423642.pdf

Research ArticleHybrid Biogeography Based Optimization for ConstrainedNumerical and Engineering Optimization

Zengqiang Mi1 Yikun Xu1 Yang Yu1 Tong Zhao1 Bo Zhao2 and Liqing Liu1

1State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources North China Electric Power UniversityBaoding 071003 China2Electronics-Controlling Lab of Construction Vehicle Jilin University Changchun 130022 China

Correspondence should be addressed to Yang Yu ncepu yy163com

Received 16 April 2014 Accepted 18 July 2014

Academic Editor Baozhen Yao

Copyright copy 2015 Zengqiang Mi 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

Biogeography based optimization (BBO) is a new competitive population-based algorithm inspired by biogeography It simulatesthe migration of species in nature to share information A new hybrid BBO (HBBO) is presented in the paper for constrainedoptimization By combining differential evolution (DE) mutation operator with simulated binary crosser (SBX) of geneticalgorithms (GAs) reasonably a newmutation operator is proposed to generate promising solution instead of the randommutationin basic BBO In addition DE mutation is still integrated to update one half of population to further lead the evolution towards theglobal optimum and the chaotic search is introduced to improve the diversity of population HBBO is tested on twelve benchmarkfunctions and four engineering optimization problems Experimental results demonstrate that HBBO is effective and efficient forconstrained optimization and in contrast with other state-of-the-art evolutionary algorithms (EAs) the performance of HBBO isbetter or at least comparable in terms of the quality of the final solutions and computational cost Furthermore the influence ofthe maximum mutation rate is also investigated

1 Introduction

With the development of science and engineering relatedoptimization problems become more and more complexOptimization methods are being confronted with great chal-lenges brought by some undesirable but unavoidable charac-teristics of optimization problems such as being high-dimen-sional nondifferentiable nonconvex noncontinuous and soon Efficient optimization methods are urgently required bythe complicated optimization problems in the real worldTherefore various evolutionary algorithms have been appliedto solve difficult optimization problems in recent decadeswhich includeGAs [1] particle swarmoptimization approach(PSO) [2] DE [3 4] ant colony optimization (ACO) [5]artificial bee colony strategy (ABC) [6 7] and BBO [8 9]

Biogeography based optimization (BBO) is a new popula-tion-based algorithm It simulates the blossom and extinctionof species in different habitats based on the mathematicalmodel of biogeography Decision variables of better solutionstend to be shared in the migration operation and decision

variables of each solution are probabilistically replaced toimprove the diversity of population in mutation operationDue to good search ability BBO has been applied to PIDparameter tuning [10] parameter estimation of chaotic sys-tem [11] complex system optimization [12] satellite imageclassification [13] and so forth

In comparison with other EAs owing to direct-copying-based migration and randommutation exploration ability ofBBO is not so efficient despite outstanding exploitation Inother words BBO can be easily trapped into local optimumand suffer from premature convergence owing to lack ofcorresponding exploration to balance its exploitation

In order to overcome the weakness of BBO lots ofimproved BBO variants have been proposed Ma [14] pre-sented six different migration mathematical models andmade basic improvements on the migration operator Gonget al [15] hybridized the DE with the migration operator ofBBO to balance the exploration and exploitation of BBOMotivated by blended crossover operator in GAs Ma andSimon [16] got decision variables of an offspring by blending

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 423642 15 pageshttpdxdoiorg1011552015423642

2 Mathematical Problems in Engineering

corresponding decision variables of two parent individualsbased on different weighting constants Li and Yin [17]proposed multiparent migration in which three consecutiveindividuals are chosen to generate three new individualsby basic BBO and then the new individuals are modifiedlike multiparent crossover in GA besides the mutationwas improved based on Gaussian operator Li et al [18]updated the decision variables not selected in migration bygenerating a perturbation from the neighborhood solutionsand Gaussian operator was integrated into mutation Wangand Xu [11] integrated DE mutation operator into migrationoperator of BBO and simplex search was introduced toimprove the searching accuracy Sayed et al [10] formednew decision variables of an offspring by combining corre-sponding decision variables from two different parents withweighted constants related to the rank of their fitness inmigration Boussaıd et al [19] proposed a new hybrid BBOin which new solutions are first generated by DE mutationand then modified by migration of original BBO Xiong etal [20] utilized four individualsrsquo features to construct a newsolution in proposed polyphyletic migration and orthogonallearning was introduced to further enhance converge speedtoward global optimum

In order to balance the exploration and exploitation ofBBO a new hybrid BBO called as HBBO is proposed inthe paper The unique points of HBBO are shown as thefollowing On one hand a new hybrid mutation operatorcombining DE mutation and SBX is presented in HBBOwhile operators of EAs are often hybridized with migrationoperator in most of BBO variants On the other handHBBO provides a new method to extend BBO to optimizeconstrained problems well due to only a few BBO variantsavailable for constrained optimization in previous literaturesIn addition DE is applied to evolve one half of populationto improve convergence speed further and chaotic search isintroduced to enhance the diversity of population Experi-ments have been conducted on twelve benchmark functionsand four engineering optimization problems and HBBO iscompared with many other state-of-the-art algorithms fromthe quality of solutions obtained and computational costFurthermore the influence of maximum mutation rate onHBBO is studied

The rest of the paper is organized as follows Constrainedoptimization basic BBO mutation strategies of DE andSBX are briefly introduced in Section 2 In Section 3the HBBO method proposed in the paper is specificallydepictedThe comparisonwith six state-of-the-art algorithmson twelve benchmark functions is presented in Section 4 InSection 5 HBBO is compared with other methods on fourwell known engineering optimization problems Section 6further demonstrates the efficiency of HBBO and presentsthe investigation on the influence ofmaximummutation rateFinally the work is concluded in Section 7

2 Preliminary

21 Constrained Optimization Constrained optimizationsare always inevitable in scientific study and engineering

design A general constrained optimization problem can bewritten as the following

Minimize 119891 (x)

subject to 119892119894 (x) le 0 119894 = 1 2 119901

ℎ119895 (x) = 0 119895 = 1 2 119902

(1)

where x represents the solution vector x = [1199091 1199092

119909119863]119879 119863 is the dimensionality of a solution in the paper 119901

is the number of inequality constraints and 119902 is the numberof equality constraints In common practice equality con-straints are often transformed to inequality constraints with agiven small tolerance 120575 For example the equality constraintabove can be converted to |ℎ

