Comparing Evolutionary Strategies on a Biobjective Cultural Algorithm

Research ArticleComparing Evolutionary Strategies ona Biobjective Cultural Algorithm

Carolina Lagos1 Broderick Crawford12 Enrique Cabrera3

Ricardo Soto14 Joseacute-Miguel Rubio15 and Fernando Paredes6

1 Escuela de Ingenierıa Informatica Pontificia Universidad Catolica de Valparaıso 2362807 Valparaıso Chile2 Universidad Finis Terrae 7500000 Santiago Chile3 CIMFAV Facultad de Ingenierıa Universidad de Valparaıso 2362735 Valparaıso Chile4Universidad Autonoma de Chile 7500138 Santiago Chile5 Departamento de Computacion e Informatica Universidad de Playa Ancha 33449 Valparaıso Chile6 Escuela de Ingenierıa Industrial Universidad Diego Portales 8370109 Santiago Chile

Correspondence should be addressed to Carolina Lagos carolinalagoscmailpucvcl

Received 9 April 2014 Accepted 27 June 2014 Published 31 August 2014

Academic Editor Xin-She Yang

Copyright copy 2014 Carolina Lagos 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

Evolutionary algorithms have been widely used to solve large and complex optimisation problems Cultural algorithms (CAs) areevolutionary algorithms that have been used to solve both single and to a less extentmultiobjective optimisation problems In orderto solve these optimisation problems CAs make use of different strategies such as normative knowledge historical knowledgecircumstantial knowledge and among others In this paper we present a comparison among CAs that make use of differentevolutionary strategies the first one implements a historical knowledge the second one considers a circumstantial knowledge andthe third one implements a normative knowledge These CAs are applied on a biobjective uncapacitated facility location problem(BOUFLP) the biobjective version of the well-known uncapacitated facility location problem To the best of our knowledge onlyfew articles have applied evolutionary multiobjective algorithms on the BOUFLP and none of those has focused on the impactof the evolutionary strategy on the algorithm performance Our biobjective cultural algorithm called BOCA obtains importantimprovements when compared to other well-known evolutionary biobjective optimisation algorithms such as PAES and NSGA-IIThe conflicting objective functions considered in this study are cost minimisation and coverage maximisation Solutions obtainedby each algorithm are compared using a hypervolume S metric

1 Introduction

Evolutionary algorithms (EAs) are an effective alternative to(approximately) solve several large and complex optimisationproblems as they are able to find good solutions for awide range of problems in acceptable computational timeAlthough less studied EAs for multiobjective optimisation(MO) problems called evolutionarymultiobjective optimisa-tion (EMO) have demonstrated to be very effective In factduring the last two decades several authors have focusedtheir efforts on the development of several EMO algorithmsto solve a wide range ofMO problems For instance Maravall

and de Lope [1] use a genetic algorithm (GA) to solve themultiobjective dynamic optimisation for automatic parkingsystem In [2] the authors propose an improvement to thewell-known NSGA algorithm (called NSGA-II) based on anelitist approach In [3] the author presents an EMO algorithmapplied on a specific variation of the well-studied capacitatedvehicle routing problem (CVRP) where the author includesin the EMO algorithm an explicit collectivememorymethodnamely the extended virtual loser (EVL) Other well-knownEMO algorithms developed during the last two decades arePAES [4] and MO particle swarm optimisation [5] Morerecently hybrid techniques have been also applied to a large

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 745921 10 pageshttpdxdoiorg1011552014745921

2 The Scientific World Journal

number of optimisation problems (see [6]) A comprehensiveliterature review related to EMO algorithms can be found in[7]

EMO algorithms have some problems that must be takeninto account though For instance they tend to fall intopremature convergence with low evolution efficiency [8]This is because of implicit information embodied in theevolution process and domain knowledge corresponding tooptimisation problems which are not fully included in thesolution approach [9] To overcome these problems onecan make use of implicit evolution information Reynolds[10] proposes an EA called cultural algorithm (CA) whichis inspired from human culture evolution process and thatmakes use of the implicit evolution information generated ateach iterationTheCAs have a dual evolution structure whichconsists of two spaces population and belief space On theone hand the population space works as in any other EAthat is using evolutionary operators such as mutation andcrossover On the other hand in the belief space implicitknowledge is extracted from selected individuals in thepopulation and stored in a different way Then they are usedto guide the whole evolution process in the population spacesuch that they can induce the population to escape fromlocal optimal solutions It has been proved that CAs caneffectively improve the evolution performance [9] Althoughless studied CAs have been also used to solve MO problemsCoello et al [11] andCoello et al [12] two remarkable surveysonCAs onlymentionCoello and Landa [13] work as exampleof a CA application solving MO problems More recentlyZhang et al [14] present a CA which is enhanced by usinga particle swarm optimisation algorithm This enhanced CAis applied to fuel distribution MO problem Srinivasan andRamakrishnan [15] present a MO cultural algorithm that isapplied on data mining domain In [16] the authors applieda CA to a biobjective portfolio selection problem usingnormative knowledge in the belief space In [17] authorspresent a formal framework to implement MO culturalalgorithms To the best of our knowledge [18] is the onlyarticle that uses CAs to solve the biobjective uncapacitatedfacility location problem (BOUFLP) Furthermore we didnot find any article which compares the performance of CAsusing different evolutionary strategies at the belief space levelThus in this paper we present an extension of the biobjectivecultural algorithm (BOCA) developed in [18] We use twodifferent strategies at the belief space level and compare theperformance of our new algorithms with the performanceof the previous one We also compare its results with otherwell-known EMO algorithms such as PAES and NSGA-IIObtained solutions were compared using the hypervolume Smetric proposed in [19]

The remaining of this paper is organised as followsSection 2 shows an overview on MO focused on EMOalgorithms Section 21 presents briefly the BOUFLP andsome of its distinctive features In Section 3 we describe ourimplementation for the BOCA algorithm We describe themain differences between our implementation and the one in[18] Section 32 presents the BOCA algorithm applied to a setof well-known instances from the literature Finally Section 4presents the conclusions of this work

