parameter tuning for local-search-based matheuristic...

Research ArticleParameter Tuning for Local-Search-Based Matheuristic Methods

Guillermo Cabrera-Guerrero1 Carolina Lagos1 Carolina Castantildeeda2 Franklin Johnson3

Fernando Paredes4 and Enrique Cabrera5

1Pontificia Universidad Catolica de Valparaıso 2362807 Valparaıso Chile2Instituto Tecnologico Metropolitano Calle 73 No 76A-374 Vıa al Volador Medellın Colombia3Universidad de Playa Ancha Casilla 34-V Valparaıso Chile4Universidad Diego Portales 8370109 Santiago Chile5CIMFAV-Facultad de Ingenierıa Universidad de Valparaıso 2374631 Valparaıso Chile

Correspondence should be addressed to Guillermo Cabrera-Guerrero guillermocabrerapucvcl

Received 20 June 2017 Revised 2 October 2017 Accepted 25 October 2017 Published 31 December 2017

Academic Editor Kevin Wong

Copyright copy 2017 Guillermo Cabrera-Guerrero et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Algorithms that aim to solve optimisation problems by combining heuristics and mathematical programming have attractedresearchersrsquo attention These methods also known asmatheuristics have been shown to perform especially well for large complexoptimisation problems that include both integer and continuous decision variables One common strategy used by matheuristicmethods to solve such optimisation problems is to divide themain optimisation problem into several subproblemsWhile heuristicsare used to seek for promising subproblems exact methods are used to solve them to optimality In general we say that bothmixed integer (non)linear programming problems and combinatorial optimisation problems can be addressed using this strategyBeside the number of parameters researchers need to adjust when using heuristic methods additional parameters arise whenusing matheuristic methods In this paper we focus on one particular parameter which determines the size of the subproblemWe show howmatheuristic performance varies as this parameter is modified We considered a well-known NP-hard combinatorialoptimisation problem namely the capacitated facility location problem for our experiments Based on the obtained results wediscuss the effects of adjusting the size of subproblems that are generated when using matheuristics methods such as the oneconsidered in this paper

1 Introduction

Solving mixed integer programming (MIP) problems as wellas combinatorial optimisation problems is in general avery difficult task Although efficient exact methods havebeen developed to solve these problems to optimality as theproblem size increases exact methods fail to solve it within anacceptable computational time As a consequence nonexactmethods such as heuristic andmetaheuristic algorithms havebeen developed to find good quality solutions In additionhybrid strategies combining different nonexact algorithmsare also promising ways to tackle complex optimisationproblems Unfortunately algorithms that do not considerexact methods cannot give us any guarantee of optimalityand thus we do not know how good (or bad) solutions foundby these methods are

One hybrid strategy that combines nonexact methodsare memetic algorithms which are population-based meta-heuristics that use an evolutionary framework integratedwith local search algorithms [1] Memetic algorithms providelifetime learning process to refine individuals in order toimprove the obtained solutions every iteration or generationtheir applications have been grown significantly over theyears in several NP-hard optimisation problems [2] Thesealgorithms are part of the paradigm of memetic computa-tion where the concept of meme is used to automate theknowledge transfer and reuse across problems [3] A largenumber of memetic algorithms can be found in the litera-ture Depending on its implementation memetic algorithmsmight (or might not) give guarantee of local optimalityroughly speaking if heuristic local search algorithms areconsidered then no optimality guarantee is given if exact

HindawiComplexityVolume 2017 Article ID 1702506 15 pageshttpsdoiorg10115520171702506

2 Complexity

methods are considered as the local optimisers then localoptimality could be ensured

To overcome the situation described above hybrid meth-ods that combine heuristics and exact methods to solveoptimisation problems have been proposed These methodsalso known asmatheuristics [4] have been shown to performbetter than both heuristic and mathematical programmingmethods when they are applied separately The idea of com-bining the power of mathematical programming with flex-ibility of heuristics has gained attention within researchersrsquocommunity We can found matheuristics attempting to solveproblems arising in the field of logistics [5ndash9] health caresystems [10ndash13] and puremathematics [14 15] among othersMatheuristics have been demonstrated to be very effectivein solving complex optimisation problems Some interestingsurveys on matheuristics are [16 17] Although there is someoverlapping between memetic algorithms and matheuristicones in this paperwe have chosen to label the studied strategyas matheuristic as we think matheuristics definition betterfits the framework we are interested in

Because of their complexity MIP problems as well ascombinatorial optimisation problems are often tackled usingmatheuristic methods One common strategy to solve thisclass of optimisation problems is to divide the main opti-misation problem into several subproblems While heuristicsare used to seek for promising subproblems exact methodsare used to solve them to optimality One advantage of thisapproach is that it does not depend on the (non)linearity ofthe resulting subproblem Instead it has been pointed out thatit is desirable that the resulting subproblem would be convex[11] Having a convex subproblem would allow us to solve itto optimality and thus comparing solutions obtained at eachsubproblem becomes more senseful This strategy has beensuccessfully applied to problems arising in fields as diverse aslogistics and radiation therapy

In this paper we aim to study the impact of parametertuning on the performance of matheuristic methods as theone described above To this end the well-known capacitatedfacility location problem is used as an application of hardcombinatorial optimisation problem To the best of ourknowledge no paper has focused on parameter tuning formatheuristic methods

This paper is organised as follows Section 2 shows thegeneral matheuristic framework we consider in this paperDetails on the algorithms that are used in this study are alsoshown in this section In Section 3 the capacitated facilitylocation problem is introduced and itsmathematicalmodel isdescribed in Section 31 The experiments performed in thisstudy are presented and the obtained results are discussedin Section 32 Finally in Section 4 some conclusions arepresented and the future work is outlined

2 Matheuristic Methods

This section is twofold We start by describing a generalmatheuristic framework that is used to solve both MIPproblems and combinatorial optimisation problems and howit is different from other commonly used approaches such

as memetic algorithms and other evolutionary approachesAfter that we present the local-search-based algorithms weconsider in this work to perform our experiments We finishthis section by introducing the parameter we will be focusedon in this study

21 General Framework Equations (1a) to (1f) show thegeneral form of MIP problems Hereafter we will refer to thisproblem as the MIP(119909 119910) problem or themain problem

MIP (119909 119910) min119909119910

119891 (119909 119910) (1a)

st 119892119895 (119909 119910) le 119887119895 for 119895 = 1 119898 (1b)

119892119895 (119909) le 119887119895 for 119895 = 1 119898 (1c)

119892119895 (119910) le 119887119895 for 119895 = 1 119898 (1d)

119910 isin 0 1119900 (1e)

119909 isin R119899 (1f)

where 119891(119909 119910) is an objective function 119898 is the numberof inequality constraints on 119909 and 119910 119898 is the number ofinequality constraints on 119909 119898 is the number of inequalityconstraints on 119910 119900 is the number of binary (ge0) decisionvariables 119910 and 119899 is the number of continuous decisionvariables Combinatorial optimisation problems as the onewe consider in this paper can be easily obtained by eitherremoving the continuous decision variable from the modelor making it integer (ie 119909 isin Z119900ge)

Although there exist a number of exact algorithms thatcan find an optimal solution for the MIP(119909 119910) as the size ofthe problem increases exact methods fail or take too longBecause of this heuristic methods are used to obtain goodquality solutions of the problem within an acceptable timeHeuristic methods cannot guarantee optimality though

During the last two decades the idea of combin-ing heuristic methods and mathematical programming hasreceived much more attention Exploiting the advantages ofeach method appears to be a senseful strategy to overcometheir inherent drawbacks Several strategies have been pro-posed to combine heuristics and exact methods to solveoptimisation problems such as the MIP(119909 119910) problem Forinstance Chen and Ting [18] and Lagos et al [8] com-bine the well-known ant colony optimisation (ACO) algo-rithm and Lagrangian relaxation method to solve the singlesource capacitated facility location problem and a distribu-tion network design problem respectively In these articlesLagrangian multipliers are updated using ACO algorithmAnother strategy to combine heuristics and exact methodsis to let heuristics seek for subproblems of MIP(119909 119910) whichin turn are solved to optimality by some exact method Onealternative to obtain subproblems of MIP(119909 119910) is to add aset of additional constraints on a subset of binary decisionvariablesThese constraints are of the form 119910119894 = 0 with 119894 isin IandI being the set of index that are restricted in subproblemMIPI(119909 119910) (see (1a) to (1f)) The portion of binary decisionvariables 119910119894 that are set to 0 is denoted by 120572 (ie I = 120572times 119900)

Complexity 3

with 119900 being the number of binary variables 119910119894 Then theobtained subproblem which we call MIPI(119909 119910) is

MIPI (119909 119910) min119909119910

119891 (119909 119910) (2a)

st 119892119895 (119909 119910) le 119887119895

for 119895 = 1 119898(2b)

119892119895 (119909) le 119887119895 for 119895 = 1 119898 (2c)

119892119895 (119910) le 119887119895 for 119895 = 1 119898 (2d)

119910119894 = 0 119894 isin I (2e)

119910 isin 0 1119900 (2f)

119909 isin R119899 (2g)

In this paper we assume that a constraint on the 119894thresource of vector 119910 of the form 119910119894 = 0 that is 119894 isin Imeans that such a resource is not available for subproblemMIPI(119909 119910) We can note that as the number of constrainedbinary decision variables 119910119894 associated withI increases thatis as (1 minus 120572) increases the subproblem MIPI(119909 119910) becomessmaller Similarly as the number of constrained binarydecision variables 119910119894 associated with I gets smaller that is(1 minus 120572) decreases the subproblem MIPI(119909 119910) gets largeras there are more available resources It is also easy to notethat as (1 minus 120572) increases we can obtain optimal solutions ofthe corresponding subproblem MIPI(119909 119910) relatively fasterHowever quality of the obtained solutions is usually impairedas the solution space is restricted by the additional constraintsassociated with the set I Similarly as (1 minus 120572) gets smallerobtaining optimal solutions of the associated subproblemsmight take longer but the quality of the obtained solutionsis in general greatly improved Finally when 120572 = 0 we havethat subproblemMIPI(119909 119910) is identical to themain problemMIP(119909 119910) We assume that optimal solution of subproblemMIPI(119909 119910) (119909 119910) is also feasible for theMIP(119909 119910) problemMoreover we can note that there must be a minimal setI for which an optimal solution (119909 119910) of the associatedsubproblem MIPI(119909 119910) is also an optimal solution (119909lowast 119910lowast)of the main problem MIP(119909 119910)

In some cases the value of 120572 might be predefined bythe problem that is being solved For instance the problemof finding the best beam angle configuration for radiationdelivery in cancer treatment (beam angle optimisation prob-lem) usually sets the number of beams to be used in a beamangle configuration (see Cabrera-Guerrero et al [11] Li etal [19] and Li et al [20]) This definition is made by thetreatment planner and it does not take into account thealgorithm performance but clinical aspects Unlike this kindof problems there are many other problems where the valueof 120572 is not predefined and then setting it to an efficient valueis important for the algorithm performance Figure 1 showsthe interaction between the heuristic and the exact method

One distinctive feature ofmatheuristics is that there existsan interaction between the heuristic method and the exactmethod In the method depicted in Figure 1 we have that

Heuristicmethod

Exactmethod

(xlowast ylowast)

Seeks for new ℐ

ℐ

３ＩＦＰＭ－）０ℐ(x y)