119894(x)| le 120575 In the paper the

feasible-based rule byDeb [21] is applied to handle constraintIn the constraint handling mechanism fitness value andconstraint violation are considered separately based on thefollowing criterions (a) any feasible solution is preferred toany infeasible solution (b) between two feasible solutionsthe one having smaller objective function value is preferred(c) between two infeasible solutions the one having smallerconstraint violation is preferred

22 Biogeography Based Optimization Biogeography is thestudy of the distribution of species on earth surface overtime BBO is proposed based on the mathematical model ofbiogeography by Simon in 2008 [8] In BBO every solution isanalogous to a habit habit suitability index (HSI) is utilizedto measure habits just like fitness function in other EAs theelements that characterize habitability are called suitabilityindex variables (SIVs) which are identical to the decisionvariables in other EAs A good solution is similar to a habitatwith high HSI which have a large number of species and viceversa The species in habitats with high HSI tend to emigrateto habitats with low HSI That is habitats with high HSI tendto share their features while habitats with lowHSI are inclinedto accept the features from good habitats

In BBO each individual evolves by immigration andmutation operator The SIVs of individuals are probabilisti-cally shared in migration operator as shown in Algorithm 1where 119883

119894119895is the jth SIV of ith individual in the population

120582119894here represents the immigration rate of 119883

119894 and 120583

119896is the

emigration rate of 119883119896 which are related to the number of

species in the corresponding habitat NP is the populationsize in the paper

The following mathematical model is applied to calculateimmigration rate and emigration rate owing to its outstand-ing performance in [22]

120582119894= 119868 (1 minus

119878119894

119899)

120583119894= 119864(

119878119894

119899)2

(2)

where 119878119894is the number of species in habitat 119883

119894 119868 and 119864

are respectively the maximum value of immigration andemigration rate 119899 is equal to 119878max

Mathematical Problems in Engineering 3

Target individual119883119894for migration

For 119895 = 1 to119863 doSelect119883

119894119895with probabilityprop 120582

119894

If119883119894119895is selected

For 119896 = 1 to NP doSelect119883

119896with probabilityprop 120583

119896

If119883119896is selected

Replace119883119894119895with119883

119896119895

End ifEnd for

End ifEnd for

Algorithm 1 Migration operator

Target individual119883119894for mutation

For 119895 = 1 to119863Select119883

119894119895based on119898

119894

If119883119894119895is selectedReplace it with a randomly generated SIV

End ifEnd for

Algorithm 2 Mutation operator

In mutation operator it is probabilistically decidedwhether or not to replace each SIV in a solution by a randomlygenerated SIV in the light of mutation rate The detail ofmutation operator is shown inAlgorithm2Themutation ratem can be calculated as follows

119898119894= 119898max (1 minus

119875119894

119875max) (3)

where 119875max = max(119875119894) 119875119894represents the priori probability of

existence for ith individual119898max is a user-defined parameterwhich represents the maximummutation rate

More details about basic BBO can be found in [8 23]

23 Differential Evolution DE algorithm is a population-based stochastic search method proposed by Storn and Pricein 1997 [3] Due to simple structure few parameters easyuse and fast convergence speed DE has obtained wideapplication in various regions DE generates new individualsby perturbing a randomly chosen individual with weighteddifferences for some couples of different individuals Onlywhen the offspring outperforms corresponding parent theoffspring survives as the parent for next generationMutationoperator is the most important part in DE In this partonly three widely applied mutation strategies are brieflyintroduced as follows

(1) rand1

119884119894119895= 1198831199031119895+ 119865 (119883

1199032119895minus 1198831199033119895) (4)

(2) best1

119884119894119895= 119883119892119895

+ 119865 (1198831199031119895minus 1198831199032119895) (5)

(3) rand to best1

119884119894119895= 1198831199031119895+ 119865 (119883

119892119895minus 1198831199031119895) + 119865 (119883

1199032119895minus 1198831199033119895) (6)

where 119883119892is the best individual in population 119903

1 1199032 1199033are

uniformly distributed different numbers in the range [1NP]119865 is mutation scaling factor119884 represents the new individualsgenerated by mutation operator

24 SBX of GA Genetic algorithms simulate the evolutionalprocess in nature to solve optimization problems In GAsome good individuals are chosen based on Debrsquos feasible-based rule Different individuals can share information incrosser operator SBX is one of the most popular crosseroperators which can explore the neighborhood region ofparent individual as follows

1198621119895

=1

2[(1 minus 120573

119895) 119875119886119903119890119899119905

1119895+ (1 + 120573

119895) 119875119886119903119890119899119905

2119895] (7)

1198622119895

=1

2[(1 + 120573

119895) 119875119886119903119890119899119905

1119895+ (1 minus 120573

119895) 119875119886119903119890119899119905

2119895] (8)

where 119862119894119895is the jth decision variable of the ith offspring

individual 119875119886119903119890119899119905119894119895is the jth decision variable of ith parent

individual selected 120573 can be obtained by the randomnumber119906 in [0 1] based on (9) where 120578 is the distribution index forcrossover The detail of SBX can be found in [24] Considerthe following

120573 (119906) =

(2119906)1(120578+1) if 119906 le 051

[2 (1 minus 119906)]1(120578+1)if 119906 gt 05

(9)

3 Proposed Algorithm HBBO

In mutation operator of basic BBO SIVs are replaced prob-abilistically by new SIVs randomly generated Although themutation of BBO can improve the diversity of populationthe random operation brings blindness to search To modifythe defect a new hybrid mutation operator is proposed inwhich DE mutation operator and SBX are mixed to generatepromising SIV as shown in Algorithm 3 From Algorithm 3it can been seen that two candidate SIVs are generated foreach SIV mutated one is gotten by DE rand1 mutation andthe other by SBX One point should be stated specially 119883

1199031

in DE rand1 mutation and 1198751198861199031198901198991199051and 119875119886119903119890119899119905

