Research Article A Biogeography-Based Optimization

Research ArticleA Biogeography-Based Optimization Algorithm Hybridized withTabu Search for the Quadratic Assignment Problem

Wee Loon Lim1 Antoni Wibowo2 Mohammad Ishak Desa3 and Habibollah Haron1

1Faculty of Computing Universiti Teknologi Malaysia (UTM) 81310 Johor Bahru Johor Malaysia2School of Quantitative Sciences UUM College of Arts and Sciences Universiti Utara Malaysia 06010 Sintok Kedah Malaysia3Faculty of Information and Communication Technology Universiti Teknikal Malaysia Melaka Hang Tuah Jaya76100 Durian Tunggal Melaka Malaysia

Correspondence should be addressed to Wee Loon Lim weeloon87gmailcom

Received 13 August 2015 Accepted 25 October 2015

Academic Editor Ezequiel Lopez-Rubio

Copyright copy 2016 Wee Loon Lim et alThis 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 quadratic assignment problem (QAP) is an NP-hard combinatorial optimization problem with a wide variety of applicationsBiogeography-based optimization (BBO) a relatively new optimization technique based on the biogeography concept usesthe idea of migration strategy of species to derive algorithm for solving optimization problems It has been shown that BBOprovides performance on a par with other optimization methods A classical BBO algorithm employs the mutation operator asits diversification strategy However this process will often ruin the quality of solutions in QAP In this paper we propose a hybridtechnique to overcome the weakness of classical BBO algorithm to solve QAP by replacing themutation operator with a tabu searchprocedure Our experiments using the benchmark instances from QAPLIB show that the proposed hybrid method is able to findgood solutions for them within reasonable computational times Out of 61 benchmark instances tested the proposed method isable to obtain the best known solutions for 57 of them

1 Introduction

The quadratic assignment problem (QAP) belongs to theclass of combinatorial optimization This problem concernsassigning a number of facilities to the same number oflocations with the objective to find a way of assignment suchthat the total cost involved is minimized The total cost ofthe QAP is the product of flow and distances between thefacilities

Sahni and Gonzalez [1] have shown that QAP is NP-hard Therefore unless P = NP it is impossible to findoptimal solutions in polynomial timeThe complexity ofQAPdraws the interest of researchers worldwide over the pastfew decades More than 300 papers have been publishedfor the theory applications and solution techniques forthe QAP [2] Despite the extensive research done QAPremains one of the most difficult combinatorial optimizationproblems Generally QAP instances with problem size gt 30cannot be solved within reasonable computational times In

the literature a lot of heuristic approaches to the QAP havebeen proposed Some of the common methods are simulatedannealing tabu search (TS) genetic algorithm (GA) and antcolony optimization [3ndash10]

Among the exact methods for QAP branch and bound(BB) algorithm is the most efficient [11 12] Currently thelargest instance solved by BB algorithm is of size 36 [4]Solving QAP instances of size 20 was impossible before1990s It was until 1994 whenMautor and Roucairol [13] gavethe exact solutions for nug16 and els19 The BB algorithmof Brixius and Anstreicher [14] solved the nug25 instanceafter 13 days of CPU time using sequential processing Thekra30a was solved by Hahn and Krarup [15] after 99 days ofwork with a sequential workstation Nystrom [16] providedthe exact solutions for the ste36b and ste36c after 200days of work using a distributed environment The nug30remained unsolved until 2002 when Anstreicher et al [17]reported the exact solution with seven days of work using650 processors Basically the exact methods require much

Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2016 Article ID 5803893 12 pageshttpdxdoiorg10115520165803893

2 Computational Intelligence and Neuroscience

more computational resources and time than the heuristicalgorithms This is the reason we did not use exact methodsin this study

TheQAP has a wide variety in the real-world applicationswith facility layout problem being the most popular appliedarea These include campus planning backboard wiringhospital planning zoning in a forest and the placementof electronic components Besides the QAP is also usedfor typewriter keyboard design development of controlboards turbine runner problem ranking of archaeology datascheduling of production lines analysis of chemical reactionsdata analysis economic problems VLSI circuit and systemdesign website structure and optimizing of Arabic keyboarddesign Comprehensive review of QAP can be referred to in[2 18]

Biogeography is a study of the geographical distributionof biological organisms The science of biogeography canbe referred to in the work of two naturalists in the nine-teenth century named Wallace [19] and Darwin [20] Inthe 1960s MacArthur and Wilson [21] started to work onmathematical models of biogeography Their study focusedon the distribution of species among neighboring habitatsand how species migrate from one habitat to another How-ever it was until 2008 when Simon [22] presented a newoptimization method based on it namely biogeography-based optimization (BBO) In his work Simon applied BBOto the problem of sensor selection for aircraft engine healthestimation and the performance of BBO is at a par with otherpopulation-based methods such as ant colony optimizationdifferential evolution GA and particle swarm optimization(PSO)The good performance of BBO on the sensor selectionproblem proves that it can be successfully applied to practicalproblems

In his next paper Simon [23] showed that BBO outper-forms GA when both of them have a low mutation rateFurthermore BBO has shown a good performance whenbeing applied to real-world optimization problems such asclassification of satellite images groundwater detection andthe solving of complex economic load dispatch [24ndash26]

The good performance of BBO on complex optimizationproblems inspired us to apply this iterative and population-based method to QAP A classical BBO algorithm usesmigration and mutation operators as its intensification anddiversification strategy respectively During the iterated pro-cess of BBO algorithm migration operator tends to improveevery solutionwithin the populationwhilemutation operatorincreases the diversity among them However quite often themutation operator will ruin the quality of solutions

In order to overcome the weakness of mutation operatorwe propose to replace it with a tabu search procedure TS isa heuristic procedure for solving the optimization problemsA typical TS starts with defining a neighborhood in whicha given solution may move to and produce a new one TSwill evaluate each of the neighboring solutions and choosethe move that improves the objective function the most Ifthere is no improving move TS will choose one that weakensthe objective function the least The main difference of TSfrom other local search techniques is that it will use memoryto store the recent moves These moves are then forbidden

and thus avoiding the search procedure to return to the localoptimum just visited

The characteristics of TS described above make it asuitable replacement for the mutation operator of classicalBBO Not only the diversity of the population is maintainedbut at the same time it prevents the quality of solutions withinthe population from being ruined We choose to incorporatethe robust tabu search (RTS) of Taillard [27] into our BBOalgorithm for QAP because it is efficient and requires fewerparameters in its implementation

