Multi-objective sequence dependent setup times permutation flowshop: A new algorithm and a...

European Journal of Operational Research xxx (2013) xxx–xxx

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European Journal of Operational Research

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Discrete Optimization

Multi-objective sequence dependent setup times permutation flowshop: A newalgorithm and a comprehensive study

Michele Ciavotta, Gerardo Minella ⇑, Rubén RuizGrupo de Sistemas de Optimización Aplicada, Instituto Tecnológico de Informática, Ciudad Politécnica de la Innovación, Edificio 8G, Acceso B, Universitat Politècnica de València,Camino de Vera s/n, 46022 Valencia, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 August 2010Accepted 30 December 2012Available online xxxx

Keywords:SchedulingPermutation flowshopMulti-objectiveSequence dependent setup timesIterated greedy

0377-2217/$ - see front matter � 2013 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.ejor.2012.12.031

⇑ Corresponding author. Tel.: +34 96 387 99 52; faxE-mail addresses: [email protected] (M. C

(G. Minella), [email protected] (R. Ruiz).

Please cite this article in press as: Ciavotta, M., eprehensive study. European Journal of Operatio

The permutation flowshop scheduling problem has been thoroughly studied in recent decades, both fromsingle objective as well as from multi-objective perspectives. To the best of our knowledge, little has beendone regarding the multi-objective flowshop with Pareto approach when sequence dependent setuptimes are considered. As setup times and multi-criteria problems are important in industry, we mustfocus on this area. We propose a simple, yet powerful algorithm for the sequence dependent setup timesflowshop problem with several criteria. The presented method is referred to as Restarted Iterated ParetoGreedy or RIPG and is compared against the best performing approaches from the relevant literature.Comprehensive computational and statistical analyses are carried out in order to demonstrate that theproposed RIPG method clearly outperforms all other algorithms and, as a consequence, it is a state-of-art method for this important and practical scheduling problem.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

The flowshop scheduling problem (FSP) is characterized by a setN of n jobs that must be processed by a set M of m machines. All mmachines are disposed in series and, without loss of generality,jobs visit machine 1 first, then machine 2 and so on until machinem. Each job needs a given, known in advance, fixed and non-nega-tive processing time at each machine. This is denoted as pij, foreach j 2 N and i 2M. A job cannot be in process at more than onemachine simultaneously and one machine can only process onejob at a time. The aim of this problem is to sequence the n jobson the m machines so that a given criterion is optimized. Basically,there are n! possible job permutations at each machine. In themost general case, each machine is associated with a differentqueue of jobs and hence, there are (n!)m possible solutions to thisproblem, where each solution is commonly referred to as asequence.

The FSP has been criticized for being too theoretical as most realindustry settings seldom fit into such a model. In part, this is attrib-utable to the absence of setup times, which are very common inindustry. Additionally, real-life problems have a multi-objectivenature. Furthermore, flowshops rarely have the inter-machine flex-ibility to manipulate jobs in the processing queues. For all these

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t al. Multi-objective sequence dnal Research (2013), http://dx.

reasons, this paper studies the multi-objective sequence depen-dent setup times permutation flowshop variant.

Setup times involve non-productive operations that have to beperformed on machines and that are not part of the job’s process-ing times. These may include, but are not limited to, cleaning, fix-ing and releasing parts to machines. Although on some occasionssetup times can be included in the processing times, in the major-ity of industrial contexts it is not possible to ignore them. We canroughly classify setups into two main categories. The first one de-fines those setups which are Sequence Independent (SIST) i.e., thesetup or changeover time for a machine only depends on the jobthat is to be processed next. The second category is the SequenceDependent setup times (SDST) case, where setup time depends bothon the current job being processed and on the next job in the se-quence. This second category is much more complex and includesthe first one as a particular case, permitting it to describe severaloperational scenarios. Furthermore, setups can be either anticipa-tory or non-anticipatory. In the former case, setups can be per-formed as soon as the machine is free and before the next job inthe sequence is loaded. In this setting, we denote by Sijk, "i 2M,"j, k 2 N, j – k the job sequence dependent setup time at machinei when processing job k after having processed job j.

A large body of research in the FSP deals with the optimizationof a single criterion. The most commonly studied objective is theminimization of the maximum completion time or makespan, de-noted to as Cmax which is calculated as maxn

j¼1fCmjg, where Cmj isthe completion time of job j, i.e., the time at which job j finishesprocessing at machine m. Often, Cmj is simply denoted as Cj. Given

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2 M. Ciavotta et al. / European Journal of Operational Research xxx (2013) xxx–xxx

a sequence p of n jobs where p(l) denotes the job occupyingposition l in the permutation with l = {1, . . . , n}, Cmax withSDST can be calculated in OðnmÞ steps with the following recur-sive formula: Ci;pðlÞ ¼maxfCi�1;pðlÞ ;Ci;pðl�1Þ þ Si;pðl�1Þ ;pðlÞ g þ pi;pðlÞ whereCi;pð0Þ ¼ 0; C0;pðlÞ ¼ 0 and Si;pð0Þ ;pðlÞ ¼ 0; 8i 2 M; l ¼ f1; . . . ;ng.

Makespan has been widely studied since a minimum valuetranslates into a high resource utilization, throughput and OverallEquipment Efficiency (OEE). A second commonly studied criterionis the total flowtime, defined as TFT ¼

Pnj¼1Cj if we assume that all

jobs are available at time 0 (i.e., the release dates or rj are all zero).TFT, albeit certainly related, is quite different from makespan. Alow TFT value reduces the Work-In-Process (WIP) inventory whichis of paramount importance in real production shops. TFT also en-sures a minimum cycle time in production environments. Cmax andTFT are production-oriented criteria and neglect an important as-pect of production which is client satisfaction. Jobs often model cli-ent orders that have a desired delivery date. This date is accountedfor in scheduling by means of a due date dj for each job. A job issaid to be tardy if Cj > dj. With this in mind, we define the tardinessof a job j as Tj = max{Cj � dj,0}. As can be expected, not all ordersfrom clients are equally important. To model this, a priority, impor-tance or weight wj is also given for each job. Considering all previ-ous definitions, the third most commonly treated objective is thetotal weighted tardiness or TWT ¼

Pnj¼1wj � Tj.

The previous objectives are the three most common, but are notthe only ones. In practice, as one can expect, a combination of objec-tives is usually sought. For example, optimizing makespan results ina very high machine utilization. However, most due dates are likelyto be violated. As a consequence, a multi-objective approach isneeded. As concluded from the multi-objective flowshop review ofMinella et al. (2008), most authors deal with several objectives inthe most simple way, which is just adding them into a singleweighted linear combination measure for example a � Cmaxþð1� aÞ � TFT , where 0 6 a 6 1. This is an example of the ‘‘a priori’’approach. The problem with this is that often, objectives are mea-sured in different scales and it is difficult to map a into a valid userpreference. Another procedure, referred to as the ‘‘a posteriori’’approach consists of finding out a set of solutions. Each solution rep-resents a trade-off in the optimization of a given set of independentobjective functions. This solution set is called the Pareto front. It isassumed that a multi-objective procedure returns this set to thedecision maker, which later picks one solution from it.

We restrict ourselves to the permutation version of the flow-shop problem where job passing is not allowed from machine tomachine, i.e., the permutation of jobs cannot change from one ma-chine to the next. This results in a smaller solution space of n! Thisversion of the problem is denoted as the permutation flowshopproblem or PFSP in short. This special case is important in practicesince in-process storage of products is very limited in most situa-tions. Note that even this simplification of the problem, with nosetups and one single objective still remains NP-hard for manycommon criteria (Garey et al., 1976) and remains intractable forlow values of n.

