Kuala Lumpur, Malaysia, March 8-10, 2016 A Modified ABC ...ieomsociety.org/ieom_2016/pdfs/333.pdf · combined to develop two modified artificial bee colony algorithms ... [6, 7],

Proceedings of the 2016 International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia, March 8-10, 2016

A Modified ABC Algorithm with Genetic and Local Search for solving QAP

Suman Samanta*, Department of Industrial and Management Engineering

Indian Institute of Technology, Kanpur Kanpur, India

E-mail (*corresponding author): [email protected]

Deepu Philip Department of Industrial and Management Engineering

Indian Institute of Technology Kanpur-208016, India.

J. Ramkumar Department of Mechanical Engineering and Design Program

Indian Institute of Technology Kanpur-208016, India

Abstract—The two most commonly used local search operators in heuristic or metaheuristic approach to solve combinatorial optimization problems are genetic search operators and neighborhood search operators. In this work both the operators are combined to develop two modified artificial bee colony algorithms (ABC) for solving combinatorial optimization problems. We used those modified ABCs to solve some standard quadratic assignment problems (QAP) available in QAP library. The performance of the algorithms are compared with some state of the art algorithms by means of percentage variation from the known optima and minimum time requirement to reach the optima.

Keywords— Combinatorial optimization problem, quadratic assignment problem, artificial bee colony algorithm.

I. INTRODUCTION

QAP is to allocate n facilities in n locations, taken from n available locations in such way that minimizes the interdepartmental traveling distance, cost or time for any facility. It is one of the fundamental combinatorial optimization problems from the category of facility layout optimization. This problem was first formalized by Koopmans and Beckmann [1]. The problem is formulated as shown in Equation 1.

( ) ( )∑∑= =∈

n

i

n

kkiik

n

baS 1 1

minϕϕϕ

(1)

Where nS is the set of all n permutations of the integers 1, 2, …..,n. Each individual value of the product ( ) ( )kiikba ϕϕ denotesthe transportation distance caused by assigning facility i to location ( )iϕ and facility k to location ( )kϕ .

QAP is an NP-hard problem by nature [2 – 4]. Because of this, heuristic and meta-heuristic approaches are widely used to solve this kind of problem. Two most commonly used local search operators these heuristics or metaheuristics commonly adopt are neighborhood search operator and genetic search operator. The neighborhood search operators try to improve the solution by recombination of locations within itself. Which means the child (new solution) is generated by exploiting only one parent. Some example of such operators are swap or exchange, insert, inverse, translocation, scramble etc. [5]. Among these search operators, ‘swap’ and ‘insert’ are most commonly used. Some of the algorithms that uses neighborhood search operator for local search are: tabu search (TS) [6, 7], robust tabu search (ROTS) [8], simulated annealing (SA) [9, 10], go with the winner (GWW) [11], variable neighborhood search (VNS) [12, 13] etc. The genetic search operator uses crossover between more than one (mostly two) initial solutions (parents) to form the offspring or child. Some example of such operators are: position based crossover (PBX), order based crossover (OBX), one point crossover (1PX), Cycle crossover operator (CX), Order crossover (OX), Linear order crossover operator (LOX), Partially mapped crossover operator (PMX), Edge recombination crossover operator (ER), Two-point crossover operator (2PX_V1, V2, and V3). Kellego¨z et. al [14] proved that PBX and OBX are better than other genetic operators by using fifteen one machine total weighted tardiness problem. Some

1207© IEOM Society International


application of genetic searches for combinatorial optimization is: genetic algorithm (GA) [15], genetic search based artificial bee colony algorithm (g-ABC) [16], best-so-far ABC [17] etc. However, to solve medium to high difficulty level problems, there has an increasing trend of combining both the genetic and local search operators; eg. Genetic hybrid algorithms [18-24].

The two strategies that various neighborhood search algorithms adopt while neighborhood search are any neighbor search and best neighbor search. Any neighbor search does not compare among the neighbors and randomly select any neighbor. Some algorithms that use the any neighbor search are VNS, GWW, genetic local search (GLS) [21] etc. On the other hand, the best neighbor strategy considers the best possible neighbor under the given conditions. TS, ROTS etc. are the examples of such algorithms which explore the best possible neighbor.

