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Atabusearchalgorithmforthesinglemachinetotalweightedtardinessproblem

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European Journal of Operational Research 176 (2007) 1423–1435

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Discrete Optimization

A tabu search algorithm for the single machine totalweighted tardiness problem

Umit Bilge *, Mujde Kurtulan, Furkan Kırac

Department of Industrial Engineering, Bogazici University, Bebek, 80815 _Izstanbul, Turkey

Received 5 March 2003; accepted 7 October 2005Available online 18 January 2006

Abstract

In this study, a tabu search (TS) approach to the single machine total weighted tardiness problem (SMTWT) is pre-sented. The problem consists of a set of independent jobs with distinct processing times, weights and due dates to bescheduled on a single machine to minimize total weighted tardiness. The theoretical foundation of single machine sched-uling with due date related objectives reveal that the problem is NP-hard, rendering it a challenging area for meta-heu-ristic approaches. This paper presents a totally deterministic TS algorithm with a hybrid neighborhood and dynamictenure structure, and investigates the strength of several candidate list strategies based on problem specific character-istics in increasing the efficiency of the search. The proposed TS approach yields very high quality results for a set ofbenchmark problems obtained from the literature.� 2005 Elsevier B.V. All rights reserved.

Keywords: Tabu search; Single machine scheduling; Total weighted tardiness minimization

1. Introduction and literature survey

One of the scheduling problems, simple to statebut not easy to solve, is the problem of sequencingn independent jobs on a single machine with theobjective of minimizing total weighted tardiness.The machine processes only one job at a time

0377-2217/$ - see front matter � 2005 Elsevier B.V. All rights reservdoi:10.1016/j.ejor.2005.10.030

* Corresponding author. Tel.: +90 212 359 6407; fax: +90 212265 1800.

E-mail address: bilge@boun.edu.tr (U. Bilge).

and without interruption. Each job i has an integerprocessing time pi, a positive weight wi and a dis-tinct due date di. For a given processing order ofthe jobs, the earliest completion time Ci and tardi-ness Ti = max{0, Ci � di} can be computed foreach job. The problem is represented as n/1/

PwiTi

and referred as single machine total weighted tar-diness problem (SMTWT).

The SMTWT problem is NP-hard (see Lawler,1977; Lenstra et al., 1977; Du and Leung, 1990)and solution approaches like Dynamic Program-ming and Branch-and-Bound are computationally

ed.

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inefficient, especially when the number of jobs isbeyond 50, as the results presented in a compara-tive study by Abdul-Razaq et al. (1990) demon-strate. Special sub-models are investigated toidentify polynomially solvable cases; some recentworks to this end are by Cheng et al. (2005) andTian et al. (2005). It is also well known that thereis no simple dispatching rule that works best for allproblem environments. If there is no more thanone tardy job, then the earliest due date (EDD)sequence is optimal, whereas weighted shortestprocessing time (WSPT) order gives the optimalsequence when all jobs are necessarily tardy.Therefore, EDD generally performs well for lightlyloaded machines while WSPT should be preferredunder heavy loading. Several heuristic dispatchingrules like those developed by Carroll (1965),Montagne Jr. (1969), Rachamadugu and Morton(1982), Morton et al. (1984), and Panwalkaret al. (1993) have been based on this idea. As thereviews by Koulamas (1994) and Chen et al.(1998) indicate developing approximation algo-rithms with good performance guarantees for thisproblem is difficult.

The quest for good and robust heuristics, there-fore, continued with sophisticated approaches likemeta-heuristics which extend neighborhood searchbeyond local optima. Matsuo et al. (1989) addressthe SMTWT by a simulated annealing algorithmwhich starts with a good initial solution and lowacceptance probability to accelerate the searchfor a near optimal solution. Potts and Van Was-senhove (1991) propose a descent heuristic and asimulated annealing method for SMTWT. Crau-wels et al. (1998) present single and multi-startversions of descent, simulated annealing, tabusearch (TS) and genetic algorithm implementa-tions for the same problem and show that whilesimulated annealing is outperformed, tabu searchdominates the other methods. Congram et al.(2002) treat SMTWT with an ‘iterated dynasearch’algorithm, which is a local search technique thatuses dynamic programming to find the best movewhich is composed of a set of independent inter-change moves and searches an exponential sizeneighborhood in polynomial time. They obtainresults that are superior to other local search pro-cedures. Laguna et al. (1991) consider a single