Figure 1 Interaction between a heuristic method and an exactmethod

on one hand the heuristic method influences the solverby passing onto it a set of constraints I that defines thesubproblem to be solved by the exact method On the otherhand we have that the solution obtained by the solver isreturned to the heuristic method such that it can be usedto influence the selection of the elements in the next set ofconstraints that is the solution of a subproblem might beused by the heuristic to obtain some useful information togenerate next subproblems

As mentioned above there is one parameter that is notpart of the set of parameters of the heuristic method norpart of the set of parameters of the exact method Thisparameter which we call 120572 only comes up after these twomethods are posed together Then in this paper we areinterested in how the choice of 120572 canmodify the performanceof the proposed strategy in terms of the quality of theobtained solutions of themain problem Sincemany differentmatheuristic frameworks might be proposed to solve theMIP(119909 119910) problem we restrict this study to local-search-based matheuristics Thus in this study three local-search-based matheuristic algorithms are implemented and theirresults compared Additionally we implement a very simplemethod which we call ldquoblind algorithmrdquo as a baseline for thisstudy Next sections explain these algorithms

22 Local Search Algorithms As mentioned in previoussections the aimof this paper is not to provide a ldquostate-of-the-artrdquo algorithm to solve MIP problems but instead to studythe effect that changing the size of subproblems has on thequality of the obtained solutions when using local-search-based matheuristic methods as the one described in Sec-tion 21Thus three local search algorithms are implementednamely steepest descent (SD) next descent (ND) and tabusearch (TS) Local search algorithms need a neighbourhoodN to be defined bymeans of a neighbourhoodmovementWedefine the same neighbourhood movement for the all threemethods

Since the local search algorithms move on the sub-problem space that is the local search algorithms look forpromising subproblemsMIPI(119909 119910) or equivalently look forpromising sets I then we need to define a neighbourhoodwithin the same search space Thus the neighbour set of Iwhich we denote by N(I) or simply I1015840 corresponds toall resources 119910119894 such that they either are not considered in

4 Complexity

Input120572 (portion of the resources included inI)Output (119909 119910) (locally optimal solution)

(1) begin(2) 119896 = 0(3) I119896 = selectBinaryVariablesRandomly(120572)(4) (119909 119910) = solve MIPI119896

(119909 119910)(5) repeat(6) localOptimum = true(7) 119896 = 119896 + 1(8) foreachI1015840119896minus1 isin N(119868119896minus1) do(9) (1199091015840 1199101015840) = solve MIPI1015840

119896minus1

(119909 119910)(10) IfMIP(1199091015840 1199101015840) lt MIP(119909 119910) then(11) (119909 119910) = (1199091015840 1199101015840)(12) I119896 = I1015840119896minus1(13) localOptimum = false(14) until localOptimum(15) return (119909 119910)

Algorithm 1 Steepest descent algorithm

the optimal solution of MIPI(119909 119910) or are actually part ofI Since the number of resources 119910119894 that meet these criteriamight be above or below the number of elements set I1015840must contain according to the parameter 120572 that is beingconsidered we randomly select among the variables thatmeet these criteria until the neighbour set I1015840 is completedThis neighbourhood movement ensures that those resources119910119894 that are part of the optimal solution of subproblemMIPI(119909 119910) that is 119910119894 notin I and 119910119894 = 1 will continue to beavailable for sets I1015840 isin N(I) Thus the set of resources 119910119894that can potentially be part of sets I1015840 isin N(I) is defined asfollows

I1015840 isin N (I) = 119910119894 119910119894 isin I or (119910119894 notin I 119910119894 = 0) (3)

where (119909 119910) is the optimal solution ofMIPI(119909 119910)The initialsetI local search methods start with is set randomly for theall three algorithms implemented in this paper

221 Steepest Descent Algorithm The steepest descent algo-rithm starts with an initial I0 for which its associatedsubproblem MIPI0

(119909 119910) is labelled as the current subprob-lem For this current subproblem its entire neighbourhoodN(I0) is generated and the neighbour that leads to the bestobjective function value is selected If the best neighboursubproblem is better than the current one then the bestneighbour is set as the new current solution If the bestneighbour is not better than the current subproblem thenthe algorithm stops and the optimal solution of the currentsubproblem is returned as a locally optimal solution of themainMIP(119909 119910) problem Algorithm 1 shows the pseudocodefor the steepest descent algorithm implemented in this paper

Although the steepest descent algorithm can be con-sidered in general a deterministic algorithm in the sensethat given an initial solution it converges to the same localoptima in our implementation the algorithm does not visitall possible neighbours and therefore it becomes a stochasticlocal search Since only one local optimum is generated at

each run we repeat the algorithm until the time limit isreached Same is done for both ND and TS algorithms weintroduce next

222 Next Descent Algorithm Asmentioned in Section 221the steepest descent might take too long to converge if thesize of the neighbourhood is too large or solving an individualsubproblem is too time-consuming Thus we include in thispaper a local search algorithm called next descent that aimsto converge faster than the steepest descent without a majorimpact on the solution quality Algorithm 2 shows the nextdescent method

Just as in the steepest descent algorithm the next descentalgorithm starts with an initial solution which is labelledas the current solution Then a random element from theneighbourhood of the current solution is selected and solvedand the objective function value of its optimal solution iscompared to the objective function value of the current solu-tion Like in the SD algorithm if the neighbour solution is notbetter than the current solution another randomly selectedset I1015840 from the neighbourhood N(I119896minus1) is generated andcompared to the current solutionUnlike the SD algorithm inthe ND algorithm if the neighbour is better than the currentsolution then the neighbour is labelled as the new currentsolution andnootherI1015840 isin N(I119896minus1) is visitedThe algorithmrepeats these steps until the entire neighbourhood has beencomputed with no neighbour resulting in a better solutionthan the current one in which case the algorithm stops

223 Tabu Search Algorithm Unlike the algorithms de-scribed in the previous sections tabu search is a localsearch technique guided by the use of adaptive or flexiblememory structures [5] and thus it is inherently a stochasticlocal search algorithm The variety of the tools and searchprinciples introduced and described in [21] are such that theTS can be considered as the seed of a general frameworkfor modern heuristic search [22] We include TS as it has

Complexity 5




119896minus1

(119909 119910)(10) ifMIP(1199091015840 1199101015840) lt MIP(119909 119910) then(11) (119909 119910) = (1199091015840 1199101015840)(12) I119896 = I1015840119896minus1(13) localOptimum = false(14) break(15) until localOptimum(16) return (119909 119910)

Algorithm 2 Next descent algorithm

been applied to several combinatorial optimisation problems(see eg [5 23ndash27]) including of course mixed integerprogramming problems as the one we consider in this study

As in the algorithms introduced above TS also startswith an initial set of constraints I0 for which its associatedsubproblem MIPI0

(119909 119910) is solved Then as in the SD algo-rithm the ldquoentirerdquo neighbourhood of the current solutionis computed As explained before since the neighbourhoodN has a stochastic component it is actually not possible togenerate the entire neighbourhood of a solution howeverwe use the term ldquoentirerdquo to stress the fact that all thegenerated neighbours of N are solved This is different fromthe ND algorithm where not all the generated neighboursare necessarily solved The number of generated neighboursns is as in the algorithms above set equal to 10 MoreoverTS implements a list called tabu list that aims to avoid cyclesduring the search Each time a neighbourhood is generatedthewarehouses removed from 119868(119896minus1) aremarked as tabuOncewe have solved the subproblems MIPI1015840

119896

(119909 119910) with I1015840119896 isinN(119868119896) its optimal solutions are ranked and the one with thebest objective function value is chosen as candidate solutionfor the next iteration If the candidate solution was generatedusing amovement within the tabu list it should be discardedand the next best neighbour of the list should be chosen as thenew candidate solution However there is one exception tothis rule if the candidate solution is within the tabu list butits objective function value is better than the best objectivefunction found so far by the algorithm then the so called aspi-ration criterion is invoked and the candidate solution is set asthe new current solution and passed on to the next iteration

Unlike the algorithms introduced above TS does notrequire next current solution to be better than the previousone This means that it is able to avoid local optimal bychoosing solutions that are more expensive as they allows thealgorithm to visit other (hopefully promising) areas in thesearch space However in case the algorithm cannot make

any improvement after a predefined number of iterationsa diversification mechanism is used to get out from low-quality neighbourhoods and ldquojumprdquo to other neighbour-hoods The diversification mechanism implemented here is arestart method which set the current solution to a randomlygenerated solution without losing the best solution found sofar Termination criterion implemented here is the time limit

The tabu search implemented in this paper is as followsAs Algorithm 3 shows tabu search requires the following

parameters(i) Time limit total time the algorithm will perform(ii) DiversBound total number of iteration without

improvements on the best solution before diversifica-tion criterion (restart method) is applied

(iii) TabuListSize (ts) size of tabuList Number of itera-tions for which a specific movement remains banned

224 Blind Algorithm We finally implement a very simpleheuristic method we call blind as it moves randomly ateach iteration Pseudocode of this method is presented inAlgorithm 4

As we can see no additional intelligence is added to theblind algorithm It is just a random search that after a pre-defined number of iterations (or any other ldquostop criterionrdquo)returns the best solution it found during its search Thus wecan consider this algorithm as a baseline of this study

We apply the all four algorithms described in this sectionto two prostate cases Details on this case and the obtainedresults are presented in the next section

23 Subproblem Sizing All the algorithms above performvery different as the value of the input parameter 120572 variesOn the one hand setting 120572 to a very small value provokesthat either the solver fails to solve the subproblem becauseof lack of memory or it takes too long to find an optimal

6 Complexity

Input120572 (portion of the resources included inI119896)Output (119909 119910) (approximately optimal solution)

(1) begin(2) 119896 = 0(3) I119896 = selectBinaryVariablesRandomly(120572)(4) (119909119888 119910119888) = solve MIP119896(119909 119910) set current sol

(5) (119909 119910) = (119909119888 119910119888) set best sol

(6) Y = 0 tabu list initially empty

(7) 119899119900Im119901119903119900V119890119898119890119899119905119862119900119906119899119905119890119903 = 0(8) repeat(9) C = 0(10) 119896 = 119896 + 1(11) foreachI1015840119896minus1 isin N(1198681015840119896minus1) do(12) (119909119899 119910119899) = solve MIPI1015840

119896minus1

(119909 119910)(13) C = C cup (119909119899 119910119899)(14) Sort(C)(15) foreach (119909119899 119910119899) isin C do(16) if isTabu((119909119899 119910119899)) then(17) ifMIP(119909119899 119910119899) lt MIP(119909 119910) then(18) Y = 119906119901119889119886119905119890119879119886119887119906119871119894119904119905(Y (119909119888 119910119888) (119909119899 119910119899) ts)(19) (119909119888 119910119888) = (119909119899 119910119899)(20) (119909 119910) = (119909119899 119910119899)(21) I119896 = I119899119896minus1(22) 119899119900Im119901119903119900V119890119898119890119899119905119862119900119906119899119905119890119903 = 0(23) break(24) else(25) Y = 119906119901119889119886119905119890119879119886119887119906119871119894119904119905(Y (119909119888 119910119888) (119909119899 119910119899))(26) (119909119888 119910119888) = (119909119899 119910119899)(27) I119896 = I119899119896minus1 if MIP(119909119899 119910119899) lt MIP(119909 119910) then(28) (119909 119910) = (119909119899 119910119899)(29) 119899119900Im119901119903119900V119890119898119890119899119905119862119900119906119899119905119890119903 = 0(30) else(31) 119899119900Im119901119903119900V119890119898119890119899119905119862119900119906119899119905119890119903 + +(32) break(33) if 119899119900Im119901119903119900V119890119898119890119899119905119862119900119906119899119905119890119903 ge 119889119894V119890119903119904119861119900119906119899119889 then(34) I119896 = selectBinaryVariablesRandomly(120572)(35) until time limit is reached(36) return (119909 119910)

