
Ac = 훽Ar + (1 minus 훽)Am
Ai = 훿Ai + A1(1 minus 훿)
Figure 5 Workflow of Modified Nelder-Mead Method
Computational Intelligence and Neuroscience 13
Multi-Objective Directed Bee Colony Optimization(1) Initialization119891(119909) is the objective function to be minimized
Initialize 119890 number of parameters and 119871119892 length of steps where 119892 = 0 to 119890Initialize initial value and the final value of the parameter as 119876119892119894 and 119876119892119891lowastlowast Solution Representation lowastlowastThe solutions are represented in the form of Binary values which can be generated as followsFor each solution 119894 = 1 119899119883119894 = 1199091198941 1199091198942 119909119894119889 | 119889 isin total days amp 119909119894119889 = rand ge 029 forall119889End for
(2) The number of steps in each step can be calculated using
119899119892 = 119876119892119894 minus 119876119892119891119871119892(3) The total number of volumes can be calculated using119873V = 119890prod
119892=1
119899119892(4) The midpoint of the volume to calculate starting point of the exploration can be calculated using
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ](5) Explore the search volume according to the Modified Nelder-Mead Method using Algorithm 2(6) The recorded value of the optimized point in vector table using[119891(1198811) 119891(1198812) 119891(119881119873V )](7) The globally optimized point is chosen based on Bee decision-making process using Consensus and Quorum
method approach 119891(119881119892) = min [119891(1198811) 119891(1198812) 119891(119881119873V )]Algorithm 3 Pseudocode of MODBCO
their best nest site for the exploration of food source Inthe multiagent system each agent after collecting individualsolution gives it to the centralized systems To select the bestoptimal solution forminimal optimal cases themathematicalformulation can be stated as
dance119894 = min (119891119894 (119881)) (32)
This mathematical formulation will find the minimaloptimal cases among the search solution where 119891119894(119881) is thesearch value calculated by the agent The search values arerecorded in the vector table 119881 119881 is the vector which consistsof 119890 number of elements The element 119890 contains the value ofthe parameter both optimal solution and parameter valuesare recorded in the vector table
522 Consensus Theconsensus is thewidespread agreementamong the group based on voting the voting pattern ofthe scout bees is monitored periodically to know whetherit reached an agreement and started acting on the decisionpattern Honey bees use the consensus method to select thebest search value the globally optimized point is chosen bycomparing the values in the vector table The globally opti-mized points are selected using themathematical formulation
523 Quorum In quorummethod the optimum solution iscalculated as the final solution based on the threshold levelobtained by the group decision-making process When thesolution reaches the optimal threshold level 120585119902 then the solu-tion is considered as a final solution based on unison decisionprocess The quorum threshold value describes the quality of
the food particle result When the threshold value is less thecomputation time decreases but it leads to inaccurate experi-mental resultsThe threshold value should be chosen to attainless computational timewith an accurate experimental result
6 Experimental Design and Analysis
61 Performance Metrics The performance of the proposedalgorithm MODBCO is assessed by comparing with fivedifferent competitor methods Here six performance metricsare considered to investigate the significance and evaluate theexperimental results The metrics are listed in this section
611 Least Error Rate Least Error Rate (LER) is the percent-age of the difference between known optimal value and thebest value obtained The LER can be calculated using
LER () = 119903sum119894=1
OptimalNRP-Instance minus fitness119894OptimalNRP-Instance
(34)
612 Average Convergence The Average Convergence is themeasure to evaluate the quality of the generated populationon average The Average Convergence (AC) is the percentageof the average of the convergence rate of solutions The per-formance of the convergence time is increased by the AverageConvergence to exploremore solutions in the populationTheAverage Convergence is calculated usingAC
= 119903sum119894=1
1 minus Avg_fitness119894 minusOptimalNRP-InstanceOptimalNRP-Instance
lowast 100 (35)
where (119903) is the number of instances in the given dataset
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
Multi-Objective Directed Bee Colony Optimization(1) Initialization119891(119909) is the objective function to be minimized
Initialize 119890 number of parameters and 119871119892 length of steps where 119892 = 0 to 119890Initialize initial value and the final value of the parameter as 119876119892119894 and 119876119892119891lowastlowast Solution Representation lowastlowastThe solutions are represented in the form of Binary values which can be generated as followsFor each solution 119894 = 1 119899119883119894 = 1199091198941 1199091198942 119909119894119889 | 119889 isin total days amp 119909119894119889 = rand ge 029 forall119889End for
(2) The number of steps in each step can be calculated using
119899119892 = 119876119892119894 minus 119876119892119891119871119892(3) The total number of volumes can be calculated using119873V = 119890prod
119892=1