2 Overview

In this section we show an overview of topics relatedto this paper In Section 21 MO concepts are presentedemphasizing EMO algorithms and its state of art Morespecifically we focus on the development of EMO algorithmsfor MO combinatorial optimisation (MOCO) problems InSection 22 we present the BOUFLP formulation based on acost-coverage approach Finally in Section 23 we present anoverview of CAs and its multiobjective extension Details ofour CA implementation are also presented at the end of thissection

21 (Evolutionary) Multiobjective Optimisation In this sec-tionwe briefly introduce themain principles ofMOproblemsand particularly MOCO problems For a comprehensivereview of this topic see [20 21] In this paper we will makeuse of the following notation for the comparison of vectors(solutions) Let 119910 and 119910

1015840isin R119901 We say that 119910 ≦ 119910

1015840 if119910119896≦ 1199101015840

119896forall119896 = 1 119901 Similarly we will say that 119910 le 119910

1015840 if119910 ≦ 119910

1015840 but 119910 = 1199101015840 Finally we say that 119910 lt 119910

1015840 if 119910119896lt 1199101015840

119896forall119896 =

1 119901 A solution 119909 isin R119899 with 119899 being equal to the numberof decision variables is called an efficient solution and itsimage 119891(119909) with 119891 R119899 rarr R119901 a nondominated point ofthe MO problem if there is no 119909 isin R119899 with 119909 = 119909 such that119891(119909) le 119891(119909) In [22] the author describes several excellencerelations These relations establish strict partial orders in theset of all nondominated points related to different aspectsof their quality Previously in [23 24] the authors considerseveral outperformance relations to address the closenessof the set of nondominated points found by an algorithmto the actual set of nondominated points called ParetoFrontier (PF) In [25] a comprehensive explanation of thedesirable features of an approximation to the PF is presentedIn this paper we choose the S metric which is properlyexplained in [24] The S metric calculates the hypervolumeof a multidimensional region [19] and allows the integrationof aspects that are individually measured by other metricsAn advantage of the S metric is that each algorithm can beassessed independently of the other algorithms involved inthe study However the S values of two setsA andB namelySA and SB respectively cannot be used to derive whethereither set entirely dominates the other Figure 1(a) shows asituation where SA gt SB and setA completely dominates setB Figure 1(b) shows a situation where SB gt SA but neitherA dominatesB norB dominatesA

In this paper we use EAs to solve the BOUFLP An EAis a stochastic search procedure inspired by the evolutionprocess in nature In this process individuals evolve and thefitter ones have a better chance of reproduction and survivalThe reproduction mechanisms favour the characteristics ofthe stronger parents and hopefully produce better childrenguaranteeing the presence of those characteristics in futuregenerations [26] EAs have been successfully applied to alarge number of both single- andmultiobjective optimisationproblems Comprehensive reviews of EMO algorithms arepresented in [11 24] andmore recently in [12] Amore general

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1 3 5 7 9 11 13 15 170

1

2

3

4

5

6

7

8

9

nds ands b

f1(x)

f2(x)

(a) Nondominated setA completely dominates nondominated setB

1 3 5 7 9 11 13 15 170

1

2

3

4

5

6

7

8

9

nds ands b

f1(x)

f2(x)

(b) NeitherA dominatesB norB dominatesA

Figure 1 S metric represented as the area that is dominated by a setof (approximately) nondominated points

review of hybrid heuristics solving MOCO problems whereEMO algorithms are also included is presented in [27]

22 Biobjective Uncapacitated Facility Location ProblemFacility location problem (FLP) is one of the most importantproblems for companies with the aim of distributing productsto their customers The problem consists of selecting sites toinstall plants warehouses and distribution centres allocatingcustomers to serving facilities and interconnecting facilitiesby flow assignment decisions Comprehensive reviews andanalysis of the FLP are presented in [28ndash30]

In this paper we consider a two-level supply chain wherea single plant serves a set of warehouses which serve a set ofend customers or retailers Figure 2 shows this configuration

Two main (conflicting) objectives can be identified in theFLP

(i) minimise the total cost associated with the facilityinstallation and customer allocation and

(ii) maximise the customers rate coverage

Several works on single-objective optimisation have beencarried out considering these two objectives separately Onthe one hand uncapacitated FLP (UFLP) is one of moststudied FLPs in the literature In the UFLP the main goalis to minimise the location-allocation cost of the networkOn the other hand 119901 median FLPs are one of the mostcommon FLPs among those that are focused on coveragemaximisationMost important FLPmodels arewell describedand formalised in [29] MO FLPs have been also well studiedin the literature during the last two decades A survey on thistopic can be found in [31]

As we mentioned before in this paper we solve theBOUFLP BOUFLP has been modelled with minisum andmaxisumobjectives (cost and coverage)The followingmodelformulation is based on [32] Let 119868 = 1 119898 be the set ofpotential facilities and 119869 = 1 119899 the set of customersLet FC

119894be the fixed cost of opening facility 119894 and 119889

119895the

demand of customer 119895 Let 119888119894119895be the cost of assigning the

customer 119895 to facility 119894 and ℎ119894119895the distance between facility 119894

and customer 119895 Let119863MAX be the maximal covering distancethat is customers within this distance to an open facility areconsidered well served Let 119876

119895= 119894 isin 119868 ℎ

119894119895le 119863MAX

be the set of facilities that could serve customer 119895 within themaximal covering distance 119863MAX Let 119909119894 be 1 if facility 119894 isopen and 0 otherwise Let 119910

119894119895be 1 if the whole demand of

customer 119895 is served by facility 119894 and 0 otherwise Consider

minimise 119891 (119909 119910) (1)

st sum

119894isin119868

119910119894119895

forall119895 isin 119869 (2)

119910119894119895le 119909119894

forall119895 isin 119869 119894 isin 119868 (3)

119910119894119895isin 0 1 forall119895 isin 119869 119894 isin 119868 (4)

119909119894isin 0 1 forall119894 isin 119868 (5)

with 119891 R119899 rarr R119901 and 119901 = 2 Objective functions 1198911and 119891