Algorithm 3 Tabu search algorithm

To find an (approximately) optimal solution for MIP(119909 119910)Input120572 (portion of the resources included inI119896)Output (119909 119910) (approximately optimal solution)

(1) begin(2) 119896 = 0(3) I119896 = selectBinaryVariablesRandomly(120572)(4) (119909 119910) = solve (MIPI119896

(119909 119910))(5) while stopCriterion do(6) 119896 = 119896 + 1(7) I119896 = selectBinaryVariablesRandomly(120572)(8) (119909119888 119910119888) = solve (MIPI119896

(119909 119910))(9) if MIP(119909119888 119910119888) lt MIP(119909 119910) then(10) (119909 119910) = (119909119888 119910119888)(11) I119896 = I119896minus1(12) return (119909 119910)

Algorithm 4 Matheuristic method using the blind algorithm

Complexity 7

solution of the subproblemOn the other hand setting120572 closeto the total number of binary decision variables provokesthat the algorithm fails to find a solution for the generatedsubproblems as there is no feasible solution to it (ie thesubproblems are too restrictive) Thus we have to find avalue of 120572 such that exact methods can solve the obtainedsubproblems within an acceptable time Further 120572 shouldallow local search methods to iterate as much as neededSolving the obtained subproblems within few seconds iscritical to matheuristic methods as they usually need severaliterations before to converge to a good quality solutionThis isespecially true for matheuristics that consider local search orpopulation-based heuristic methods Then there is a trade-off between the quality of the solution of subproblems andthe time that is needed to generate such solutions Thereforefinding a value of 120572 that gives us a good compromise betweenthese two aspects is critical for the overall performance of thematheuristic methods explained before It is interesting thatthe problem of finding efficient values of 120572might be seen as amultiobjective optimisation problem

In next section we explain the experiments that weperform to study how the choice of 120572 hits local-search-basedmatheuristics performance Based on the results we draftsome guidelines to set value of parameter 120572 at the end of nextsection

3 Computational Experiments

This section starts briefly introducing the problem we con-sider in this paperThen the experiments performed here arepresented and their results are discussed

31 The Capacitated Facility Location Problem The capac-itated facility location problem (CFLP) is a well-knownproblem in combinatorial optimisation The CFLP has beenshown to be NP-hard [28] The problem consists of selectingspecific sites at which to install plants warehouses anddistribution centres while assigning customers to servicefacilities and interconnecting facilities using flow assignmentdecisions [23] In this study we consider the CFLP problemto evaluate the performance of a very simple matheuristicalgorithm We consider a two-level supply chain in which asingle plant serves a set of warehouses which in turn servea set of end customers or retailers Figure 2 shows the basicconfiguration of our supply chain Thus the goal is to find aset of locations that serves the entire set of customers in anoptimal way As Figure 2 shows each customer (or cluster) isserved only by one warehouse

The optimisation model considers the installation cost(ie the cost associated with opening a specific warehouse)and transportation or allocation cost (ie the cost of trans-porting one item from a warehouse to a customer) Themathematical model for the CFLP is

CFLP (119909 119910) min119909119910

119899

sum119894=1

(119891119894119910119894) +119899

sum119894=1

119898

sum119895=1

(119888119894119895119889119895119909119894119895) (4a)

st119898

sum119895=1

119889119895119909119894119895 le 119868cap119894 119910119894 forall119894 = 1 119899 (4b)

Customers(clusters)

Warehouse 1

Warehouse 2

Warehouse 3

warehousePlant or central

Figure 2 Distribution network structure considered in this studyIt consists of one central plant a set of potential warehouses and aset of customers or retailers [23]

119899

sum119894=1

119909119894119895 = 1 forall119895 = 1 119898 (4c)

119909119894119895 le 119910119894

forall119894 = 1 119899 forall119895 = 1 119898(4d)

119910119894 119909119894119895 isin 0 1

forall119894 = 1 119899 forall119895 = 1 119898(4e)

Equation (4a) is the total system costThe first term is thefixed setup and operating cost when openingwarehousesThesecond term is the daily transport cost between warehouseand customers which depends on the customer demand119889 and distance 119888119894119895 between warehouse 119894 and customer 119895Inequality (4b) ensures that total demand of warehouse 119894 willnever be greater than its capacity 119868cap119894 Equation (4c) ensuresthat customers are served by only one warehouse Equation(4d) makes sure that customers are only allocated to availablewarehouses Finally (4e) states integrality (0minus1) for the binaryvariables 119909119894119895 and 119910119894 Other versions of the CFLP relax thisconstraints by making 119909 isin R119899times119898 with 0 le 119909119894119895 le 1 In thatcase the CFLP would be just as the MIP(119909 119910) problem

32 Experiments In this paper three benchmarks for theCFLP are considered The first benchmark corresponds toproblem sets AB and C from the OR Library [29]Instances in problem setsAB andC consider 1000 clientsand 100 warehouses Warehouses capacity for instances A1A2 A3 and A4 are equal to 8000 10000 12000 and14000 respectively Warehouses capacity for instances B1B2 B3 and B4 are equal to 5000 6000 7000 and 8000respectively Warehouses capacity for instances C1 C2 C3and C4 are equal to 5000 5750 6500 and 7250 respec-tively Table 1 shows these instances and their correspondingoptimal values obtained by the MILP solver Column 119905 shows

8 Complexity

0

1

Cluster centreCustomers

12

08

06

04

02

0 1 1208060402

(a) Clients distributed in clusters

Customers

0

1

12

08

06

04

02

0 1 1208060402

(b) Clients uniformly distributed

Figure 3 Example of the instances considered in this study

Table 1 Instances for the first benchmark used in this study (OR Library)

OR LibraryInstances |119868| times |119869| 119868cap 119891(119909lowast) 119905 (sec)A1 1000 times 100 8000 1924082245 1745A2 1000 times 100 10000 1843804654 1778A3 1 000 times 100 12000 1776520195 199A4 1000 times 100 14000 1716043901 8B1 1000 times 100 5000 1365637958 136B2 1000 times 100 6000 1336192745 151B3 1000 times 100 7000 1319855643 256B4 1000 times 100 8000 1308251650 27C1 1000 times 100 5000 1164659697 155C2 1000 times 100 5750 1157034029 70C3 1000 times 100 6500 1151874374 31C4 1000 times 100 7250 1150576739 24

the time needed by the MILP solver to reach the optimalsolution

The second benchmark is a set of instances where clientsand warehouses are uniformly distributed over an imaginarysquare of 100times100[distance units]2 (see Figure 3(a))We callthis set DS1119880 The number of clients considered in instancesbelonging to setDS1119880 ranges from 500 to 1000 (500 600 700800 900 and 1000) while the number of warehouses consid-ered varies between 500 and 1000Thus set DS1119880 consists of12 problem classes (500times500 500times600 500times900 500times1000 1000 times 500 1000 times 1000) For each problem class10 instances are randomly generated using the procedureproposed in [30] and that was also used in [5 23] We do thisin order to minimise any instance dependant effect Table 2shows the average values for each class of problems

Finally a third benchmark consisting on clients that areorganised in clusters is considered We call this benchmark

DS1119862 (see Figure 3(b)) Instances in DS1119862 are generatedvery similar to the ones in DS1119880 The only difference is thatclients in DS1119862 are not uniformly distributed and thus thosewarehouses that are within (or very close to) a cluster have aninstallation cost slightly higher than those warehouses thatare far away from the clusters

Table 2 shows the results obtained by theMIP solver fromGurobi when solving each instance of DS1119862 As we can seethe MIP solver is able to solve all the instances to optimalityFurther the solver finds the optimal solution for almost allthe instances in less than 1800 secs Columns 119905avg 119905min and119905max in Table 2 show the average time and both minimumand maximum times respectively This is because we solve10 different instances for each instance class As mentionedbefore we do this to avoid any instance dependent effect

After we have solved the problem using the MIP solverwe apply the all three local-search-based matheuristics and

Complexity 9

Table 2 Instances for the DS1119880 and DS1119862 benchmarks

Instances DS1 119880 DS1 119862|119868| times |119869| 119891(119909lowast) 119905avg 119905min 119905max 119891(119909lowast) 119905avg 119905min 119905max

(1) 500 times 500 1206703 170 95 287 1239007 191 99 410(2) 500 times 1000 1105345 267 194 356 1135271 262 201 410(3) 600 times 500 1495399 261 105 424 1493432 287 120 522(4) 600 times 1000 1418477 399 309 568 1374514 331 255 501(5) 700 times 500 1824349 427 128 674 1811493 292 96 645(6) 700 times 1000 1618827 726 263 1515 1618729 596 306 1673(7) 800 times 500 2093195 455 227 748 2070369 343 131 648(8) 800 times 1000 1936477 761 399 1679 1888597 1040 590 1828(9) 900 times 500 2434037 391 179 681 2362877 428 178 1024(10) 900 times 1000 2113024 1228 672 1787 2154107 1158 507 1564(11) 1000 times 500 2704499 448 189 1484 2658059 407 154 642(12) 1000 times 1000 2436250 1187 670 1791 2395198 1419 588 2653

0

1

2

3

4

5

GA

P va

lue (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

(a) Evolution of the GAP for119874119877119871119894119887A instances

200

400

600

800

1000Ti

me (

sec)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

(b) Evolution of the time for119874119877119871119894119887A instances

Figure 4 Average results obtained by the local search algorithms for each value of (1 minus 120572) for 119874119877119871119894119887A instances

the blind algorithm to each instance and allow them to runfor 2000 secs The proposed algorithms solve each instance10 times for each value of (1 minus 120572) Table 3 shows the resultsobtained by the local-search-based matheuristic methods forinstances A B and C for four different values of (1 minus120572) As expected the blind algorithm consistently obtainsGAP values much higher than the other three methodsFurther local-search-based algorithms are able to find theoptimal solution for the majority of the instances While theaverage GAP for the blind algorithm is 165 the averageGAP for the SD ND and TS methods is 033 036and 015 respectively As expected the best average valueis obtained by the TS algorithm while difference betweenSD and ND algorithms is negligible Regarding the timeneeded by the algorithms to converge the blind algorithm

is the one that takes longer (770 secs) As mentioned beforethe ND algorithm is in average the fastest one with 299seconds while the SD algorithm almost doubles this timewith 547 seconds in average TS algorithm takes 371 secondsin average before convergence

Figures 4(a) 5(a) and 6(a) show the evolution of theGAP as parameter (1 minus 120572) increases As we expected thelarger the value of (1 minus 120572) is set to the smaller the GAPAlso it is clear that the blind algorithms obtain higher GAPvalues than the other three algorithms considered in thisstudy As we mentioned before the blind algorithm worksas our baseline algorithm It is interesting to note thatfor all algorithms the worst performance is obtained for(1 minus 120572) = 01 This is mainly because for this value notenough facilities are available and the algorithms open a set