119899119892(4) The midpoint of the volume to calculate starting point of the exploration can be calculated using
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ](5) Explore the search volume according to the Modified Nelder-Mead Method using Algorithm 2(6) The recorded value of the optimized point in vector table using[119891(1198811) 119891(1198812) 119891(119881119873V )](7) The globally optimized point is chosen based on Bee decision-making process using Consensus and Quorum
method approach 119891(119881119892) = min [119891(1198811) 119891(1198812) 119891(119881119873V )]Algorithm 3 Pseudocode of MODBCO
their best nest site for the exploration of food source Inthe multiagent system each agent after collecting individualsolution gives it to the centralized systems To select the bestoptimal solution forminimal optimal cases themathematicalformulation can be stated as
dance119894 = min (119891119894 (119881)) (32)
This mathematical formulation will find the minimaloptimal cases among the search solution where 119891119894(119881) is thesearch value calculated by the agent The search values arerecorded in the vector table 119881 119881 is the vector which consistsof 119890 number of elements The element 119890 contains the value ofthe parameter both optimal solution and parameter valuesare recorded in the vector table
522 Consensus Theconsensus is thewidespread agreementamong the group based on voting the voting pattern ofthe scout bees is monitored periodically to know whetherit reached an agreement and started acting on the decisionpattern Honey bees use the consensus method to select thebest search value the globally optimized point is chosen bycomparing the values in the vector table The globally opti-mized points are selected using themathematical formulation
523 Quorum In quorummethod the optimum solution iscalculated as the final solution based on the threshold levelobtained by the group decision-making process When thesolution reaches the optimal threshold level 120585119902 then the solu-tion is considered as a final solution based on unison decisionprocess The quorum threshold value describes the quality of
the food particle result When the threshold value is less thecomputation time decreases but it leads to inaccurate experi-mental resultsThe threshold value should be chosen to attainless computational timewith an accurate experimental result
6 Experimental Design and Analysis
61 Performance Metrics The performance of the proposedalgorithm MODBCO is assessed by comparing with fivedifferent competitor methods Here six performance metricsare considered to investigate the significance and evaluate theexperimental results The metrics are listed in this section
611 Least Error Rate Least Error Rate (LER) is the percent-age of the difference between known optimal value and thebest value obtained The LER can be calculated using
LER () = 119903sum119894=1
OptimalNRP-Instance minus fitness119894OptimalNRP-Instance
(34)
612 Average Convergence The Average Convergence is themeasure to evaluate the quality of the generated populationon average The Average Convergence (AC) is the percentageof the average of the convergence rate of solutions The per-formance of the convergence time is increased by the AverageConvergence to exploremore solutions in the populationTheAverage Convergence is calculated usingAC
= 119903sum119894=1
1 minus Avg_fitness119894 minusOptimalNRP-InstanceOptimalNRP-Instance
lowast 100 (35)
where (119903) is the number of instances in the given dataset
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
Multi-Objective Directed Bee Colony Optimization(1) Initialization119891(119909) is the objective function to be minimized
Initialize 119890 number of parameters and 119871119892 length of steps where 119892 = 0 to 119890Initialize initial value and the final value of the parameter as 119876119892119894 and 119876119892119891lowastlowast Solution Representation lowastlowastThe solutions are represented in the form of Binary values which can be generated as followsFor each solution 119894 = 1 119899119883119894 = 1199091198941 1199091198942 119909119894119889 | 119889 isin total days amp 119909119894119889 = rand ge 029 forall119889End for
(2) The number of steps in each step can be calculated using
119899119892 = 119876119892119894 minus 119876119892119891119871119892(3) The total number of volumes can be calculated using119873V = 119890prod
119892=1
119899119892(4) The midpoint of the volume to calculate starting point of the exploration can be calculated using
[1198761198941 minus 11987611989112 1198761198942 minus 11987611989122 119876119894119890 minus 1198761198911198902 ](5) Explore the search volume according to the Modified Nelder-Mead Method using Algorithm 2(6) The recorded value of the optimized point in vector table using[119891(1198811) 119891(1198812) 119891(119881119873V )](7) The globally optimized point is chosen based on Bee decision-making process using Consensus and Quorum
method approach 119891(119881119892) = min [119891(1198811) 119891(1198812) 119891(119881119873V )]Algorithm 3 Pseudocode of MODBCO
their best nest site for the exploration of food source Inthe multiagent system each agent after collecting individualsolution gives it to the centralized systems To select the bestoptimal solution forminimal optimal cases themathematicalformulation can be stated as
dance119894 = min (119891119894 (119881)) (32)