2

are as follows

1198911(119909 119910) = sum

119894isin119868

FC119894119909119894+ sum

119894isin119868

sum

119895isin119869

119888119894119895119910119894119895 (6)

1198912(119909 119910) = minussum

119895isin119869

119889119895sum

119894isin119876119895

119910119894119895 (7)

Equation (6) represents total operating cost the first termcorresponds to location cost that is the sum of the fixed costsof all the open facilities and the second term represents theallocation cost that is the cost of attending customer demandby an open facility Equation (7) measures coverage as thesum of the demand of customers attended by open facilitieswithin the maximal covering distance Equations (2) and (3)

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Plant or centralwarehouse

Warehouse 1

Warehouse 2

Warehouse 3

Customers(clusters)

Figure 2 A two-level supply chain network configuration

ensure that each customer is attended by only one facilityEquation (3) also forces customer to be assigned to an openfacility Finally equations (4) and (5) set decision variables asbinary

23 Biobjective Cultural Algorithm The experience andbeliefs accepted by a community in a social system are themain motivations for the creation of the CAs Originally pro-posed by Reynolds [10] CAs model the evolution of culturalsystems based on the principles of human social evolutionIn this case evolution is seen as an optimisation process [10]The CAs guide the evolution of the population based on theknowledge Knowledge acquired during previous iterationsis provided to future generations allowing accelerating theconvergence of the algorithm to good solutions [33] Domainknowledge is modelled separately from the populationbecause there is certain independence between them whichallows us to work and model them separately in order toenhance the overall algorithm performance Figure 3 showsthis interaction

CAs are mainly characterised by presenting two inher-itance systems one at population level called populationspace and the other at knowledge level called belief spaceThis key feature is designed to increase the learning ratesand convergence of the algorithm and thus to do a moreresponsive system for a number of problems [34] Moreoverit allows us to identify two significant levels of knowledge

a microevolutionary level (represented by the populationspace) andmacroevolutionary level (represented by the spaceof beliefs) [35]

CAs have the following components population space(set of individuals who have independent features) [35] beliefspace (stored individuals acquired in previous generations)[34] computer protocol connecting the two spaces anddefining the rules on the type of information to be exchangedbetween them by using the acceptance and influence func-tions and finally knowledge sources which are describedin terms of their ability to coordinate the distribution ofindividuals depending on the nature of a problem instance[35] These knowledge sources can be of the followingtypes circumstantial normative domain topographic andhistorical

The most distinctive feature of CAs is the use of thebelief space which through an influence function affectsfuture generations For this reason in this paper we focuson the effect on the algorithm performance of changes insuch an influence function To do this we have consideredresults obtained previously in [18] where the authors usedan influence function based on historical knowledge andwe compare those results with our BOCA implementationwhich considers two influence functions the first one basedon circumstantial knowledge and the second one basedon normative knowledge Algorithm 1 shows the generalprocedure of our BOCA algorithm

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Belief space

Population space

Update ()

Accept ()Influence ()

New generation ()

Best individuals ()

2

2

2

1

1

1

n minus 1 n

n minus 1 n

n minus 1 n

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Figure 3 CA general diagram

begin119896 = 0initialise Popuation 119875

119896

initialise BeliefSpace 119861119896

while 119896 lt 119894119905119890119903119886119905119894119900119899119873119906119898119887119890119903 doEvaluate(119875

119896)

119875best119896

= bestIndividuals(119875119896)

updateBeliefSpace(119861119896 accept(119875best

119896))

influence(119875119896 119861119896)

119896 = 119896 + 1119875119896= newPopulation(119875

119896minus1)

end

Algorithm 1 BOCA general algorithmic frame

To initialise the population we use a semirandom func-tion In its first phase this function defines in a stochasticway the set of facilities that will be opened (selected facili-ties) Then we allocate each customer to a selected facilityminimising the cost function 119891

1while avoiding minimising

the coverage function 1198912 This strategy provides better results

than using completely random initial populations and itscomputational time additional cost is marginal

To obtain the next generation two parents are used in arecombination process To avoid local optimal values we donot overuse the culture Thus a parent is selected from thepopulation to obtain diversity and the other parent is selected

from the belief space to influence the next generation Thebelief space keeps a list of all the individuals which meetsome criteria These criteria depend on what knowledgethe algorithm implements In this paper the circumstantialknowledge selects the best individuals found so far foreach objective function Thus one individual will give usinformation on the best value found for 119891

1and the other will

do the same for 1198912 The historical knowledge stores a list of

individuals with the best fitness value found so farThe fitnessvalue is calculated as the hypervolume S that is covered by anindividual Finally normative knowledge considers a list ofindividuals which are pairwise nondominated with respect tothe other individuals of their generation

Let |119869| be the number of available facilities and let |119868|

be the number of customers of our BOUFLP In this paperdecisions variables 119909 and 119910 are represented by a binary |119869|-length vector and |119869| times |119868| matrix respectively Comparingtwo different solutions (individuals) needs an evaluationcriterion In this paper we use the same criterion explainedin [18]

3 Computational Experiments

In this section we present the set of instances that are usedin this study as well as results obtained by our BOCAimplementation

31 Instances Presentation The instances that were used inthis paper correspond to random instances using a problemgenerator that follows the methodology from UflLib [36]Previous works in the literature have also used this problemgenerator to create their test instances [18 26]

The BOCA algorithm has several parameters that need tobe set As in [18] the number of generations considered inthis paper is equal to 100 Population size 119871 is set equal to 100mutation probability in the population space 119875ps is equal to02 and probability of mutation in the belief space 119875bs is 004Both 119871 and 119875ps values are different from the values used in[18] These values are chosen as they all together yield to thebest performance of the algorithm given some test instancesThus although resulting values are different from that used in[18] the method we use to set them is the same as that usedin that work This is important in order to fairly compare thedifferent BOCA algorithms

32 Results and Discussion In this section we compare theresults obtained by the previous BOCAalgorithm (see [18] forfurther details) and our approach Moreover a comparisonbetween results obtained by well-known EMO algorithmssuch as NSGA-II and PAES and our BOCA algorithm is alsopresented in this section