Following the well known classification scheme of Graham et al.(1979) and the extension of the notation for the multi-objectiveproblems by T’kindt and Billaut (2006), the problem studied in thispaper is denoted as F/prmu, Sijk/#(c1,c2) where c1 and c2 are thetwo objectives that are considered in a Pareto approach. The twocombinations of objectives that we consider in this paper are(Cmax,TFT) and (Cmax,TWT). To the best of our knowledge, this prob-lem (even with a different set of objectives) has not been studied inthe scientific literature and this paper presents the first attempt tosolve it. We approach this problem with a recently proposed meta-heuristic strategy, specially tailored for multi-objective problems.

The remainder of this paper is organized as follows: Section 2presents a review of the literature on multi-objective optimization

Please cite this article in press as: Ciavotta, M., et al. Multi-objective sequence dprehensive study. European Journal of Operational Research (2013), http://dx.

as well as existing results for the PFSP with setup times. Section 3details the proposed algorithm which is later tested in Section 4 bycarrying out a wide campaign of experiments and the results arestatistically analyzed in detail. Finally, in Section 5, some conclu-sions and further research topics are given.

2. Literature review

To the best of our knowledge, no paper has been published thatconsiders all characteristics of the problem studied in this work,i.e., multi-objective permutation flowshop problem with sequencedependent setup times. Hence, in the following subsections wepresent first a brief review of multi-objective flowshop and seconda review about SDST flowshop with one single objective.

2.1. Multi-objective flowshop

The literature on multi-objective optimization is extremely rich.However, the multi-objective PFSP field is relatively scarce, spe-cially in comparison with the large number of papers publisheddealing with the single criterion flowshop problem. The few pro-posed multi-objective methods for the PFSP are mainly based onevolutionary optimization and on local search techniques like sim-ulated annealing (SA) or tabu search. In Minella et al. (2008), theauthors carefully reviewed the literature related to this problem.Thus, here we restrict ourselves to only the most significant worksand to some other more recent published material.

Methods belonging to the ‘‘a priori’’ multi-objective approach(weighted objective functions, lexicographical and goal optimiza-tion, etc.), in general, return a single solution, the closest one todecision-maker’s desires. Differently, methods belonging to the‘‘a posteriori’’ approaches return several equivalent solutions (Par-eto set) among which the decision maker can choose.

Focusing only on the ‘‘a posteriori’’ approach, the number of pub-lications in the flowshop literature is reduced to a little set. A geneticalgorithm (GA) was proposed by Murata et al. (1996) which wascapable of obtaining a Pareto front for makespan and total tardiness.This algorithm, referred to as MOGA (Multi Objective Genetic Algo-rithm), applies elitism and the selection phase employs a fitness va-lue assigned to each solution as a function of the weighted sum ofthe objectives. The weights for each objective are randomly as-signed at each iteration of the algorithm. Later, in Ishibuchi andMurata (1998), the authors extended this algorithm by means of alocal search procedure applied to every newly generated solution.

A genetic algorithm is shown by Bagchi (2001), which is basedon the Srinivas and Deb (1994) NSGA method. Some short experi-ments are given for a single flowshop instance with flowtime andmakespan objectives. Murata et al. (2001) improve the earlierMOGA algorithm by Murata et al. (1996). This new method, calledCMOGA, refines the weight assignment.

Ishibuchi et al. (2003) present a comprehensive study about theeffect of adding local search to their previous algorithm (Ishibuchiand Murata, 1998). The local search is only applied to good individ-uals and by specifying search directions. This form of local searchwas shown to give better solutions for many different multi-objec-tive genetic algorithms. In Loukil et al. (2000), several schedulingproblems are solved with different combinations of objectives.The main technique used is a multi-objective tabu search, referredto as MOTS. Later, in Loukil et al. (2005), a similar study is carriedout. In this case the multi-objective simulated annealing (MOSA)approach is employed.

Suresh and Mohanasundaram (2004) propose a Pareto-basedsimulated annealing algorithm for makespan and total flowtimecriteria. Experiments are conducted and the proposed method iscompared against that of Ishibuchi et al. (2003) and against an



M. Ciavotta et al. / European Journal of Operational Research xxx (2013) xxx–xxx 3

early unpublished version of the SA later presented in Varadhara-jan and Rajendran (2005). Arroyo and Armentano (2004) studiedheuristics for several two and three objective combinations amongmakespan, flowtime and maximum tardiness. For the general mmachine case, the authors compare the results against those ofFraminan et al. (2002). The results favor the proposed method that,when used as a seed sequence, also improves the results of the GAof Murata et al. (1996). The same authors developed a tabu searchfor the makespan and maximum tardiness objectives in Armentanoand Arroyo (2004). The algorithm includes several advanced fea-tures like diversification and local search in several neighborhoods.The proposed method is shown to be competitive in numericalexperiments. In a more recent paper, Arroyo and Armentano(2005) carry out a similar study but in this case using genetic algo-rithms. The makespan and total flowtime objectives are studied byVaradharajan and Rajendran (2005) with the help of simulatedannealing methods. These algorithms start from heuristic solutionsthat are further enhanced by improvement schemes. Two versionsof these SA (MOSA and MOSA-II) are shown to outperform the GAof Ishibuchi and Murata (1998). According to the comprehensivecomputational evaluation of Minella et al. (2008), where 23 methodswere tested for the multi-objective flowshop, an enhanced version ofMOSA_Varadharajan algorithm (named MOSA-II in the original paper)is shown to consistently outperform all other methods.

Pasupathy et al. (2006) proposed a Pareto-archived geneticalgorithm with local search and have tested it with the makespanand flowtime objectives.

Geiger (2007) has published an interesting study where thetopology of the multi-objective flowshop problem search space isexamined. Using several local search algorithms, the author ana-lyzes the distribution of several objectives and tests several combi-nations of criteria. Yanda and Tamura (2007) presented a variant ofthe NSGAII of Deb (2002), referred to as hMGA, which uses a work-ing population with dynamic size made of only heterogeneoussolutions. According to the authors, this choice prevents the algo-rithm from getting stalled in local optima. Framinan and Leisten(2008) presented an iterated greedy (IG) procedure based on theNEH heuristic. This algorithm is an evolution of the IG basic prin-ciple for the multi-objective PFSP. Recently, Rajendran and Ziegler(2009) proposed an ant-colony algorithm for the flowshop scedul-ing problem (MOACA) with the objective of minimizing the make-span and total flowtime. The authors presented 20 variants of thealgorithm and some of them turned out to be highly competitivefor the considered benchmark.

2.2. SDST flowshop

Hundreds of papers dealing with the permutation flowshopproblem have been published in the literature but only a relativelyminor fraction of them consider sequence dependent setup times.Exact techniques for the SDST permutation flowshop have shownrather limited results. The latest reference and most advancedstudy is that of Ríos-Mercado and Bard (2003) which studied thepolyhedral structure of two mixed-integer programs for the SDSTflowshop in order to generate more effective cuts to use in abranch-and-cut framework. Some heuristics and metaheuristicalgorithms for the F/Sijk, prmu/Cmax have been proposed. For exam-ple, Ríos-Mercado and Bard (1998) presented a modification of thewell known NEH heuristic for the regular flowshop from Nawazet al. (1983) that takes into account setup times. In the same papera GRASP algorithm is also proposed. In a later work, the sameauthors presented a modification of the heuristic of Simons(1992) resulting in a new method called HYBRID (Ríos-Mercadoand Bard, 1999). Ruiz et al. (2005) proposed a genetic and amemetic algorithm for the F/Sijk, prmu/Cmax. They carried out an


comprehensive experimental study comparing their proposalsagainst several methods adapted from the F//Cmax problem.