This mathematical formulation will find the minimaloptimal cases among the search solution where 119891119894(119881) is thesearch value calculated by the agent The search values arerecorded in the vector table 119881 119881 is the vector which consistsof 119890 number of elements The element 119890 contains the value ofthe parameter both optimal solution and parameter valuesare recorded in the vector table
522 Consensus Theconsensus is thewidespread agreementamong the group based on voting the voting pattern ofthe scout bees is monitored periodically to know whetherit reached an agreement and started acting on the decisionpattern Honey bees use the consensus method to select thebest search value the globally optimized point is chosen bycomparing the values in the vector table The globally opti-mized points are selected using themathematical formulation
523 Quorum In quorummethod the optimum solution iscalculated as the final solution based on the threshold levelobtained by the group decision-making process When thesolution reaches the optimal threshold level 120585119902 then the solu-tion is considered as a final solution based on unison decisionprocess The quorum threshold value describes the quality of
the food particle result When the threshold value is less thecomputation time decreases but it leads to inaccurate experi-mental resultsThe threshold value should be chosen to attainless computational timewith an accurate experimental result
6 Experimental Design and Analysis
61 Performance Metrics The performance of the proposedalgorithm MODBCO is assessed by comparing with fivedifferent competitor methods Here six performance metricsare considered to investigate the significance and evaluate theexperimental results The metrics are listed in this section
611 Least Error Rate Least Error Rate (LER) is the percent-age of the difference between known optimal value and thebest value obtained The LER can be calculated using
LER () = 119903sum119894=1
OptimalNRP-Instance minus fitness119894OptimalNRP-Instance
(34)
612 Average Convergence The Average Convergence is themeasure to evaluate the quality of the generated populationon average The Average Convergence (AC) is the percentageof the average of the convergence rate of solutions The per-formance of the convergence time is increased by the AverageConvergence to exploremore solutions in the populationTheAverage Convergence is calculated usingAC
= 119903sum119894=1
1 minus Avg_fitness119894 minusOptimalNRP-InstanceOptimalNRP-Instance
lowast 100 (35)
where (119903) is the number of instances in the given dataset
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
613 Standard Deviation Standard deviation (SD) is themeasure of dispersion of a set of values from its meanvalue Average Standard Deviation is the average of the
standard deviation of all instances taken from the datasetThe Average Standard Deviation (ASD) can be calculatedusing
ASD = radic 119903sum119894=1
(value obtained in each instance119894 minusMean value of the instance)2 (36)
where (119903) is the number of instances in the given dataset
614 Convergence Diversity The Convergence Diversity(CD) is the difference between best convergence rate andworst convergence rate generated in the population TheConvergence Diversity can be calculated using
CD = Convergencebest minus Convergenceworst (37)
where Convergencebest is the convergence rate of best fitnessindividual and Convergenceworst is the convergence rate ofworst fitness individual in the population
615 Cost Diversion Cost reduction is the differencebetween known cost in the NRP Instances and the costobtained from our approach Average Cost Diversion (ACD)is the average of cost diversion to the total number of instan-ces taken from the datasetThe value ofACRcan be calculatedfrom
ACR = 119903sum119894=1
Cost119894 minus CostNRP-InstanceTotal number of instances
(38)
where (119903) is the number of instances in the given dataset
62 Experimental Environment Setup The proposed Direct-ed Bee Colony algorithm with the Modified Nelder-MeadMethod to solve the NRP is illustrated briefly in this sectionThe main objective of the proposed algorithm is to satisfymultiobjective of the NRP as follows
(a) Minimize the total cost of the rostering problem(b) Satisfy all the hard constraints described in Table 1(c) Satisfy as many soft constraints described in Table 2(d) Enhance the resource utilization(e) Equally distribute workload among the nurses
The Nurse Rostering Problem datasets are taken fromthe First International RosteringCompetition (INRC2010) byPATAT-2010 a leading conference inAutomated Timetabling[38]The INRC2010 dataset is divided based on its complexityand size into three tracks namely sprint medium andlong datasets Each track is divided into four types as earlylate hidden and hint with reference to the competitionINRC2010 The first track sprint is the easiest and consistsof 10 nurses 33 datasets which are sorted as 10 early types10 late types 10 hidden types and 3 hint type datasets Thescheduling period is for 28 days with 3 to 4 contract types 3to 4 daily shifts and one skill specification The second track