In this paper we propose a BBO algorithm hybridizedwith tabu search (BBOTS) for QAP We apply it to thebenchmark instances obtained from QAPLIB [28] with sizeranging from 12 to 80 The experimental results demonstratethat the proposed algorithm outperforms the classical BBOalgorithm with mutation operator Our hybrid algorithm isable to solve most of the benchmark instances of QAP tested

2 Problem Definition

The QAP is an assignment model in which a set of facilitiesis supposed to be allocated to a set of locations with eachlocation having exactly one facility The objective is to finda way of assignment such that the total cost involved isminimizedThe total cost is calculated bymultiplying the flowand the distances between facilities

Let 119899 be the number of facilities and locations given two119899 times 119899matrices as follows

119865 = (119891119894119896) where119891

119894119896is the flow from facility 119894 to facility

119896

119863 = (119889119895119897) where 119889

119895119897is the distance from location 119895 to

location 119897

The Koopmans-Beckmann [29] form of the QAP can beformulated as follows

min120593isin119878119899

[

119899

sum

119894=1

119899

sum

119896=1

119891119894119896119889120593(119894)120593(119896)

] (1)

where 119878119899is the set of all permutations of the integers

1 2 119899 Each product of119891119894119896119889120593(119894)120593(119896)

is the cost for assigningfacility 119894 to location 120593(119894) and facility 119896 to location 120593(119896)There are several ways in mathematics to represent anassignmentThroughout this paper we use the representationof assignment by permutation

3 Methods

In this section we will first introduce BBO as general Thisincludes explaining the terminology used and its differ-ence with another popular population-based optimizationmethod the GA Next we will explain the proposed algo-rithm in detail We developed two BBO algorithms for QAP

(i) a classical BBO algorithm with mutation operator

(ii) a BBO algorithm hybridized with tabu search

Computational Intelligence and Neuroscience 3

Computational experiments were carried out to show thedifference in performance between them Note that theimplementation of both algorithms is identical except for themutation stepTherefore all the other steps of the algorithmsare explained only once

31 Biogeography-Based Optimization BBO is a new algo-rithm inspired by biogeography which studies the geograph-ical distribution of biological organisms MacArthur andWilson [21] worked together on the mathematical modelsof biogeography in the 1960s They focused primarily onthe distribution of species among neighboring habitats andhow species migrate from one habitat to another Since thenbiogeography has become a major area of research Howeverit was until 2008 when Simon [22] generalized it to obtain ageneral-purpose optimization algorithm

Just like other biology-based algorithms for example GAand PSO BBO is a population-based algorithm in which apopulation of candidate solutions is used in the searchingprocedure for global optima BBO has certain commonfeatures with another popular optimization algorithm theGA In GA an individual within the population is called achromosome and has its own fitness value Likewise in BBOeach individual is termed as a habitat and has its habitatsuitability index (HSI) to evaluate its quality as a solutionSince we are dealing with aminimization problem a low-HSIhabitat represents a good solution and a high-HSI habitat isa poor solution instead Each chromosome in GA consistsof genes while for BBO each habitat is characterized bySuitability Index Variables (SIVs) There are two main oper-ators in GA which are crossover and mutation Meanwhilein BBO the main operators are migration and mutation Themigration operator consists of emigration and immigrationIt is used to improve and evolve the habitats (solutionsto the optimization problem) in the population Solutionfeatures (SIVs) emigrate from low-HSI habitats (emigratinghabitats) to high-HSI habitats (immigrating habitats) Inother words high-HSI habitats accept new features fromlow-HSI habitats through the immigration process Thereare different alternatives for migration and mutation processof BBO The way we implement these two operators areexplained in detail in the later part Table 1 compares thecharacteristics of BBO and GA

The classical BBO algorithm proposed by Simon [22] canbe described with the algorithm in the following

A Classical BBO Algorithm

(1) Initialize the BBO parameters (this includes derivinga representation scheme for habitats which is prob-lems dependent and also initializing the maximummigration rate maximum mutation rate and elitismparameter)

(2) Initialize a random set of habitats corresponding tothe potential solutions

(3) Associate each habitat with immigration and emigra-tion rate based on their HSI

Table 1 Comparison of characteristics for BBO and GA

BBO GAPopulation-based Population-basedHabitat ChromosomeSIV GeneHSI FitnessMigration operator Crossover operatorMutation operator Mutation operator

(4) Probabilistically perform migration to modify eachone-elite habitat Then recompute each HSI

(5) Associate each habitat with mutation rate based ontheir species count

(6) Probabilistically perform mutation to modify eachnonelite habitat Then recompute each HSI

(7) Go to step (3) for the next iteration Repeat untila predefined number of generations are reached orafter an acceptable solution is found

32 The Proposed Algorithm

321 Representing Scheme of Habitats As mention earlier ahabitat is represented by a permutation of integer 1 2 119899in which 119899 is the number of facilities and locations that is theproblem size For example (2 5 1 4 3) is a possible habitatfor an instance of QAPwith problem size of 5 In this case ldquo2rdquoof (2 5 1 4 3) means facility 1 is placed at location 2 and it isa SIV of this habitat Likewise ldquo5rdquo means facility 2 is placedat location 5 and so on

322 Initialization of the BBO Algorithm The BBO algo-rithm starts with a population of randomly generated habi-tats In our algorithm a habitat is represented by a permuta-tion of integers A permutation is randomly generated andwill only be inserted if it does not exist in the populationyet This is to avoid having duplicate habitats in the initialpopulation and hence enhancing the diversity

323 Selection Strategy forMigration In BBO a good habitat(solution) is one with low HSI Good habitats tend to sharetheir features with poor habitat This is done by migratingSIVs from emigrating habitats to immigrating habitats Inorder to perform migration we will first use immigrationrates (120582

119896) of a habitat to decide whether to modify it then we

use emigration rates (120583119896) of other habitats to decide which of

them should migrate a SIV to the first habitatAccording to the biogeography the SIVs of a good habitat

(with low HSI) tend to emigrate to a poor habitat (with highHSI)Therefore a good habitat has relatively high 120583