2in SBX are

all randomly selected from the first half of parent populationwhich is sorted based on Debrsquos feasible rule (better one infront)119883

1199032

1198831199033

in DE rand1mutation are randomly selectedfrom the whole populationThe core idea of hybrid mutationis based on the following considerations First owing towell-known performance in locating the region of globaloptimum DE mutation can explore new search space withmore clear direction towards global optimum instead of therandom mutation in the original BBO Second SBX canexplore the neighbor region of parent individual so that itcan be combined with DE to explore search space efficientlyThird the combination of DE mutation and SBX can balancethe exploitation ability of BBO

4 Mathematical Problems in Engineering

Target individual119883119894for mutation

For 119895 = 1 to119863If119883119894119895is selected for mutation as basic BBOGet two candidate SIVs of offspring(1) Get a temp SIV by DE rand1mutation(2) If rand lt 05 (rand is random number in [0 1])

Get another temp SIV by (7)Else

Get another temp SIV by (8)End if

Elsethe 119895th SIV of 119894th individual in population Island survives as SIV of offspring(population Island contains new individuals gotten by migration operator)

End ifEnd forTwo temp offspring individuals are gotten for119883

119894 and the better one survives as offspring

Algorithm 3 Hybrid mutation operator

In order to speed up convergence DE is furtherhybridized with BBO The first half of parent populationalso evolves by two DE mutation strategies (rand1 and randto best1) and two new individuals are generated for eachone in the first half The best one among these two newindividuals and corresponding parent individual survivesto replace corresponding one in the second half of parentpopulation

For convenience and easy use self-adaption mechanismfor mutation scaling factor of DE proposed in [25] is appliedin which each individual is given an independent mutationscaling factor The self-adaption mechanism is written asfollows

119865119866+1119894

= 119865119897+ 119903119886119899119889

1sdot 119865119906

if 1199031198861198991198892lt 03

119865119866119894

otherwise(10)

where 119865119866+1119894

represents mutation scaling factor for ith indi-vidual in (119866+1)th generation119865119866

119894representsmutation scaling

factor for ith individual inGth generation119865119897119865119906are the lower

boundary and the change range of mutation scaling factorrespectively and 119903119886119899119889

1and 119903119886119899119889

2are uniformly distributed

numbers in [0 1]Based on unique ergodicity inherent stochastic property

and irregular chaos chaotic search can reach each situationin given space so that it can contribute to the escape from thelocal optimum and is often integrated into EAs to enhanceglobal search ability Hence the chaotic search is brought infor the first half of population In the paper logistic maps areused to generate chaotic sequences as follows

119882119894+1119895

= 120576 sdot 119882119894119895sdot (1 minus 119882

119894119895) 0 lt 119882

1119895lt 1 (11)

where1198821119895119882119894119895119882119894+1119895

are respectively the first ith (119894 + 1)thelement of sequence 119882

11198951198822119895

119882119873119895

119873 represents thesize of population for chaotic search 119895 = 1 2 119863 120576is the control parameter sequence 119882

11198951198822119895

119882119873119895

ischaotic when 120576 = 4 and 119882

1119895= 025 05 075 We can

apply the following equation to perform chaotic search for

i-th individual 119883119894in parent population by vector 119882

119894=

[11988211989411198821198942 119882

119894119863]119879

1198831015840119894= 119883119894+ 119877119866sdotlowast (2119882

119894minus 1) (12)

where1198831015840119894is the new individual generated by chaotic search for

119883119894119877119866 represents the search radium vector inGth generation

each dimension in 119877119866 represents the search radium for thevariables in corresponding dimension of solutions in parentpopulation

In the initial phase large chaotic search radium is helpfulfor escape from the local optimum small chaotic searchradium can improve the accuracy of search at the later stageof evolution The search radium 119877119866

119895for jth decision variable

is adapted as follows

119877119866119895= 119861119900119906119899119889

119895sdot (0005 minus 0004

119866

119866max) (13)

where 119861119900119906119899119889119895is the initial search radium for jth decision

variable 119866max is the maximum number of generationsIn order to maintain solutions feasible any new decision

variable generated in evolution process should be repaired if itviolates boundary Suppose that 119905

119895is the jth decisions variable

in certain new individual generated during evolution processIf 119905119895violates given boundaries it can be modified as follows

119905119895= 119897119887119895+ (119906119887119895minus 119897119887119895) sdot 119903119886119899119889119899119906119898

119895 (14)

where 119897119887119895 119906119887119895are respectively the lower boundary and upper

boundary of jth decision variable randnumj is uniformlydistributed number in [0 1] in each dimension

The whole procedure of HBBO is described in Algo-rithm 4 in detail From Algorithm 4 it can be seen thatoperations are mainly concentrated on the first half ofpopulation It can be explained as follows First the con-vergence speed can be improved by focusing operations onthe first half Second the information of the second half

Mathematical Problems in Engineering 5

Generate the initial population POP and vector 119865 of mutation scaling factorsEvaluate the fitness and constraint violations of each individual in POPFor each generation do

Sort the individuals in POP based on Debrsquos feasibility-based rule (better in front)For each one in POPrsquos first half

Get two new individuals by two DE mutation strategies (rand1 rand to best1)Evaluate the fitness value and constraint violations of these two new individualsAmong these two new individuals and corresponding parent individual the best one is stored into populationTempbest

End forUpdate the vector 119875 of prior probabilityFor each one in the first half of POP

Generate a new individual by Algorithm 1 and store it into population IslandEnd forFor each one in the first half of POP

Get one offspring by Algorithm 3 and replace the corresponding individual in population Island with itEnd forGo on chaotic search for the first half of POP and the new individuals generated are stored into population tempIslandMake a contrast between the corresponding ones in Island and tempIsland and the first half of POP the best one survivesas the corresponding one in POP for next generationThe population Tempbest replace the second half of POP as the parent ones for next generationUpdate 119865 119877 by (10) (13) respectively