machine scheduling problem for minimizing thesum of setup costs and linear delay penalties, andpropose a TS algorithm that uses hybrid neighbor-hood consisting of both swap and insertion moves.Bilge et al. (2003) develop a totally deterministicTS algorithm for parallel machine total tardinessproblem (PMTT), and report results that aremuch superior on the benchmarks available inthe literature. They use a hybrid neighborhoodgeneration technique with different context-depen-dent candidate list strategies and report that thetime performance and the quality of the resultsobtained under candidate list strategies are supe-rior to the case when no candidate list strategy isemployed.

TS algorithm proposed in this paper alsoemploys a hybrid neighborhood along with adynamic tenure structure, however both the hybridneighborhood and the dynamic tenure structuresare different from the PMTT case; and the theoret-ical foundation of single machine scheduling isexploited to develop new effective candidate liststrategies. The next section presents the details ofthe tabu search algorithm implemented. The per-formance of the proposed algorithm is tested overthe problem set used by Crauwels et al. (1998) andpresented in Section 3. Finally, conclusions are dis-cussed in Section 4.

2. Tabu search approach to SMTWT

Tabu search (Glover and Laguna, 1997) is ameta-heuristic that guides a local search procedureto explore the solution space beyond local opti-mality. TS allows intelligent problem solving bythe incorporation of adaptive memory and respon-sive exploration. Key elements of the search pathare selectively remembered and strategic choicesare made to guide the search out of local optimaand into diverse regions. The adaptive memoryusage is a clever compromise between the rigidmemory structure of exact techniques likeBranch-and-Bound and the memoryless heuristicslike local search procedures.

The basic procedure of TS can be summarizedas follows. Starting from an initial solution, TSiteratively moves from the current solution to its

U. Bilge et al. / European Journal of Operational Research 176 (2007) 1423–1435 1425

best neighbor, even if this new solution is worsethan the one available, until a pre-specified stop-ping criterion becomes true. However, consideringevery possible move from the current solutionmay be extremely time consuming and computa-tionally expensive and devising a candidate list

strategy to isolate good candidates from theneighborhood may be helpful. In order to avoidcycling and becoming trapped in local optima,certain moves (or solution attributes) that lead topreviously explored regions are forbidden orclassified as tabu, forming the short-term memoryof TS. The tabu status of a move may be cancelledmaking it an allowable move if an aspiration

criterion is satisfied (if, for instance, the tabumove leads to a new best solution). The length oftime during which a certain move is classified astabu, tabu tenure, is an important parameter fortabu search. Tabu tenure, can be kept constantor varied dynamically throughout the search. TScannot always succeed to direct the search intothe regions where the best solutions are found bysolely employing short-term memory strategies.Hence, long-term memory strategies to lead thesearch into unexplored regions of the solutionspace (diversification) or to perform a more thor-ough examination in some good or promisingregions (intensification) may be needed. Thesekey aspects of the totally deterministic TS algo-rithm tailored to SMTWT in this paper aredescribed below.

2.1. Solution representation and initial solution

generation

A solution for the SMTWT is represented as apermutation of integers 1, . . . , n, which definesthe processing order of jobs on the single machine.Four methods are used to generate starting solu-tions. These are:

1. EDD based list scheduling, where jobs areordered with respect to their earliest due dates.

2. WSPT, where the jobs are sequenced such that(w1/p1) P (w2/p2) P � � �P (wn/pn).

3. R&M heuristic developed by Rachamaduguand Morton (1982) (also known as ApparentUrgency Rule), which is based on sorting the

jobs in order of non-increasing priorities, pi,where

pi ¼ ðwi=piÞ½expf�ðSiÞþ=kpavg�.Here, Si = di � pi � t is the slack time of job i

at time t. If the slack is negative, the job is sureto be tardy and receives full WSPT priority.Typical value of the factor k is suggested ask = 2, and pav is the average processing timeof all the jobs to be scheduled.