119896and low

120582119896 while a poor solution has relatively low 120583

119896and high 120582

119896


(1) for 119894 = 1 to ℎ do(2) if rand isin [0 1] lt 120582

119894then ⊳ solution119867

119894is selected



119895is selected

(5) randomly select a SIV 120590 from119867119895

(6) replace a random SIV in119867119894with 120590

Algorithm 1 Selection strategy of migration operator

E = I

Rate

Immigration120582

Emigration120583

S1 S2Smax

Number of species

Figure 1 Model for immigration and emigration rates

The immigration rate 120582119896and the emigration rates 120583

119896are

calculated with the following equations respectively

120582119896= 119868(1 minus

119896

ℎ

) (2)

120583119896=

119864119896

ℎ

(3)

In the equations 119896 represents the rank of a habitat aftersorting them in accordance to their HSI Habitats with highHSI (a poor solution) have lower rank while habitats with lowHSI will have higher rank In other words the habitats aresorted from the worst to the best In the equations ℎ is thenumber of habitats in the population while 119868 is themaximumimmigration rate and 119864 the maximum emigration rate whichare both usually set to 1

Figure 1 illustrates the model for immigration and emi-gration rates [22] In the figure 119878

1is a relatively poor solution

while 1198782is a relatively good solution Although in the figure

the immigration and emigration rate are considered linearthey can actually be replaced with curves in which betterperformancemight be attained In fact Mussetta and Pirinoli

[30] have shown that a variation of BBO with quadraticmigration model and restart procedure outperforms theclassical BBO when being applied to several benchmarkfunctions The selection strategy is summarized with thealgorithm in Algorithm 1

324 Selection Strategy for Mutation In BBO each habitathas an associated probability for them to exist as a solutionto the given problem The probability that whether mutationoccurs in a habitat is called the mutation rate To determinethe mutation rate for each habitat we must first evaluate thespecies count probability with the following equation

P = ksum

ℎ+1

119894=1V119894

(4)

in which V and V119894are evaluated by

k = [V1 V2sdot sdot sdot Vℎ+1]119879

V119894

=

ℎ

(ℎ + 1 minus 119894) (119894 minus 1)

(119894 = 1 2 119894

1015840)

Vℎ+2minus119894

(119894 = 119894

1015840+ 1 119894

1015840+ 2 ℎ + 1)

(5)

where 1198941015840 = ceil((119899+1)2) Themutation rate119898(119878) is inverselyproportional to the species count probability therefore wehave

119898(119878) = 119898max (1 minus 119875119904

119875max) (6)

where 119898max is the maximum mutation rate and 119875max thelargest species count probabilityThe details on how to derivethe above formulae can be referred to in [22]

325 Migration Operator As mentioned earlier the SIVsfrom a good habitat tend to migrate into a poor habitat Thismigration operator is performed probabilistically based onimmigration and emigration rates In this section we willexplain how the migration is implemented in our BBOTSalgorithm

Consider dealing with an instance of QAP with problemsize of 5 Suppose based on immigration and emigrationrates that an emigrating habitat 119867

119890= (2 4 3 5 1) and


Emigrating habitat He

Immigrating habitat Hi

New habitat Hn

1

1

2

2

3

3

4

4

5

1 23 45

5

25

(1) SIV ldquo2rdquo from He migrates into Hi and replaces ldquo5rdquo(2) SIV ldquo5rdquo then takes the place of SIV ldquo2rdquo(3) SIVs ldquo3rdquo ldquo1rdquo and ldquo4rdquo remain at original locations

Figure 2 Migration operator of BBOTS algorithm

an immigrating habitat 119867119894= (5 3 1 4 2) are selected

Therefore a SIV of119867119890will be randomly selected and replaces

a randomly selected SIV of119867119894

Assuming the first element of119867119890 ldquo2rdquo is selected to replace

the first SIV of 119867119894 ldquo5rdquo Therefore the new habitat 119867

119899=

(2 3 1 4 2) is produced However it is not a feasible solutionfor our problem In order to maintain the feasibility of asolution once a new SIV migrates into a habitat the old SIVwill replace the SIV with the same value of newly emigratingSIV Therefore the new habitat 119867

119899 will be (2 3 1 4 5)

Figure 2 illustrates this step clearlyOnce a new habitat is produced it will only be accepted

into the population only if it is not the same with any existinghabitat This is to enhance the diversity throughout thepopulation Besides we use the concept of elitism to preventthe best solutions from being corrupted by immigrationThisis done by setting the immigration rate for best solutions to 0

326 Mutation Operator According to biogeography cata-clysmic events may happen from time to time drasticallychanging the characteristics of a natural habitat In BBOalgorithm this is imitated through a mutation operatorThis process is important to increase the diversity amongpopulation

In classical BBO algorithm a mutation is performed bysimply replacing a selected SIV of a habitat with a randomlygenerated SIV Remember in BBOmigration operator servesas intensification strategy while mutation operator is usedto maintain the diversity Habitats within the population areimproved by migration operator throughout the iteration ofBBO algorithm However this effort might be ruined by themutation operator because the quality of themutated habitats

is not guaranteed Quite often the mutation process willresult in a poor habitat

A simple solution for this drawback comes in mind suchthat after performing the mutation the mutated habitats arekept in the population only if the quality is better than theoriginal habitats However this is not practical when solvinga complex optimization problem such as QAP Most of thetime the resulting habitats from a simple mutation operatorare unlikely to be better than the original habitats especiallyas the algorithm converges

To overcome this weakness of classical BBO algorithmwe propose to replace the mutation operator with a tabusearch procedure Proposed by Glover [31] TS is a meta-heuristic which performs local search based on the infor-mation in the memory TS is both neighborhood-based anditerative At each iteration current solution will make a moveto the neighborhood solutionwith the best objective functionvalue To avoid trapping in local optima the move that hasbeen made will be stored in a tabu list and a reverse moveto previous solutions is forbidden The performance of tabusearch highly depends on the neighborhood type used andtabu list implementation

The advantages of replacing the mutation operator ofclassical BBOalgorithmwithTS come in two points First theoriginal aim of mutation process is maintained which is toincrease the diversity of the population Besides at the sametime the quality of the resulting habitats is prevented frombeing ruined

In order to prove the above statement we developed twoBBO algorithms for QAP



For the classical BBO algorithm the SIV of a habitat willbe chosen to undergo mutation based on the mutation rateThe mutation process is then performed by replacing the oldSIV with another randomly generated SIV After that SIV ofthe habitat which is of the same value with the randomlygenerated SIV will inherit the original value of the mutatedSIVThus the resulting habitat will remain a feasible solutionFigure 3 illustrates the mutation process of the classical BBOalgorithm for QAP