About the F=Sijk; prmu=Pn

j¼1wjTj, little has been published. InParthasarathy and Rajendran (1997a,b), a Simulated Annealingheuristic was proposed for the SDST flowshop problem with thegoal of minimizing the maximum weighted tardiness and the totalweighted tardiness, respectively. Rajendran and Ziegler (1997)introduced an algorithm formed by a new heuristic and a localsearch improvement scheme for the weighted flowtime objective.Another similar work is that of Rajendran and Ziegler (2003) werea combined objective of total weighted flow-time and tardiness isconsidered. Ruiz and Stntzle (2008) proposed two iterated greedyalgorithms for the PFSP with sequence dependent setup times.The first one follows the guidelines of the IG framework adaptedto setup times and the second incorporates a simple descent localsearch. More details can be found in Ruiz et al. (2005) where theauthors carried out an extensive literature survey about this prob-lem and in Allahverdi et al. (2008), an updated and comprehensivereview of scheduling research with setup times.

3. Restarted Iterated Pareto Greedy

The iterated greedy methodology (IG) belongs to the stochasticlocal search techniques (SLS) and the basic scheme was first pre-sented by Jacobs and Brusco (1995) for the set covering problem.Afterwards, Schrimpf et al. (2000) named as ‘‘Ruin and Recreate’’ avery similar algorithm for the vehicle routing problem. Iteratedgreedy method was applied by Ruiz and Stntzle (2007) to the regu-lar permutation flowshop problem with the makespan minimiza-tion objective. The results have encouraged several other authorsto propose variants and adaptations to other problems, includingthe already cited paper of Ruiz and Stntzle (2008) for the SDST flow-shop, Pan et al. (2007) for the no-wait flowshop or Ying (2009) forthe hybrid flowshop. In all these problems, IG has produced state-of-the-art results. Framinan and Leisten (2008) proposed a multi-objetive IG for the regular permutation flowshop problem whichis basically an evolution of the NEH heuristic of Nawaz et al.(1983) modified to use Pareto dominance. As a result, it seems plau-sible to attempt an extension for the multi-objective flowshop withsetup times. Note that IG works with a single solution and was pro-posed for a single objective, therefore, a just simple adaptation is notpossible if high quality results have to be achieved.

Similarly to Minella et al. (2011), we propose an extension ofthe original Iterated Greedy algorithm named Restarted IteratedPareto Greedy (RIPG), which is now presented. The rationale of thisalgorithm is very simple: a greedy multi-objective strategy is iter-atively applied over a set of non-dominated solutions.

The proposed RIPG is broken into five phases: In the first phase(Initialization), an initial set of good solutions is generated using dif-ferent heuristics, each one designed to attain good values for a spe-cific criterion. The remaining four phases are iteratively repeatedand constitute the main loop of the algorithm. The second phase,called Selection, chooses one solution from the current working setfor the next phase: the Greedy one in which the selected solutionis disrupted by means of a Destruction operator, by removing someelements and then a greedy procedure, called Construction, is ap-plied. Afterwards, a Local search phase is applied over a selected ele-ment of the current working set. Lastly, a Restart procedure isimplemented to prevent the algorithm from getting stuck in localoptima. The following sections describe each phase in detail.

3.1. Algorithm initialization and selection phase

As is of common knowledge, initial solutions expressly gener-ated to have opportune features often play an underlying role in




creating a high performing algorithm. On the other hand, since ourfirst concern is to create an algorithm capable of performing wellfor different pairs of obejectives, we chose heuristics that returngood enough solutions for many different objectives ensuring alsoa sufficient degree of diversity. In order to generate a good InitialSolution Set (ISS) we make use of the initialization procedure pro-posed by Varadharajan and Rajendran (2005), which demonstratedto return high quality initial solutions in the review of Minella et al.(2008). This procedure uses the NEH heuristic of Nawaz et al.(1983) and the heuristic of Rajendran (1995), both designed forthe optimization of a single criterion. Such heuristics are used togenerate two distinct solutions for each objective to optimize.

In a first step, all initial solutions are processed by the Greedyphase one by one. The resulting frontiers of this process are addedto the ISS and then, the dominated solutions are removed and theinitial current working set is conformed. The aim behind this policyis to avoid that a likely large improvement during the initial itera-tions might generate a set of solutions that dominate the remain-ing initial solutions, impoverishing the quality and diversity of theworking set too early. At each iteration of the algorithm, the selec-tion phase is responsible for pointing the search towards promisingdirections. Selection achieves this goal by choosing one solutionfrom the current working set on the basis of considerations relatedto their quality. In this way, the algorithm focuses on only thosesolutions that are more likely to increase the quality of the currentworking set, speeding up the whole search process.

A modified version of the Crowding Distance Assignment (CDA)procedure, originally presented in (Deb, 2002), has been developedin order to carry out the selection process. This procedure first di-vides the working set into several dominance levels. Each solutionof one level strictly dominates all the solutions in the next level.The CDA then assigns to each solution a value (Crowding Distance)

Fig. 1. Pseudocode of MCD


dependent on the normalized Euclidean distances between it andthe solutions that precede and follow it in the same dominance le-vel. The main difference resides in the fact that the modified pro-cedure considers the number of times each solution has beenalready selected in previous iterations (Selection Counter), and usesthis information to calculate the Modified Crowding Distance(MCD). The element with the highest value of MCD is selected asthe starting point for the Greedy or local search phases.

The aim of this MCD procedure is to select a candidate solutionbelonging to a less crowded region of the Pareto front and at thesame time has already been selected a small number of times.The use of such an operator demonstrated, in preliminary experi-ments, to significantly improve the Pareto front in terms of qualityand spread of its solutions. The pseudocode of this procedure ispresented in Fig. 1a.

3.2. Greedy phase

This is the main and most innovative part of the algorithm eventhough the original IG structure with two phases: Destruction andConstruction respectively, is preserved. However, this greedy phasein the RIPG is radically different from the original IG where onlyone partial solution is maintained and a NEH-like greedy heuristicis applied in one unique step at each iteration of the algorithm. Inour case, the Greedy phase becomes an iterative process, thatworks with a set of partial solutions and returns a set of non-dom-inated permutations.

The Destruction step chooses a random starting position and ablock of k consecutive elements (jobs) are removed from the se-lected solution. Note that in the original IG algorithm, the removalof jobs is not carried out in blocks.

A and LS procedures.




For the Construction step, a variation of the NEH insertionscheme is used. The main difference from that heuristic lies inthe use of Pareto dominance to maintain not just one incompletepartial solution at each iteration (as in NEH), but a whole set ofnon-dominated partial solutions generated during the insertionprocess. Actually, the Construction procedure inserts, one by one,all removed elements from the block back into each partial solu-tion from the non-dominated partial solution set. This insertingschema was already effectively used in Arroyo and Armentano(2004). At each step, a new set of partial solutions is generated.More specifically, let n be the length of the initial solution and kthe size of the block of removed elements. At the end of the firststep, after the first removed element is inserted into all positionsof the partial solution, we have n � k + 1 partial solutions of lengthn � k + 1. In the second iteration, the procedure inserts the secondremoved element in all positions of each one of the n � k + 1 partialsolutions generated in the previous step. Then, at the end of thissecond iteration, the number of partial solutions is: (n � k + 1) �(n � k + 2). Following the same reasoning, at the end of the con-struction phase, a set of

Qki¼1ðn� kþ iÞ of complete solutions is

generated. This defines an upper bound for the number of solutionsgenerated by the greedy phase of the algorithm. Regardless of this,the bound is very far from being tight because, at each iteration, allthe dominated incomplete sequences are removed. The greedyphase therefore returns a set of non-dominated solutions, whichis added to the current working set, and then, dominated solutionsare removed. Finally, the MCD selection procedure is applied to thecurrent working set and a solution is selected to be processed bythe local search phase, which is explained next.