In this work, the genetic and neighborhood search operators are used in the two basic local search steps (viz. employed step and onlooker step) correspondingly of artificial bee colony algorithm to solve QAP. Two kinds of neighborhood search strategies, viz. any and best, are utilized. The two resulted algorithms are named as m-ABC1 and m-ABC2 here after. The algorithms are used to solve various standard QAPs available in the QAP library [4]. The performance of the algorithms are compared with other state of the algorithms viz. ROTS [8], GWW [10], VNS –II [13], GA [15], g-ABC [16], best-so-far ABC (b-s-f ABC) [17], Genetic local search (GLS) [21] by two criteria: the optimum value found by the algorithm under similar parametric settings and minimum time to reach the optimum value. The result shows the developed algorithms performs significantly better than most of the other state of the art algorithms.

II. MODIFIED ABC WITH GENETIC AND NEIGHBORHOOD SEARCH OPERATOR FOR QAP Artificial bee colony algorithm (ABC) is a non-conventional optimization algorithm developed by Karaboga [25]. With its

two-step local search (employed and onlooker) and one step global search (scout) methodology, it became a very popular choice in solving various optimization problems; viz. power flow optimization and optimal sizing of photovoltaic systems [26, 27], control engineering [28], image processing [29], design optimization [30], parametric optimization non-traditional manufacturing processes[31] etc. In this work, we used the main essence of ABC algorithm while using a genetic operator in the employed step and neighborhood search operator in the onlooker step for solving QAP. We also have used two different search strategy in the neighborhood search (onlooker step) two form two variations of the modified ABC – m-ABC1, and m-ABC2. The various steps involved in both the variation are discussed below. The pseudocodes of the algorithm is described in Fig. 1-5.

A. Initialization This step selects random food source (feasible solution) within the search space for the formation of the initial population.

Each food source is assigned to an employed bee for further improvement.

B. Employed step In this step, each employed bee selects a neighborhood position randomly and performs a local search by using genetic

search operator PBX [13]. Greedy selection approach is applied to select better food source among the newly explored food sources and old food sources.

C. Onlooker step The onlooker selects the food sources based on their fitness. The formulae for calculation of the fitness and finding out the

probability of selection of a food source is similar to as described by Karaboga [24]. For m-ABC1, we use neighborhood search operator SWAP or INSERT [5], each with a probability of 0.5 for each selected food source to do the required local search. While for m-ABC2, we use the best neighbor selection approach instead of any neighbor. The nectar amount of various neighbors is identified by the way described by Taillard [8].

D. Scoutstep Similar to the ABC described by Karaboga [24], here also a food source is abandoned for further search when it does not

show any improvement for a predefined number of trials (limit). This kind of food source is replaced by new randomly selected food source. However, the insertion of a new food source to the population is restricted to maximum one in a cycle.



Fig. 1. Flow for m-ABC algorithms

Fig. 2.Pseudocode for send employed bees

Send Employed Bees (colony) For i= 1 to CS

For each food source xi Select a neighbor xk randomly from the colony (neighbor position ≠ i) do PBX between xi and xk to identify new neighboring solution x2i

end For do CalculateFitness (xi and x2i) to calculate fitness value of each food source do GreedySelection between xi and x2i

Algorithm: m-ABC for QAP o Input

Colony size (CS): the size of the initial population limit: the abandonment criterion MaxCycle: maximum cycle number runtime: number of runs

o Output: Best solution found

1. Initialization: Initialize the population

2. For r=1 to runtime while t <= MaxCycle send employed bees (colony) send onlooker bees (colony, fitness) send scout bee (colony, limit)

3. Return Best solution found



Fig. 3.Pseudocode for send onlooker bees for m-ABC1

III. SIMULATION RESULTS AND ANALYSIS

A. Benchmarks In this work, we have used 20 benchmark problems starting from n=12 to n=64 having difficulty level low to medium from

the QAP Library [4] for accessing the performance of each algorithm. The problems are: Chr12c, Had14, Nug15, Had16, Nug17, Had18, Els19, Scr20, Nug21, Nug22, Nug24, Nug25, Bur26h, Nug27, Nug28, Lipa30a, Esc32h, Lipa40b, Lipa50b, and Esc64a.