End for

Algorithm 4 Hybrid biogeography based optimization

Table 1 Main characteristics of the twelve selected benchmark functions

Benchmark function 119863 Type of objective function 120588 LI NI NE 119886

G01 13 Quadratic 00003 9 0 0 6G02 20 Nonlinear 999970 1 1 0 1G03 10 Nonlinear 00000 0 0 1 1G04 5 Quadratic 269668 0 6 0 2G05 4 Nonlinear 00000 2 0 3 3G06 2 Nonlinear 00064 0 2 0 2G07 10 Quadratic 00002 3 5 0 6G08 2 Nonlinear 08575 0 2 0 2G09 7 Nonlinear 05235 0 4 0 2G10 8 Linear 00007 3 3 0 3G11 2 Quadratic 00000 0 0 1 1G12 3 Quadratic 4774 0 93 0 0

is also utilized in migration and hybrid mutation operatorto generate promising solutions Third the risk of trappinginto stagnation brought by concentration of operations canbe relieved by chaotic search Consequently the focus ofoperations can make the search of HBBO efficient

4 Simulation Tests on Benchmark Functions

41 Parameter Setting and Statistical Results Obtained byHBBO In order to validate the performance of the proposedHBBO on numerical optimization twelve benchmark testfunctions are adopted The selected benchmark problemspropose a good challenge and measure for constrainedoptimization techniques Main characteristics of the selectedbenchmark functions are shown in detail in Table 1 where

119863 is the dimensionality of a solution for test function 120588represents the ratio of feasible region to search space NIis the number of nonlinear inequality constraints LI is thenumber of linear inequality constraints NE is the numberof nonlinear equality constraints and 119886 is the number ofconstraints active at the optimal solution The 120588 metric canbe computed as the following

120588 =|119865|

|119878| (15)

where |119878| is the number of solutions generated randomly(|119878| = 1000000 in the paper) and |119865| is the number of feasiblesolutions found in all the solutions randomly generated Allthe benchmark functions selected are depicted explicitly inAppendix A

6 Mathematical Problems in Engineering

Table 2 Statistic results for twelve benchmark functions obtained by HBBO

Function Optimal Best Mean Median Worst SD FFEsG01 minus15 minus15 minus14799953 minus15 minus13 610119864 minus 1 50100G02 minus0803619 minus08036179 minus07805965 minus07852652 minus07330360 1870119864 minus 2 75200G03 minus1 minus10050100 minus10050100 minus10050100 minus10050100 288119864 minus 12 150100G04 minus30665539 minus3066553867 minus306655387 minus306655387 minus306655387 111119864 minus 11 37600G05 51264981 51264842 51264842 51264842 51264842 923119864 minus 4 375100G06 minus696181388 minus696181388 minus696181388 minus696181388 minus696181388 370119864 minus 12 50100G07 243062091 243062091 243062091 243062091 243062091 649119864 minus 10 125400G08 minus0095825 minus0095825 minus0095825 minus0095825 minus0095825 106119864 minus 17 10100G09 680630057 680630057 680630057 680630057 680630057 516119864 minus 13 75100G10 7049248021 7049248021 7049248021 7049248021 7049248021 108119864 minus 7 137600G11 075 074990 074990 074990 074990 113119864 minus 16 75100G12 minus1 minus1 minus1 minus1 minus1 0 12600

For each test function we performed 30 independentruns in matlab 70 The parameters of HBBO for experimentsare set as follows 119864 = 119868 = 1 is chosen as recommendedin [8] 119898max is set to be 08 which is much bigger than thecorresponding value in basic BBO because big 119898max canimprove mutation probabilities of individuals in populationand enhance population diversity based on the suggestionsof mutation factor in DE in [3] and numerous experiments119865119897= 075 and 119865

119906= 015 are chosen 120578 = 20 is chosen in the

light of the effect of 120578 on the search ability of SBX [24]Through various tests an appropriate set of population

size NP for all the selected functions is found with whichHBBO can present desirable performance In the set foundpopulation size NP for each benchmark function is given asthe following 200 for G02 150 for G07 and 100 for the rest ofbenchmark functions In each run themaximumgenerationsare given as the following 200 for G01 and G06 150 for G02and G04 600 for G03 1500 for G05 334 for G07 40 for G08300 for G09 and G11 550 for G10 and 50 for G12 In G03 andG05 the toleration value for equation constraint equals 0001as recommended in [19] the toleration value for equationconstraint of G11 is set to be 00001 as suggested in [26]

Table 2 summarizes the statistical features of results fortwelve test functions obtained by HBBO and number offitness function evaluations (FFEs) required From Table 2we can see that HBBO can get optimal solution in all 30 runsfor seven benchmark functions (G04 G06 G07 G08 G09G10 and G12) for G01 HBBO can get the optimal solutionsin some runs the best results obtained by HBBO are veryclose to the knownbest solution forG02 for three benchmarkfunctions (G03 G05 and G11) the results gained by HBBOare very close to the optimal solutions or the known bestsolution

42 Comparison with Other State-of-the-Art Methods In thispart the proposed approach HBBO is compared with othersix state-of-the-art optimization technologies

The following are the six state-of-the art optimizationtechnologies conventional BBO with DE mutation technol-ogy (CBO-DM) [19] hybrid PSO with DE strategy (PSO-DE) [27] coevolutionaryDE algorithm (CDE) [28] changing

range genetic algorithm (CRGA) [26] self-adaptive penaltyfunction based algorithm (SAPF) [29] and simple mul-timembered evolution strategy (SMES) [30] The statisticresults of other six approaches are compared with thatof HBBO in Table 3 which are gotten from the originalreferences The ldquoNArdquo in tables of the paper indicates theresults of compared algorithms are not available It should benoted that the best results obtained by algorithms are markedin boldface in the following tables As far as computationalcost is concerned CBO-DM SAPF CDE SMES respectivelyneed 350000 500000 248000 and 240000 FFEs for allthe test functions PSO-DE needs 70100 FFEs for G0417600 FFEs for G12 and 140100 FFEs for the rest of testfunctions 1350 to 68000 FFEs are required for CRGA thecomputational cost for CRGA is given in detail in [26]