10 Complexity

Table3Re

sults

obtained

bythelocalsearch

metho

dsforthe

ORLibraryrsquos

insta

nces

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

A1

01

522

622

965

248

945

1424

673

785

03

163

1124

000

1240

000

489

000

736

05

032

664

000

1095

000

274

024

700

07

010

745

000

366

000

138

000

1476

A2

01

431

463

000

833

000

1415

000

866

03

122

893

000

366

019

1290

019

615

05

052

987

183

52009

956

000

745

07

011

282

009

378

000

1562

000

743

A3

01

181

841

000

341

031

65000

686

03

023

503

000

602

000

693

000

1083

05

012

880

000

431

000

442

000

659

07

000

272

000

151

000

211

000

211

A4

01

156

990

000

25000

84000

8403

000

411

000

8000

31000

3105

000

111

000

2000

14000

1407

000

23000

1000

7000

7

B1

01

522

622

003

639

023

960

001

319

03

163

1124

001

588

008

436

003

1452

05

032

664

000

707

003

216

000

605

07

010

745

001

347

000

1630

000

1192

B2

01

431

463

144

806

275

222

009

714

03

122

893

000

135

000

69000

1165

05

052

987

000

186

000

196

005

530

07

011

282

000

112

005

315

000

261

B3

01

181

841

040

403

043

1270

030

142

03

023

503

000

96000

181

000

267

05

012

880

000

588

000

236

000

564

07

000

272

000

179

000

316

000

316

B4

01

156

990

000

1309

032

144

032

144

03

000

411

000

24000

360

000

360

05

000

111

000

57000

381

000

381

07

000

23000

28000

349

000

349

C1

01

809

712

970

1220

814

921027

133

03

320

919

000

657

000

486

000

394

05

199

1105

000

173

000

496

000

1469

07

048

863

000

239

000

355

000

247

Complexity 11

Table3Con

tinued

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

C2

01

574

937

142

1254

232

1397

070

8403

164

934

000

136

000

107

000

172

05

035

562

000

1225

000

248

000

537

07

002

766

000

1294

000

164

000

169

C3

01

517

979

111

1254

154

819

020

585

03

104

993

000

73000

38000

272

05

022

906

000

587

000

64000

1387

07

004

740

000

551

000

149

000

779

C4

01

486

663

135

153

034

523

085

313

03

065

711

003

7003

35003

107

05

008

821

000

18003

11003

1307

002

228

003

8003

22000

29

12 Complexity

0

1

2

3

4

5G

AP

valu

e (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

(a) Evolution of the GAP for119874119877119871119894119887B instances

200

400

600

800

Tim

e (se

c)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

(b) Evolution of the time for119874119877119871119894119887B instances

Figure 5 Average results obtained by the local search algorithms for each value of (1 minus 120572) for 119874119877119871119894119887B instances

GA

P va

lue (

)

0

1

2

3

4

5

6

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

(a) Evolution of the GAP for119874119877119871119894119887C instances

Tim

e (se

c)

200

400

600

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

(b) Evolution of the time for119874119877119871119894119887 C instances

Figure 6 Average results obtained by the local search algorithms for each value of (1 minus 120572) for 119874119877119871119894119887C instances

of facilities randomly so the problem has a feasible solutionHowever this repairing process is completely random andthus algorithms performance is impaired

Since the problems from theORLibrary are onlymediumsize instances local-search-based matheuristics consistentlyfind the optimal solution for almost all instances for (1minus120572) =03 05 07 In fact even the blind algorithm finds solutionsthat are very close to the optimal ones when (1 minus 120572) = 07

Figures 4(b) 5(b) and 6(b) show the time needed byour algorithms to find its best solution as parameter (1 minus 120572)increases Unlike we expected the time needed to find thebest solution does not increase as the parameter (1 minus 120572)gets larger In fact both algorithms SD and blind convergefaster when (1 minus 120572 = 07) the larger value we triedin our experiments This can be explained because of theproblems features We note that optimal solutions for all the

Complexity 13

GA

P va

lue (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

(a) Evolution of the GAP for DS1119880 instancesG

AP

valu

e (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

(b) Evolution of the GAP for DS1119862 instances

Figure 7 Average GAP values obtained by the local search algorithms for each value of (1 minus 120572) for DS1 instances

problems in the benchmark of the OR Library need not toomany warehouses to be open Thus when a large portionof the potential warehouses are available (as for the (1 minus120572) = 07 case) the algorithm is likely to find such optimalsolutions in early iterations In fact for the vast majority ofthe experiments optimal solution is found within the first 2to 3 iterations

We now move on the DS1119880 and the DS1119862 benchmarksAs we noted before these sets of instances are much largerand harder to solve than the problems in the OR Library wediscussed above Further as the optimal solutions of theseinstances do not require too many warehouses to be openthe repairing procedure applied to the OR Library instancesfor (1 minus 120572) = 01 does not have a great impact on theresults Thus local-search-based matheuristics outperformthe results obtained by the blind algorithm for all values ofparameter 120572 Figures 7(a) and 7(b) show the GAP valuesobtained by the all four algorithms for instances DS1119880 andDS1119862 respectively

Just as in theORLibrary instances as the parameter (1minus120572)gets larger the GAP approximates to 0 for the all three local-search-based matheuristic proposed in this paper While forthe DS1119880 instances both the TS and the ND algorithms reachGAP values very close to 0 for (1 minus 120572) = 05 and (1 minus 120572) =07 for the DS1119862 instances the best value obtained by theTS and the ND algorithms is 071 and 061 respectively for(1 minus 120572) = 07 Notably the ND algorithm performs slightlybetter than the TS for the DS1119862 instance when (1 minus 120572) =05 obtaining a best average value of 066 against the 142obtained by the TS algorithm In spite of that we can notethat the differences between the GAP obtained by both theND and the TS algorithms when using (1 minus 120572) = 05 and(1minus120572) = 07 are negligible just as in theORLibrary instances

Figures 8(a) and 8(b) shows the time needed by the allfour algorithms to find their best solution for instances DS1119880and DS1119862 respectively As expected as the parameter (1 minus120572) gets larger longer times are needed by the algorithms toconverge This is important as one would like to use a valuefor (1minus120572) such thatminimumGAP values are reached withinfew minutes

Results in Figures 7 and 8 show clearly that there is acompromise between the quality of the solution found by thealgorithms and the time they need to do so Further we knowthat this compromise can be managed by adjusting the valueof parameter (1minus120572) For the case of instancesDS1119880 andDS1119862such a value should be between 04 and 06 Moreover for thecase of the instances of the OR Library parameter (1 minus 120572)should be set around 03 It is interesting to note that whilethe size of instances DS1119880 and DS1119862 is equivalent instancesof the OR Library are much smaller

We can also note that instances that include clusterstend to take longer to converge This is especially true asparameter (1 minus 120572) gets larger Further for these instances asthe parameter (1minus120572) gets larger less iterations are performedby algorithms although each of these iterations takes muchlonger As mentioned before these should be taken intoaccount when using population-based algorithms within theframework presented in this study as such kind of heuristicalgorithms needs to perform several iterations before toconverge to good quality solutions

4 Conclusions and Future Work

In this paper we show the impact of parameter tuningon a local-search-based matheuristic framework for solving

14 ComplexityTi

me (

sec)

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

1200

1400

1600

1800

2000

(a) Evolution of the time for DS1119880 instances

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

Tim

e (se

c)

800

1000

1200

1400

1600

1800

2000

(b) Evolution of the time for DS1119862 instances

Figure 8 Average times needed by the local search algorithms for each value of (1 minus 120572) for DS1 instances

mixed integer (non)linear programming problems In partic-ular matheuristics that combine local search methods anda MIP solver are tested In this study we focus on the sizeof the subproblem generated by the local search methodthat is passed on to the MIP solver As expected the size ofthe subproblems that are solved in turn by the matheuristicmethod has a big impact on the behaviour of thematheuristicand consequently on its obtained results as the size ofthe subproblem increases (ie more integerbinary decisionvariables are considered) the results obtained by the MIPsolver are closer to the optimal solution The time requiredby the algorithms tested in this paper to find its best solutionalso increases as the subproblem gets larger Further as thesubproblem gets larger fewer iterations can be performedwithin the allowed timeThis is important as other heuristicssuch as evolutionary algorithms and swarm intelligencewhere many iterations are needed before converging to agood quality solution might be not able to deal with largesubproblems We also note that the improvement in the GAPvalues after certain value of parameter (1 minus 120572) is negligibleand that this value depends to some extent on the size of theproblem for medium size instances such as the ones in theOR Library parameter (1minus120572) should be set to a value around03 while for larger instances (DS1119880 and DS1119862) it shouldbe set to a value between 04 and 06 The specific valueswill depend on both the accuracy level requested by usersand the time available to perform the algorithm Thereforethe challenge when designing matheuristic frameworks asthe one presented in this paper is to find a value forparameter (1 minus 120572) that allows the heuristic algorithm toperform as many iterations as needed and that provides agood compromise between solution quality and run timesAlthough the values provided here are only valid for the

problem and the algorithms considered in this paper wethink that the obtained results can be used as a guide byother researchers using similar frameworks andor dealingwith similar problems

As a future work strategies such as evolutionary algo-rithms and swarm intelligence will be tested within thematheuristic framework considering the results obtained inthis study We expect that intelligent methods such as theones named before greatly improve the results obtained bythe local search methods considered in this study Moreoverthe matheuristic framework used in this paper might alsobe applied to other MILP and MINLP problems such as forinstance the beam angle optimisation problem in radiationtherapy for cancer treatment

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Guillermo Cabrera-Guerrero wishes to acknowledge Inge-nierıa 2030 DI 0394162017 and CONICYTFONDECYTINICIACION11170456 projects for partially supporting thisresearch

References

[1] F Neri and C Cotta ldquoMemetic algorithms and memeticcomputing optimization a literature reviewrdquo Swarm and Evo-lutionary Computation vol 2 pp 1ndash14 2012

Complexity 15

[2] Y-s Ong M H Lim and X Chen ldquoMemetic Computa-tionmdashPast Present amp Futurerdquo IEEE Computational IntelligenceMagazine vol 5 no 2 pp 24ndash31 2010

[3] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011

[4] V Maniezzo T Stutzle and S Voszlig Eds Matheuristics -Hybridizing Metaheuristics and Mathematical Programmingvol 10 of Annals of Information Systems Springer 2010

[5] G Guillermo Cabrera E Cabrera R Soto L J M Rubio BCrawford and F Paredes ldquoA hybrid approach using an artificialbee algorithm with mixed integer programming applied to alarge-scale capacitated facility location problemrdquoMathematicalProblems in Engineering vol 2012 Article ID 954249 14 pages2012

[6] M Caserta and S Voszlig ldquoA math-heuristic Dantzig-Wolfe algo-rithm for capacitated lot sizingrdquo Annals of Mathematics andArtificial Intelligence vol 69 no 2 pp 207ndash224 2013

[7] L C Coelho J-F Cordeau and G Laporte ldquoHeuristics fordynamic and stochastic inventory-routingrdquoComputers ampOper-ations Research vol A no 0 pp 55ndash67 2014

[8] C Lagos F Paredes S Niklander and E Cabrera ldquoSolving adistribution network design problem by combining ant colonysystems and lagrangian relaxationrdquo Studies in Informatics andControl vol 24 no 3 pp 251ndash260 2015

[9] B Raa W Dullaert and E-H Aghezzaf ldquoA matheuristicfor aggregate production-distribution planning with mouldsharingrdquo International Journal of Production Economics vol 145no 1 pp 29ndash37 2013

[10] H Allaoua S Borne L Letocart and R Wolfler Calvo ldquoAmatheuristic approach for solving a home health care problemrdquoElectronic Notes in Discrete Mathematics vol 41 pp 471ndash4782013

[11] G Cabrera-Guerrero M Ehrgott A J Mason and A RaithldquoA matheuristic approach to solve the multiobjective beamangle optimization problem in intensity-modulated radiationtherapyrdquo International Transactions in Operational Research2016