B. Performance criteria Two performance criteria are used to compare among the algorithms:

1) Percentage difference to the optima (dif): The percentage difference between the optima found by the algorithm and known optima [4].

2) The minimum time is taken by the algorithm to reach the optima (Mintime).

Send Onlooker Bees (colony, fitness)(m-ABC1) Selection probability of each food source i, probi is

∑=

= CS

ii

ii

fit

fitprob

1

i = 1, t =0; while t<CS

if rand<probi t=t+1; xt = xi; if rand < 0.5

do SWAP on xt to generate new food source xt'

else, do INSERT on xt to generate new food source xt

' end if

end if i=i+1 if i==CS+1 i=1 end if

do GreedySelection between xi and xt'

end while do CalculateFitness (xi)



Fig. 4. Pseudocode send onlooker bees for m-ABC2

Fig. 5.Pseudocode for send scout bees

C. Parameter settings Since we are calculating the minimum time requirement to reach the optima as a comparison criterion, it is important to

maintain equivalent settings for each algorithms and also to run on the same platform. To do that, the initial population size for all the population based algorithms are made similar. All the algorithm codes are reproduced in matlab, and run them on Matlab R2014a interface in an Intel i7 -4770s CPU with cloak speed 3.10 GHz and 8GB DDR3 ram system. All the parametric settings for different algorithms are summarized in TABLE I.1) Fig. 6 and 7 shows the convergence of various algorithms for Nug15 and Nug28 respectively.

Send scout bee (colony, limit) Find limitk==max(limt) If max(limit)>MaxLimit

limitk=0; xk= randomly selected new food source evaluate and include xk in the colony

end if

Send Onlooker Bees (colony, fitness)(m-ABC2) Selection probability of each food source i, probi is

∑=

= CS

ii

ii

fit

fitprob

1

i = 1, t =0, pi=0; while t<CS

if rand<probi t=t+1; pi= pi+1; xt = xi; xt

' = best neighbor of xt end if i=i+1 if i==CS+1 i=1 end if

do GreedySelection between xi and xt'

end while do CalculateFitness (xi)



TABLE I. PARAMETER SETTING FOR VARIOUS ALGORITHMS

Algorithm Colony size/population size

Limit/mutation rate/random walk (L)

Maximum cycle number

m-ABC1 2n for n<28 4n for n>= 28 Limit: 100 100n

g-ABC -do- -do- -do- b-s-f ABC -do- -do- -do-

GLS -do- -NA- -do- m-ABC2 2n Limit:100 -do-

GWW n for n <=20 2n for n>20

L=50 for n<=16 L=100 for n>16

Till the algorithm reaches to the local optima

ROTS -NA- -NA- 100n

GA 2n for n<30 4n for n>= 30 Mutation rate: 5% 100n

VNS-II -do- Shaking limit:100 -do-

D. Simulation Results for the Benchmarked Problems All the algorithms were run for 30 times with the above mentioned parametric settings for each problem instances.

TABLEII summarizes the run results. The red colored timings show the best found timings and green colored timings show the second best timing to reach the optima of a specific problem instance.

E. Analysis of the results Simulation results of the various algorithms under the stated parametric settings reveals various interesting insights.

1) With the lower value of n (<17) all the algorithms have converged to the optima, whereas, with the increase of n, all the algorithms could not reach to the optima.

2) Four algorithms have provided optimum results in all the benchmarked problems considered, viz. m-ABC1 and 2, GWW, and ROTS.

3) ROTS scored minimum time in total 16 problems, however, m-ABC2 scored the minimum time in four problems (Chrl12c, Nug15, Bur26h, Nug28).m-ABC1 could not score the fastest time in any of the instances.

4) The developed m-ABC2 gives 2nd fastest to reach optima after ROTS in all the other instances

5) The convergence curves in Fig. 6 and 7 depict that m-ABC2 and ROTS have very similar rate of convergence at the beginning, however, the m-ABC2 has stiffer rate of convergence afterwards.