For the proposed BBOTS algorithm we chose to replacethe mutation operator with robust tabu search (RTS) ofTaillard [27] Despite being developed long time ago (1991)RTS is still one of the best performing algorithms for QAP

The tabu list of RTS consists of pairs of facilities thatcannot be exchanged In the tabu list the latest iteration atwhich a pair of facilities is placed at certain locations is storedA swap of a pair of facilities is taboo for a number of iterationsif they were swapped in the last iteration However a taboomove will be allowed if the new solution has a better objectivefunction value than the current best solutionThe tabu tenure


12 34 5

12 34 4

12 3 55

Habitat

Mutated habitat

(1) SIV ldquo4rsquorsquo is chosen to mutate(2) Assume that the new SIV which is randomly

generated is ldquo5rsquorsquo SIV ldquo4rsquorsquo is replaced with ldquo5rsquorsquo(3) The original SIV ldquo5rsquorsquo takes the original value of

mutated SIV ldquo4rsquorsquo(4) The resulting mutated habitat is thus feasible

Figure 3Themutation process of classical BBO algorithm for QAP

of RTS changes between 09119899 and 11119899 dynamically during theprocedure

The advantages of RTS as compared to other adaptions oftabu search for QAP are that RTS is both efficient and robustThe robustness of RTS means that it requires less complexityand fewer parameters in its implementation Therefore as weincorporate RTS into our BBOTS algorithm we do not haveto alter the parameter values for every benchmark instance ofQAP tested

In BBOTS algorithm the mutation rate is now theprobability that a habitat in the population will undergo RTSprocedure The number of iterations of RTS is set to 119899 whichis the size of a problem As compared to the original RTS thenumber of iterations we use is relatively much lower This isbecause the use of RTS in our BBOTS algorithm is limitedto a quick search only The resulting habitat after the RTSprocedure will replace the original habitat if and only if thepopulation does not contain a same habitat yet

327 BBOTS Algorithm for QAP The BBOTS algorithm forQAP is shown in Algorithm 2

4 Results and Discussion

Because of the nature of heuristic algorithms in order toevaluate the performance of a new algorithm it is a commonpractice to test it with the benchmark instances of theproblem and compare its results with other existingmethodsFor QAP there are benchmark instances of various sizesavailable from the QAPLIBThese instances are used in most

Table 2 Parameter setting of classical BBO and BBOTS algorithmsfor QAP

Parameter ValuePopulation size 100Number of iterations 300Maximum immigration rate 1Maximum emigration rate 1Maximum mutation rate 01Number of elites 2

of the literature The benchmark instances of QAP can beclassified into four types

(i) Type I real-life instances obtained from practicalapplications of QAP

(ii) Type II unstructured randomly generated instancesforwhich the distance andflowmatrices are randomlygenerated based on a uniform distribution

(iii) Type III randomly generated instances with structurethat is similar to that of real-life instances

(iv) Type IV instances in which distances are based on theManhattan distance on a grid

The computational results are reported through 3 stagesFirst we compare the results of classical BBO algorithm withmutation operator and the proposed BBOTS by applyingthem to selected benchmark instances from QAPLIB Afterthat we further the experiments of BBOTS algorithm bytesting with even more benchmark instances of QAP Lastlywe compare the BBOTS algorithmwith other state-of-the-artmethods

Both the classical BBO algorithmwith mutation operatorand BBOTS for QAP are programmed in MATLAB runningon the same machine with an Intel Core i3-2120 processorat 33 GHz In order to fairly compare their performancethe parameter setting for both of the algorithms is the samethroughout all the instances tested The values of parametersused are indicated in Table 2

We compare the performance of classical BBO andBBOTS algorithm for QAP by applying them on 37 bench-mark instances The computational results are shown inTable 3 The performance of the algorithms is evaluated withthe following criteria

(i) the best solution found over 10 runs

(ii) the average deviation from the optimal or best knownsolution 120575 = 100(119911 minus 119911)119911 () where 119911 is the averageobjective function value and 119911 is the best knownsolution value


(1) Parameter setting Popsize = 100 NumIteration = 300 119864 = 1 119868 = 1119898max = 01 elite = 2(2) Initialization(3) repeat(4) generate a habitat(5) if habitat does not exist yet then(6) insert it into population(7) else(8) discard it(9) until population size is reached(10) calculate immigration emigration and mutation rates(11) compute HSI of each habitat(12) associate each habitat with immigration and emigration rate(13) for 119905 = 1 to NumIteration do(14) Migration(15) for 119894 = 1 to Popsize do(16) generate Rand isin [0 1](17) if Rand lt 120582

119894then

(18) for 119895 = 1 to Popsize do(19) generate Rand isin [0 1](20) if Rand lt 120583

119894then

(21) perform migration operator(22) if new habitat does not exist in the population then(23) new habitat replaces old habitat(24) else(25) keeps the old habitat(26) recompute HSI for each habitat(27) associate each habitat with mutation rate(28) Mutation(29) for 119894 = 1 to Popsize do(30) generate Rand isin [0 1](31) if Rand lt 119898

119894then

(32) perform tabu search(33) if new habitat does not exist in the population then(34) new habitat replace the old habitat(35) else(36) keeps the old habitat(37) recompute HSI of each habitat(38) associate each habitat with immigration and emigration rate(39) if best known solution is found or NumIteration is reached then(40) terminate the algorithm

Algorithm 2 BBOTS algorithm for QAP

(iii) the number of times () in which the optimal or bestknown solution is reached

The benchmark instances used are of different typesRegardless of the types of instances tested BBOTS algorithmclearly performs much better than the classical BBO withmutation operator BBOTS algorithm is able to find theoptimal or best known solutions for 36 instances out of a totalof 37 while the classical BBO algorithm only manages to get2 The average deviations from the best known solutions ofBBOTS are also much lower for all the benchmark instancestested

We further the experiments of our BBOTS algorithm byapplying it to more benchmark instances of QAP Table 4shows the results obtained by BBOTS algorithm for 61benchmark instances with size ranging from 12 to 80 Theparameter values used are the same as those used previouslyThe BBOTS algorithm is terminated at 300 iterations or whenthe best known solutions are found whichever comes first