[12] Y Li D Yao J Zheng and J Yao ldquoA modified genetic algo-rithm for the beam angle optimization problem in intensity-modulated radiotherapy planningrdquo in Artificial Evolution E-G Talbi P Liardet P Collet E Lutton and M SchoenauerEds vol 3871 of Lecture Notes in Computer Science pp 97ndash106Springer Berlin Germany 2006

[13] Y Li D Yao W Chen J Zheng and J Yao ldquoAnt colony systemfor the beam angle optimization problem in radiotherapyplanning a preliminary studyrdquo in Proceedings of the 2005 IEEECongress on Evolutionary Computation volume 2 pp 1532ndash1538 IEEE Edinburgh Scotland UK September 2005

[14] S Hanafi J Lazic N Mladenovic C Wilbaut and I CrevitsldquoNew hybrid matheuristics for solving the multidimensionalknapsack problemrdquo Lecture Notes in Computer Science (includ-ing subseries Lecture Notes in Artificial Intelligence and LectureNotes in Bioinformatics) Preface vol 6373 pp 118ndash132 2010

[15] E-G Talbi ldquoCombining metaheuristics with mathematicalprogramming constraint programming andmachine learningrdquo4OR vol 11 no 2 pp 101ndash150 2013

[16] C Archetti and M G Speranza ldquoA survey on matheuristics forrouting problemsrdquo EURO J in Computer Optimization vol 2no 4 pp 223ndash246 2014

[17] M A Boschetti V Maniezzo M Roffilli and A Bolufe RohlerldquoMatheuristics Optimization simulation and controlrdquo Lecture

Notes in Computer Science (including subseries Lecture Notesin Artificial Intelligence and Lecture Notes in Bioinformatics)Preface vol 5818 pp 171ndash177 2009

[18] C-H Chen and C-J Ting ldquoCombining lagrangian heuristicand ant colony system to solve the single source capacitatedfacility location problemrdquo Transportation Research Part ELogistics and Transportation Review vol 44 no 6 pp 1099ndash1122 2008

[19] Y Li J Yao and D Yao ldquoAutomatic beam angle selection inIMRT planning using genetic algorithmrdquo Physics in Medicineand Biology vol 49 no 10 pp 1915ndash1932 2004

[20] Y Li D Yao J Yao and W Chen ldquoA particle swarm optimiza-tion algorithm for beam angle selection in intensity-modulatedradiotherapy planningrdquo Physics inMedicine and Biology vol 50no 15 pp 3491ndash3514 2005

[21] F Glover and M Laguna Tabu Search Kluwer AcademicPublishers Norwell MA USA 1997

[22] M Pirlot ldquoGeneral local search methodsrdquo European Journal ofOperational Research vol 92 no 3 pp 493ndash511 1996

[23] C Lagos G Guerrero E Cabrera et al ldquoA Matheuris-tic Approach Combining Local Search and MathematicalProgrammingrdquo Scientific Programming vol 2016 Article ID1506084 2016

[24] V Vails M A Perez and M S Quintanilla ldquoA tabu searchapproach to machine schedulingrdquo European Journal of Opera-tional Research vol 106 no 2-3 pp 277ndash300 1998

[25] S Hanafi and A Freville ldquoAn efficient tabu search approach forthe 0-1 multidimensional knapsack problemrdquo European Journalof Operational Research vol 106 no 2-3 pp 659ndash675 1998

[26] J Brandao and A Mercer ldquoA tabu search algorithm for themulti-trip vehicle routing and scheduling problemrdquo EuropeanJournal of Operational Research vol 100 no 1 pp 180ndash191 1997

[27] K S Al-Sultan and M A Al-Fawzan ldquoA tabu search approachto the uncapacitated facility location problemrdquo Annals of Oper-ations Research vol 86 pp 91ndash103 1999

[28] P B Mirchandani and R L Francis Eds Discrete LocationTheory Wiley-Interscience New York NY USA 1990

[29] J E Beasley ldquoOR-Library distributing test problems by elec-tronic mailrdquo Journal of the Operational Research Society vol 41no 11 pp 1069ndash1072 1990

[30] J G Villegas F Palacios and A Medaglia ldquoSolution methodsfor the bi-objective (cost-coverage) unconstrained facility loca-tion problemwith an illustrative examplerdquoAnnals of OperationsResearch vol 147 pp 109ndash141 2006

Submit your manuscripts athttpswwwhindawicom

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MathematicsJournal of


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Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Complexity

methods are considered as the local optimisers then localoptimality could be ensured

To overcome the situation described above hybrid meth-ods that combine heuristics and exact methods to solveoptimisation problems have been proposed These methodsalso known asmatheuristics [4] have been shown to performbetter than both heuristic and mathematical programmingmethods when they are applied separately The idea of com-bining the power of mathematical programming with flex-ibility of heuristics has gained attention within researchersrsquocommunity We can found matheuristics attempting to solveproblems arising in the field of logistics [5ndash9] health caresystems [10ndash13] and puremathematics [14 15] among othersMatheuristics have been demonstrated to be very effectivein solving complex optimisation problems Some interestingsurveys on matheuristics are [16 17] Although there is someoverlapping between memetic algorithms and matheuristicones in this paperwe have chosen to label the studied strategyas matheuristic as we think matheuristics definition betterfits the framework we are interested in

Because of their complexity MIP problems as well ascombinatorial optimisation problems are often tackled usingmatheuristic methods One common strategy to solve thisclass of optimisation problems is to divide the main opti-misation problem into several subproblems While heuristicsare used to seek for promising subproblems exact methodsare used to solve them to optimality One advantage of thisapproach is that it does not depend on the (non)linearity ofthe resulting subproblem Instead it has been pointed out thatit is desirable that the resulting subproblem would be convex[11] Having a convex subproblem would allow us to solve itto optimality and thus comparing solutions obtained at eachsubproblem becomes more senseful This strategy has beensuccessfully applied to problems arising in fields as diverse aslogistics and radiation therapy

In this paper we aim to study the impact of parametertuning on the performance of matheuristic methods as theone described above To this end the well-known capacitatedfacility location problem is used as an application of hardcombinatorial optimisation problem To the best of ourknowledge no paper has focused on parameter tuning formatheuristic methods

This paper is organised as follows Section 2 shows thegeneral matheuristic framework we consider in this paperDetails on the algorithms that are used in this study are alsoshown in this section In Section 3 the capacitated facilitylocation problem is introduced and itsmathematicalmodel isdescribed in Section 31 The experiments performed in thisstudy are presented and the obtained results are discussedin Section 32 Finally in Section 4 some conclusions arepresented and the future work is outlined

2 Matheuristic Methods

This section is twofold We start by describing a generalmatheuristic framework that is used to solve both MIPproblems and combinatorial optimisation problems and howit is different from other commonly used approaches such

as memetic algorithms and other evolutionary approachesAfter that we present the local-search-based algorithms weconsider in this work to perform our experiments We finishthis section by introducing the parameter we will be focusedon in this study

21 General Framework Equations (1a) to (1f) show thegeneral form of MIP problems Hereafter we will refer to thisproblem as the MIP(119909 119910) problem or themain problem

MIP (119909 119910) min119909119910

119891 (119909 119910) (1a)

st 119892119895 (119909 119910) le 119887119895 for 119895 = 1 119898 (1b)

119892119895 (119909) le 119887119895 for 119895 = 1 119898 (1c)

119892119895 (119910) le 119887119895 for 119895 = 1 119898 (1d)

119910 isin 0 1119900 (1e)

119909 isin R119899 (1f)

where 119891(119909 119910) is an objective function 119898 is the numberof inequality constraints on 119909 and 119910 119898 is the number ofinequality constraints on 119909 119898 is the number of inequalityconstraints on 119910 119900 is the number of binary (ge0) decisionvariables 119910 and 119899 is the number of continuous decisionvariables Combinatorial optimisation problems as the onewe consider in this paper can be easily obtained by eitherremoving the continuous decision variable from the modelor making it integer (ie 119909 isin Z119900ge)

Although there exist a number of exact algorithms thatcan find an optimal solution for the MIP(119909 119910) as the size ofthe problem increases exact methods fail or take too longBecause of this heuristic methods are used to obtain goodquality solutions of the problem within an acceptable timeHeuristic methods cannot guarantee optimality though

During the last two decades the idea of combin-ing heuristic methods and mathematical programming hasreceived much more attention Exploiting the advantages ofeach method appears to be a senseful strategy to overcometheir inherent drawbacks Several strategies have been pro-posed to combine heuristics and exact methods to solveoptimisation problems such as the MIP(119909 119910) problem Forinstance Chen and Ting [18] and Lagos et al [8] com-bine the well-known ant colony optimisation (ACO) algo-rithm and Lagrangian relaxation method to solve the singlesource capacitated facility location problem and a distribu-tion network design problem respectively In these articlesLagrangian multipliers are updated using ACO algorithmAnother strategy to combine heuristics and exact methodsis to let heuristics seek for subproblems of MIP(119909 119910) whichin turn are solved to optimality by some exact method Onealternative to obtain subproblems of MIP(119909 119910) is to add aset of additional constraints on a subset of binary decisionvariablesThese constraints are of the form 119910119894 = 0 with 119894 isin IandI being the set of index that are restricted in subproblemMIPI(119909 119910) (see (1a) to (1f)) The portion of binary decisionvariables 119910119894 that are set to 0 is denoted by 120572 (ie I = 120572times 119900)

Complexity 3


MIPI (119909 119910) min119909119910

119891 (119909 119910) (2a)

st 119892119895 (119909 119910) le 119887119895

for 119895 = 1 119898(2b)

119892119895 (119909) le 119887119895 for 119895 = 1 119898 (2c)

119892119895 (119910) le 119887119895 for 119895 = 1 119898 (2d)

119910119894 = 0 119894 isin I (2e)

119910 isin 0 1119900 (2f)

119909 isin R119899 (2g)




Heuristicmethod

Exactmethod

(xlowast ylowast)

Seeks for new ℐ

ℐ







4 Complexity




119896minus1













Complexity 5




119896minus1






119896












6 Complexity



(5) (119909 119910) = (119909119888 119910119888) set best sol



119896minus1








Complexity 7







CFLP (119909 119910) min119909119910

119899

sum119894=1

(119891119894119910119894) +119899

sum119894=1

119898

sum119895=1

(119888119894119895119889119895119909119894119895) (4a)

st119898

sum119895=1

119889119895119909119894119895 le 119868cap119894 119910119894 forall119894 = 1 119899 (4b)

Customers(clusters)

Warehouse 1

Warehouse 2

Warehouse 3



119899

sum119894=1

119909119894119895 = 1 forall119895 = 1 119898 (4c)

119909119894119895 le 119910119894

forall119894 = 1 119899 forall119895 = 1 119898(4d)

119910119894 119909119894119895 isin 0 1

forall119894 = 1 119899 forall119895 = 1 119898(4e)