4. R&M+ heuristic, which is obtained by search-ing for the k value in R&M yielding the lowestP

wiTi. Hence, a schedule is generated usingeach of the k values in the range [0.5, 4.0] withincrements of 0.1, and the best schedule thusobtained is used as the initial solution.

2.2. Neighborhood generation

Insert moves and pairwise exchanges (swaps)are frequently used move types for neighborhoodgeneration in permutation problems. An insertmove identifies two jobs i and j in the current solu-tion and moves job i to be between job j and itsdirect predecessor. A swap move exchanges thelocations of two jobs so that each job is placedin the location previously occupied by the other.A swap move can be considered as a move thatcombines two insert moves. The neighborhoodused in this study is a hybrid one that consists ofthe complete ‘‘insert neighborhood’’ and the com-plete ‘‘swap neighborhood’’.

2.3. Tabu classification

Attribute-based tabu classifications containmore information about an already visited solu-tion and generally considered more effective in pre-venting cycling as opposed to move-based (i.e.prohibit moving a job directly involved in a recentmove) tabu classifications (Glover and Laguna,1997). This is the case for SMTWT as well, as ver-ified by our preliminary experiments. Therefore,the tabu classification used is based on the arcsthat appear in a solution. If in a given permutationjob i is directly preceded by job i � 1 then this isreferred as arc ((i � 1), i) as shown in Fig. 1. The

j-1 j+1j i-1 i i+1

Fig. 1. Arc-based solution representation.

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tabu classification aims to prohibit some of thearcs broken by recent moves to be constructedagain during tabu duration. The swap and insertmoves are handled separately and the arcs to bedeclared as tabu active after each type of moveare determined through some experimentation.The resulting tabu classification strategy worksas follows: If the move just performed is to insertjob i between job j � 1 and job j, all permutationsin which arc ((j � 1), j) or arc ((i � 1), i) appear areclassified as tabu. When jobs i and j are swapped,three of the arcs that are broken, namely arcs((j � 1), j), ((i � 1), i), and (j, (j + 1)) become tabuactive. This requires all the new arcs formed by acandidate insert or swap move to be checked andthe move is tabu if any of these arcs are tabu active.This arc-based tabu classification is stricter in itsclassification of tabu compared to the one used inBilge et al. (2003), and therefore called Arc CheckStrict Tabu Classification (ARC+). The tabu statusof a move is to be overridden if it yields a solutionbetter than the best obtained so far.

2.4. Candidate list strategies

The candidate list strategies (CLS) developedare based on problem specific characteristics. Inthis study, a CLS can use two basic types of con-straints in screening the neighborhood. The firstis based on the nature of the job to be selectedfor an insert or swap move. This is called the job

heuristic. If only tardy jobs are allowed to initiatea move, the job heuristic is called ‘‘TARDY’’and the CLS works as follows: Take a tardy jobi and consider all inserts and swaps with jobs thatcurrently precede it (not necessarily directly). If onthe other hand tardiness restriction does not apply,the job heuristic is ‘‘NONE’’.

The second type of constraint applies to the setof allowable moves for a given job. This is calledthe move heuristic. The move heuristics are basedon theoretical results on SMTWT, namely on thefollowing two corollaries by Rinnooy Kan (1976):

Corollary 1. For the SMTWT problem, n/1/PwiTi, if for any two jobs i and j, the three

conditions stating that {dj 6 di, wj P wi, pj 6 pi}are all satisfied, then only those schedules in which

‘‘job j precedes job i’’ need to be considered.

Corollary 2. For the SMTWT problem, n/1/PwiTi, if for any two jobs i and j, the three condi-

tions stating that {dj 6 Ci, wj P wi, pj 6 pi} are allsatisfied, then only those schedules in which ‘‘job j

precedes job i’’ need to be considered.

The move heuristic is categorized as type A ortype B depending on whether it is based on Corol-laries 1 or 2. The candidate list strategy A-ORworks as follows: Take a job i (as in Fig. 1) andconsider all inserts and swaps with job j whichcurrently precedes it (not necessarily directly) aslong as at least one of the conditions in Corollary1 does not hold. Hence, if (di < dj OR wi > wj

OR pi < pj) is true then job j does not have to

precede job i and job i can be inserted beforejob j or swapped with job j. The candidate liststrategy B-OR is the counterpart of this forCorollary 2.