3.3. Local search phase

A simple and fast local search procedure has been demonstratedto be very helpful in improving the quality of solutions in the singleas well as in the multi-objective cases. Hence, we added to ouralgorithm a simple local search phase aimed at refining the workof the greedy phase. In the original IG, the local search procedureuses as an input the outcome of the greedy phase. For the multi-objective case, the greedy phase returns a set of non-dominatedelements, and it is likely that the current working set changes afteradding it. To better tackle the multi-objective nature of the prob-lem, thereby, it is not trivial to decide which solutions undergothe local search procedure. Therefore, we decided to entail oncemore the selection procedure previously described to choose asolution from the current working set for feeding the local searchphase.

In order to maintain the algorithm as simple and fast as possiblewe focused our effort in obtaining a simple and fast local searchprocedure. The rationale of this phase is quite straightforward: nsel

elements belonging to the selected solution are randomly chosen,removed and re-inserted into nneigh consecutive positions, half ofwhich usually precede and half follow the original position of theelement. In fact, depending on the distance of the original positionfrom the beginning or from the end of the sequence, the neighbor-hood, still having the same amount of moves, may or may not besymmetric. Local search in a multi-objective setting is not as sim-ple as one might think. As a matter of fact, the concept of firstimprovement or best improvement does not directly apply. Inour case, all movements, i.e., nsel � nneigh insertions are carriedout and all solutions are evaluated. Afterwards, Pareto dominanceis checked and a final non-dominated set is generated as a result.Note that is much faster than generating, one by one, all neighborsand checking each time for dominance. In order to further speed upthis local search, we employ the well known accelerations of Tail-lard (1990). Finally, this set is included into the working set anddominance is applied again. During the initial design phase we


studied the algorithm performance by varying nsel and nneigh. Weobtained the best results for nneigh = 5 and for nsel by dynamicallychanging its value with the ncount value (Selection Counter intro-duced in Section 3.1) of the selected solution, in accordance withthe following formula:

nsel ¼ncount if ncount 6 n=2n=2 otherwise

�

The pseudocode of the local search procedure is presented inFig. 1b.

3.4. Restart phase

IG methods have one main drawback: they are prone to getstuck in local optimum solutions. The reason lies behind their verynature as they are greedy methods. RIPG is no different. To avoidthis potential problem, we have included a simple, yet reliable re-start phase. This procedure merely consists of storing all the ele-ments of the current working set in a separate archive and thencreating a new random working set of 100 elements. The mainadvantage of this restart procedure is that it is a very fast way tointroduce diversification inside our metaheuristic scheme, whereasits main inconvenience consists of the difficulty in choosing of asuitable restarting criterion. The general idea is to execute a restartwhen the working set has not been changed during a sufficientlylarge number of iterations. To accomplish with our objective ofsimplicity and reliability, we use the simple approach of checkingwhether the size of the current working set changes after each iter-ation. Of course, this strategy is sometimes inaccurate because itcannot detect a change in the working set that does not affect itscardinality, but has the unquestioned advantage of being very fast.Another important issue to take into account while designing aneffective restart is to establish the minimum number of iterationsto safely assert that the search process is in a stalemate. In fact, ifthe restart condition is not carefully designed, it might either beapplied too often, thus preventing reaching a steady state in thesearch or be too seldom applied, wasting, in this way, valuableCPU time. We carried out short initial tests considering severalfixed as well as dynamic restarting points and the best results wereachieved by applying a restart after n � 2 iterations withoutchanges in the cardinality of the current working set. Note that thisrule is effective since it takes into account the size of the instanceallowing for more iterations to bigger ones for which the algorithmneeds a larger amount of time to reach a steady state.

4. Experimental phase

This section is aimed at introducing the reader to all the ele-ments needed in order to fully understand the experiments carriedout, the results and their implications. First we deal with thedescription of the state-of-the-art algorithms the RIPG is comparedto within the experimental phase. Later on, we describe in detailthe test bed instance sets used and discuss the performance assess-ment methodologies considered. Then, the design process of ourproposed algorithm is described and ultimately, we describe thetest campaign and analyze the results.

4.1. Adaptation of existing metaheuristics

In the previously cited review work of Minella et al. (2008), wecarried out a comprehensive analysis of the performances of themost well known multi-objective algorithms. We carefully re-implemented 23 algorithms and made an exhaustive test cam-paign with three couples of objectives, three different stoppingtimes and a large set of instances expressly created for the




multi-objective flowshop problem. Recall that in this paper we ex-tend this problem with the presence of setup times. A preliminaryexperiment in which all the algorithms have been tested using areduced set of instances has been carried out and on the basis ofthe results obtained we have selected the best algorithms. Manyof these are specifically designed to tackle flowshop problemswithout setups while the other three are general purpose multi-criteria optimization procedures. Note that, to the best of ourknowledge, no multi-objective methods have been specifically pro-posed for the setup times flowshop with Pareto approach. The gen-eric methods are now briefly explained. A genetic algorithm isproposed by Corne et al. (2000). This method, called PESA uses aselection and replacement procedure based on a crowding mea-sure. Later, in Corne et al. (2001) an enhanced PESAII method isprovided. This algorithm differs from the preceding one only inthe selection technique in which the fitness value is assignedaccording to a hyperbox calculation in the objective space.

Kollat and Reed (2005) proposed a variation of NSGAII proposedby Deb (2002) referred to as e-NSGAII by adding e-dominancearchiving and adaptive population sizing. The reader can find amore detailed description of these algorithms in Minella et al.(2008).

We decided to include another seven algorithms recently pre-sented in the literature that did not make it for Minella et al.(2008) evaluation. Five of them have been modified in order touse up all the available time unlike the original versions whichended after a fixed number of iterations. Those algorithms arehighlighted by adding a ‘‘_M’’ at the end of the name. Note thatone of them, a modified version of the multi-objective simulatingannealing of Varadharajan and Rajendran (2005), referred to asMOSA_Varad_M has been already presented in Minella et al.(2011) where it was shown to clearly outperform the original ver-sion. For space reasons we included at http://soa.iti.es as on-linematerial the pseudo-codes of the original as well as the modifiedversions of these algorithms. The selected algorithms, seventeenin total, either specific for the PFSP or for the general multi-objec-tive version, are summarized in Table 1. Table 2 shows the valuesfor the parameters used for all the algorithms in the comparison.Those values are those reported in their respective original works.

4.2. Benchmark description

In the experiments presented in this paper we make use ofthree different instance sets based on the original work of Taillard(1993) for the regular flowshop and on the papers of Ruiz et al.(2005) and Ruiz and Stntzle (2008) for the setup times version.

Table 1Re-implemented methods for the SDST multi-objective flowshop.