Computational results show that the proposed BBOTSalgorithm is able to find optimal or best known solutionsfor 57 QAP benchmark instances out of a total of 61 Mostof the times BBOTS is able to solve the instances tested onevery single runThe average deviations from the best known


Table 3 Comparative results between classical BBO and BBOTS algorithms for QAP

Instance Best known solution BBO BBOTSBest solution 120575 Best solution 120575

chr12a 9552 9988 33654 0 9552 0000 10chr12b 9742 9942 28337 0 9742 0000 10chr12c 11156 12336 32854 0 11156 0000 10chr15a 9896 14496 97409 0 9896 0000 10chr15b 7990 15298 136683 0 7990 0298 9chr15c 9504 18392 112822 0 9504 0000 10chr18a 11098 29870 190483 0 11098 0079 8chr18b 1534 2170 51356 0 1534 0000 10els19 17212548 21315378 34293 0 17212548 0000 10esc16a 68 70 8235 0 68 0000 10esc16b 292 292 0000 10 292 0000 10esc16c 160 164 5875 0 160 0000 10esc16d 16 18 22500 0 16 0000 10had12 1652 1662 1877 0 1652 0000 10had14 2724 2762 2349 0 2724 0000 10had16 3720 3820 3468 0 3720 0000 10had18 5358 5500 3763 0 5358 0000 10had20 6922 7156 4143 0 6922 0000 10nug12 578 590 6574 0 578 0000 10nug14 1014 1086 9310 0 1014 0000 10nug15 1150 1250 11078 0 1150 0000 10nug16a 1610 1762 11503 0 1610 0000 10nug16b 1240 1374 12694 0 1240 0000 10nug17 1732 1914 12125 0 1732 0012 9nug18 1930 2130 12114 0 1930 0000 10nug20 2570 2868 13416 0 2570 0000 10rou12 235528 247850 7057 0 235528 0000 10rou15 354210 389802 12105 0 354210 0000 10rou20 725522 810284 12490 0 725522 0062 4scr12 31410 31410 8243 1 31410 0000 10scr15 51140 58958 22004 0 51140 0000 10scr20 110030 146230 37559 0 110030 0000 10tai10a 135028 137362 2974 0 135028 0000 10tai12a 224416 230704 8210 0 224416 0000 10tai15a 388214 404108 8843 0 388214 0000 10tai17a 491812 540308 11651 0 491812 0093 8tai20a 703482 789348 13658 0 705622 0677 0

solutions over 10 runs are at most 2788 Even on largeinstance like tai80a the average deviation is less than 3

For indicative purpose we compare the results of ourBBOTS algorithm with 4 state-of-the-art QAP approaches inthe literature

(i) Iterated Tabu Search (ITS) algorithm by Misevicius(2012) [6] the reported results are obtained using anIntel Pentium 900MHz single-core processor

(ii) hybrid metaheuristics combining greedy randomizedadaptive search procedure and simulated anneal-ing and tabu search (SA-TS) by Gunawan et al(2014) [32] the reported results are obtained using a267GHz Intel Core 2 Duo CPU

(iii) information combination based evolutionary algo-rithm (ICEA) by Sun et al (2014) [33]


Table 4 Computational results of BBOTS algorithm on benchmarkinstances of QAP

Instance Best known solution Best solution 120575 bur26a 5426670 5426670 0028 5bur26b 3817852 3817852 0000 10bur26c 5426795 5426795 0000 10bur26d 3821225 3821225 0000 10bur26e 5386879 5386879 0000 10bur26f 3782044 3782044 0000 10bur26g 10117172 10117172 0000 10bur26h 7098658 7098658 0000 10chr12a 9552 9552 0000 10chr12b 9742 9742 0000 10chr12c 11156 11156 0000 10chr15a 9896 9896 0000 10chr15b 7990 7990 0298 9chr15c 9504 9504 0000 10chr18a 11098 11098 0079 8chr18b 1534 1534 0000 10chr20a 2192 2192 0876 3chr20c 14142 14142 0604 9els19 17212548 17212548 0000 10esc16a 68 68 0000 10esc16b 292 292 0000 10esc16c 160 160 0000 10esc16d 16 16 0000 10had12 1652 1652 0000 10had14 2724 2724 0000 10had16 3720 3720 0000 10had18 5358 5358 0000 10had20 6922 6922 0000 10kra30a 88900 88900 0090 9kra30b 91420 91420 0060 6kra32 88700 88700 0311 7nug12 578 578 0000 10nug14 1014 1014 0000 10nug15 1150 1150 0000 10nug16a 1610 1610 0000 10nug16b 1240 1240 0000 10nug17 1732 1732 0012 9nug18 1930 1930 0000 10nug20 2570 2570 0000 10nug21 2438 2438 0000 10nug22 3596 3596 0000 10nug24 3488 3488 0000 10nug25 3744 3744 0000 10nug27 5234 5234 0000 10nug28 5166 5166 0209 4nug30 6124 6124 0065 2rou12 235528 235528 0000 10

Table 4 Continued

Instance Best known solution Best solution 120575 rou15 354210 354210 0000 10rou20 725522 725522 0062 4scr12 31410 31410 0000 10scr15 51140 51140 0000 10scr20 110030 110030 0000 10sko42 15812 15812 0028 9tai10a 135028 135028 0000 10tai12a 224416 224416 0000 10tai15a 388214 388214 0000 10tai17a 491812 491812 0093 8tai20a 703482 705622 0677 0tai30a 1818146 1843224 1795 0tai80a 13499184 13841214 2788 0wil50 48816 48848 0117 0

(iv) genetic algorithmwith a new recombination operator(GAR) by Tosun (2014) [8] the reported results areobtained using a 221 GHz AMD Athlon (TM) 64 times 2dual processor

Since there are differences in computing hardware ter-mination criterion and result reporting method comparingthe results of each algorithm is not a straightforward taskTherefore the comparison in Table 5 should be interpretedcautiously

The average deviations from the optimal or best knownsolutions are over 10 runs except for SA-TS and ICEA whichwere executed for 20 and 30 times respectively The numberof times in which the optimal or best known solutions arereached is indicated in the parentheses if known Shouldthe particular authors not report their results on certainbenchmark instances the cells are left empty