Another type of move heuristic is devised forbringing in some diversifying effect against theaggressive nature of TS which usually leads thesearch into a local optimum in the most directway. The Candidate List Strategy A-Diverse (A-DIV) works as follows: Take a job i (as inFig. 1) and consider all inserts and swaps withjob j which currently precedes it (not necessarilydirectly) as long as the conditions {di 6 dj,wi P wj, pi 6 pj} are not all satisfied simulta-neously. Here, if job i does need to precede job j

according to Corollary A, this move is not allowedto be performed explicitly (other moves may leadinto it indirectly). Although this may seem some-what unintuitive at first glance, it does create a dif-ferent search path that turns out to perform well inmany cases. Again, the counterpart for Corollary 2is called Candidate List Strategy B-Diverse (B-DIV).

U. Bilge et al. / European Journal of Operational Research 176 (2007) 1423–1435 1427

It is possible to exploit any combination of thejob/move heuristics, since they can be superim-posed on one another, e.g. TARDY-A-OR. If onlya job heuristic is applied, and no move heuristic isused, then the CLS imposed becomes ‘‘Tardy Can-didates List Strategy’’ and is denoted as TARDY-NONE. The choice of not using any of thesecandidate list strategies leads to the considerationof the entire neighborhood. This case is denotedas ‘‘No CLS’’ or ‘‘NONE’’ in short and is usedmainly for benchmarking the other CLSs devel-oped in terms of computational speed and solutionquality.

2.5. Tabu tenure

In this study, both the single and dynamic ten-ure approaches are tested. The dynamic tenurestrategy used creates a sequence of small (S), med-ium (M) and large (L) tabu tenure values repeatedin a sequence called cycle string, throughout thesearch. Systematically varying the tabu tenure inthis way results in a balance between intensifica-tion and diversification, since short tabu tenuresallow fine-tuning of neighborhood search andclose examination of regions around a local opti-mum, while long tenures tend to direct the searchto different parts of the solution space (Gloverand Laguna, 1997). Due to the dynamic tabutenure and nature of the CLSs, no other divers-ification strategies are developed. However, anintensification strategy is employed after theshort-term memory TS is completed.

2.6. Intensification strategy

The intensification strategy consists of record-ing a fixed length sequential list of elite solutionsduring the short-term-memory search phase, then,after clearing the memory, restarting and carryingout a search of given duration from each of thoseelite solutions, with a given CLS and a set oftenure values that depend on the CLS. An elitesolution is defined as a solution that can surviveas the best-so-far for a given number of iterations(Bilge et al., 2003). All of the CLS combinationsdescribed in Section 2.5 can be employed in theintensification phase; however, complementing

CLSs should be employed in the short-term mem-ory and intensification phases since it is not desir-able to have too much screening in both phases.

3. Computational studies

The performance of the various TS strategiesdescribed above are compared through extensiveexperimentation on a set of benchmark problemsobtained from literature, to come up with a robustTS algorithm for SMTWT. The experimentation isperformed using a new version of ‘‘WinMeta’’, aninteractive software program developed previouslyin C++ to solve parallel machine scheduling prob-lems with tardiness objectives via meta-heuristicapproaches (Bilge et al., 2003). As the genericstructure of WinMeta allows extensions, the sin-gle-machine TS strategies designed in this studyare implemented and incorporated in the software.The particular solution strategy to be employedfor each problem instance can be set by the userby selecting the proper combination from themenus and specifying the necessary parameters.The user friendly graphical interface of WinMetaallows ease of experimentation and flexibility inthe strategies to employ as well as ease of analysisof results via detailed output report files. Theexperiments are conducted on a Pentium 4–1.6 GHz CPU, Host Bus 133 MHz with 512 MBRAM.