8 Complexity

0

1


12

08

06

04

02

0 1 1208060402


Customers

0

1

12

08

06

04

02

0 1 1208060402











Complexity 9




0

1

2

3

4

5

GA

P va

lue (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

1000Ti

me (

sec)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807






10 Complexity

Table3Re

sults

obtained

bythelocalsearch

metho

dsforthe

ORLibraryrsquos

insta

nces

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

A1

01

522

622

965

248

945

1424

673

785

03

163

1124

000

1240

000

489

000

736

05

032

664

000

1095

000

274

024

700

07

010

745

000

366

000

138

000

1476

A2

01

431

463

000

833

000

1415

000

866

03

122

893

000

366

019

1290

019

615

05

052

987

183

52009

956

000

745

07

011

282

009

378

000

1562

000

743

A3

01

181

841

000

341

031

65000

686

03

023

503

000

602

000

693

000

1083

05

012

880

000

431

000

442

000

659

07

000

272

000

151

000

211

000

211

A4

01

156

990

000

25000

84000

8403

000

411

000

8000

31000

3105

000

111

000

2000

14000

1407

000

23000

1000

7000

7

B1

01

522

622

003

639

023

960

001

319

03

163

1124

001

588

008

436

003

1452

05

032

664

000

707

003

216

000

605

07

010

745

001

347

000

1630

000

1192

B2

01

431

463

144

806

275

222

009

714

03

122

893

000

135

000

69000

1165

05

052

987

000

186

000

196

005

530

07

011

282

000

112

005

315

000

261

B3

01

181

841

040

403

043

1270

030

142

03

023

503

000

96000

181

000

267

05

012

880

000

588

000

236

000

564

07

000

272

000

179

000

316

000

316

B4

01

156

990

000

1309

032

144

032

144

03

000

411

000

24000

360

000

360

05

000

111

000

57000

381

000

381

07

000

23000

28000

349

000

349

C1

01

809

712

970

1220

814

921027

133

03

320

919

000

657

000

486

000

394

05

199

1105

000

173

000

496

000

1469

07

048

863

000

239

000

355

000

247

Complexity 11

Table3Con

tinued

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

C2

01

574

937

142

1254

232

1397

070

8403

164

934

000

136

000

107

000

172

05

035

562

000

1225

000

248

000

537

07

002

766

000

1294

000

164

000

169

C3

01

517

979

111

1254

154

819

020

585

03

104

993

000

73000

38000

272

05

022

906

000

587

000

64000

1387

07

004

740

000

551

000

149

000

779

C4

01

486

663

135

153

034

523

085

313

03

065

711

003

7003

35003

107

05

008

821

000

18003

11003

1307

002

228

003

8003

22000

29

12 Complexity

0

1

2

3

4

5G

AP

valu

e (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

Tim

e (se

c)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000



GA

P va

lue (

)

0

1

2

3

4

5

6

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


Tim

e (se

c)

200

400

600

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000






Complexity 13

GA

P va

lue (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


AP

valu

e (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807











14 ComplexityTi

me (

sec)

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

1200

1400

1600

1800

2000


BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

Tim

e (se

c)

800

1000

1200

1400

1600

1800

2000








Acknowledgments


References


Complexity 15






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 3


MIPI (119909 119910) min119909119910

119891 (119909 119910) (2a)

st 119892119895 (119909 119910) le 119887119895

for 119895 = 1 119898(2b)

119892119895 (119909) le 119887119895 for 119895 = 1 119898 (2c)

119892119895 (119910) le 119887119895 for 119895 = 1 119898 (2d)

119910119894 = 0 119894 isin I (2e)

119910 isin 0 1119900 (2f)

119909 isin R119899 (2g)




Heuristicmethod

Exactmethod

(xlowast ylowast)

Seeks for new ℐ

ℐ







4 Complexity




119896minus1













Complexity 5




119896minus1






119896












6 Complexity



(5) (119909 119910) = (119909119888 119910119888) set best sol



119896minus1








Complexity 7







CFLP (119909 119910) min119909119910

119899

sum119894=1

(119891119894119910119894) +119899

sum119894=1

119898

sum119895=1

(119888119894119895119889119895119909119894119895) (4a)

st119898

sum119895=1

119889119895119909119894119895 le 119868cap119894 119910119894 forall119894 = 1 119899 (4b)

Customers(clusters)

Warehouse 1

Warehouse 2

Warehouse 3



119899

sum119894=1

119909119894119895 = 1 forall119895 = 1 119898 (4c)

119909119894119895 le 119910119894

forall119894 = 1 119899 forall119895 = 1 119898(4d)

119910119894 119909119894119895 isin 0 1

forall119894 = 1 119899 forall119895 = 1 119898(4e)



8 Complexity

0

1


12

08

06

04

02

0 1 1208060402


Customers

0

1

12

08

06

04

02

0 1 1208060402











Complexity 9




0

1

2

3

4

5

GA

P va

lue (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

1000Ti

me (

sec)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807






10 Complexity

Table3Re

sults

obtained

bythelocalsearch

metho

dsforthe

ORLibraryrsquos

insta

nces

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

A1

01

522

622

965

248

945

1424

673

785

03

163

1124

000

1240

000

489

000

736

05

032

664

000

1095

000

274

024

700

07

010

745

000

366

000

138

000

1476

A2

01

431

463

000

833

000

1415

000

866

03

122

893

000

366

019

1290

019

615

05

052

987

183

52009

956

000

745

07

011

282

009

378

000

1562

000

743

A3

01

181

841

000

341

031

65000

686

03

023

503

000

602

000

693

000

1083

05

012

880

000

431

000

442

000

659

07

000

272

000

151

000

211

000

211

A4

01

156

990

000

25000

84000

8403

000

411

000

8000

31000

3105

000

111

000

2000

14000

1407

000

23000

1000

7000

7

B1

01

522

622

003

639

023

960

001

319

03

163

1124

001

588

008

436

003

1452

05

032

664

000

707

003

216

000

605

07

010

745

001

347

000

1630

000

1192

B2

01

431

463

144

806

275

222

009

714

03

122

893

000

135

000

69000

1165

05

052

987

000

186

000

196

005

530

07

011

282

000

112

005

315

000

261

B3

01

181

841

040

403

043

1270

030

142

03

023

503

000

96000

181

000

267

05

012

880

000

588

000

236

000

564

07

000

272

000

179

000

316

000

316

B4

01

156

990

000

1309

032

144

032

144

03

000

411

000

24000

360

000

360

05

000

111

000

57000

381

000

381

07

000

23000

28000

349

000

349

C1

01

809

712

970

1220

814

921027

133

03

320

919

000

657

000

486

000

394

05

199

1105

000

173

000

496

000

1469

07

048

863

000

239

000

355

000

247

Complexity 11

Table3Con

tinued

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

C2

01

574

937

142

1254

232

1397

070

8403

164

934

000

136

000

107

000

172

05

035

562

000

1225

000

248

000

537

07

002

766

000

1294

000

164

000

169

C3

01

517

979

111

1254

154

819

020

585

03

104

993

000

73000

38000

272

05

022

906

000

587

000

64000

1387

07

004

740

000

551

000

149

000

779

C4

01

486

663

135

153

034

523

085

313

03

065

711

003

7003

35003

107

05

008

821

000

18003

11003

1307

002

228

003

8003

22000

29

12 Complexity

0

1

2

3

4

5G

AP

valu

e (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

Tim

e (se

c)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000



GA

P va

lue (

)

0

1

2

3

4

5

6

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


Tim

e (se

c)

200

400

600

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000






Complexity 13

GA

P va

lue (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


AP

valu

e (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807











14 ComplexityTi

me (

sec)

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

1200

1400

1600

1800

2000


BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

Tim

e (se

c)

800

1000

1200

1400

1600

1800

2000








Acknowledgments


References


Complexity 15






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




4 Complexity




119896minus1













Complexity 5




119896minus1






119896












6 Complexity



(5) (119909 119910) = (119909119888 119910119888) set best sol



119896minus1








Complexity 7







CFLP (119909 119910) min119909119910

119899

sum119894=1

(119891119894119910119894) +119899

sum119894=1

119898

sum119895=1

(119888119894119895119889119895119909119894119895) (4a)

st119898

sum119895=1

119889119895119909119894119895 le 119868cap119894 119910119894 forall119894 = 1 119899 (4b)

Customers(clusters)

Warehouse 1

Warehouse 2

Warehouse 3



119899

sum119894=1

119909119894119895 = 1 forall119895 = 1 119898 (4c)

119909119894119895 le 119910119894

forall119894 = 1 119899 forall119895 = 1 119898(4d)

119910119894 119909119894119895 isin 0 1

forall119894 = 1 119899 forall119895 = 1 119898(4e)



8 Complexity

0

1


12

08

06

04

02

0 1 1208060402


Customers

0

1

12

08

06

04

02

0 1 1208060402











Complexity 9




0

1

2

3

4

5

GA

P va

lue (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

1000Ti

me (

sec)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807






10 Complexity

Table3Re

sults

obtained

bythelocalsearch

metho

dsforthe

ORLibraryrsquos

insta

nces

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

A1

01

522

622

965

248

945

1424

673

785

03

163

1124

000

1240

000

489

000

736

05

032

664

000

1095

000

274

024

700

07

010

745

000

366

000

138

000

1476

A2

01

431

463

000

833

000

1415

000

866

03

122

893

000

366

019

1290

019

615

05

052

987

183

52009

956

000

745

07

011

282

009

378

000

1562

000

743

A3

01

181

841

000

341

031

65000

686

03

023

503

000

602

000

693

000

1083

05

012

880

000

431

000

442

000

659

07

000

272

000

151

000

211

000

211

A4

01

156

990

000

25000

84000

8403

000

411

000

8000

31000

3105

000

111

000

2000

14000

1407

000

23000

1000

7000

7

B1

01

522

622

003

639

023

960

001

319

03

163

1124

001

588

008

436

003

1452

05

032

664

000

707

003

216

000

605

07

010

745

001

347

000

1630

000

1192

B2

01

431

463

144

806

275

222

009

714

03

122

893

000

135

000

69000

1165

05

052

987

000

186

000

196

005

530

07

011

282

000

112

005

315

000

261

B3

01

181

841

040

403

043

1270

030

142

03

023

503

000

96000

181

000

267

05

012

880

000

588

000

236

000

564

07

000

272

000

179

000

316

000

316

B4

01

156

990

000

1309

032

144

032

144

03

000

411

000

24000

360

000

360

05

000

111

000

57000

381

000

381

07

000

23000

28000

349

000

349

C1

01

809

712

970

1220

814

921027

133

03

320

919

000

657

000

486

000

394

05

199

1105

000

173

000

496

000

1469

07

048

863

000

239

000

355

000

247

Complexity 11

Table3Con

tinued

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

C2

01

574

937

142

1254

232

1397

070

8403

164

934

000

136

000

107

000

172

05

035

562

000

1225

000

248

000

537

07

002

766

000

1294

000

164

000

169

C3

01

517

979

111

1254

154

819

020

585

03

104

993

000

73000

38000

272

05

022

906

000

587

000

64000

1387

07

004

740

000

551

000

149

000

779

C4

01

486

663

135

153

034

523

085

313

03

065

711

003

7003

35003

107

05

008

821

000

18003

11003

1307

002

228

003

8003

22000

29

12 Complexity

0

1

2

3

4

5G

AP

valu

e (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

Tim

e (se

c)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000



GA

P va

lue (

)

0

1

2

3

4

5

6

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


Tim

e (se

c)

200

400

600

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000






Complexity 13

GA

P va

lue (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


AP

valu

e (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807











14 ComplexityTi

me (

sec)

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

1200

1400

1600

1800

2000


BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

Tim

e (se

c)

800

1000

1200

1400

1600

1800

2000








Acknowledgments


References


Complexity 15






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 5




119896minus1






119896












6 Complexity



(5) (119909 119910) = (119909119888 119910119888) set best sol



119896minus1








Complexity 7







CFLP (119909 119910) min119909119910

119899

sum119894=1

(119891119894119910119894) +119899

sum119894=1

119898

sum119895=1

(119888119894119895119889119895119909119894119895) (4a)

st119898

sum119895=1

119889119895119909119894119895 le 119868cap119894 119910119894 forall119894 = 1 119899 (4b)