With respect to CBO-DM a variant of BBO similarresults are obtained by HBBO for five functions (G04 G06G08 G09 and G12) in two functions (G07 G10) HBBO hasbetter performance in the respect of consideredmetrics (bestmean and worst) in G02 HBBO gets better best value withgreater variability the results of HBBO are obviously inferiorbut comparable for G01 the results obtained by HBBO areonly lightly inferior for three test functions (G03 G05 andG11) In addition the computational cost is far less than thatof CBO-DM for all selected benchmark functions exceptG05Consequently HBBO is powerful competitor for CBO-DMon constrained optimization

In contrast with other five state-of-the-art methods theperformance of HBBO is obviously inferior for function G01HBBO can get better or similar solutions for the selectedtest functions except for G01 G03 and G11 In G03 and G11the results obtained by HBBO are only lightly inferior tothose of SMES Furthermore the computational cost is verycompetitive with respect to othermethods for all selected testfunctions except G05

5 Simulation Tests on EngineeringOptimization Problems

In this part four well-known engineering optimization prob-lems are utilized to validate the performance of HBBO on

Mathematical Problems in Engineering 7

Table 3 Statistical features of results for twelve benchmark functions obtained by HBBO and other six state-of-the-art algorithms

Function Metrics HBBO CBO-DM PSO-DE CRGA SAPF SMES CDE

G01Best minus15 minus15000 15000000 minus149977 minus15000 minus15000 minus150000Mean minus14799953 minus15000 minus15000000 minus149850 minus14552 minus15000 minus150000Worst minus13 minus15000 minus15000000 minus149467 minus13097 minus15000 minus150000

G02Best minus08036179 minus0803557 minus08036145 minus0802959 minus0803202 minus0803601 minus0794669Mean minus07805965 minus0802774 minus0756678 minus0764494 minus0755798 minus0785238 minus0785480Worst minus07330360 minus0792576 minus06367995 minus0722109 minus0745712 minus0751322 minus0779837

G03Best minus10050100 minus1000 minus10050100 minus09997 minus1000 minus1000 NAMean minus10050100 minus1000 minus10050100 minus09972 minus0964 minus1000 NAWorst minus10050100 minus1000 minus10050100 minus09931 minus0887 minus1000 NA

G04Best minus3066553867 minus30665539 minus30665539 minus30665520 minus30665401 minus30665539 minus30665539Mean minus306655387 minus30665539 minus30665539 minus30664398 minus30665922 minus30665539 minus30665536Worst minus306655387 minus30665539 minus30665539 minus30660313 minus30656471 minus30665539 minus30665509

G05Best 51264842 5126498 NA 5126500 5126907 5126599 NAMean 51264842 5126498 NA 5507041 5214232 5174492 NAWorst 51264842 5126498 NA 6112075 5564642 5304167 NA

G06Best minus696181388 minus6961814 minus69618139 minus6956251 minus6961046 minus6961814 minus6961814Mean minus696181388 minus6961814 minus69618139 minus6740288 minus6953061 minus6961284 minus6960603Worst minus696181388 minus6961814 minus69618139 minus6077123 minus6943304 minus6952482 minus6901285

G07Best 243062091 24326 243062091 24882 24838 24327 NAMean 243062091 24345 24306210 25746 27328 24475 NAWorst 243062091 24378 243062172 27381 33095 24843 NA

G08Best minus0095825 minus0095825 minus0095826 minus0095825 minus0095825 minus0095825 NAMean minus0095825 minus0095825 minus0095826 minus0095819 minus0095635 minus0095825 NAWorst minus0095825 minus0095825 minus0095826 minus0095808 minus0092697 minus0095825 NA

G09Best 680630057 680630 68063006 680726 680773 680632 680771Mean 680630057 680630 68063006 681347 681246 680643 681503Worst 680630057 680630 68063006 682965 682081 680719 685144

G10Best 7049248021 7059802 70492480 7114743 7069981 7051903 NAMean 7049248021 7075832 70492480 8785149 7238964 7253047 NAWorst 7049248021 7098254 70492482 1082609 7489406 7638366 NA

G11Best 074990 075 0749999 0750 0749 075 NAMean 074990 075 0749999 0752 0751 075 NAWorst 074990 075 0749999 0757 0757 075 NA

G12Best minus1 minus1000000 1000000 minus1000000 minus1000000 minus1000 minus1000000Mean minus1 minus1000000 minus1000000 minus1000000 minus099994 minus1000 minus1000000Worst minus1 minus1000000 minus1000000 minus1000000 minus0999548 minus1000 minus1000000

solving real-world optimization problems The four engi-neering optimization problems contain welded beam designproblem tensioncompression spring design problem speedreducer design problem and three-bar truss design problemwhich are listed in Appendix B Parameters in HBBO forthese four engineering optimization problems are as followspopulation size and maximum generations are respectively50 and 200 for welded beam design problem 50 and 350 fortensioncompression spring design problem 100 and 100 forspeed reducer design problem and 50 and 60 for three-bartruss design problem other parameters for HBBO are set inthe sameway as Section 4 For each engineering optimizationproblem 30 independent runs are performed Table 4 showedthe statistic results for the four engineering optimization

problems solved by HBBO We will evaluate performance ofHBBO in respect of the quality of results and computationalcost

In order to demonstrate the superiority of HBBO it iscompared with other state-of-the-art algorithms on the fourengineering problems Welded beam and tensioncompres-sion spring design problems are also attempted by PSO-DE[27] CDE [28] coevolutionary particle swarm optimization(CPSO) [31] (120583 + 120582)-evolutionary strategy ((120583 + 120582)-ES) [32]unified particle swarm optimization (UPSO) [33] and ABC[7] PSO-DE [27] (120583 + 120582)-ES [32] and ABC [7] have alsoalready performed on speed reducer design problem PSO-DE [27] and Ray and Liew [34] have also been applied tosolve three-bar truss design problem The comparison of

8 Mathematical Problems in Engineering

Table 4 Statistic results for four engineering optimization problems solved by HBBO

Engineering optimizationproblem Best Mean Median Worst SD FFEs

Welded beam design 1724852309 1724852309 1724852309 1724852309 114119864 minus 15 25050Tensioncompressionspring design 0012665233 0012665393 0012665234 0012666698 318119864 minus 07 43800

Speed reducer design 2996348165 2996348165 2996348165 2996348165 552119864 minus 12 25100Three-bar truss design 26389584338 26389584338 26389584338 26389584338 708119864 minus 14 7550