Acronym Year Author/s

MOGA_Murata 1996 Murata et alPESA 2000 Corne et al.PESAII 2001 Corne et al.CMOGA 2001 Murata et alMOTS 2004 Armentanoe-NSGAII 2005 Kollat and RMOGALS_Arroyo 2005 Arroyo andMOSA_Varad 2005 VaradharajaPGA_ALS 2006 Pasupathy ePILS 2007 GeigerhMGA 2007 Yanda and TMOIGS 2008 Framinan anMOACA17_M 2009 Rajendran aMOACA18_M 2009 Rajendran aMOACA19_M 2009 Rajendran aMOACA20_M 2009 Rajendran aMOSA_Varad_M 2011 Minella et a


Each set contains instances with several combinations for thenumber of jobs (n) and number of machines (m). The n �m combi-nations are: {20,50,100} � {5,10,20} and 200 � {10,20} for a totalof 11 different groups of instances. The first two sets, referred to asSSD50 and SSD125, respectively, have ten instances for each group,resulting in a total of 110 instances per set. Setup times in SSD50and SSD125 are generated to be respectively the 50% and the125% of the processing times (pij). This means that since the pij inTaillard’s instances are generated using a uniform distribution inthe range [0–99] in the first set, setup times are uniformly distrib-uted in the range [0–49], while in the second, their range is [0–124]. Each instance is assigned with a set of weights and due dates.The weights are drawn from a uniform U[1,10] distribution whilethe due dates dj are generated by means of the expression: dj = Pj �(1 + random � 3) where Pj ¼

Pmi¼1pij is the sum of the processing

times over all machines for jobs j 2 N and random is a randomnumber uniformly distributed in [0,1]. Finally, we created fromscratch a third set of instances, referred to as SSDTest, borrowingthe structure from the first two. This set is used to calibrate ouralgorithm and contains only four instances for each group, half ofwhich have setup times in [0–49] and half in [0–124]. All bench-marks as well as all solutions obtained in this paper are availableat http://soa.iti.es.

4.3. Performance assessment methodologies

In single objective optimization, the concept of a better solutionis straightforward. However, when multiple objectives are present,deciding when a given result A is better than B is far from simple.As mentioned, the outcome of a multi-objective optimizer is notjust one solution, but a set of several different solutions. Usually,the outcome is already processed so that only non-dominated solu-tions are given. This is commonly referred to as a frontier. It is obvi-ous, of course, that a frontier is better than another if all the pointsof the former (strongly) dominate all the points of the latter. In thecase that this condition does not hold, an unambiguous way toestablish which frontier outperforms the other does not exist tothe best of our knowledge.

Many indices have been introduced in the literature to over-come this impasse but this is still an open issue so much so that,recently, first Knowles et al. (2006) and later Zitzler et al. (2008)demonstrated that many frequently used metrics are non-Pareto-compliant i.e., in some cases, they can assign a better value to aPareto frontier B respect to frontier A even if A dominates B. Thosemetrics therefore, can often give wrong and misleading results.Therefore, special attention must be given to the choice of qualitymeasures to ensure fair and generalizable results.

Type

. Genetic algorithm. SpecificGenetic algorithm. GeneralGenetic algorithm. General

. Genetic algorithm. Specificand Arroyo Tabu search. Specificeed Genetic algorithm. GeneralArmentano Genetic algorithm. Specificn and Rajendran Simulated annealing. Specifict al. Genetic algorithm. Specific

Iterated local search. Specificamura Genetic algorithm. Specificd Leisten Iterated Greedy. Specific

nd Ziegler Ant-colony algorithm. Specific.nd Ziegler Ant-colony algorithm. Specific.nd Ziegler Ant-colony algorithm. Specific.nd Ziegler Ant-colony algorithm. Specific.l. MOSA_Varad improved version


http://soa.iti.es

http://soa.iti.es


Table 2Table of parameter values for the re-implemented methods.

Algorithm Parameter Value

MOSA_Varad_M Non-Dominated Solutions Archive Size 500Initial Temperature 575Final Temperature When time is

overRatio of Temperature Decreasing 0.9

MOSA_Varad Non-Dominated Solutions Archive Size 500Initial Temperature 575Final Temperature 20Ratio of Temperature Decreasing 0.9

MOTS Population Size 10PILS Number of environments 3MOIGS Destruction Block Size 7PESA Population Size 100

Crossover Probability 0.7Mutation Probability 1/Number of

jobsNon-Dominated Solutions Archive Size 100Number of Divisions in the Grid 32

PESAII Population Size 100Crossver Probability 0.7Mutation Probability 1/Number of

jobsCardinality of the Non-DominatedSolutions set

100

Number of Divisions in the Grid 32MOGALS_Arroyo Number of elite solutions 20

Population size 100Crossover probability 0.9Mutation probability 0.6Max Paths 10Max Iterations of Local Search 15Iterations between each 2 searchexecutions

100

PGA_ALS Population size 100Crossver probability 1Mutation probability 0.1

MOGA_MURATA Number of elite solutions 5Population size 100Crossover probability 0.9Mutation probability 0.1

e-NSGAII Population 10Crossover probability 1Mutation probability 1/Number of

jobsEpsilon value 0.001

CMOGA Number of elite solutions 3Population size 101Crossover probability 0.8Mutation probability 0.3Maximum cell distance 20

hMGA Population size 100Crossover probability 0.9Mutation probability 0.01

MOACA17_M q 0.7C 2P 1.5K 50Population size 500






In this paper, we choose the hypervolume IH and the unary Epsi-lon I1

e indicators that Knowles et al. (2006) and Zitzler et al. (2008)demonstrated as being Pareto-compliant and, at the moment, canbe considered as the state-of-the-art indicators for the evaluationof multi-objective algorithms. Additionally, considering two qual-ity indicators instead of one, increases the soundness of our con-clusions. Given two multi-objective optimizers, these twoindicators can, sometimes, give conflicting results, indicating thatneither one can be considered superior.

The hypervolume indicator IH, first introduced by Zitzler andThiele (1999) measures the normalized (hyper) volume of thesolution space dominated by the Pareto front approximation gener-ated by one algorithm. A reference point is needed for closing thehypervolume. We obtained this reference point by consideringthe worst value for each objective over the whole set of Pareto frontgenerated by all the methods for a certain instance and multiplyingthem by 20%. More formally, the IH indicator can be defined asfollows: Given a set of Pareto frontiers F , being F 2 F a frontier,pt 2 F a point belonging to the frontier, NObj the number of objec-tives and NSol the number of points in F, then the hypervolumefor F can be calculated as IHðFÞ ¼

P16i6NSol

P16j6NObj

pti;j�1:2�minjF1:2�ðmaxjF�minjFÞ

,where minjF and maxjF are the best and worst values, respec-tively, for objective j over all the frontiers in F . Notice that, as theobjective values are normalized, the maximum IH value can by ob-tained by the product of the reference point values: 1.2 � 1.2 = 1.44.

As far as the Unary Epsilon Indicator I1e is concerned, it was pro-

posed initially by Zitzler et al. (2003) and was later extended inKnowles et al. (2006) and in Zitzler et al. (2008). It measures theminimum distance between a given Pareto front and the optimalone or an approximation of it. Since for our problem the optimalfront for each instance is not known, a reference set, constitutedby gathering all non-dominated solutions obtained by all the testedalgorithms, is used. As said, the objectives values are normalizedand additionally translated by one unit in order to avoid divisionby zero errors in the calculation of the indicator. This approachwas used with success by Minella et al. (2008) and by Minellaet al. (2011). In this way the indicator varies between 1 and 2. Avalue close to 1 means that the considered frontier is close to thereference set, whereas a value close to 2 means that the set of solu-tions is distant. It is formally calculated as follows. P is the Paretofront or a reference set and S is an approximation to the Paretofront. Actually, I1

e ¼ IeðS; PÞ where IeðS; PÞ ¼maxx2 minx1 maxjfjðx1Þfjx2 .

The main drawback of using unary indicators when comparingdifferent frontiers is that such measures cannot supply us withinformation about the spatial behavior of the considered algo-rithms, i.e., in which region of the objectives space one algorithmsbehaves better or worse than the other. This knowledge is espe-cially important during the design phase of a new method. In fact,having at our disposal a technique that permits us to analyze theoutcomes and highlights potential problems of convergence wouldhighly simplify and accelerate the development processes.