Customers(clusters)

Warehouse 1

Warehouse 2

Warehouse 3



119899

sum119894=1

119909119894119895 = 1 forall119895 = 1 119898 (4c)

119909119894119895 le 119910119894

forall119894 = 1 119899 forall119895 = 1 119898(4d)

119910119894 119909119894119895 isin 0 1

forall119894 = 1 119899 forall119895 = 1 119898(4e)



8 Complexity

0

1


12

08

06

04

02

0 1 1208060402


Customers

0

1

12

08

06

04

02

0 1 1208060402











Complexity 9




0

1

2

3

4

5

GA

P va

lue (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

1000Ti

me (

sec)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807






10 Complexity

Table3Re

sults

obtained

bythelocalsearch

metho

dsforthe

ORLibraryrsquos

insta

nces

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

A1

01

522

622

965

248

945

1424

673

785

03

163

1124

000

1240

000

489

000

736

05

032

664

000

1095

000

274

024

700

07

010

745

000

366

000

138

000

1476

A2

01

431

463

000

833

000

1415

000

866

03

122

893

000

366

019

1290

019

615

05

052

987

183

52009

956

000

745

07

011

282

009

378

000

1562

000

743

A3

01

181

841

000

341

031

65000

686

03

023

503

000

602

000

693

000

1083

05

012

880

000

431

000

442

000

659

07

000

272

000

151

000

211

000

211

A4

01

156

990

000

25000

84000

8403

000

411

000

8000

31000

3105

000

111

000

2000

14000

1407

000

23000

1000

7000

7

B1

01

522

622

003

639

023

960

001

319

03

163

1124

001

588

008

436

003

1452

05

032

664

000

707

003

216

000

605

07

010

745

001

347

000

1630

000

1192

B2

01

431

463

144

806

275

222

009

714

03

122

893

000

135

000

69000

1165

05

052

987

000

186

000

196

005

530

07

011

282

000

112

005

315

000

261

B3

01

181

841

040

403

043

1270

030

142

03

023

503

000

96000

181

000

267

05

012

880

000

588

000

236

000

564

07

000

272

000

179

000

316

000

316

B4

01

156

990

000

1309

032

144

032

144

03

000

411

000

24000

360

000

360

05

000

111

000

57000

381

000

381

07

000

23000

28000

349

000

349

C1

01

809

712

970

1220

814

921027

133

03

320

919

000

657

000

486

000

394

05

199

1105

000

173

000

496

000

1469

07

048

863

000

239

000

355

000

247

Complexity 11

Table3Con

tinued

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

C2

01

574

937

142

1254

232

1397

070

8403

164

934

000

136

000

107

000

172

05

035

562

000

1225

000

248

000

537

07

002

766

000

1294

000

164

000

169

C3

01

517

979

111

1254

154

819

020

585

03

104

993

000

73000

38000

272

05

022

906

000

587

000

64000

1387

07

004

740

000

551

000

149

000

779

C4

01

486

663

135

153

034

523

085

313

03

065

711

003

7003

35003

107

05

008

821

000

18003

11003

1307

002

228

003

8003

22000

29

12 Complexity

0

1

2

3

4

5G

AP

valu

e (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

Tim

e (se

c)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000



GA

P va

lue (

)

0

1

2

3

4

5

6

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


Tim

e (se

c)

200

400

600

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000






Complexity 13

GA

P va

lue (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


AP

valu

e (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807











14 ComplexityTi

me (

sec)

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

1200

1400

1600

1800

2000


BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

Tim

e (se

c)

800

1000

1200

1400

1600

1800

2000








Acknowledgments


References


Complexity 15






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




6 Complexity



(5) (119909 119910) = (119909119888 119910119888) set best sol



119896minus1








Complexity 7







CFLP (119909 119910) min119909119910

119899

sum119894=1

(119891119894119910119894) +119899

sum119894=1

119898

sum119895=1

(119888119894119895119889119895119909119894119895) (4a)

st119898

sum119895=1

119889119895119909119894119895 le 119868cap119894 119910119894 forall119894 = 1 119899 (4b)

Customers(clusters)

Warehouse 1

Warehouse 2

Warehouse 3



119899

sum119894=1

119909119894119895 = 1 forall119895 = 1 119898 (4c)

119909119894119895 le 119910119894

forall119894 = 1 119899 forall119895 = 1 119898(4d)

119910119894 119909119894119895 isin 0 1

forall119894 = 1 119899 forall119895 = 1 119898(4e)



8 Complexity

0

1


12

08

06

04

02

0 1 1208060402


Customers

0

1

12

08

06

04

02

0 1 1208060402











Complexity 9




0

1

2

3

4

5

GA

P va

lue (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

1000Ti

me (

sec)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807






10 Complexity

Table3Re

sults

obtained

bythelocalsearch

metho

dsforthe

ORLibraryrsquos

insta

nces

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

A1

01

522

622

965

248

945

1424

673

785

03

163

1124

000

1240

000

489

000

736

05

032

664

000

1095

000

274

024

700

07

010

745

000

366

000

138

000

1476

A2

01

431

463

000

833

000

1415

000

866

03

122

893

000

366

019

1290

019

615

05

052

987

183

52009

956

000

745

07

011

282

009

378

000

1562

000

743

A3

01

181

841

000

341

031

65000

686

03

023

503

000

602

000

693

000

1083

05

012

880

000

431

000

442

000

659

07

000

272

000

151

000

211

000

211

A4

01

156

990

000

25000

84000

8403

000

411

000

8000

31000

3105

000

111

000

2000

14000

1407

000

23000

1000

7000

7

B1

01

522

622

003

639

023

960

001

319

03

163

1124

001

588

008

436

003

1452

05

032

664

000

707

003

216

000

605

07

010

745

001

347

000

1630

000

1192

B2

01

431

463

144

806

275

222

009

714

03

122

893

000

135

000

69000

1165

05

052

987

000

186

000

196

005

530

07

011

282

000

112

005

315

000

261

B3

01

181

841

040

403

043

1270

030

142

03

023

503

000

96000

181

000

267

05

012

880

000

588

000

236

000

564

07

000

272

000

179

000

316

000

316

B4

01

156

990

000

1309

032

144

032

144

03

000

411

000

24000

360

000

360

05

000

111

000

57000

381

000

381

07

000

23000

28000

349

000

349

C1

01

809

712

970

1220

814

921027

133

03

320

919

000

657

000

486

000

394

05

199

1105

000

173

000

496

000

1469

07

048

863

000

239

000

355

000

247

Complexity 11

Table3Con

tinued

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

C2

01

574

937

142

1254

232

1397

070

8403

164

934

000

136

000

107

000

172

05

035

562

000

1225

000

248

000

537

07

002

766

000

1294

000

164

000

169

C3

01

517

979

111

1254

154

819

020

585

03

104

993

000

73000

38000

272

05

022

906

000

587

000

64000

1387

07

004

740

000

551

000

149

000

779

C4

01

486

663

135

153

034

523

085

313

03

065

711

003

7003

35003

107

05

008

821

000

18003

11003

1307

002

228

003

8003

22000

29

12 Complexity

0

1

2

3

4

5G

AP

valu

e (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

Tim

e (se

c)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000



GA

P va

lue (

)

0

1

2

3

4

5

6

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


Tim

e (se

c)

200

400

600

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000






Complexity 13

GA

P va

lue (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


AP

valu

e (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807











14 ComplexityTi

me (

sec)

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

1200

1400

1600

1800

2000


BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

Tim

e (se

c)

800

1000

1200

1400

1600

1800

2000








Acknowledgments


References


Complexity 15






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 7







CFLP (119909 119910) min119909119910

119899

sum119894=1

(119891119894119910119894) +119899

sum119894=1

119898

sum119895=1

(119888119894119895119889119895119909119894119895) (4a)

st119898

sum119895=1

119889119895119909119894119895 le 119868cap119894 119910119894 forall119894 = 1 119899 (4b)

Customers(clusters)

Warehouse 1

Warehouse 2

Warehouse 3



119899

sum119894=1

119909119894119895 = 1 forall119895 = 1 119898 (4c)

119909119894119895 le 119910119894

forall119894 = 1 119899 forall119895 = 1 119898(4d)

119910119894 119909119894119895 isin 0 1

forall119894 = 1 119899 forall119895 = 1 119898(4e)



8 Complexity

0

1


12

08

06

04

02

0 1 1208060402


Customers

0

1

12

08

06

04

02

0 1 1208060402











Complexity 9




0

1

2

3

4

5

GA

P va

lue (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

1000Ti

me (

sec)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807






10 Complexity

Table3Re

sults

obtained

bythelocalsearch

metho

dsforthe

ORLibraryrsquos

insta

nces

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

A1

01

522

622

965

248

945

1424

673

785

03

163

1124

000

1240

000

489

000

736

05

032

664

000

1095

000

274

024

700

07

010

745

000

366

000

138

000

1476

A2

01

431

463

000

833

000

1415

000

866

03

122

893

000

366

019

1290

019

615

05

052

987

183

52009

956

000

745

07

011

282

009

378

000

1562

000

743

A3

01

181

841

000

341

031

65000

686

03

023

503

000

602

000

693

000

1083

05

012

880

000

431

000

442

000

659

07

000

272

000

151

000

211

000

211

A4

01

156

990

000

25000

84000

8403

000

411

000

8000

31000

3105

000

111

000

2000

14000

1407

000

23000

1000

7000

7

B1

01

522

622

003

639

023

960

001

319

03

163

1124

001

588

008

436

003

1452

05

032

664

000

707

003

216

000

605

07

010

745

001

347

000

1630

000

1192

B2

01

431

463

144

806

275

222

009

714

03

122

893

000

135

000

69000

1165

05

052

987

000

186

000

196

005

530

07

011

282

000

112

005

315

000

261

B3

01

181

841

040

403

043

1270

030

142

03

023

503

000

96000

181

000

267

05

012

880

000

588

000

236

000

564

07

000

272

000

179

000

316

000

316

B4

01

156

990

000

1309

032

144

032

144

03

000

411

000

24000

360

000

360

05

000

111

000

57000

381

000

381

07

000

23000

28000

349

000

349

C1

01

809

712

970

1220

814

921027

133

03

320

919

000

657

000

486

000

394

05

199

1105

000

173

000

496

000

1469

07

048

863

000

239

000

355

000

247

Complexity 11

Table3Con

tinued

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

C2

01

574

937

142

1254

232

1397

070

8403

164

934

000

136

000

107

000

172

05

035

562

000

1225

000

248

000

537

07

002

766

000

1294

000

164

000

169

C3

01

517

979

111

1254

154

819

020

585

03

104

993

000

73000

38000

272

05

022

906

000

587

000

64000

1387

07

004

740

000

551

000

149

000

779

C4

01

486

663

135

153

034

523

085

313

03

065

711

003

7003

35003

107

05

008

821

000

18003

11003

1307

002

228

003

8003

22000

29

12 Complexity

0

1

2

3

4

5G

AP

valu

e (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

Tim

e (se

c)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000



GA

P va

lue (

)

0

1

2

3

4

5

6

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


Tim

e (se

c)

200

400

600

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000






Complexity 13

GA

P va

lue (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


AP

valu

e (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807











14 ComplexityTi

me (

sec)

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

1200

1400

1600

1800

2000


BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

Tim

e (se

c)