3.1. Test problems and performance measures

The problem set used for experimentation con-sists of single machine scheduling problems of 40,50 and 100 jobs, developed and tested by Crauwelset al. (1998), who adopted the problem generationscheme proposed by Potts and Van Wassenhove(1985). There are 125 instances in each set andthese instances are available in the OR libraryrun by Beasley (2001), which is a collection of testdata sets for a variety of Operations Researchproblems. As presented in the electronic ORlibrary, the optimal values of solutions are avail-able for 124 of the 40 job problem instances andfor 115 of the 50 job problem instances, with theunsolved problem being instance number 19 in

Table 1Best performing strategies revealed by the preliminary exper-imentation over 50-job problem set

Initial solutionheuristic

CLS Best performingfixed tenuresJob Move

R&M NONE NONE 50, 60R&M+ TARDY NONE 25, 30R&M+ TARDY A-OR 25, 30R&M+ TARDY A-DIV 25, 30R&M+ TARDY B-DIV 25, 30

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the 40 job problem set and instances 11, 12, 14, 19,36, 44, 66, 87, 88, and 111, in the 50 job problemset. As for the 100 job problem instances, someof the best known solution values reported byCrauwels et al. (1998) were improved by Congramet al. (2002) using the ‘iterated dynasearch’ algo-rithm, hence these modified values are used forbenchmarking.

In each experiment, the performance evaluationis based on two criteria: the average relative per-cent deviation from the optimal (or best known)value over all the 125 instances in the set (ARPD),and the total number of instances out of 125 forwhich the optimal (or best known) solution couldnot be obtained, namely the number of non-opti-mal (NO) solutions.

3.2. Parameter and strategy selection

The TS algorithm presented in this studyemploys various strategies and parameters. Thestrategy selection and parameter calibration pro-cess is performed in a number of stages. The firststep consists of exploratory testing to assess thebest set of strategies and good ranges of parame-ters; and this step is conducted only on 50-jobset of problems. Fixing the strategies and relativelyinsensitive parameters obtained as such, the nextstep involves systematic testing over the more sen-sitive parameters, namely the tenure parameters,to arrive at an empirical pattern based on problemsize. At this step, the 40-job problem set is usedtogether with the 50-job set. Finally, the empiricalparameter pattern thus obtained, which is a func-tion of problem size, is directly applied over the100-job set. Thus the parameter calibration taskis overcome without further tedious experimenta-tion over the large problems. As a result, a robustTS algorithm complete with parameter settingstrategies is obtained. The basic results of thesealgorithmic design and parameter calibration stepsare summarized in this section.

3.2.1. Strategy selection

This first stage of experimentation is performedonly on the 50-job set of problems and aims toselect the best TS strategies for SMTWT. Thestopping criterion for the tabu search method is

set to be 5000 non-improving iterations for eachof the instances.

As the first step, four initial solution generationheuristics (EDD, WSPT, R&M, R&M+) and tencandidate list strategies (two job type vs. five movetype CLS) are tested by taking all combinations. Ineach of these experiments the tenure is set equal tothe problem size (n = 50). After eliminating thosestrategies that are dominated, the most promisingfive candidate list strategy-initial solution heuristiccombinations are retained for further analysis.These are presented in Table 1. It can be seen that,R&M and R&M+ perform best among the initialsolution heuristics.

The next step is to determine the best fixed tabutenure values for the specific strategies chosenabove. The tenure search is performed over differ-ent ranges depending on the specific strategyemployed, i.e. over [40, 100] for no CLS and[20, 50] for a CLS. A CLS that reduces the searchneighborhood considerably necessitates the selec-tion of a smaller tenure value. On the contrary,selecting a large search neighborhood requiresthe selection of larger tabu tenure. The outcomeof this phase of experimentation is summarizedin the last column of Table 1.

Next, experimentation for devising a systematicdynamic tenure strategy is performed. Several setsof small (S), medium (M) and large (L) tenure val-ues are applied for a duration of [2 · medium ten-ure] iterations. The outcome of this step is depictedin Table 2. The value of M turns out to be close tothe best performing tenures in the previous step,and cycle string is selected to be SMLM. As clearlyseen in Table 2, applying a dynamic tenure struc-ture rather than a single tenure value improvesthe quality of the solutions remarkably.