Tables 1 and 2 show the results obtained by the BOCAimplementations using historical [18] circumstantial andnormative knowledge respectively In the same way Tables 3and 4 present the results obtained by the well-known NSGA-II and PAES algorithms For each algorithm 119878 value () time119905 (in seconds) and the number of efficient solutions 119909 isin 119883

have been included in these tables As we mentioned before

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Table 1 Results obtained by the BOCA implementations for instance of class 119860

Instance BOCA (historical) BOCA (circumstantial) BOCA (normative)1199051(sec) S

1 |119883

1| 119905

2(sec) S

2 |119883

2| 119905

3(sec) S

3 |119883

3|

11986010ndash251198621 252 07057 13 167 07058 13 37 06963 11

11986010ndash251198622 197 06136 11 176 06136 11 27 06135 10

11986010ndash251198623 186 06993 9 184 06992 9 33 06992 9

11986010ndash251198624 115 05729 5 193 05729 5 28 05728 3

11986010ndash251198625 152 06351 8 168 06352 8 32 06352 8

11986010ndash251198626 183 05982 9 186 05961 8 34 05944 8

11986030ndash751198621 464 08071 18 793 07828 16 1908 07300 12

11986030ndash751198622 1247 08013 47 598 07798 23 1201 07090 30

11986030ndash751198623 1177 07552 50 611 07374 21 1305 07098 29

11986030ndash751198624 450 07471 21 960 06307 10 1450 06048 15

11986030ndash751198625 940 07790 46 610 07628 32 1725 07078 28

11986030ndash751198626 740 07245 37 536 06656 14 446 07010 28

11986050ndash1501198621 1473 07878 52 2386 06685 27 3481 07140 42

11986050ndash1501198622 1043 07611 37 4266 06418 27 5412 06200 32

11986050ndash1501198623 1165 07883 53 1644 07171 24 3941 07160 27

11986050ndash1501198624 735 06345 17 1437 05849 17 3419 06010 20

11986050ndash1501198625 1459 08039 53 2333 06725 21 2745 07158 35

11986050ndash1501198626 1434 07903 43 2399 06535 26 2811 07040 48

Table 2 Results obtained by the BOCA implementations for instance of class 119861

Instance BOCA (historical) BOCA (circumstantial) BOCA (normative)1199051(sec) S

1 |119883

1| 119905

2(sec) S

2 |119883

2| 119905

3(sec) S

3 |119883

3|

11986110ndash251198621 135 07795 7 196 07795 7 36 07721 8

11986110ndash251198622 130 05856 5 128 05813 4 48 05807 4

11986110ndash251198623 238 07410 14 229 07278 12 101 07170 13

11986110ndash251198624 260 07261 14 222 07246 14 182 07218 12

11986110ndash251198625 115 08117 6 185 07250 4 25 08020 5

11986110ndash251198626 285 06517 18 245 06340 15 31 05967 12

11986130ndash751198621 683 07650 38 736 06945 26 1208 07119 20

11986130ndash751198622 944 07323 59 882 07133 30 1014 07115 24

11986130ndash751198623 1351 06335 72 683 06263 28 1076 06106 28

11986130ndash751198624 618 05214 32 857 04947 27 472 04824 29

11986130ndash751198625 905 07473 45 917 06869 37 1820 06647 25

11986130ndash751198626 800 08775 42 565 08502 24 781 08310 35

11986150ndash1501198621 802 08345 23 885 08105 18 2371 08090 20

11986150ndash1501198622 1221 07634 53 1405 07134 24 2678 07050 35

11986150ndash1501198623 1410 07915 58 1706 07169 13 2791 06940 45

11986150ndash1501198624 567 07498 10 604 06720 17 3841 06420 15

11986150ndash1501198625 939 06393 37 482 05470 27 5712 05870 25

11986150ndash1501198626 1904 08016 67 1745 07560 34 4958 07150 54

we want to produce a set with a large number of efficientsolutions 119909 isin 119883 a 119878 value close to 100 (ideal) and a small 119905For the sake of easy reading we have split the set of instancesinto two subsets (instances type 119860 and 119861)

We then compare our BOCA implementations with theone presented in [18] Tables 5 and 6 show a comparisonbetween those algorithms As we can see when compared interms of its 119878 value (the bigger the better) BOCA algorithm

using historical knowledge (BOCAH) performs consistentlybetter than the ones using circumstantial (BOCAC) andnormative (BOCAN) knowledge In fact BOCAH obtains a 119878

value that is in average 58 bigger than the one obtainedby BOCAC and 65 bigger than the 119878 value obtained byBOCAN When compared in terms of the CPU time neededto reach the number of iterations (generations) BOCAH isin average faster than both BOCAC and BOCAN algorithms

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Table 3 Results obtained bywell-knownMOEA algorithmsNSGA-II and PAES for instances of class 119860

Instance NSGA-II PAES1199054(sec) S

4 |119883

4| 1199055(sec) S

5 |119883

5|

11986010ndash251198621 1344 07057 13 372 07057 13

11986010ndash251198622 1511 06136 11 394 06136 11

11986010ndash251198623 1859 06993 9 326 06993 9

11986010ndash251198624 3186 05729 5 305 05729 5

11986010ndash251198625 1748 06351 8 378 06351 8

11986010ndash251198626 1650 05982 9 384 05982 9

11986030ndash751198621 1633 08071 18 499 07795 16

11986030ndash751198622 1345 08012 43 622 07915 41

11986030ndash751198623 1394 07538 43 603 07384 29

11986030ndash751198624 1874 07508 20 525 07342 20

11986030ndash751198625 1413 07783 40 595 07572 33

11986030ndash751198626 1474 06676 36 511 05300 12

11986050ndash1501198621 2385 07913 43 1386 07391 25

11986050ndash1501198622 2522 07597 39 1231 06729 23

11986050ndash1501198623 2298 07665 34 1269 06900 26

11986050ndash1501198624 2575 06344 17 1233 06106 15

11986050ndash1501198625 2446 08072 40 1251 07590 23

11986050ndash1501198626 2259 07862 38 1218 06461 27

Table 4 Results obtained bywell-knownMOEAalgorithmsNSGA-II and PAES for instances of class 119861