800

1000

1200

1400

1600

1800

2000








Acknowledgments


References


Complexity 15






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




8 Complexity

0

1


12

08

06

04

02

0 1 1208060402


Customers

0

1

12

08

06

04

02

0 1 1208060402











Complexity 9




0

1

2

3

4

5

GA

P va

lue (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

1000Ti

me (

sec)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807






10 Complexity

Table3Re

sults

obtained

bythelocalsearch

metho

dsforthe

ORLibraryrsquos

insta

nces

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

A1

01

522

622

965

248

945

1424

673

785

03

163

1124

000

1240

000

489

000

736

05

032

664

000

1095

000

274

024

700

07

010

745

000

366

000

138

000

1476

A2

01

431

463

000

833

000

1415

000

866

03

122

893

000

366

019

1290

019

615

05

052

987

183

52009

956

000

745

07

011

282

009

378

000

1562

000

743

A3

01

181

841

000

341

031

65000

686

03

023

503

000

602

000

693

000

1083

05

012

880

000

431

000

442

000

659

07

000

272

000

151

000

211

000

211

A4

01

156

990

000

25000

84000

8403

000

411

000

8000

31000

3105

000

111

000

2000

14000

1407

000

23000

1000

7000

7

B1

01

522

622

003

639

023

960

001

319

03

163

1124

001

588

008

436

003

1452

05

032

664

000

707

003

216

000

605

07

010

745

001

347

000

1630

000

1192

B2

01

431

463

144

806

275

222

009

714

03

122

893

000

135

000

69000

1165

05

052

987

000

186

000

196

005

530

07

011

282

000

112

005

315

000

261

B3

01

181

841

040

403

043

1270

030

142

03

023

503

000

96000

181

000

267

05

012

880

000

588

000

236

000

564

07

000

272

000

179

000

316

000

316

B4

01

156

990

000

1309

032

144

032

144

03

000

411

000

24000

360

000

360

05

000

111

000

57000

381

000

381

07

000

23000

28000

349

000

349

C1

01

809

712

970

1220

814

921027

133

03

320

919

000

657

000

486

000

394

05

199

1105

000

173

000

496

000

1469

07

048

863

000

239

000

355

000

247

Complexity 11

Table3Con

tinued

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

C2

01

574

937

142

1254

232

1397

070

8403

164

934

000

136

000

107

000

172

05

035

562

000

1225

000

248

000

537

07

002

766

000

1294

000

164

000

169

C3

01

517

979

111

1254

154

819

020

585

03

104

993

000

73000

38000

272

05

022

906

000

587

000

64000

1387

07

004

740

000

551

000

149

000

779

C4

01

486

663

135

153

034

523

085

313

03

065

711

003

7003

35003

107

05

008

821

000

18003

11003

1307

002

228

003

8003

22000

29

12 Complexity

0

1

2

3

4

5G

AP

valu

e (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

Tim

e (se

c)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000



GA

P va

lue (

)

0

1

2

3

4

5

6

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


Tim

e (se

c)

200

400

600

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000






Complexity 13

GA

P va

lue (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


AP

valu

e (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807











14 ComplexityTi

me (

sec)

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

1200

1400

1600

1800

2000


BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

Tim

e (se

c)

800

1000

1200

1400

1600

1800

2000








Acknowledgments


References


Complexity 15






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 9




0

1

2

3

4

5

GA

P va

lue (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

1000Ti

me (

sec)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807






10 Complexity

Table3Re

sults

obtained

bythelocalsearch

metho

dsforthe

ORLibraryrsquos

insta

nces

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

A1

01

522

622

965

248

945

1424

673

785

03

163

1124

000

1240

000

489

000

736

05

032

664

000

1095

000

274

024

700

07

010

745

000

366

000

138

000

1476

A2

01

431

463

000

833

000

1415

000

866

03

122

893

000

366

019

1290

019

615

05

052

987

183

52009

956

000

745

07

011

282

009

378

000

1562

000

743

A3

01

181

841

000

341

031

65000

686

03

023

503

000

602

000

693

000

1083

05

012

880

000

431

000

442

000

659

07

000

272

000

151

000

211

000

211

A4

01

156

990

000

25000

84000

8403

000

411

000

8000

31000

3105

000

111

000

2000

14000

1407

000

23000

1000

7000

7

B1

01

522

622

003

639

023

960

001

319

03

163

1124

001

588

008

436

003

1452

05

032

664

000

707

003

216

000

605

07

010

745

001

347

000

1630

000

1192

B2

01

431

463

144

806

275

222

009

714

03

122

893

000

135

000

69000

1165

05

052

987

000

186

000

196

005

530

07

011

282

000

112

005

315

000

261

B3

01

181

841

040

403

043

1270

030

142

03

023

503

000

96000

181

000

267

05

012

880

000

588

000

236

000

564

07

000

272

000

179

000

316

000

316

B4

01

156

990

000

1309

032

144

032

144

03

000

411

000

24000

360

000

360

05

000

111

000

57000

381

000

381

07

000

23000

28000

349

000

349

C1

01

809

712

970

1220

814

921027

133

03

320

919

000

657

000

486

000

394

05

199

1105

000

173

000

496

000

1469

07

048

863

000

239

000

355

000

247

Complexity 11

Table3Con

tinued

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

C2

01

574

937

142

1254

232

1397

070

8403

164

934

000

136

000

107

000

172

05

035

562

000

1225

000

248

000

537

07

002

766

000

1294

000

164

000

169

C3

01

517

979

111

1254

154

819

020

585

03

104

993

000

73000

38000

272

05

022

906

000

587

000

64000

1387

07

004

740

000

551

000

149

000

779

C4

01

486

663

135

153

034

523

085

313

03

065

711

003

7003

35003

107

05

008

821

000

18003

11003

1307

002

228

003

8003

22000

29

12 Complexity

0

1

2

3

4

5G

AP

valu

e (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

Tim

e (se

c)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000



GA

P va

lue (

)

0

1

2

3

4

5

6

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


Tim

e (se

c)

200

400

600

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000






Complexity 13

GA

P va

lue (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


AP

valu

e (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807











14 ComplexityTi

me (

sec)

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

1200

1400

1600

1800

2000


BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

Tim

e (se

c)

800

1000

1200

1400

1600

1800

2000








Acknowledgments


References


Complexity 15






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




10 Complexity

Table3Re

sults

obtained

bythelocalsearch

metho

dsforthe

ORLibraryrsquos

insta

nces

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

A1

01

522

622

965

248

945

1424

673

785

03

163

1124

000

1240

000

489

000

736

05

032

664

000

1095

000

274

024

700

07

010

745

000

366

000

138

000

1476

A2

01

431

463

000

833

000

1415

000

866

03

122

893

000

366

019

1290

019

615

05

052

987

183

52009

956

000

745

07

011

282

009

378

000

1562

000

743

A3

01

181

841

000

341

031

65000

686

03

023

503

000

602

000

693

000

1083

05

012

880

000

431

000

442

000

659

07

000

272

000

151

000

211

000

211

A4

01

156

990

000

25000

84000

8403

000

411

000

8000

31000

3105

000

111

000

2000

14000

1407

000

23000

1000

7000

7

B1

01

522

622

003

639

023

960

001

319

03

163

1124

001

588

008

436

003

1452

05

032

664

000

707

003

216

000

605

07

010

745

001

347

000

1630

000

1192

B2

01

431

463

144

806

275

222

009

714

03

122

893

000

135

000

69000

1165

05

052

987

000

186

000

196

005

530

07

011

282

000

112

005

315

000

261

B3

01

181

841

040

403

043

1270

030

142

03

023

503

000

96000

181

000

267

05

012

880

000

588

000

236

000

564

07

000

272

000

179

000

316

000

316

B4

01

156

990

000

1309

032

144

032

144

03

000

411

000

24000

360

000

360

05

000

111

000

57000

381

000

381

07

000

23000

28000

349

000

349

C1

01

809

712

970

1220

814

921027

133

03

320

919

000

657

000

486

000

394

05

199

1105

000

173

000

496

000

1469

07

048

863

000

239

000

355

000

247

Complexity 11

Table3Con

tinued

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

C2

01

574

937

142

1254

232

1397

070

8403

164

934

000

136

000

107

000

172

05

035

562

000

1225

000

248

000

537

07

002

766

000

1294

000

164

000

169

C3

01

517

979

111

1254

154

819

020

585

03

104

993

000

73000

38000

272

05

022

906

000

587

000

64000

1387

07

004

740

000

551

000

149

000

779

C4

01

486

663

135

153

034

523

085

313

03

065

711

003

7003

35003

107

05

008

821

000

18003

11003

1307

002

228

003

8003

22000

29

12 Complexity

0

1

2

3

4

5G

AP

valu

e (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

Tim

e (se

c)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000



GA

P va

lue (

)

0

1

2

3

4

5

6

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


Tim

e (se

c)

200

400

600

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000






Complexity 13

GA

P va

lue (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


AP

valu

e (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807











14 ComplexityTi

me (

sec)

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

1200

1400

1600

1800

2000


BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

Tim

e (se

c)

800

1000

1200

1400

1600

1800

2000








Acknowledgments


References


Complexity 15






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 11

Table3Con

tinued

Inst

(1minus120572)

Blind

SDND

TSGAP

Time

GAP

Time

GAP

Time

GAP

Time

C2

01

574

937

142

1254

232

1397

070

8403

164

934

000

136

000

107

000

172

05

035

562

000

1225

000

248

000

537

07

002

766

000

1294

000

164

000

169

C3

01

517

979

111

1254

154

819

020

585

03

104

993

000

73000

38000

272

05

022

906

000

587

000

64000

1387

07

004

740

000

551

000

149

000

779

C4

01

486

663

135

153

034

523

085

313

03

065

711

003

7003

35003

107

05

008

821

000

18003

11003

1307

002

228

003

8003

22000

29

12 Complexity

0

1

2

3

4

5G

AP

valu

e (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

Tim

e (se

c)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000



GA

P va

lue (

)

0

1

2

3

4

5

6

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


Tim

e (se

c)

200

400

600

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000






Complexity 13

GA

P va

lue (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


AP

valu

e (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807











14 ComplexityTi

me (

sec)

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

1200

1400

1600

1800

2000


BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

Tim

e (se

c)

800

1000

1200

1400

1600

1800

2000








Acknowledgments


References


Complexity 15






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




12 Complexity

0

1

2

3

4

5G

AP

valu

e (

)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


200

400

600

800

Tim

e (se

c)

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000



GA

P va

lue (

)

0

1

2

3

4

5

6

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


Tim

e (se

c)

200

400

600

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000






Complexity 13

GA

P va

lue (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


AP

valu

e (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807











14 ComplexityTi

me (

sec)

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

1200

1400

1600

1800

2000


BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

Tim

e (se

c)

800

1000

1200

1400

1600

1800

2000








Acknowledgments


References


Complexity 15






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 13

GA

P va

lue (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807


AP

valu

e (

)

0

5

10

15

20

25

30

35

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807











14 ComplexityTi

me (

sec)

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

1200

1400

1600

1800

2000


BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

Tim

e (se

c)

800

1000

1200

1400

1600

1800

2000








Acknowledgments


References


Complexity 15






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




14 ComplexityTi

me (

sec)

800

BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

1000

1200

1400

1600

1800

2000


BlindSD

NDTS

0(1 minus )

01 02 03 04 05 06 0807

Tim

e (se

c)

800

1000

1200

1400

1600

1800

2000








Acknowledgments


References


Complexity 15






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 15






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




parameter tuning for local-search-based matheuristic...

Documents