Fonseca et al. (2001) proposed a probabilistic measure, calledAttainment Function, which is able to provide useful insight intothe spatial behavior of an algorithm. To be more precise, letX 2 Rd be an arbitrary point of a d-objective solution space andF ¼ fvp 2 Rd; p ¼ 1; . . . ; Pg be a frontier generated in a single runof and algorithm referred to as a. P is the total number of pointspresent at the frontier F. The attainment function is defined byAFaðXÞ ¼ Pð9vp 2 F : vpEXÞ which describes the probability themethod a has to generate, in a single run, a Pareto front approxi-mation F in which at least one element weakly dominates (E)the arbitrary point X . In the case of stochastic algorithms, it isnot possible to express this function in a closed form but it canbe empirically approximated by employing the outcomes of




several algorithm runs. This approximation is called EmpiricalAttainment Function and is defined in Fonseca et al. (2001) as fol-lows: EAFðXÞ ¼ 1

q

Pqh¼1IðF hEXÞ where F 1;F 2; . . . ;F q represent q

Pareto set approximations obtained in q independent algorithm’sruns and

IðF hEXÞ ¼1 if F hEX0 otherwise

�

The attainment function has one inconvenience. The problem is thatit has to be calculated for each pair of compared methods. When thenumber of evaluated methods grows, the number of pairwise com-parisons grows quadratically. In order to overcome this drawback,López-Ibáñez et al. (2006) presented the Diff-EAF that is a probabil-ity function defined as the difference between two EAFs. Let a and bbe two algorithms. The Diff-EAF is then defined as follows:

Diff-EAFða;bÞðXÞ ¼1q

Xq

h¼1

IðF ahEXÞ � IðF b

hEXÞ� �

This function represents the probability of X to be dominated by afrontier of a but not of b. Note that the Diff-EAF(a,b) may have posi-tive as well as negative values. A positive value indicates that thealgorithm a prevails over b in X , the other way around in case ofa negative value.

Finally, another important issue related to the comparisonamong algorithms is the choice of a suitable termination criterion.We believe that the fairest way to confront different methods isassigning the same amount of CPU time to each one of them andmore time to larger instances with respect to smaller ones. There-fore, we assign to each algorithm the same elapsed CPU time limitthat depends on the size of the considered instance. The algorithmsare then stopped after n �m/2 � t milliseconds of CPU time, where tis an input parameter that will be tested at different values. In thisway, we assign more time to larger instances that are obviouslymore time consuming to solve.

4.4. Design and calibration of the RIPG

The RIPG is the result of an thorough engineering process inwhich all parts that constitute it have been compared againstseveral possible alternatives. The aim is to create an efficient andeffective algorithm. To do this we employ of a sound statisticalmethodology called Design of Experiments (DoE) (see for moredetails Montgomery (2009)). In order to prevent a possible over-

1.16

1.17

1.18

1.19

1.2

(a)

RIPG

_LS

RIPG

_B

IPG_L

S

IPG_B

IεIH

Fig. 2. Means plots and Tukey HSD confidence intervals (a = 0.05) in the ANOVA test ft = 100 millisecond CPU time stopping criterion.


calibration of our proposed algorithm, we decided to use, in thisphase, an instance set (SSDTest) different to those used in the com-parison of RIPG against other algorithms (SSD50 and SSD125). Ini-tially, we studied the influence of the local search and the restartmechanism on the quality of the produced solutions. Thereby, weassembled 4 algorithms called IPG_B, IPG_LS, RIPG_B and RIPG_LS,respectively. B and LS stand for Basic and Local Search and the prefixR indicates the presence of the restart mechanism. Each algorithmwas run ten independent times (replicates) against all 44 instancesof the SSDTest set and both hypervolume and epsilon indicatorsare calculated, leading to a total of 4 � 44 � 10 = 1760 results.

Only the (Cmax,TWT) pair of objectives was considered for thecalibration and the stopping time was fixed at t = 100 milliseconds.Fig. 2a and b depict the means plots with Tukey confidence inter-vals with a 95% confidence level (a = 0.05) from the ANOVA testand report on the interaction between the response variable,respectively IH and I1

e , and the type of algorithm which is testedas a controlled factor with four levels. Due to reasons of space,we only highlight here the most important findings. Overall, themethod with restart and local search phases (RIPG_LS) achievesbetter average results, that are, even for the grand average, statis-tically better than the non-restart versions, with and without localsearch. The local search phase improves the quality of the resultsfor all the instances while the restart phase only affects the smallerones. This behavior is mainly attributable to the huge number ofelements in the solution space that prevents the algorithm fromreaching a steady state in the available computation time. On thecontrary, small instances have Pareto fronts with few solutionsand the search soon gets trapped. In this scenario, the restart pro-cedure operates by increasing the diversification in the process,thus enhancing the probability of improving the solutionsgenerated.

4.5. Computational analysis

In this section we detail the campaign of experiments we havecarried out and analyze the outcomes by means of statistical tests.We run the 18 tested algorithms ten times each (replicates) foreach one of the 220 instances of the sets SSD50 and SSD125, forboth pairs of criteria. Furthermore, two stopping times have beenconsidered (t = 150 milliseconds and t = 200 milliseconds) raisingup the total number of data samples to 158,400. All the testedmethods are coded in Delphi XE language with all the optimization

1.11

1.12

1.13

1.14

RIPG

_LS

RIPG

_B

IPG_L

S

IPG_B

(b)1

or the calibration of the RIPG. Makespan and total weighted tardiness criteria and



Table 3Results for the Cmax–TWT criteria.

SSD50 SSD125

150 millisecond 200 millisecond 150 millisecond 200 millisecond

Method IH I1e

Eval (103) Method IH I1e



Eval (103)

RIPG 1.296 1.067 2,548,679 RIPG 1.313 1.057 3,423,776 RIPG 1.318 1.064 2,762,511 RIPG 1.336 1.054 3,615,022MOSA_Varad_M 1.272 1.102 3,587,690 MOSA_Varad_M 1.282 1.096 4,772,831 MOSA_Varad_M 1.237 1.146 3,623,574 MOSA_Varad_M 1.248 1.140 4,713,809MOSA_Varad 1.232 1.127 1,339,338 MOSA_Varad 1.232 1.127 1,339,338 MOIGS 1.225 1.147 2,698,486 MOIGS 1.241 1.134 3,392,280MOIGS 1.186 1.164 2,023,348 MOIGS 1.202 1.150 2,510,204 MOSA_Varad 1.182 1.178 1,338,261 MOSA_Varad 1.182 1.178 1,338,261MOGALS_Arroyo 1.179 1.132 920,041 MOGALS_Arroyo 1.189 1.127 1,148,871 MOGALS_Arroyo 1.154 1.157 912,086 MOGALS_Arroyo 1.163 1.153 1,150,823MOTS 1.151 1.136 685,987 MOTS 1.163 1.130 798,745 MOTS 1.126 1.162 627,080 MOTS 1.135 1.158 732,265PESAII 1.106 1.201 583,685 PESAII 1.123 1.189 757,140 PESA 1.075 1.216 710,123 PESA 1.092 1.205 908,765PESA 1.104 1.202 552,473 PESA 1.121 1.191 712,579 PESAII 1.067 1.221 707,241 PESAII 1.086 1.208 907,282MOACA17_M 1.087 1.189 2,201,857 MOACA17_M 1.095 1.185 2,751,410 MOACA17_M 1.060 1.228 2,206,634 MOACA17_M 1.065 1.226 2,686,160MOACA18_M 1.087 1.188 2,208,651 MOACA18_M 1.095 1.185 2,750,188 MOACA18_M 1.059 1.227 2,205,608 MOACA18_M 1.065 1.225 2,684,479MOACA19_M 1.083 1.192 2,211,429 MOACA19_M 1.091 1.187 2,750,047 MOACA20_M 1.056 1.231 2,203,806 MOACA20_M 1.063 1.227 2,685,790MOACA20_M 1.082 1.191 2,211,536 MOACA20_M 1.091 1.187 2,748,924 MOACA19_M 1.056 1.232 2,202,398 MOACA19_M 1.062 1.228 2,686,138PGA_ALS 1.024 1.250 617,776 PGA_ALS 1.034 1.246 798,703 PGA_ALS 0.955 1.323 656,617 PGA_ALS 0.967 1.318 834,233MOGA_Murata 0.980 1.276 1,366,304 MOGA_Murata 1.004 1.263 1,826,234 e-NSGAII 0.913 1.296 1,219,000 MOGA_Murata 0.934 1.297 1,822,611e-NSGAII 0.969 1.266 1,215,814 e-NSGAII 0.989 1.255 1,629,181 MOGA_Murata 0.908 1.313 1,398,489 e-NSGAII 0.934 1.284 1,593,860CMOGA 0.897 1.332 1,155,755 CMOGA 0.930 1.313 1,537,986 CMOGA 0.810 1.380 1,167,108 CMOGA 0.843 1.360 1,522,397hMGA 0.804 1.356 498,624 hMGA 0.815 1.348 667,361 PILS 0.729 1.448 448,360 PILS 0.774 1.410 581,475PILS 0.741 1.441 426,402 PILS 0.791 1.401 565,870 hMGA 0.699 1.425 509,212 hMGA 0.709 1.417 665,950