Table 5 Statistic results for welded beam design obtained byHBBOand other six state-of-the-art methods

Method Best Mean Worst FFEsHBBO 17248523 17248523 17248523 25050PSO-DE 17248531 17248579 17248811 33000CDE 1733461 1768158 1824105 240000CPSO 1728024 1748831 1782143 200000(120583 + 120582)-ES 1724852 1777692 NA 30000UPSO 192199 283721 NA 100000ABC 1724852 1741913 NA 30000

Table 6 Statistic results of HBBO and other six state-of-the-artmethods for tensioncompression spring design

Method Best Mean Worst FFEsHBBO 0012665233 0012665393 0012666698 43800PSO-DE 0012665233 0012665233 0012665233 42100CDE 00126702 0012703 0012790 240000CPSO 00126747 001273 0012924 200000(120583 + 120582)-ES 0012689 0013165 NA 30000UPSO 001312 002294 NA 100000ABC 0012665 0012709 NA 30000

Table 7 Statistic results of HBBO and other six state-of-the-artmethods for speed reducer design

Method Best Mean Worst FFEsHBBO 2996348165 2996348165 2996348165 25100PSO-DE 2996348165 2996348165 2996348166 70100(120583 + 120582)-ES 2996348 2996348 NA 30000ABC 2997058 2997058 NA 30000

statistical results and computational cost for four engineeringoptimization problems between HBBO and other algorithmsis shown in Tables 5 6 7 and 8

From Tables 5 6 7 and 8 it can be seen that HBBOoutperforms other compared algorithms for the given engi-neering optimization problems except tensioncompressionspring design problem for which PSO-DE has best per-formance For tensioncompression spring design problemHBBO get similar best result and the mean and worst resultsof it are just lightly inferior in contrast with PSO-DE

Table 8 Statistic results of HBBO and other six state-of-the-artmethods for three-bar truss design

Method Best Mean Worst FFEsHBBO 26389584338 26389584338 26389584338 7550PSO-DE 26389584338 26389584338 26389584338 17600Ray andLiew 26389584654 26390335672 26396975638 17610

6 Discussions

In this part HBBO is compared with the original BBOand self-adapting DE (SADE) to demonstrate the searchingefficiency of HBBO further In addition the influence ofmaximum mutation rate on searching efficiency of HBBO isinvestigated

61 Comparison with the Original BBO and SADE Thedetail of the original BBO can be gotten from [8] andSADE proposed in [25] is compared Specific parametersetting of these two algorithms is the same as the originalreferences while the parameters related to test functions suchas population size and constraint tolerance are in accordancewith Section 4Debrsquos feasible rule is applied inBBOand SADEto handle constraint Here HBBO is adopted in an identicalway as described in Section 4

Figure 1 illustrates typical evolution processes of objectivefunction value of best solution in population when fourbenchmark functions (G02 G03 G07 and G09) are respec-tively solved by HBBO BBO and SADE From Figure 1it can be seen that HBBO have fastest convergence speedwhile BBO is often trapped into stagnation So it can beconcluded that the exploration and exploitation of BBO arewell enhanced and balanced by new mutation operator andfurther hybridization with DE and chaotic search

62 Influence of Maximum Mutation Rate on HBBO Themaximum value of mutation rate 119898max is related to theprobability that individuals mutate by new hybrid mutationoperator so that it affects the balance degree of explorationand exploitation of HBBO In order to investigate the effect of119898max on search efficiency of HBBO119898max is set to be differentvalues including 005 01 04 08 and 1 The investigationexperiments are performed on five benchmark functions(G01 G02 G03 G07 and G10)

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8NFFEs

Valu

e of o

bjec

tive f

unct

ion

G02

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

times10

4

(a) G02

0 2 4 6 8 10 12 14 16

0

NFFEs

Valu

e of o

bjec

tive f

unct

ion

G03

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

times10

4

(b) G03

2 4 6 8 10 12 1420

25

30

35

40

45

50

55

60

NFFEs

Loga

rithm

ic v

alue

of o

bjec

tive f

unct

ion

G07

HBBOSADEBBO

times10

4

(c) G07

1 2 3 4 5 6 7 8680

685

690

695

700

705

710

715

720

725

730

NFFEs

Valu

e of o

bjec

tive f

unct

ion

G09

HBBOSADEBBO

times10

4

(d) G09

Figure 1 Objective function value curves of four test functions solved by HBBO SADE and BBO

The other parameters are in accordance with descriptionin Section 4 For each value of119898max and test function we per-form 30 independent runs The statistical features of resultsobtained by HBBO with different 119898max are summarized inTable 9

From Table 9 we can see that the value of 119898max hassignificant influence on search efficiency of HBBO Theinfluence can be stated from three respects as the followingFirst HBBO with different value of 119898max has differentperformance on the five test functions Second HBBO withtoo small or too big 119898max could not solve the selected testfunctions well while HBBO with middle value of 119898max hasbetter comprehensive performance Third the fittest 119898maxfor each test function is different The influence above may

be explained as follows Too small 119898max cannot balance theexploitation of BBO well while too big 119898max will destroy theexploitation of BBO different test functions have differentcharacteristics so that they have different requirements ofexploration and exploitation of optimization algorithm

7 ConclusionsThe paper proposes a new hybrid biogeography based opti-mization (HBBO) for constrained optimization For thepresented algorithm HBBO a new mutation operator wasproposed to generate promising solutions by merging DEmutation with SBX a half of the population also evolved bytwomutation strategies ofDE Chaotic searchwas introduced

10 Mathematical Problems in Engineering

Table 9 Statistical features of the results obtained by HBBO with different119898max

Function Metrics HBBO119898max = 005 119898max = 01 119898max = 04 119898max = 08 119898max = 1

G01

Best minus15 minus15 minus15 minus15 minus15Mean minus14666665 minus14633610 minus14600000 minus14799953 minus14632939Worst minus12999949 minus12008289 minus13 minus13 minus12SD 758119864 minus 1 849119864 minus 1 814119864 minus 1 610E minus 1 850119864 minus 1