6) m-ABC1 has less rate of convergence than ROTS and m-ABC2. However considering both the figure, it can easily be said that this algorithm has third best convergence rate.

7) GWW has very less convergence rate at the beginning, however, the rate changes drastically with increase in cycle number, which results in very stiff convergence curve.

8) GA, GLS, and VNS-II also provide optimum value in most of the cases and reaches very near to the optima in almost all the cases under the tested parameters (TABLE II). The convergence of these algorithms are also seen to be good (especially GA and VNS-II) and lies very near to m-ABC1 (Fig.6 and 7). However, the least time taken by these algorithms to reach the optima is higher in almost all the cases in comparison with m-ABC1 (exception: Had18).

9) TABLE II and Fig.6 and 7 also reveals that g-ABC and b-s-f ABC performed worse under the given conditions and problem instances.


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Fig. 6. Convergence of various algorithms for Nug15

Fig. 7. Convergence of various algorithms for Nug28

IV. CONCLUSIONS This paper suggests two new variations of artificial bee colony algorithm named as m-ABC1 and m-ABC2 for solving

quadratic assignment problems with low to medium difficulty level. In this developed algorithm, two well-known local search approaches namely, genetic search and neighborhood search were used in two local search steps of ABC viz. employed step and onlooker step. The variation is made in the onlooker step by means of any neighborhood search and best neighborhood search, to form two variety of ABC.

The m-ABC1 and m-ABC2 were evaluated on 20 well-known test problems available in QAP library and also compared with other seven known algorithms for combinatorial optimization. The test result proves the superiority of the developed algorithms over other six algorithms than ROTS in terms of finding the optima in minimum time. However the convergence rate of m-ABC2 is found to be very similar and in some cases more stiff than ROTS.

In conclusion, it can be said that the results are very much encouraging for a further improvement of this algorithm for difficult QAPs. Although the algorithm is used here to solve QAP, it can easily be used for solving other combinatorial optimization problems with very less modification. How this algorithm works for solving other combinatorial optimization problems remain a future study of this work.

1100

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BIOGRAPHY

Suman Samantais Doctoral student in the department of Industrial and Management Engineering, Indian Institute of Technology, Kanpur. He completed is masters of production engineering (with university medal) in the year 2010 from the department of Production Engineering, Jadavpur University, Kolkata. He completed his graduation B.Tech (Hons) in Production Engineering and Management in the year 2007 from the National Institute of Technology, Jamshedpur. His research interests includes process simulation modeling, optimization, operations management etc. He is presently working in the areas of facility layout optimization, process simulation modeling and optimization, productivity improvement of various SMEs nearer to Kanpur, India.

Deepu Philip is currently an Associate Professor and teaches Production and Operations Management, Systems Simulation, and Automated Analytics. He has a bachelors in Industrial Engineering with his doctorate in Industrial & Management Engineering from MSU Bozeman. He has an outstanding combination of academic and industrial experience with leading organizations of the world. His research and teaching focus is on innovation solutions using cutting-edge digital technologies that employ the highest standards of application development. He has experience in designing and implementing complex system of systems in different fields including defense, aviation, transportation, banking, automation, health care, energy, communication, and decision making. In addition, he works on planning and scheduling, local search algorithms, systems simulation, decision support systems, distributed systems, and Fuzzy databases.

J.Ramkumar is currently a Professor and teaches manufacturing science, micro/Nano technology, and new product development. He has a bachelors inProduction Engineering with his doctorate in Defect quantification in drilling of composites from IIT Madras, India with a best thesis award. Over the yearshis contribution in teaching and research is remarkable. He has worked for BOSCH group and improved the productivity of the company. His research andteaching focus is on Nano technology and inclusive design. He has several international and national patents in his credit and has published more than 100journal papers.


Kuala Lumpur, Malaysia, March 8-10, 2016 A Modified ABC ...ieomsociety.org/ieom_2016/pdfs/333.pdf · combined to develop two modified artificial bee colony algorithms ... [6, 7],

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