5 Conclusion

In this paper we presented a biogeography-based optimiza-tion algorithm hybridized with tabu search for QAP Aclassical BBO algorithm uses the mutation operator as itsdiversification strategyThis step often destroys the quality ofhabitats within the population In the proposed BBOTS algo-rithm we replaced the mutation operator with robust tabusearch The diversity among the population is maintainedand at the same time the quality of habitats is prevented frombeing ruined

The comparative results showed that BBOTS algorithmoutperforms the classical BBO with mutation operator whentested with benchmark instances of QAP The proposedBBOTS algorithm is able to obtain the optimal or best knownsolutions for many of the benchmark instances drawn fromthe QAPLIB In particular out of a total of 61 benchmarkinstances evaluated BBOTS is able to reach the current bestknown solutions for 57 of them


Table 5 Comparative results between BBOTS algorithm and state-of-the-art QAP approaches

Instance Best known solution BBOTS ITS SA-TS ICEA GARbur26a 5426670 0028 (5) 0000 092bur26b 3817852 0000 (10) 065bur26c 5426795 0000 (10) 131bur26d 3821225 0000 (10) 056bur26e 5386879 0000 (10) 108bur26f 3782044 0000 (10) 056bur26g 10117172 0000 (10) 074bur26h 7098658 0000 (10)chr12a 9552 0000 (10) 000chr12b 9742 0000 (10) 000chr12c 11156 0000 (10) 000chr15a 9896 0000 (10) 000chr15b 7990 0298 (9) 000chr15c 9504 0000 (10) 000chr18a 11098 0079 (8) 000chr18b 1534 0000 (10) 000chr20a 2192 0876 (3) 150chr20c 14142 0604 (9) 000els19 17212548 0000 (10) 000 (10)esc16a 68 0000 (10) 000esc16b 292 0000 (10) 000esc16c 160 0000 (10) 000esc16d 16 0000 (10) 000had12 1652 0000 (10) 000 000had14 2724 0000 (10) 040 007had16 3720 0000 (10) 003 038had18 5358 0000 (10) 000 056had20 6922 0000 (10) 008 139kra30a 88900 0090 (9) 001 (8) 074 0000kra30b 91420 0060 (6) 000 (10) 000 0000kra32 88700 0311 (7) 000nug12 578 0000 (10) 000nug14 1014 0000 (10) 000nug15 1150 0000 (10) 000nug16a 1610 0000 (10)nug16b 1240 0000 (10)nug17 1732 0012 (9)nug18 1930 0000 (10) 497nug20 2570 0000 (10) 000nug21 2438 0000 (10) 000nug22 3596 0000 (10) 000nug24 3488 0000 (10) 000nug25 3744 0000 (10) 000nug27 5234 0000 (10) 000nug28 5166 0209 (4) 002nug30 6124 0065 (2) 000 (10) 001 0000rou12 235528 0000 (10) 000rou15 354210 0000 (10) 000rou20 725522 0062 (4) 003scr12 31410 0000 (10) 000


Table 5 Continued

Instance Best known solution BBOTS ITS SA-TS ICEA GARscr15 51140 0000 (10) 000scr20 110030 0000 (10) 000sko42 15812 0028 (9) 000 (10) 014 0000tai10a 135028 0000 (10) 000tai12a 224416 0000 (10) 000tai15a 388214 0000 (10) 000tai17a 491812 0093 (8) 000tai20a 703482 0677 (0) 000 (10) 016 0168tai30a 1818146 1795 (0) 000 (10) 151 0276tai80a 13499184 2788 (0) 036 (1) 357 0998 1094wil50 48816 0117 (0) 006 (4) 011 718

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research is funded by Universiti Teknologi Malaysiaunder Fundamental ResearchGrant Scheme (Vote Project noRJ13000078284F497)

References

[1] S Sahni and T Gonzalez ldquoP-complete approximation prob-lemsrdquo Journal of the Association for Computing Machinery vol23 no 3 pp 555ndash565 1976

[2] E M Loiola N M M de Abreu P O Boaventura-Netto PHahn and T Querido ldquoA survey for the quadratic assignmentproblemrdquoEuropean Journal ofOperational Research vol 176 no2 pp 657ndash690 2007

[3] J-C Wang ldquoA multistart simulated annealing algorithm forthe quadratic assignment problemrdquo in Proceedings of the 3rdIEEE International Conference on Innovations in Bio-InspiredComputing and Applications (IBICA rsquo12) pp 19ndash23 KaohsiungTaiwan September 2012

[4] U Tosun ldquoOn the performance of parallel hybrid algorithms forthe solution of the quadratic assignment problemrdquo EngineeringApplications of Artificial Intelligence vol 39 pp 267ndash278 2015

[5] M Czapinski ldquoAn effective parallel multistart tabu search forquadratic assignment problem on CUDA platformrdquo Journal ofParallel andDistributedComputing vol 73 no 11 pp 1461ndash14682013

[6] A Misevicius ldquoAn implementation of the iterated tabu searchalgorithm for the quadratic assignment problemrdquoOR Spectrumvol 34 no 3 pp 665ndash690 2012

[7] U Tosun T Dokeroglu and A Cosar ldquoA robust island parallelgenetic algorithm for the quadratic assignment problemrdquo Inter-national Journal of Production Research vol 51 no 14 pp 4117ndash4133 2013

[8] U Tosun ldquoA new recombination operator for the genetic algo-rithm solution of the quadratic assignment problemrdquo ProcediaComputer Science vol 32 pp 29ndash36 2014

[9] M Lstiburek J Stejskal A Misevicius J Korecky and Y A El-Kassaby ldquoExpansion of the minimum-inbreeding seed orchard

design to operational scalerdquo Tree Genetics and Genomes vol 11no 1 pp 1ndash8 2015

[10] A S Ramkumar S G Ponnambalam and N Jawahar ldquoApopulation-based hybrid ant system for quadratic assignmentformulations in facility layout designrdquo International Journal ofAdvanced Manufacturing Technology vol 44 no 5-6 pp 548ndash558 2009

[11] R E Burkard E Cela P M Pardalos and L Pitsoulis ldquoThequadratic assignment problemrdquo in Handbook of CombinatorialOptimization P M Pardalos and D Z Du Eds pp 241ndash338Kluwer Academic Publishers 1998

[12] P C Gilmore ldquoOptimal and suboptimal algorithms for thequadratic assignment problemrdquo Journal of the Society for Indus-trial and Applied Mathematics vol 10 no 2 pp 305ndash313 1962