G02

Best minus08036071 minus08036171 minus08036185 minus08036179 minus08036173Mean minus07780832 minus07787019 minus07799297 minus07805965 minus07775819Worst minus071807066 minus07501355 minus07248475 minus07330360 minus07225584SD 203119864 minus 2 158E minus 2 183119864 minus 2 187119864 minus 2 219119864 minus 2

G03

Best minus10044781 minus10050100 10050100 10050100 minus10050100Mean minus07515293 minus10050100 10050100 10050100 minus10050100Worst minus02971747 minus10050100 10050100 10050100 minus10050100SD 23119864 minus 1 253119864 minus 10 410119864 minus 10 288E minus 12 539119864 minus 12

G07

Best 243062091 243062091 243062091 243062091 243062091Mean 243062100 243062091 243062091 243062091 243062091Worst 243062351 243062091 243062091 243062091 243062091SD 474119864 minus 6 700119864 minus 10 235E minus 10 649119864 minus 10 797119864 minus 10

G10

Best 7049248021 7049248021 7049248021 7049248021 7049248021Mean 7054178611 7049248022 7049248021 7049248021 7049248027Worst 7194608835 7049248076 7049248021 7049248021 7049248200SD 2652 102119864 minus 5 420E minus 8 108119864 minus 7 327119864 minus 5

for escape from stagnation and Debrsquos feasibility-based rulewas applied to handle constraints Furthermore self-adaptionmechanism for themutation scaling factor of DE was utilizedto avoid bothering of choosing an appropriate parameter

Simulation experiments were performed on twelvebenchmark test functions and four well-known engineeringoptimization problems HBBO can obtain better or compara-ble results in contrast with other state-of-the-art optimizationtechnologies At the same time the low computation cost isthe obvious advantage of our HBBO In short HBBO is aneffective and efficient method for constrained optimizationIn addition the influence of maximum mutation rate wasinvestigated and the results demonstrate HBBO with maxi-mummutation rate ofmiddle value has better comprehensiveperformance

Maximum immigration rate 119868 and emigration rate 119864affect the migration probability and the prior probability ofexistence for individuals in population How to select thefittest 119864 and 119868 for HBBO will be one possible focus of ourresearch in the future Besides HBBO will be extended tomultiobjectives optimization

Appendices

A Benchmark Test Functions

A1 G01

Minimize 119891 () = 54

sum119894=1

119909119894minus 54

sum119894=1

1199092119894minus13

sum119894=5

119909119894

subject to 1198921 () = 2119909

1+ 21199092+ 11990910

+ 11990911

minus 10 le 0

1198922 () = 2119909

1+ 21199093+ 11990910

+ 11990912

minus 10 le 0

1198923 () = 2119909

2+ 21199093+ 11990911

+ 11990912

minus 10 le 0

1198924 () = minus8119909

1+ 11990910

le 0

1198925 () = minus8119909

2+ 11990911

le 0

1198926 () = minus8119909

3+ 11990912

le 0

1198927 () = minus2119909

4minus 1199095+ 11990910

le 0

1198928 () = minus2119909

6minus 1199097+ 11990911

le 0

1198929 () = minus2119909

8minus 1199099+ 11990912

le 0

(A1)

where the bounds are 0 le 119909119894le 1 (119894 = 1 9) 0 le 119909

119894le 100

(119894 = 10 11 12) and 0 le 11990913

le 1 The global optimum is atlowast = (1 1 1 1 1 1 1 1 1 3 3 3 1) with

119891 (lowast) = minus15 (A2)

A2 G02

Minimize 119891 () = minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum119899

119894=1cos4 (119909

119894) minus 2prod

119899

119894=1cos2 (119909

119894)

radicsum119899

119894=11198941199092119894

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

subject to 1198921 () = 075 minus

119899

prod119894=1

119909119894le 0

1198922 () =

119899

sum119894=1

119909119894minus 75119899 le 0

(A3)

Mathematical Problems in Engineering 11

where 119899 = 20 and 0 le 119909119894

le 10 (119894 = 1 119899) Theglobal optimum is unknown the best objective function valuereported is 119891(lowast) = minus0803619

A3 G03

Maximize 119891 () = (radic119899)119899119899

prod119894=1

119909119894

subject to ℎ () =119899

sum119894=1

1199092119894minus 1 = 0

(A4)

where 119899 = 10 and 0 le 119909119894le 1 (119894 = 1 2 119899) The optimum

solution is 119909lowast119894= 1radic119899 (119894 = 1 2 119899) with 119891(lowast) = 1

A4 G04

Maximize 119891 () = 5357854711990923+ 08356891119909

11199095

+ 372932391199091minus 40792141

subject to 1198921 () = 85334407 + 00056858119909

21199095

+ 0000626211990911199094

minus 0002205311990931199095minus 92 le 0

1198922 () = minus85334407 minus 00056858119909

21199095

minus 0000626211990911199094

+ 0002205311990931199095le 0

1198923 () = 8051249 + 00071317119909

21199095

+ 0002995511990911199092

+ 0002181311990923minus 110 le 0

1198924 () = minus8051249 minus 00071317119909

21199095

minus 0002995511990911199092

minus 0002181311990923+ 90 le 0

1198925 () = 9300961 + 00047026119909

31199095

+ 0001254711990911199093

+ 0001908511990931199094minus 25 le 0

1198926 () = minus9300961 minus 00047026119909

31199095

minus 0001254711990911199093

minus 0001908511990931199094+ 20 le 0

(A5)

where 78 le 1199091

le 102 33 le 1199092

le 45 and 27 le119909119894le 45 (119894 = 3 4 5) The optimum solution is lowast = (78

33 29995256025682 45 36775812905788) with 119891(lowast) =30665539

A5 G05

Minimize 119891 () = 31199091+ 00000011199093

1+ 21199092

+ (0000002

3) 11990932

subject to 1198921 () = minus119909

4+ 1199093minus 055 le 0

1198922 () = minus119909

3+ 1199094minus 055 le 0

ℎ3 () = 1000 sin (minus119909

3minus 025)

+ 1000 sin (minus1199094minus 025) + 8948

minus 1199091= 0

ℎ4 () = 1000 sin (119909

3minus 025)