is a medium which is more complex than sprint track andit consists of 30 to 31 nurses 18 datasets which are sorted as5 early types 5 long types 5 hidden types and 3 hint typesThe scheduling period is for 28 days with 3 to 4 contracttypes 4 to 5 daily shifts and 1 to 2 skill specifications Themost complicated track is long with 49 to 40 nurses andconsists of 18 datasets which are sorted as 5 early types 5 longtypes 5 hidden types and 3 hint typesThe scheduling periodfor this track is 28 days with 3 to 4 contract types 5 dailyshifts and 2 skill specifications The detailed description ofthe datasets available in the INRC2010 is shown in Table 3The datasets are classified into twelve cases based on the sizeof the instances and listed in Table 4
Table 3 describes the detailed description of the datasetscolumns one to three are used to index the dataset to tracktype and instance Columns four to seven will explain thenumber of available nurses skill specifications daily shifttypes and contracts Column eight explains the number ofunwanted shift patterns in the roster The nurse preferencesare managed by shift off and day off in columns nine and tenThe number of weekend days is shown in column elevenThelast column indicates the scheduling period The symbol ldquo119909rdquoshows there is no shift off and day off with the correspondingdatasets
Table 4 shows the list of datasets used in the experimentand it is classified based on its size The datasets presentin case 1 to case 4 are smaller in size case 5 to case 8 areconsidered to be medium in size and the larger sized datasetis classified from case 9 to case 12
The performance of MODBCO for NRP is evaluatedusing INRC2010 dataset The experiments are done on dif-ferent optimization algorithms under similar environmentconditions to assess the performance The proposed algo-rithm to solve the NRP is coded using MATLAB 2012platform under Windows on an Intel 2GHz Core 2 quadprocessor with 2GB of RAM Table 3 describes the instancesconsidered by MODBCO to solve the NRP The empiricalevaluations will set the parameters of the proposed systemAppropriate parameter values are determined based on thepreliminary experiments The list of competitor methodschosen to evaluate the performance of the proposed algo-rithm is shown in Table 5 The heuristic parameter and thecorresponding values are represented in Table 6
63 Statistical Analysis Statistical analysis plays a majorrole in demonstrating the performance of the proposedalgorithm over existing algorithms Various statistical testsand measures to validate the performance of the algorithmare reviewed byDemsar [39]The authors used statistical tests
16 Computational Intelligence and Neuroscience
Table 3 The features of the INRC2010 datasets
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
Track Type Instance Nurses Skills Shifts Contracts Unwanted pattern Shift off Day off Weekend Time period
Sprint
Early 01ndash10 10 1 4 4 3 2 1-01-2010 to 28-01-2010
Hidden
01-02 10 1 3 3 4 2 1-06-2010 to 28-06-201003 05 08 10 1 4 3 8 2 1-06-2010 to 28-06-201004 09 10 1 4 3 8 2 1-06-2010 to 28-06-201006 07 10 1 3 3 4 2 1-01-2010 to 28-01-201010 10 1 4 3 8 2 1-01-2010 to 28-01-2010
Late
01 03ndash05 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 3 3 4 2 1-01-2010 to 28-01-2010
06 07 10 10 1 4 3 0 2 1-01-2010 to 28-01-201008 10 1 4 3 0 times times 2 1-01-2010 to 28-01-201009 10 1 4 3 0 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 03 10 1 4 3 8 2 1-01-2010 to 28-01-201002 10 1 4 3 0 2 1-01-2010 to 28-01-2010
Medium
Early 01ndash05 31 1 4 4 0 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 30 2 5 4 9 times times 2 1-06-2010 to 28-06-201005 30 2 5 4 9 times times 2 1-06-2010 to 28-06-2010
Late
01 30 1 4 4 7 2 1-01-2010 to 28-01-201002 04 30 1 4 3 7 2 1-01-2010 to 28-01-201003 30 1 4 4 0 2 1-01-2010 to 28-01-201005 30 2 5 4 7 2 1-01-2010 to 28-01-2010
Hint 01 03 30 1 4 4 7 2 1-01-2010 to 28-01-201002 30 1 4 4 7 2 1-01-2010 to 28-01-2010
Long
Early 01ndash05 49 2 5 3 3 2 1-01-2010 to 28-01-2010
Hidden 01ndash04 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-201005 50 2 5 3 9 times times 2 3 1-06-2010 to 28-06-2010
Late 01 03 05 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 04 50 2 5 4 9 times times 2 3 1-01-2010 to 28-01-2010
Hint 01 50 2 5 3 9 times times 2 3 1-01-2010 to 28-01-201002 03 50 2 5 3 7 times times 2 1-01-2010 to 28-01-2010
Table 4 Classification of INRC2010 datasets based on the size
SI number Case Track Type1 Case 1 Sprint Early2 Case 2 Sprint Hidden3 Case 3 Sprint Late4 Case 4 Sprint Hint5 Case 5 Middle Early6 Case 6 Middle Hidden7 Case 7 Middle Late8 Case 8 Middle Hint9 Case 9 Long Early10 Case 10 Long Hidden11 Case 11 Long Late12 Case 12 Long Hint
like ANOVA Dunnett test and post hoc test to substantiatethe effectiveness of the proposed algorithm and help todifferentiate from existing algorithms
631 ANOVA Test To validate the performance of theproposed algorithm ANOVA (Analysis of Variance) is usedas the statistical analysis tool to demonstrate whether oneor more solutions significantly vary [40] The authors usedone-way ANOVA test [41] to show significance in proposedalgorithm One-way ANOVA is used to validate and compare
Table 5 List of competitors methods to compare
Type Method ReferenceM1 Artificial Bee Colony Algorithm [14]M2 Hybrid Artificial Bee Colony Algorithm [15]M3 Global best harmony search [16]M4 Harmony Search with Hill Climbing [17]M5 Integer Programming Technique for NRP [18]
Table 6 Configuration parameter for experimental evaluation
Type MethodNumber of bees 100Maximum iterations 1000Initialization technique BinaryHeuristic Modified Nelder-Mead MethodTermination condition Maximum iterationsRun 20Reflection coefficient 120572 gt 0Expansion coefficient 120574 gt 1Contraction coefficient 0 gt 120573 gt 1Shrinkage coefficient 0 lt 120575 lt 1differences between various algorithms The ANOVA testis performed with 95 confidence interval the significantlevel of 005 In ANOVA test the null hypothesis is testedto show the difference in the performance of the algorithms