[13] T Mautor and C Roucairol ldquoA new exact algorithm for thesolution of quadratic assignment problemsrdquo Discrete AppliedMathematics vol 55 no 3 pp 281ndash293 1994

[14] N W Brixius and K M Anstreicher ldquoSolving quadraticassignment problems using convex quadratic programmingrelaxationsrdquo Optimization Methods and Software vol 16 no 1ndash4 pp 49ndash68 2001

[15] P M Hahn and J Krarup ldquoA hospital facility layout problemfinally solvedrdquo Journal of Intelligent Manufacturing vol 12 no5 pp 487ndash496 2001

[16] M Nystrom Solving Certain Large Instances of the QuadraticAssignment Problem Steinbergrsquos Examples California Instituteof Technology Pasadena Calif USA 1999

[17] K Anstreicher N Brixius J-P Goux and J Linderoth ldquoSolvinglarge quadratic assignment problems on computational gridsrdquoMathematical Programming vol 91 no 3 pp 563ndash588 2002

[18] Z Drezner P M Hahn and E D Taillard ldquoRecent advancesfor the quadratic assignment problem with special emphasis oninstances that are difficult for meta-heuristic methodsrdquo Annalsof Operations Research vol 139 pp 65ndash94 2005

[19] A Wallace The Geographical Distribution of Animals (TwoVolumes) Adamant Media Corporation Boston Mass USA2005

[20] C Darwin On the Origin of Species Gramercy New York NYUSA 1995

[21] R MacArthur and E Wilson The Theory of BiogeographyPrinceton University Press Princeton NJ USA 1967

[22] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008


[23] D Simon ldquoAprobabilistic analysis of a simplified biogeography-based optimization algorithmrdquo Evolutionary Computation vol19 no 2 pp 167ndash188 2011

[24] V Panchal P Singh N Kaur and H Kundra ldquoBiogeographybased satellite image classificationrdquo International Journal ofInformation Security vol 6 no 2 pp 269ndash274 2009

[25] V Panchal H Kundra and A Kaur ldquoAn integrated approachto biogeography based optimization with case based reasoningfor retrieving groundwater possibilityrdquo International Journal ofComputer Applications vol 1 no 8 pp 975ndash8887 2010

[26] A Bhattacharya and P K Chattopadhyay ldquoSolving complexeconomic load dispatch problems using biogeography-basedoptimizationrdquo Expert Systems with Applications vol 37 no 5pp 3605ndash3615 2010

[27] E Taillard ldquoRobust taboo search for the quadratic assignmentproblemrdquo Parallel Computing Theory and Applications vol 17no 4-5 pp 443ndash455 1991

[28] R E Burkard E Cela S E Karisch and F Rendl QAPLIBHome Page 2004 httpanjosmgipolymtlcaqaplib

[29] T C Koopmans andM Beckmann ldquoAssignment problems andthe location of economic activitiesrdquo Econometrica vol 25 no 1pp 53ndash76 1957

[30] M Mussetta and P Pirinoli ldquoMmCn-BBO schemes for elec-tromagnetic problem optimizationrdquo in Proceedings of the 7thEuropeanConference onAntennas and Propagation (EuCAP rsquo13)pp 1058ndash1059 Gothenburg Sweden April 2013

[31] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[32] A Gunawan K M Ng K L Poh and H C Lau ldquoHybridmetaheuristics for solving the quadratic assignment problemand the generalized quadratic assignment problemrdquo in Proceed-ings of the IEEE International Conference on Automation Scienceand Engineering (CASE rsquo14) pp 119ndash124 Taipei Taiwan August2014

[33] J Sun Q Zhang and X Yao ldquoMeta-heuristic combining prioronline and offline information for the quadratic assignmentproblemrdquo IEEE Transactions on Cybernetics vol 44 no 3 pp429ndash444 2014

Submit your manuscripts athttpwwwhindawicom

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more computational resources and time than the heuristicalgorithms This is the reason we did not use exact methodsin this study

TheQAP has a wide variety in the real-world applicationswith facility layout problem being the most popular appliedarea These include campus planning backboard wiringhospital planning zoning in a forest and the placementof electronic components Besides the QAP is also usedfor typewriter keyboard design development of controlboards turbine runner problem ranking of archaeology datascheduling of production lines analysis of chemical reactionsdata analysis economic problems VLSI circuit and systemdesign website structure and optimizing of Arabic keyboarddesign Comprehensive review of QAP can be referred to in[2 18]

Biogeography is a study of the geographical distributionof biological organisms The science of biogeography canbe referred to in the work of two naturalists in the nine-teenth century named Wallace [19] and Darwin [20] Inthe 1960s MacArthur and Wilson [21] started to work onmathematical models of biogeography Their study focusedon the distribution of species among neighboring habitatsand how species migrate from one habitat to another How-ever it was until 2008 when Simon [22] presented a newoptimization method based on it namely biogeography-based optimization (BBO) In his work Simon applied BBOto the problem of sensor selection for aircraft engine healthestimation and the performance of BBO is at a par with otherpopulation-based methods such as ant colony optimizationdifferential evolution GA and particle swarm optimization(PSO)The good performance of BBO on the sensor selectionproblem proves that it can be successfully applied to practicalproblems

In his next paper Simon [23] showed that BBO outper-forms GA when both of them have a low mutation rateFurthermore BBO has shown a good performance whenbeing applied to real-world optimization problems such asclassification of satellite images groundwater detection andthe solving of complex economic load dispatch [24ndash26]

The good performance of BBO on complex optimizationproblems inspired us to apply this iterative and population-based method to QAP A classical BBO algorithm usesmigration and mutation operators as its intensification anddiversification strategy respectively During the iterated pro-cess of BBO algorithm migration operator tends to improveevery solutionwithin the populationwhilemutation operatorincreases the diversity among them However quite often themutation operator will ruin the quality of solutions

In order to overcome the weakness of mutation operatorwe propose to replace it with a tabu search procedure TS isa heuristic procedure for solving the optimization problemsA typical TS starts with defining a neighborhood in whicha given solution may move to and produce a new one TSwill evaluate each of the neighboring solutions and choosethe move that improves the objective function the most Ifthere is no improving move TS will choose one that weakensthe objective function the least The main difference of TSfrom other local search techniques is that it will use memoryto store the recent moves These moves are then forbidden

and thus avoiding the search procedure to return to the localoptimum just visited