+ 1000 sin (1199093minus 1199094minus 025) + 8948

minus 1199092= 0

ℎ5 () = 1000 sin (119909

4minus 025)

+ 1000 sin (1199094minus 1199093minus 025)

+ 12948 = 0

(A6)

where 0 le 1199091

le 1200 0 le 1199092

le 1200 minus055 le 1199093

le055 and minus055 le 119909

4le 055 The best known solution

is lowast = (6799453 1026067 01188764 03962336) where119891(lowast) = 51264981

A6 G06

Minimize 119891 () = (1199091minus 10)3+ (1199092minus 20)3

subject to 1198921 () = minus(119909

1minus 5)2minus (1199092minus 5)2+ 100 le 0

1198922 () = minus(119909

1minus 6)2minus (1199092minus 5)2minus 8281 le 0

(A7)

where 13 le 1199091

le 100 and 0 le 1199092

le 100 Theoptimum solution is lowast = (14095 084296) with 119891(lowast) =minus696181388

A7 G07

Minimize 119891 () = 11990921+ 11990922+ 11990911199092minus 14119909

1minus 16119909

2

+ (1199093minus 10)2+ 4(1199094minus 5)2+ (1199095minus 3)2

+ 2(1199096minus 1)2+ 511990927

+ 7(1199098minus 11)2+ 2(1199099minus 10)2

+ (11990910

minus 7)2+ 45

12 Mathematical Problems in Engineering

subject to 1198921 () = minus105 + 4119909

1+ 51199092minus 31199097+ 91199098le 0

1198922 () = 10119909

1minus 81199092minus 17119909

7+ 21199098le 0

1198923 () = minus8119909

1+ 21199092+ 51199099minus 211990910

minus 12 le 0

1198924 () = 3(119909

1minus 2)2+ 4(1199092minus 3)2

+ 211990923minus 71199094minus 120 le 0

1198925 () = 51199092

1+ 81199092+ (1199093minus 6)2minus 21199094minus 40 le 0

1198926 () = 1199092

1+ 2(1199092minus 2)2minus 211990911199092+ 14119909

5

minus 61199096le 0

1198927 () = 05(119909

1minus 8)2+ 2(1199092minus 4)2

+ 311990925minus 1199096minus 30 le 0

1198928 () = minus3119909

1+ 61199092+ 12(119909

9minus 8)2minus 711990910

le 0

(A8)

where minus10 le 119909119894

le 10 (119894 = 1 2 10) The optimumsolution is lowast = (2171996 2363683 8773926 509598409906548 1430574 1321644 9828726 8280092 8375927)with 119891(lowast) = 243062091

A8 G08

Maximize 119891 () =sin3 (2120587119909

1) sin (2120587119909

2)

11990931(1199091+ 1199092)

subject to 1198921 () = 1199092

1minus 1199092+ 1 le 0

1198922 () = 1 minus 119909

1+ (1199092minus 4)2le 0

(A9)

where 0 le 1199091le 10 and 0 le 119909

2le 10 The optimal solution is

lowast = (12279713 42453733) where 119891(lowast) = 0095825

A9 G09

Minimize 119891 () = (1199091minus 10)2+ 5(1199092minus 12)2+ 11990943

+ 3(1199094minus 11)2+ 101199096

5+ 711990926+ 11990947

minus 411990961199097minus 10119909

6minus 81199097

subject to 1198921 () = minus127 + 21199092

1+ 311990942+ 1199093

+ 411990924+ 51199095le 0

1198922 () = minus282 + 7119909

1+ 31199092+ 101199092

3

+ 1199094minus 1199095le 0

1198923 () = minus196 + 23119909

1+ 11990922+ 611990926minus 81199097le 0

1198924 () = 41199092

1+ 11990922minus 311990911199092+ 211990923

+ 51199096minus 11119909

7le 0

(A10)

where minus10 le 119909119894

le 10 (119894 = 1 2 7) The optimumsolution is lowast = (2330499 1951372 minus04775414 4365726minus06244870 1038131 1594227) with 119891(lowast) = 680630057

A10 G10

Minimize 119891 () = 1199091+ 1199092+ 1199093

subject to 1198921 () = minus1 + 00025 (119909

4+ 1199096) le 0

1198922 () = minus1 + 00025 (119909

5+ 1199097minus 1199094) le 0

1198923 () = minus1 + 001 (119909

8minus 1199095) le 0

1198924 () = minus119909

11199096+ 83333252119909

4+ 100119909

1

minus 83333333 le 0

1198925 () = minus119909

21199097+ 1250119909

5+ 11990921199094

minus 12501199094le 0

1198926 () = minus119909

31199098+ 1250000 + 119909

31199095

minus 25001199095le 0

(A11)

where 100 le 1199091le 10 000 1000 le 119909

119894le 10 000 (119894 = 2 3) and

100 le 119909119894le 10000 (119894 = 4 5 8) The optimum solution

is lowast = (5793066 13599707 51099707 1820177 295601217928 286165 3956012) with 119891(lowast) = 7049248021

A11 G11

Minimize 119891 () = 11990921+ (1199092minus 1)2

subject to ℎ () = 1199092minus 11990921= 0

(A12)

where minus1 le 1199091le 1 and minus1 le 119909

2le 1 The optimum solution

is lowast = (plusmn1radic2 12) with 119891(lowast) = 075

A12 G12

Maximize 119891 () = (100 minus (1199091minus 5)2minus (1199092minus 5)2

minus(1199093minus 5)2) times (100)

minus1

subject to 119892 () = (1199091minus 119901)2+ (1199092minus 119902)2+ (1199093minus 119903)2

minus 00625 le 0

(A13)

where 0 le 119909119894le 10 (119894 = 1 2 3) and 119901 119902 119903 = 1 2 9 A

point (1199091 1199092 1199093) is feasible if and only if there exist 119901 119902 119903

such that the above inequality holds The optimum solutionis lowast = (5 5 5) with 119891(lowast) = 1

B Engineering Design Problems

B1 Welded Beam Design Problem A welded beam isdesigned for the minimum cost subject to constraints onshear stress (120591) bending stress in the beam (120579) bucking load