Instance NSGA-II PAES1199054(sec) S

4 |119883

4| 1199055(sec) S

5 |119883

5|

11986110ndash251198621 1439 07795 7 397 07795 7

11986110ndash251198622 1430 05856 5 405 05856 5

11986110ndash251198623 1288 0741 13 433 07410 13

11986110ndash251198624 1747 07261 14 440 07261 14

11986110ndash251198625 1766 08117 6 363 08117 6

11986110ndash251198626 1261 06517 17 394 06517 17

11986130ndash751198621 1566 07609 30 562 07134 27

11986130ndash751198622 1525 07285 49 563 06899 34

11986130ndash751198623 1345 06330 51 578 05991 53

11986130ndash751198624 2562 05333 23 502 05211 21

11986130ndash751198625 1433 07631 31 564 07360 26

11986130ndash751198626 1460 08088 47 585 07602 32

11986150ndash1501198621 2698 08447 23 1280 06925 10

11986150ndash1501198622 2289 07515 41 1241 04970 13

11986150ndash1501198623 2284 07988 52 1279 06836 26

11986150ndash1501198624 2603 07500 10 1192 07253 14

11986150ndash1501198625 2301 06387 35 1212 05021 16

11986150ndash1501198626 3178 08029 43 1260 07152 32

We can note that for 119861 instances times required by BOCAH

and BOCAC are in average quite similar (only 16 ofdifference) Finally when we look at the number of efficientsolutions found by each algorithm (|119883| column) we can seethat again BOCAH outperforms both BOCAC and BOCAN

algorithms In this case the average number of efficient

Table 5 Comparison among our BOCA implementations (119860instances)

Instance ΔS1S2 Δ

11990511199052

Δ|1 |

|2 |ΔS1S3 Δ

11990511199053

Δ|1 |

|3 |

11986010minus25

1198621

minus0014 3373 000 1332 8532 153811986010minus25

1198622

0000 1066 000 0016 8629 90911986010minus25

1198623

0014 108 000 0014 8226 00011986010minus25

1198624

0000 minus6783 000 0017 7565 400011986010minus25

1198625

minus0016 minus1053 000 minus0016 7895 00011986010minus25

1198626

0351 minus164 1111 0635 8142 111111986030minus75

1198621

3011 minus7091 1111 9553 minus31121 333311986030minus75

1198622

2683 5204 5106 11519 369 361711986030minus75

1198623

2357 4809 5800 6012 minus1088 420011986030minus75

1198624

1558 minus11333 5238 19047 minus22222 285711986030minus75

1198625

2080 3511 3043 9140 minus8351 391311986030minus75

1198626

8130 2757 6216 3244 3973 243211986050minus150

1198621

15143 minus6198 4808 9368 minus13632 192311986050minus150

1198622

15675 minus30901 2703 18539 minus41889 135111986050minus150

1198623

90320 minus4112 5472 9172 minus23828 490611986050minus150

1198624

78170 minus9551 000 5280 minus36517 minus176511986050minus150

1198625

16345 minus5990 6038 10959 minus8814 339611986050minus150

1198626

17310 minus6729 3953 10920 minus9603 minus1163

Table 6 Comparison among our BOCA implementations

Instance ΔS1S2 Δ

11990511199052

Δ|1 |

|2 |ΔS1S3 Δ

11990511199053

Δ|1 |

|3 |

11986110ndash251198621 0000 minus4519 000 0949 7333 minus1429

11986110ndash251198622 0734 154 2000 0837 6308 2000

11986110ndash251198623 1781 378 1429 3239 5756 714

11986110ndash251198624 0207 1462 000 0592 3000 1429

11986110ndash251198625 10681 minus6087 3333 1195 7826 1667

11986110ndash251198626 2716 1404 1667 8439 8912 3333

11986130ndash751198621 9216 minus776 3158 6941 minus7687 4737

11986130ndash751198622 2595 657 4915 2840 minus742 5932

11986130ndash751198623 1137 4944 6111 3615 2036 6111

11986130ndash751198624 5121 minus3867 1563 7480 9992 938

11986130ndash751198625 8082 minus133 1778 11053 minus10110 4444

11986130ndash751198626 3111 2938 4286 5299 238 1667

11986150ndash1501198621 2876 minus1035 2174 3056 minus19564 1304

11986150ndash1501198622 6550 minus1507 5472 7650 minus11933 3396

11986150ndash1501198623 9425 minus2099 7759 12318 minus9794 2241

11986150ndash1501198624 10376 minus653 minus7000 14377 minus57743 minus5000

11986150ndash1501198625 14438 4867 2703 8181 minus50831 3243

11986150ndash1501198626 5689 835 4925 10803 minus16040 1940

solutions found by the BOCAH algorithm is about 20biggerthan the one obtained by the other two approaches

Results above are consistent with the good performanceobtained by the BOCAH approach in [18] Moreover resultsshow that performance of the BOCA algorithm dependslargely on the selected knowledge and it can make thedifference in terms of 119878 value time and number of efficientsolutions found by the algorithmThis is an important finding

8 The Scientific World Journal

Table 7 Comparison among our BOCAwith circumstantial knowl-edge and NSGA-II and PAES

Instance ΔS2S4 Δ

11990521199054

Δ|2 |

|4|ΔS2S5 Δ

11990521199055

Δ|2 |

|5 |

11986010ndash251198621 minus135 minus353243 minus1818 minus135 minus90541 minus1818

11986010ndash251198622 minus002 minus549630 minus1000 minus002 minus135926 minus1000

11986010ndash251198623 minus001 minus553333 000 minus001 minus88788 000

11986010ndash251198624 minus002 minus1127857 minus6667 minus002 minus98929 minus6667

11986010ndash251198625 002 minus536250 000 002 minus108125 000

11986010ndash251198626 minus064 minus475294 minus1250 minus064 minus102941 minus1250