The characteristics of TS described above make it asuitable replacement for the mutation operator of classicalBBO Not only the diversity of the population is maintainedbut at the same time it prevents the quality of solutions withinthe population from being ruined We choose to incorporatethe robust tabu search (RTS) of Taillard [27] into our BBOalgorithm for QAP because it is efficient and requires fewerparameters in its implementation

In this paper we propose a BBO algorithm hybridizedwith tabu search (BBOTS) for QAP We apply it to thebenchmark instances obtained from QAPLIB [28] with sizeranging from 12 to 80 The experimental results demonstratethat the proposed algorithm outperforms the classical BBOalgorithm with mutation operator Our hybrid algorithm isable to solve most of the benchmark instances of QAP tested

2 Problem Definition

The QAP is an assignment model in which a set of facilitiesis supposed to be allocated to a set of locations with eachlocation having exactly one facility The objective is to finda way of assignment such that the total cost involved isminimizedThe total cost is calculated bymultiplying the flowand the distances between facilities

Let 119899 be the number of facilities and locations given two119899 times 119899matrices as follows

119865 = (119891119894119896) where119891

119894119896is the flow from facility 119894 to facility

119896

119863 = (119889119895119897) where 119889

119895119897is the distance from location 119895 to

location 119897

The Koopmans-Beckmann [29] form of the QAP can beformulated as follows

min120593isin119878119899

[

119899

sum

119894=1

119899

sum

119896=1

119891119894119896119889120593(119894)120593(119896)

] (1)

where 119878119899is the set of all permutations of the integers

1 2 119899 Each product of119891119894119896119889120593(119894)120593(119896)

is the cost for assigningfacility 119894 to location 120593(119894) and facility 119896 to location 120593(119896)There are several ways in mathematics to represent anassignmentThroughout this paper we use the representationof assignment by permutation

3 Methods

In this section we will first introduce BBO as general Thisincludes explaining the terminology used and its differ-ence with another popular population-based optimizationmethod the GA Next we will explain the proposed algo-rithm in detail We developed two BBO algorithms for QAP


























119896and low


119896and high 120582

119896




119894is selected



119895is selected




E = I

Rate

Immigration120582

Emigration120583

S1 S2Smax

Number of species



119896are


120582119896= 119868(1 minus

119896

ℎ

) (2)

120583119896=

119864119896

ℎ

(3)








P = ksum

ℎ+1

119894=1V119894

(4)



V119894

=

ℎ

(ℎ + 1 minus 119894) (119894 minus 1)

(119894 = 1 2 119894

1015840)

Vℎ+2minus119894

(119894 = 119894

1015840+ 1 119894

1015840+ 2 ℎ + 1)

(5)


119898(119878) = 119898max (1 minus 119875119904

119875max) (6)




119890= (2 4 3 5 1) and




New habitat Hn

1

1

2

2

3

3

4

4

5

1 23 45

5

25








119899=


119899 will be (2 3 1 4 5)
















12 34 5

12 34 4

12 3 55

Habitat

Mutated habitat

























119894then


119894then


119894then



















Table 4 Continued





5 Conclusion







Table 5 Continued




Acknowledgment


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in


























119896and low


119896and high 120582

119896




119894is selected



119895is selected




E = I

Rate

Immigration120582

Emigration120583

S1 S2Smax

Number of species



119896are


120582119896= 119868(1 minus

119896

ℎ

) (2)

120583119896=

119864119896

ℎ

(3)








P = ksum

ℎ+1

119894=1V119894

(4)



V119894

=

ℎ

(ℎ + 1 minus 119894) (119894 minus 1)

(119894 = 1 2 119894

1015840)

Vℎ+2minus119894

(119894 = 119894

1015840+ 1 119894

1015840+ 2 ℎ + 1)

(5)


119898(119878) = 119898max (1 minus 119875119904

119875max) (6)




119890= (2 4 3 5 1) and




New habitat Hn

1

1

2

2

3

3

4

4

5

1 23 45

5

25








119899=


119899 will be (2 3 1 4 5)
















12 34 5

12 34 4

12 3 55

Habitat

Mutated habitat

























119894then


119894then


119894then



















Table 4 Continued





5 Conclusion







Table 5 Continued




Acknowledgment


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






119894is selected



119895is selected




E = I

Rate

Immigration120582

Emigration120583

S1 S2Smax

Number of species



119896are


120582119896= 119868(1 minus

119896

ℎ

) (2)

120583119896=

119864119896

ℎ

(3)








P = ksum

ℎ+1

119894=1V119894

(4)



V119894

=

ℎ

(ℎ + 1 minus 119894) (119894 minus 1)

(119894 = 1 2 119894

1015840)

Vℎ+2minus119894

(119894 = 119894

1015840+ 1 119894

1015840+ 2 ℎ + 1)

(5)


119898(119878) = 119898max (1 minus 119875119904

119875max) (6)




119890= (2 4 3 5 1) and




New habitat Hn

1

1

2

2

3

3

4

4

5

1 23 45

5

25








119899=


119899 will be (2 3 1 4 5)
















12 34 5

12 34 4

12 3 55

Habitat

Mutated habitat

























119894then


119894then


119894then



















Table 4 Continued





5 Conclusion







Table 5 Continued




Acknowledgment


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






New habitat Hn

1

1

2

2

3

3

4

4

5

1 23 45

5

25








119899=


119899 will be (2 3 1 4 5)
















12 34 5

12 34 4

12 3 55

Habitat

Mutated habitat

























119894then


119894then


119894then



















Table 4 Continued





5 Conclusion







Table 5 Continued




Acknowledgment


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




12 34 5

12 34 4

12 3 55

Habitat

Mutated habitat

























119894then


119894then


119894then



















Table 4 Continued





5 Conclusion







Table 5 Continued




Acknowledgment


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119894then


119894then


119894then



















Table 4 Continued





5 Conclusion







Table 5 Continued




Acknowledgment


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in















Table 4 Continued





5 Conclusion







Table 5 Continued




Acknowledgment


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






Table 4 Continued





5 Conclusion







Table 5 Continued




Acknowledgment


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







Table 5 Continued




Acknowledgment


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Table 5 Continued




Acknowledgment


References











































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Documents

Research Article A Biogeography-Based Optimization