Table 4Results for the Cmax–TFT criteria.

SSD50 SSD125

150 millisecond 200 millisecond 150 millisecond 200 millisecond

Method IH I1e




Eval (103)

RIPG 1.314 1.062 2,886,704 RIPG 1.333 1.053 3,899,048 RIPG 1.322 1.063 2,963,529 RIPG 1.339 1.055 3,913,874MOSA_Varad_M 1.217 1.138 3,190,094 MOSA_Varad_M 1.232 1.133 4,271,434 MOIGS 1.196 1.124 3,490,327 MOIGS 1.218 1.114 4,607,469MOIGS 1.197 1.121 3,144,601 MOIGS 1.219 1.109 4,182,232 MOSA_Varad_M 1.148 1.168 3,207,332 MOSA_Varad_M 1.166 1.161 4,231,502MOSA_Varad 1.151 1.170 1,337,774 MOSA_Varad 1.151 1.170 1,337,774 MOSA_Varad 1.067 1.208 1,336,762 MOSA_Varad 1.067 1.208 1,336,762MOGALS_Arroyo 1.131 1.166 890,481 MOGALS_Arroyo 1.143 1.159 1,213,891 MOGALS_Arroyo 1.043 1.210 893,042 MOGALS_Arroyo 1.056 1.203 1,192,328MOTS 1.093 1.188 528,608 MOTS 1.108 1.181 654,054 MOTS 1.029 1.220 487,054 MOTS 1.041 1.213 603,787PESA 1.023 1.212 847,249 PESA 1.044 1.201 1,131,626 MOACA18_M 0.946 1.280 2,269,013 PESA 0.961 1.248 1,242,884PESAII 1.002 1.222 811,593 PESAII 1.023 1.211 1,083,591 MOACA17_M 0.943 1.281 2,266,877 MOACA18_M 0.952 1.277 2,805,182MOACA18_M 0.995 1.253 2,271,125 MOACA18_M 0.999 1.249 2,808,732 MOACA20_M 0.941 1.283 2,269,028 MOACA17_M 0.950 1.277 2,803,932MOACA17_M 0.992 1.254 2,266,600 MOACA17_M 0.998 1.249 2,806,738 PESA 0.940 1.259 945,584 MOACA20_M 0.948 1.279 2,803,418MOACA20_M 0.990 1.255 2,271,435 MOACA20_M 0.996 1.251 2,809,364 MOACA19_M 0.937 1.284 2,224,286 MOACA19_M 0.945 1.281 2,784,915MOACA19_M 0.989 1.256 2,225,466 MOACA19_M 0.993 1.253 2,775,174 PESAII 0.916 1.272 897,434 PESAII 0.937 1.261 1,183,023PGA_ALS 0.921 1.306 661,826 PGA_ALS 0.934 1.300 867,378 PGA_ALS 0.835 1.349 681,036 PGA_ALS 0.847 1.342 877,647MOGA_Murata 0.821 1.336 1,434,468 MOGA_Murata 0.850 1.319 1,935,043 e-NSGAII 0.705 1.400 1,227,838 MOGA_Murata 0.731 1.385 1,910,583e-NSGAII 0.799 1.345 1,231,011 e-NSGAII 0.827 1.328 1,660,687 MOGA_Murata 0.702 1.404 1,443,474 e-NSGAII 0.729 1.385 1,627,731CMOGA 0.753 1.374 1,188,429 CMOGA 0.790 1.350 1,602,032 CMOGA 0.626 1.450 1,191,594 CMOGA 0.663 1.426 1,581,127PILS 0.661 1.476 486,066 PILS 0.713 1.429 654,566 PILS 0.607 1.508 508,728 PILS 0.650 1.467 673,374hMGA 0.585 1.505 512,607 hMGA 0.598 1.495 691,926 hMGA 0.468 1.584 517,613 hMGA 0.479 1.575 687,022

M.Ciavotta

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options activated. The experiments have been executed on a clus-ter of 12 Core Duo 2.4 gigahertz computers running with Win-dows� XP SP3 O.S. and 2 gigabytes of RAM memory. The resultsare summarized in Table 3, for the Cmax–TWT case and Table 4for Cmax–TFT. Although each cell in the tables is the average of noless than 2200 data points, it is still necessary to carry out a carefulstatistical experiment in order to assess whether the observed dif-ferences in the average values are statistically meaningful. We didparametric ANOVA analyses as well as non-parametric Friedmanrank-based tests for both instance sets, for the two performanceindicators, the two pairs of objectives and for the two stoppingtimes. This results in a total of 32 different experiments. Thereby,we carried out 16 multi-factor ANOVAS where the size of the in-stance and the algorithm are the controlled factors. One half ofthe experiments have been carried out employing the hypervo-lume as a response variable, and then we repeated the same testsconsidering the epsilon indicator. All the tests have been executedwith a confidence level of 95% (a = 0.05). Note that since we arecarrying out four tests over the same results (parametric andnon-parametric, epsilon and hypervolume), we employ the Bonfer-roni adjustment for the a level, i.e., we use an adjusted as of 0.01for a real a of 0.05. In order to safely apply ANOVA, it is necessaryto check three main hypothesis: normality, homogeneity of vari-ance (or homoscedasticity) and independence of residuals. Theresiduals resulting from the experimental data have been analyzedand all three hypothesis can be accepted.

We make use of parametric as well as non-parametric tests tostrengthen the soundness of our conclusions. Therefore we com-

2.10.1

MOGALS_Arroyo

MOSA_Varad

MOIGS

MOSA_Varad_M

RIPG

2.10.1

(a)

MOGALS_Arroyo

MOSA_Varad

MOIGS

MOSA_Varad_M

1.0 1.2

(c)

RIPG

(d)

MOGALS_Arroyo

MOSA_Varad

MOSA_Varad_M

RIPG

(b)

MOIGS

MOGALS_Arroyo

MOSA_Varad

MOSA_Varad_M

MOIGS

RIPG

2.10.1

(a) Hypervolume indicator response variable. Makespan and totalflowtime criteria.