11986030ndash751198621 minus1056 1441 minus5000 minus678 7385 minus3333

11986030ndash751198622 minus1300 minus1199 minus4333 minus1164 4821 minus3667

11986030ndash751198623 minus620 minus682 minus4828 minus403 5379 000

11986030ndash751198624 minus2414 minus2924 minus3333 minus2140 6379 minus3333

11986030ndash751198625 minus996 1809 minus4286 minus698 6551 minus1786

11986030ndash751198626 476 minus23049 minus2857 2439 minus1457 5714

11986050ndash1501198621 minus1083 3149 minus238 minus352 6018 4048

11986050ndash1501198622 minus2253 5340 minus2188 minus853 7725 2813

11986050ndash1501198623 minus705 4169 minus2593 363 6780 370

11986050ndash1501198624 minus556 2469 1500 minus160 6394 2500

11986050ndash1501198625 minus1277 1089 minus1429 minus604 5443 3429

11986050ndash1501198626 minus1168 1964 2083 822 5667 4375

Table 8 Comparison among our BOCAwith circumstantial knowl-edge and NSGA-II and PAES

Instance ΔS2S4 Δ

11990521199054

Δ|2 |

|4 |ΔS2S5 Δ

11990521199055

Δ|2 |

|5|

11986110ndash251198621 minus096 minus389722 1250 minus096 minus100278 1250

11986110ndash251198622 minus084 minus287917 minus2500 minus084 minus74375 minus2500

11986110ndash251198623 minus335 minus117525 000 minus335 minus32871 000

11986110ndash251198624 minus060 minus85989 minus1667 minus060 minus14176 minus1667

11986110ndash251198625 minus121 minus696400 minus2000 minus121 minus135200 minus2000

11986110ndash251198626 minus922 minus396774 minus4167 minus922 minus117097 minus4167

11986130ndash751198621 minus688 minus2964 minus5000 minus021 5348 minus3500

11986130ndash751198622 minus239 minus5039 minus10417 304 4448 minus4167

11986130ndash751198623 minus367 minus2500 minus8214 188 4628 minus8929

11986130ndash751198624 minus1055 minus44280 2069 minus802 minus636 2759

11986130ndash751198625 minus1480 2126 minus2400 minus1073 6901 minus400

11986130ndash751198626 267 minus8694 minus3429 852 2510 857

11986150ndash1501198621 minus441 minus1379 minus1500 1440 4601 5000

11986150ndash1501198622 minus660 1453 minus1714 2950 5366 6286

11986150ndash1501198623 minus1510 1817 minus1556 150 5417 4222

11986150ndash1501198624 minus1682 3223 3333 minus1298 6897 667

11986150ndash1501198625 minus881 5972 minus4000 1446 7878 3600

11986150ndash1501198626 minus1229 3590 2037 minus003 7459 4074

as it points out the relevance of the choice of a specific type ofknowledge

We now compare BOCAC and BOCAN algorithms tothe well-known NSGA-II and PAES algorithms Tables 7 and8 show a comparison between our BOCAC algorithm andthe NSGA-II and PAES algorithms As we can see althoughBOCAC obtains in average a 119878 value 68 lower than the

Table 9 Comparison among our BOCAwith normative knowledgeand NSGA-II and PAES

Instance ΔS3S4 Δ

11990531199054

Δ|3 |

|4 |ΔS3S5 Δ

11990531199055

Δ|3 |

|5 |

11986010ndash251198621 001 minus70479 000 001 minus12275 000

11986010ndash251198622 000 minus75852 000 000 minus12386 000

11986010ndash251198623 minus001 minus91033 000 minus001 minus7717 000

11986010ndash251198624 000 minus155078 000 000 minus5803 000

11986010ndash251198625 002 minus94048 000 002 minus12500 000

11986010ndash251198626 minus035 minus78710 minus1250 minus035 minus10645 minus1250

11986030ndash751198621 minus310 minus10593 minus1250 042 3707 000

11986030ndash751198622 minus274 minus12492 minus8696 minus150 minus401 minus7826

11986030ndash751198623 minus222 minus12815 minus10476 minus014 131 minus3810

11986030ndash751198624 minus1904 minus9521 minus10000 minus1641 4531 minus10000

11986030ndash751198625 minus203 minus13164 minus2500 073 246 minus313

11986030ndash751198626 minus030 minus17500 minus15714 2037 466 1429

11986050ndash1501198621 minus1837 004 minus5926 minus1056 4191 741

11986050ndash1501198622 minus1837 4088 minus4444 minus485 7114 1481

11986050ndash1501198623 minus689 minus3978 minus4167 378 2281 minus833

11986050ndash1501198624 minus846 minus7919 000 minus439 1420 1176

11986050ndash1501198625 minus2003 minus484 minus9048 minus1286 4638 minus952

11986050ndash1501198626 minus2031 584 minus4615 113 4923 minus385

one obtained by the NSGA-II algorithm it is more thanthree times faster Moreover when BOCAC is comparedto PAES algorithm the obtained 119878 values are in averageequivalent while BOCAC is around 30 faster than PAESPAES obtains in average more efficient points than BOCAC

though (932)Finally Tables 9 and 10 show a comparison between

BOCAN and NSGA-II and PAES algorithms BOCAN per-forms quite similar to PAES algorithm with respect to both119878 value and the number of obtained efficient solutionsHowever BOCAN is faster than PAES Similar situationoccurs when BOCAN is compared to NSGA-II algorithmAlthough NSGA-II obtains better values for both 119878 and |119883|BOCAN is much faster than NSGA-II This situation canbe explained by the very fast performance that our BOCAN

algorithm obtains for the set of small instances When welook further at the results we can note that if we onlyconsider both medium and large size instances executiontimes obtained by both algorithms are quite similar to eachotherThis result confirmswhat is outlined in [18] in the senseof the good performance that the BOCA algorithm showsFurthermore our results confirm this good performancewith respect to other well-known EMO algorithms does notdepend on which type of knowledge is considered Howeveras we mentioned before the choice of the knowledge usedon the BOCA algorithm is an important issue and it has animpact on the algorithm performance

4 Conclusions and Future Work

Evolutionary algorithms are a very good alternative to solvecomplex combinatorial optimisation problems In this paper