Computational Intelligence and Neuroscience 17
Table 7 Experimental result with respect to best value
Instances Optimal value MODBCO M1 M2 M3 M4 M5Best Worst Best Worst Best Worst Best Worst Best Worst Best Worst
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
If the obtained significance value is less than the criticalvalue (005) then the null hypothesis is rejected and thusthe alternate hypothesis is accepted Otherwise the nullhypothesis is accepted by rejecting the alternate hypothesis
632 Duncanrsquos Multiple Range Test After the null hypothesisis rejected to explore the group differences post hoc ormultiple comparison test is performed Duncan developed aprocedure to test and compare all pairs in multiple ranges[42] Duncanrsquos multiple range test (DMRT) classifies thesignificant and nonsignificant difference between any twomethods This method ranks in terms of mean values inincreasing or decreasing order and group method which isnot significant
64 Experimental and Result Analysis In this section theeffectiveness of the proposed algorithm MODBCO is com-pared with other optimization algorithms to solve the NRPusing INRC2010 datasets under similar environmental setupusing performance metrics as discussed To compare theresults produced byMODBCO seems to bemore competitivewith previous methods The performance of MODBCO iscomparable with previous methods listed in Tables 7ndash18The computational analysis on the performance metrics is asfollows
641 Best Value The results obtained by MODBCO withcompetitive methods are shown in Table 7 The performanceis compared with previous methods the number in the tablerefers to the best solution obtained using the correspondingalgorithm The objective of NRP is the minimization ofcost the lowest values are the best solution attained In theevaluation of the performance of the algorithm the authors
Table 8 Statistical analysis with respect to best value
have considered 69 datasets with diverse size It is apparentlyshown that MODBCO accomplished 34 best results out of 69instances
The statistical analysis tests ANOVA and DMRT forbest values are shown in Table 8 It is perceived that thesignificance values are less than 005 which shows the nullhypothesis is rejected The significant difference between
Computational Intelligence and Neuroscience 19
Table 9 Experimental result with respect to error rate
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
Figure 7 Performance analysis with respect to error rate
various optimization algorithms is observed The DMRT testshows the homogenous group two homogeneous groups forbest values are formed among competitor algorithms
642 Error Rate The evaluation based on the error rateshows that our proposed MODBCO yield lesser error ratecompared to other competitor techniques The computa-tional analysis based on error rate () is shown in Table 9 andout of 33 instances in sprint type 18 instances have achievedzero error rate For sprint type dataset 88 of instances have
attained a lesser error rate For medium and larger sizeddatasets the obtained error rate is 62 and 44 respectivelyA negative value in the column indicates correspondinginstances have attained lesser optimum valve than specifiedin the INRC2010
TheCompetitorsM2 andM5 generated better solutions atthe initial stage as the size of the dataset increases they couldnot be able to find the optimal solution and get trapped inlocal optimaThe error rate () obtained by usingMODBCOwith different algorithms is shown in Figure 7
20 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
0
20
40
60
80
100Av
erag
e Con
verg
ence
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
0
20
40
60
80
100
Aver
age C
onve
rgen
ce
MODBCOM1M2
M3M4M5
Figure 8 Performance analysis with respect to Average Convergence
Table 10 Statistical analysis with respect to error rate
(a) ANOVA test
Source factor error rateSum ofsquares df Mean square 119865 Sig
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
The statistical analysis on error rate is presented inTable 10 InANOVA test the significance value is 0000whichis less than 005 showing rejection of the null hypothesisThus there is a significant difference in value with respectto various optimization algorithmsThe DMRT test indicatestwo homogeneous groups formed from different optimiza-tion algorithms with respect to the error rate
643 Average Convergence The Average Convergence ofthe solution is the average fitness of the population to thefitness of the optimal solutionThe computational results withrespect to Average Convergence are shown in Table 11MOD-BCO shows 90 convergence rate in small size instances and82 convergence rate in medium size instances For longerinstances it shows 77 convergence rate Negative values inthe column show the corresponding instances get deviatedfrom optimal solution and trapped in local optima It isobserved that with increase in the problem size convergencerate reduces and becomesworse inmany algorithms for largerinstances as shown in Table 11The Average Convergence rateattained by various optimization algorithms is depicted inFigure 8
The statistical test result for Average Convergence isobserved in Table 12 with different optimization algorithmsFrom the table it is clear that there is a significant difference