M

MO

M

MO

(

Fig. 3. Means plot and Tukey HSD confidence intervals (as = 0.01, a = 0.05) for the ANOt = 150 millisecond (a and b) and t = 200 millisecond (c and d) CPU time stopping criteri


pare the results of the first group of tests against those of a secondgroup of non-parametric experiments. Non-parametric Friedmanrank-based tests have been carried out. Since there are 18 algo-rithms and 10 different replicates, the results for each instanceare ranked between 1 and 180. All these tests substantially validateand strengthen the results shown in the tables. In Table 3, themean values concerning both instance sets and Cmax–TWT criteriaare reported. Both indicators IH and I1

e are considered and themethods are sorted decreasingly according to the value of IH. RIPGturns out to be the best performing algorithm for each combinationof indicators, stopping time and instance set. In comparison to MO-SA_Varad_M, the second method in the ranking, the RIPG’s per-centage improvement is between 2% and 7% of the hypervolume.

Please note that the global ranking of algorithms remains sub-stantially unchanged for the different stopping times and instancesets, whereas there are few differences if we compare the rankingof IH and I1

e . This is mainly due to the fact that the unary epsilon indi-cator is the average minimum distance between the frontier gener-ated in a single run and the best Pareto front known. Also it is muchmore conservative than the hypervolume, therefore it is likely thatby using this indicator, closer values are assigned to the algorithmsand there is the possibility that they might have a different rank.Since at the moment it is not clear which indicator is the most reli-able, when this anomaly happens, the affected methods are consid-ered incomparable. It is worth to note that RIPG turns out to be morecompetitive in the SSD125 set respect to SSD50.

In Table 4 results concerning Cmax–TFT criteria are reported andagain RIPG comes out as being the best method in the set, but

(d)(c)

(a) (b)

50

RIPG

OSA_Varad_M

MOSA_Varad

MOTS

GALS_Arroyo

0

RIPG

OSA_Varad_M

MOSA_Varad

MOTS

GALS_Arroyo

500

RIPG

MOSA_Varad_M

MOIGS

MOGALS_Arroyo

MOTS

0 50

RIPG

MOSA_Varad_M

MOIGS

MOGALS_Arroyo

MOTS

0 50

b) Epsilon indicator response variable. Makespan and total weightedtardiness criteria.

VA and Friedman Rank-based tests for SSD50 (a and c) and SSD125 (b and d) withon.




unlike the Cmax–TWT case, MOIGS performs much better andachieves the second position of the rank for the SSD125 set. Thishappens because it is an improved version of an algorithm spe-cially designed to tackle the multi-objective flowshop problemwith Cmax–TFT criteria.

Note also how Tables 3 and 4, contain additional columns withthe total number of evaluated solutions for each algorithm, combi-nation of objectives, instance set and termination criterion. It isinteresting to see that there are enormous differences. Take forexample algorithms PILS and MOSA_Varad_M for instance setSSD50 and termination criterion 150 milliseconds. PILS evaluatedabout 426 thousand solutions on average whereas MOSA_Varad_Mevaluated more than 3.5 million. This is, on average, MOSA_Var-ad_M evaluates almost 8.5 more solutions than PILS for the sameCPU time. This highlights the importance of stopping algorithmsafter the same elapsed CPU time and not after the same numberof iterations. Stopping both algorithms after the same number ofiterations would result in wildly different employed CPU times.

Fig. 3a and b show some means plots for ANOVA and Friedmantests where only the first five best algorithms are depicted. It has tobe stressed that ranking tests neglect the real differences in theindicators and therefore, differences may appear smaller of greaterrespect to the equivalent ANOVA test.

Due to reasons of limited space, we cannot reproduce here thecomplete 32 plots, each one with all the 14 algorithms. These areavailable as part of the on-line material.

Lastly, we present six Fig. 4a–f) which represent the EAF forMOSA_Varad_M Fig. 4a), MOIGS Fig. 4b) and RIPG Fig. 4c) and dif-ferences of EAFs Fig. 4d–f) for instance 71 of SSD50 (100 jobs and10 machines) and (Cmax–TWT) criteria. Each image is elaboratedemploying 50 replicates for each algorithm. Although these pic-tures give us information only for a single instance, during a designphase one can use such knowledge to understand by and large the

Fig. 4. Empirical attainment functions (a–c) and differences between such functions (machines belonging to SSD50 is analyzed against makespan and total weighted tardine


behavior of the involved algorithms. Let us focus for example onpicture Fig. 4a, it is clear that MOSA_Varad_M method is worsethan MOIGS only in a central part of the objective space while itachieves better results than MOIGS at the extremes of the frontier(see Fig. 4d). Finally, also using this tool we confirm that RIPGwidely outperforms its competitors. For more details, the readeris referred to the on-line material where similar figures are re-ported for other instances.

5. Conclusions and future research

There have been several methods proposed in the literature forthe a posteriori multi-objective flowshop problem. However, asimportant as they are in practice, setup times have not been con-sidered, as far as we know, for this setting. This paper representsa first attempt to tackle this problem.

We have presented two main contributions to the field of themulti-objective flowshop. First, we have adapted the best perform-ing algorithms for the multi-objective flowshop by adding antici-pative sequence dependent setup times to the problem. Wecarried out a study of these algorithms to establish which onesshow better performance for the problem with setups. Second,we have extended a new strategy which achieved state-of-the-art results for the single objective flowshop, the Iterated Greedymetaheuristic, in order to deal with several objectives and setuptimes simultaneously. The extended IG method has been referredto as RIPG. A thorough algorithm engineering process, along withstatistical calibration led to a refined proposal which has shownto reach state-of-the-art results for this problem. This has beenconfirmed by a wide campaign of tests where the results have beenanalyzed by means of parametric as well as non-parametric statis-tical tests. We employed two Pareto compliant performance indi-cators, two stopping criteria based on the elapsed CPU time and

d–f) for MOSA_Varad_M, MOIGS and RIPG. A single instance of 100 jobs and 10ss criteria and t = 150 millisecond termination criterion.




two combinations of scheduling objectives. As a consequence, RIPGcan be considered the state-of-art procedure for this schedulingproblem.

Future research lines stem from the possibility of applying thisscheme to solve different or more constrained scheduling prob-lems such as, the hybrid flowshop or the parallel machines prob-lems. Another interesting area of future research relates to theEAF, a rather new statistical tool employed here, to study the prob-ability that one algorithm has to cover a certain zone of the objec-tive space. Its main drawback is that only one instance at a time isrepresented. An extension of the EAF that takes into account awhole instance set would be a useful tool for a deeper understand-ing of algorithm’s behavior and, furthermore, it would also lead tonew and more reliable performance indicators.

Acknowledgments

The authors thank the anonymous referees for their careful anddetailed comments which have helped improve this manuscriptconsiderably. This work is partially financed by the Spanish Minis-try of Science and Innovation, under the projects ‘‘SMPA-AdvancedParallel Multiobjective Sequencing: Practical and Theorerical Ad-vances’’ with reference DPI2008-03511/DPI and ‘‘RESULT-RealisticExtended Scheduling Using Light Techniques’’ with referenceDPI2012-36243-C02-01 and by the Small and Medium Industryof the Generalitat Valenciana (IMPIVA) and by the European Unionthrough the European Regional Development Fund (FEDER) insidethe R+D program ‘‘Ayudas dirigidas a Institutos Tecnológicos de laRed IMPIVA’’ during the year 2011, with project numbers IMDEEA/2011/142 and IMDEEA/2012/143.

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