The Scientific World Journal 9

Table 10 Comparison among our BOCA with normative knowl-edge and NSGA-II and PAES

Instance ΔS3S4 Δ

11990531199054

Δ|3 |

|4 |ΔS3S5 Δ

11990531199055

Δ|3 |

|5 |

11986110ndash251198621 000 minus63418 000 000 minus10255 000

11986110ndash251198622 minus074 minus101719 minus2500 minus074 minus21641 minus2500

11986110ndash251198623 minus181 minus46245 minus833 minus181 minus8908 minus833

11986110ndash251198624 minus021 minus68694 000 minus021 minus9820 000

11986110ndash251198625 minus1196 minus85459 minus5000 minus1196 minus9622 minus5000

11986110ndash251198626 minus279 minus41469 minus1333 minus279 minus6082 minus1333

11986130ndash751198621 minus956 minus11277 minus1538 minus272 2364 minus385

11986130ndash751198622 minus213 minus7290 minus6333 328 3617 minus1333

11986130ndash751198623 minus107 minus9693 minus8214 434 1537 minus8929

11986130ndash751198624 minus780 minus19895 1481 minus534 4142 2222

11986130ndash751198625 minus1109 minus5627 1622 minus715 3850 2973

11986130ndash751198626 487 minus15841 minus9583 1059 minus354 minus3333

11986150ndash1501198621 minus422 minus20486 minus2778 1456 minus4463 4444

11986150ndash1501198622 minus534 minus6292 minus7083 3033 1167 4583

11986150ndash1501198623 minus1142 minus3388 minus30000 464 2503 minus10000

11986150ndash1501198624 minus1161 minus33096 4118 minus793 minus9735 1765

11986150ndash1501198625 minus1676 minus37739 minus2963 821 minus15145 4074

11986150ndash1501198626 minus620 minus8212 minus2647 540 2779 588

we have implemented a biobjective cultural algorithm to solvethe well-known BOUFLP We have considered two differentsources of knowledge namely circumstantial and normativeand compare them with a previously implemented historicalknowledge Furthermore we compare our BOCAapproacheswith two well-known EAs namely NSGA-II and PAES

Although BOCA approaches using both normative andcircumstantial knowledge could not improve the resultsobtained by the BOCA algorithm with the historical knowl-edge results pointed out that performance of the BOCAalgorithm depends largely on the selected knowledge andit can make the difference in terms of 119878 value time andnumber of efficient solutions found by the algorithm Thisis an important finding as it points out the relevance of thechoice of a specific type of knowledge Moreover our resultsalso confirm the good performance showed by the BOCAalgorithmwith respect to other well-known EMO algorithmssuch as NSGA-II and PAES algorithmsThe BOCA algorithmis very competitive when compared to those EMOalgorithmsindependently of the type of knowledge implemented

As a future work we think that more investigation isneeded in order to find patterns that allow us to get theright knowledge implemented depending on the problemfeatures As we mentioned before the knowledge choice hasan impact on the performance of the BOCA algorithm andtherefore it must be studied in depth Also as future workhybrid knowledge could be implemented in order to exploitthe advantages of each kind of knowledge at the same timeMoreover our BOCA algorithm can be used to solve other

interesting MOPs arising in the logistic field such as routingor scheduling problems

Appendix

Result Tables

In this appendix section obtained results are presentedColumns 119878

sdotshow the 119878 value obtained by algorithm sdot as

Algorithms are indexed as follows The original BOCAalgorithm is indexed by 1 BOCA algorithms using cir-cumstantial and normative knowledge are indexed by 2

and 3 respectively Finally the other EAs considered inthis paper namely NSGA-II and PAES are indexed by 4

and 5 respectively Columns 119905sdotshow the time obtained by

each algorithm in seconds Columns |119883sdot| show the number

of efficient solutions found by the corresponding algorithmFinally operator Δsdot

sdotsdotshows a value that is equivalent to (sdot minus

sdotsdot)sdot times 100

Conflict of Interests

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

References

[1] D Maravall and J de Lope ldquoMulti-objective dynamic opti-mization with genetic algorithms for automatic parkingrdquo SoftComputing vol 11 no 3 pp 249ndash257 2007

[2] K Deb S Agrawal A Pratap and T Meyarivan ldquoA fast elitistnon-dominated sorting genetic algorithm for multiobjectiveoptimization Nsga IIrdquo in Parallel Problem Solving from NaturePPSN VI M Schoenauer K Deb G Rudolph et al Edsvol 1917 of Lecture Notes in Computer Science pp 849ndash858Springer Berlin Germany 2000

[3] I Borgulya ldquoAn algorithm for the capacitated vehicle routingproblem with route balancingrdquo Central European Journal ofOperations Research vol 16 no 4 pp 331ndash343 2008

[4] P J Angeline Z Michalewicz M Schoenauer X Yao and AZalzala Eds The Pareto Archived Evolution Strategy A NewBaseline Algorithm for Pareto Multiobjective Optimisation vol1 IEEE Press 1999

[5] C A Coello Coello and M Lechuga ldquoMOPSO a proposal formultiple objective particle swarm optimizationrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo02) vol 2pp 1051ndash1056 2002

[6] W K Mashwani ldquoComprehensive survey of the hybrid evolu-tionary algorithmsrdquo International Journal of Applied Evolution-ary Computation vol 4 pp 1ndash19 2013

[7] R Bhattacharya and S Bandyopadhyay ldquoSolving conflicting bi-objective facility location problem by NSGA II evolutionaryalgorithmrdquo International Journal of Advanced ManufacturingTechnology vol 51 no 1ndash4 pp 397ndash414 2010

[8] B Crawford C Lagos C Castro and F Paredes ldquoA culturalalgorithm for solving the set covering problemrdquo inAnalysis andDesign of Intelligent Systems using Soft Computing TechniquesP Melin O Castillo E Ramırez J Kacprzyk and W PedryczEds vol 41 of Advances in Soft Computing pp 408ndash415Springer Berlin Germany 2007