Computational Intelligence and Neuroscience 21
Table 11 Experimental result with respect to Average Convergence
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
Figure 9 Performance analysis with respect to Average Standard Deviation
in mean values of convergence in different optimizationalgorithms The ANOVA test depicts the rejection of the nullhypothesis since the value of significance is 0000 The posthoc analysis test shows there are two homogenous groupsamong different optimization algorithms with respect to themean values of convergence
644 Average Standard Deviation The Average StandardDeviation is the dispersion of values from its mean valueand it helps to deduce features of the proposed algorithm
The computed result with respect to the Average StandardDeviation is shown in Table 13 The Average Standard Devia-tion attained by various optimization algorithms is depictedin Figure 9
The statistical test result for Average Standard Deviationis shown in Table 14 with different types of optimizationalgorithms There is a significant difference in mean valuesof standard deviation in different optimization algorithmsThe ANOVA test proves the null hypothesis is rejected sincethe value of significance is 000 which is less than the critical
22 Computational Intelligence and Neuroscience
Case 1 Case 2 Case 3NRP Instance
Con
verg
ence
0
20
40
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 7 Case 8 Case 9NRP Instance
Con
verg
ence
0
10
20
30
40
50
60
Div
ersit
y
MODBCOM1M2
M3M4M5
Con
verg
ence
Case 10 Case 11 Case 12NRP Instance
0
20
40
60
80
100
Div
ersit
y
MODBCOM1M2
M3M4M5
Case 4 Case 5 Case 6NRP Instance
Con
verg
ence
0
10
20
30
40
Div
ersit
y
MODBCOM1M2
M3M4M5
Figure 10 Performance analysis with respect to Convergence Diversity
Table 12 Statistical analysis with respect to Average Convergence
(a) ANOVA test
Source factor Average ConvergenceSum ofsquares df Mean square 119865 Sig
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
value 005 InDMRT test there are three homogenous groupsamong different optimization algorithms with respect to themean values of standard deviation
645 Convergence Diversity The Convergence Diversity ofthe solution is to calculate the difference between best con-vergence and worst convergence generated in the populationThe Convergence Diversity and error rate help to infer theperformance of the proposed algorithm The computationalanalysis based on Convergence Diversity for MODBCO withanother competitor algorithm is shown in Table 15 TheConvergence Diversity for smaller and medium datasets is58 and 50 For larger datasets the Convergence Diversityis 62 to yield an optimum value Figure 10 shows thecomparison of various optimization algorithms with respectto Convergence Diversity
The statistical test of ANOVA and DMRT is observed inTable 16 with respect to Convergence Diversity There is asignificant difference in the mean values of the ConvergenceDiversity with various optimization algorithms For post hocanalysis test the significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected FromDMRT test the grouping of various algorithms based onmean value is shown there are three homogenous groups
Computational Intelligence and Neuroscience 23
Table 13 Experimental result with respect to Average Standard Deviation
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
Figure 11 Performance analysis with respect to Average CostDiversion
among the various optimization algorithms with respect tothe mean values of the cost diversity
646 Average Cost Diversion The computational analysisbased on cost diversion shows proposed MODBCO yieldsless diversion in cost compared to other competitor tech-niques The computational analysis with respect to AverageCost Diversion is shown in Table 17 For smaller andmediumdataset 13 and 38 of instances got diverged out of whichmany instances yield optimum value The larger dataset got56 of cost divergence A negative value in the table indicatescorresponding instances have achieved new optimized val-ues Figure 11 depicts the comparison of various optimizationalgorithms with respect to Average Cost Diversion
The statistical test of ANOVA and DMRT is observed inTable 18 with respect to Average Cost Diversion From thetable it is inferred that there is a significant difference in themean values of the cost diversion with various optimizationalgorithms The significance value is 0000 which is less thanthe critical value Thus the null hypothesis is rejected TheDMRT test reveals there are two homogenous groups among
Table 14 Statistical analysis with respect to Average StandardDeviation
(a) ANOVA test
Source factor Average Standard DeviationSum ofsquares df Mean square 119865 Sig
the various optimization algorithms with respect to the meanvalues of the cost diversion
7 Discussion
The experiments to solve NP-hard combinatorial NurseRostering Problem are conducted by our proposed algorithmMODBCO Various existing algorithms are chosen to solvethe NRP and compared with the proposed MODBCO algo-rithm The results of our proposed algorithm are comparedwith other competitor methods and the best values are tabu-lated in Table 6 To evaluate the performance of the proposed
24 Computational Intelligence and Neuroscience
Table 15 Experimental result with respect to Convergence Diversity
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
algorithm various performance metrics are considered toevaluate the efficiency of the MODBCO Tables 7ndash18 showthe outcome of our proposed algorithm and other existingmethods performance From Tables 7ndash18 and Figures 7ndash11it is evidently shown that MODBCO has more ability toattain the best value on performance metrics compared tocompetitor algorithms which use the INRC2010
Compared with other existing methods the mean valueof MODBCO is 19 reduced towards optimum value withother competitor methods and it attained lesser worst valuein addition to the best solution The datasets are dividedbased on their size as smaller medium and large datasetthe standard deviation of MODBCO is reduced to 49
222 and 413 respectivelyThe error rate of our proposedapproach when compared with other competitor methodswith various sized datasets reduces to 106 for the smallerdataset 945 for the medium datasets and 704 for thelarger datasets The convergence rate of MODBCO hasachieved 90 for the smaller dataset 82 for the mediumdataset and 7737 for the larger dataset The error rate ofour proposed algorithm is reduced by 77 when comparedwith other competitor methods
Theproposed system is tested on larger sized datasets andit is working astoundingly better than the other techniquesIncorporation of Modified Nelder-Mead in Directed BeeColony Optimization Algorithm increases the exploitationstrategy within the given exploration search space Thismethod balances the exploration and exploitation withoutany biased natureThusMODBCO converges the populationtowards an optimal solution at the end of each iteration Bothcomputational and statistical analyses show the significantperformance over other competitor algorithms in solving theNRP The computational complexity is greater due to theuse of local heuristic Nelder-Mead Method However theproposed algorithm is better than exact methods and otherheuristic approaches in solving the NRP in terms of timecomplexity
8 Conclusion
This paper tackles solving the NRP using MultiobjectiveDirected Bee Colony Optimization Algorithm namedMOD-BCO To solve the NRP effectively Directed Bee Colonyalgorithm is chosen for global search and Modified Nelder-MeadMethod for local best searchTheproposed algorithm isevaluated using the INRC2010 dataset and the performanceof the proposed algorithm is compared with other fiveexisting methods To assess the performance of our proposedalgorithm 69 different cases of various sized datasets arechosen and 34 out of 69 instances got the best resultThus our algorithm contributes with a new deterministicsearch and effective heuristic approach to solve the NRPThus MODBCO outperforms with classical Bee Colony
Computational Intelligence and Neuroscience 25
Table 17 Experimental result with respect to Average Cost Diversion
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
Optimization for solving NRP by satisfying both hard andsoft constraints
The future work can be projected to
(a) adapting proposed MODBCO for various schedulingand timetabling problems
(b) exploring unfeasible solution to imitate optimal solu-tion
(c) further tuning the parameters of the proposed algo-rithm andmeasuring the exploitation and explorationstrategy
(d) investigating for applying Second International INRC2014 datasets
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is a part of the Research Projects sponsoredby the Major Project Scheme UGC India Referencenos FNo2014-15NFO-2014-15-OBC-PON-3843(SA-IIIWEBSITE) dated March 2015 The authors would like toexpress their thanks for their financial support offered by theSponsored Agencies
References
[1] M Crepinsek S-H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACM Com-puting Surveys vol 45 no 3 article 35 2013
[2] R Bai E K BurkeG Kendall J Li andBMcCollum ldquoAhybridevolutionary approach to the nurse rostering problemrdquo IEEETransactions on Evolutionary Computation vol 14 no 4 pp580ndash590 2010
[3] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons 2009
[4] E Goldberg David Genetic Algorithm in Search Optimizationand Machine Learning vol 3 Pearson Education 1988
[5] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2011
[6] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[7] D Teodorovic P Lucic G Markovic and M DellrsquoOrco ldquoBeecolony optimization principles and applicationsrdquo in Proceed-ings of the 8th Seminar on Neural Network Applications inElectrical Engineering pp 151ndash156 September 2006
[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
[9] R Kumar ldquoDirected bee colony optimization algorithmrdquoSwarm and Evolutionary Computation vol 17 pp 60ndash73 2014
26 Computational Intelligence and Neuroscience
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955
[10] T Osogami and H Imai ldquoClassification of various neigh-borhood operations for the nurse scheduling problemrdquo inProceedings of the International Symposium on Algorithmsand Computation Taipei Taiwan December 2000 pp 72ndash83Springer Berlin Germany 2000
[11] H H Millar and M Kiragu ldquoCyclic and non-cyclic schedulingof 12 h shift nurses by network programmingrdquoEuropean Journalof Operational Research vol 104 no 3 pp 582ndash592 1998
[12] J Van den Bergh J Belien P De Bruecker E Demeulemeesterand L De Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013
[13] B Cheang H Li A Lim and B Rodrigues ldquoNurse rosteringproblemsmdasha bibliographic surveyrdquo European Journal of Opera-tional Research vol 151 no 3 pp 447ndash460 2003