10 The Scientific World Journal

[9] Y Guo J Cheng Y Cao and Y Lin ldquoA novel multi-populationcultural algorithm adopting knowledge migrationrdquo Soft Com-puting vol 15 no 5 pp 897ndash905 2011

[10] R G Reynolds ldquoAn introduction to cultural algorithmsrdquo inProceedings of the 3rd Annual Conference on EvolutionaryProgramming pp 131ndash139 World Scientic 1994

[11] C A C Coello G B Lamont and D A V VeldhuizenEvolutionary Algorithms for Solving Multi-Objective Problems(Genetic and Evolutionary Computation) Springer SecaucusNJ USA 2006

[12] C A Coello C Dhaenens and L Jourdan Advances in Multi-Objective Nature Inspired Computing Springer 1st edition 2010

[13] C A Coello and R Landa ldquoEvolutionary multiobjective opti-mization using a cultural algorithmrdquo in Proceedings of the IEEESwarm Intelligence Symposium pp 6ndash13 IEEE Service CenterPiscataway NJ USA 2003

[14] R Zhang J Zhou L Mo S Ouyang and X Liao ldquoEconomicenvironmental dispatch using an enhanced multi-objectivecultural algorithmrdquo Electric Power Systems Research vol 99 pp18ndash29 2013

[15] S Srinivasan and S Ramakrishnan ldquoA social intelligent systemfor multi-objective optimization of classification rules usingcultural algorithmsrdquo Computing vol 95 no 4 pp 327ndash3502013

[16] G G Cabrera C Vasconcellos R Soto J M Rubio F Paredesand B Crawford ldquoAn evolutionary multi-objective optimiza-tion algorithm for portfolio selection problemrdquo InternationalJournal of Physical Sciences vol 6 no 22 pp 5316ndash5327 2011

[17] R Reynolds and D Liu ldquoMulti-objective cultural algorithmsrdquoinProceedings of the IEEECongress of EvolutionaryComputation(CEC rsquo11) pp 1233ndash1241 June 2011

[18] G Cabrera J M Rubio D Dıaz B Fernandez C Cubillosand R Soto ldquoA cultural algorithm applied in a BiObjectiveuncapacitated facility location problemrdquo in Evolutionary Multi-Criterion Optimization R Takahashi K Deb EWanner and SGreco Eds vol 6576 of Lecture Notes in Computer Science pp477ndash491 Springer Berlin Germany 2011

[19] J Knowles and D Corne ldquoOn metrics for comparing non-dominated setsrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo02) vol 1 pp 711ndash716 Honolulu HawaiiUSA May 2002

[20] I Kaliszewski Soft Computing for Complex Multiple CriteriaDecision Making vol 85 of International Series in OperationsResearch amp Management Science Springer 2006

[21] M Ehrgott Multicriteria Optimization Springer Berlin Ger-many 2nd edition 2005

[22] A Farhang-mehr and S Azarm ldquoMinimal sets of qualitymetricsrdquo in Proceedings of the 2nd International Conference onEvolutionary Multi-Criterion Optimization (EMO rsquo03) LectureNotes in Computer Science pp 405ndash417 Springer 2003

[23] M P Hansen and A Jaszkiewicz ldquoEvaluating the quality ofapproximations to the non-dominated setrdquo Tech Rep IMM-REP-1998-7 Institute of Mathematical Modelling TechnicalUniversity of Denmark 1998

[24] E Zitzler Evolutionary algorithms for multiobjective optimiza-tion methods and applications [PhD thesis] Swiss FederalInstitute of Technology (ETH) Zurich Switzerland 1999

[25] E Zitzler K Deb and LThiele ldquoComparison of multiobjectiveevolutionary algorithms empirical resultsrdquo Evolutionary Com-putation vol 8 no 2 pp 173ndash195 2000

[26] J G Villegas F Palacios andA LMedaglia ldquoSolutionmethodsfor the bi-objective (cost-coverage) unconstrained facility loca-tion problemwith an illustrative examplerdquoAnnals of OperationsResearch vol 147 pp 109ndash141 2006

[27] M Ehrgott and X Gandibleux ldquoHybrid metaheuristics formulti-objective combinatorial optimizationrdquo in Hybrid Meta-heuristics C Blum M J B Aguilera A Roli and M SampelsEds vol 114 of Studies in Computational Intelligence pp 221ndash259 Springer Berlin Germany 2008

[28] J Bramel and D Simchi-Levi The Logic of Logistics The-ory Algorithms and Applications for Logistics ManagementSpringer New York NY USA 1997

[29] M S Daskin Network and Discrete Location Models Algo-rithms and Applications Wiley-Interscience New York NYUSA 1st edition 1995

[30] Z Drezner and H Hamacher Facility Location Applicationsand Theory Springer Berlin Germany 2002

[31] R Z Farahani M SteadieSeifi and N Asgari ldquoMultiple criteriafacility location problems a surveyrdquo Applied MathematicalModelling vol 34 no 7 pp 1689ndash1709 2010

[32] C S Revelle and G Laporte ldquoThe plant location problem newmodels and research prospectsrdquo Operations Research vol 44no 6 pp 864ndash874 1996

[33] R G Reynolds New Ideas in Optimization McGraw-HillMaidenhead UK 1999

[34] R Landa Becerra and C A Coello Coello ldquoA cultural algorithmwith differential evolution to solve constrained optimizationproblemsrdquo in Advances in Artificial Intelligence (IBERAMIArsquo04) C Lemaıtre C Reyes and J A Gonzalez Eds vol 3315 ofLectureNotes inComputer Science pp 881ndash890 Springer BerlinGermany 2004

[35] C Soza R Landa M Riff and C Coello ldquoA cultural algo-rithm with operator parameters control for solving timetablingproblemsrdquo in Foundations of Fuzzy Logic and Soft ComputingP Melin O Castillo L Aguilar J Kacprzyk and W PedryczEds vol 4529 of Lecture Notes in Computer Science pp 810ndash819 Springer Berlin Germany 2007

[36] M Hoefer ldquoUflLib Benchmark Instances for the UncapacitatedFacility Location Problemrdquo 2014

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