[14] L B Asaju M A Awadallah M A Al-Betar and A T KhaderldquoSolving nurse rostering problem using artificial bee colonyalgorithmrdquo in Proceedings of the 7th International Conference onInformation Technology (ICIT rsquo15) pp 32ndash38 Amman JordanMay 2015
[15] M A Awadallah A L Bolaji and M A Al-Betar ldquoA hybridartificial bee colony for a nurse rostering problemrdquo Applied SoftComputing vol 35 pp 726ndash739 2015
[16] M A Awadallah A T Khader M A Al-Betar and A L BolajildquoGlobal best harmony search with a new pitch adjustmentdesigned for nurse rosteringrdquo Journal of King Saud University-Computer and Information Sciences vol 25 no 2 pp 145ndash1622013
[17] M A Awadallah M A Al-Betar A T Khader A L Bolajiand M Alkoffash ldquoHybridization of harmony search withhill climbing for highly constrained nurse rostering problemrdquoNeural Computing and Applications vol 28 no 3 pp 463ndash4822017
[18] H G Santos T A M Toffolo R A M Gomes and SRibas ldquoInteger programming techniques for the nurse rosteringproblemrdquoAnnals of Operations Research vol 239 no 1 pp 225ndash251 2016
[19] I Berrada J A Ferland and P Michelon ldquoA multi-objectiveapproach to nurse scheduling with both hard and soft con-straintsrdquo Socio-Economic Planning Sciences vol 30 no 3 pp183ndash193 1996
[20] E K Burke J Li and R Qu ldquoA Pareto-based search methodol-ogy for multi-objective nurse schedulingrdquo Annals of OperationsResearch vol 196 pp 91ndash109 2012
[21] K A Dowsland and J MThompson ldquoSolving a nurse schedul-ing problemwith knapsacks networks and tabu searchrdquo Journalof the Operational Research Society vol 51 no 7 pp 825ndash8332000
[22] K A Dowsland ldquoNurse scheduling with tabu search andstrategic oscillationrdquo European Journal of Operational Researchvol 106 no 2-3 pp 393ndash407 1998
[23] E Burke P De Causmaecker and G VandenBerghe ldquoA hybridtabu search algorithm for the nurse rostering problemrdquo in Pro-ceedings of the Asia-Pacific Conference on Simulated Evolutionand Learning vol 1585 pp 187ndash194 Springer Berlin Germany1998
[24] E K Burke G Kendall and E Soubeiga ldquoA tabu-search hyper-heuristic for timetabling and rosteringrdquo Journal of Heuristicsvol 9 no 6 pp 451ndash470 2003
[25] E Burke P Cowling P De Causmaecker and G V BergheldquoA memetic approach to the nurse rostering problemrdquo AppliedIntelligence vol 15 no 3 pp 199ndash214 2001
[26] M Hadwan and M Ayob ldquoA constructive shift patternsapproach with simulated annealing for nurse rostering prob-lemrdquo in Proceedings of the International Symposium on Infor-mation Technology (ITSim rsquo10) pp 1ndash6 IEEE Kuala LumpurMalaysia June 2010
[27] E Sharif M Ayob andM Hadwan ldquoHybridization of heuristicapproach with variable neighborhood descent search to solvenurse Rostering problem at Universiti Kebangsaan MalaysiaMedical Centre (UKMMC)rdquo in Proceedings of the 3rd Confer-ence on Data Mining and Optimization (DMO rsquo11) pp 178ndash183June 2011
[28] U Aickelin and K A Dowsland ldquoAn indirect genetic algorithmfor a nurse-scheduling problemrdquo Computers and OperationsResearch vol 31 no 5 pp 761ndash778 2004
[29] S Asta E Ozcan and T Curtois ldquoA tensor based hyper-heuristic for nurse rosteringrdquoKnowledge-Based Systems vol 98pp 185ndash199 2016
[30] K Anwar M A Awadallah A T Khader and M A Al-BetarldquoHyper-heuristic approach for solving nurse rostering prob-lemrdquo in Proceedings of the IEEE Symposium on ComputationalIntelligence in Ensemble Learning (CIEL rsquo14) pp 1ndash6 December2014
[31] N Todorovic and S Petrovic ldquoBee colony optimization algo-rithm for nurse rosteringrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 43 no 2 pp 467ndash473 2013
[32] X-S Yang Nature-Inspired Meta-Heuristic Algorithms LuniverPress 2010
[33] S Goyal ldquoThe applications survey bee colonyrdquo IRACST-Engineering Science and Technology vol 2 no 2 pp 293ndash2972012
[34] T D Seeley P Kirk Visscher and K M Passino ldquoGroupdecision-making in honey bee swarmsrdquoAmerican Scientist vol94 no 3 pp 220ndash229 2006
[35] KM Passino T D Seeley and P K Visscher ldquoSwarm cognitionin honey beesrdquo Behavioral Ecology and Sociobiology vol 62 no3 pp 401ndash414 2008
[36] W Jiao and Z Shi ldquoA dynamic architecture for multi-agentsystemsrdquo in Proceedings of the Technology of Object-OrientedLanguages and Systems (TOOLS 31 rsquo99) pp 253ndash260 NanjingChina November 1999
[37] W Zhong J Liu M Xue and L Jiao ldquoA multi-agent geneticalgorithm for global numerical optimizationrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol34 no 2 pp 1128ndash1141 2004
[38] S Haspeslagh P De Causmaecker A Schaerf and M StoslashlevikldquoThe first international nurse rostering competition 2010rdquoAnnals of Operations Research vol 218 no 1 pp 221ndash236 2014
[39] J Demsar ldquoStatistical comparisons of classifiers over multipledata setsrdquo Journal of Machine Learning Research vol 7 pp 1ndash302006
[40] A Costa F A Cappadonna and S Fichera ldquoA dual encoding-basedmeta-heuristic algorithm for solving a constrained hybridflow shop scheduling problemrdquo Computers and Industrial Engi-neering vol 64 no 4 pp 937ndash958 2013
[41] G Gonzalez-Rodrıguez A Colubi and M A Gil ldquoFuzzy datatreated as functional data a one-way ANOVA test approachrdquoComputational Statistics and Data Analysis vol 56 no 4 pp943ndash955 2012
[42] D B Duncan ldquoMultiple range and multiple 119865 testsrdquo Biometricsvol 11 pp 1ndash42 1955