Table 2The effects of dynamic tenure and intensification strategies over the best CLS combinations (before fine-tuning tenure parameters)

Initialsolutionheuristic

Short term Intensification ARPD % Impr. inARPDa

NO % Impr.in NOa

CLSb Tenure CLSb Tenuremultiplier

Job Move S M L Cycle string Job Move

R&M NONE NONE – 50 – – – – – 0.099 – 22 –R&M NONE NONE 40 50 100 SMLM – – – 0.074 25.25 15 31.82R&M NONE NONE 40 50 100 SMLM TARDY NONE 0.5 0.001 98.65 3 80.00

R&M+ TARDY NONE – 25 – – – – – 0.035 – 28 –R&M+ TARDY NONE 20 30 40 SMLM – – – 0.026 25.71 24 14.29R&M+ TARDY NONE 20 30 40 SMLM NONE NONE 1.7 0.004 84.61 7 70.83

R&M+ TARDY A-OR – 25 – – – – – 0.072 – 50 –R&M+ TARDY A-OR 20 30 40 SMLM – – – 0.059 18.06 44 12.00R&M+ TARDY A-OR 20 30 40 SMLM NONE NONE 1.7 0.008 86.44 13 70.45

R&M+ TARDY A-DIV – 25 – – – – – 0.054 – 37 –R&M+ TARDY A-DIV 20 30 40 SMLM – – – 0.033 38.88 30 18.92R&M+ TARDY A-DIV 20 30 40 SMLM NONE NONE 1.5 0.006 81.82 10 66.67

R&M+ TARDY B-DIV – 25 – – – – – 0.055 – 42 –R&M+ TARDY B-DIV 20 30 40 SMLM – – – 0.041 25.45 36 14.29R&M+ TARDY B-DIV 20 30 40 SMLM NONE NONE 1.5 0.007 82.93 10 72.22

a With respect to the previous row.b Strategies selected as best combinations as a result of experimentation.

U. Bilge et al. / European Journal of Operational Research 176 (2007) 1423–1435 1429

Before attempting to further fine-tune thedynamic tenure structure, appropriateness of theintensification strategy is tested. The intensificationphase is applied over short-term memory TS of5000 non-improving iterations. The nature of theelite solutions is defined by ‘‘elite window’’, whichis the number of iterations that a solution shouldsurvive as the best solution found so far in orderto identify it as an elite solution. Elite windows of200 or 300 iterations and recording two or threeelites worked well in general. The duration of theintensification phase around each elite is set to beequal to 1500 non-improving iterations. Finally,in order to regulate the tenure in the intensificationphase, a tenure factor is incorporated in the inten-sification strategy. This factor is a multiplier bywhich the tenures (S, M, L) for the short-term TSphase are multiplied before being used in the inten-sification phase. This tenure multiplier is regulatedaccording to the nature of the CLSs employed bothin the short-term TS phase and the intensificationphase. The experiments revealed that the CLS dur-ing intensification must be selected in accord with

the strategy used in the short-term memory TSphase. The best strategy combinations resultingfrom these experiments are also summarized inTable 2 along with the overall effect of applyingintensification.

A direct observation from Table 2 is that inten-sification has a major effect on all of the strategiesemployed, if the appropriate complementary strat-egy is applied. The highest response is observedwhen ‘‘No CLS’’ in the short-term memory TSstrategy is followed by an intensification phaseemploying the TARDY-NONE combination ofcandidate list strategies. The quality of the solu-tions obtained in this manner is also relativelyhigher than the other cases, both in terms ofARPD and the resulting number of non-optimalsolutions. This is a reasonable match in that afterthe detailed search with no screening of candi-dates, the search procedure goes to the elites andperforms a more directed and smarter search,which is faster than the ‘‘No CLS’’ case. Hence,the search re-originates at those local optima witha completely different attitude and goes in unvisited

Table 3Dynamic tenure pattern with respect to problem size

Problemsize

CLS Dynamic tenure

Job Move S M L Cycle string

40-Job set NONE NONE 15 40 90 SMLMTARDY * 15 20 60 SMLM

50-Job set NONE NONE 15 50 110 SMLMTARDY * 15 25 75 SMLM

100-Job set NONE NONE 15 100 225 SMLMTARDY * 15 50 150 SMLM

* NONE, A-DIV or B-DIV.

1430 U. Bilge et al. / European Journal of Operational Research 176 (2007) 1423–1435

yet good regions of the search space. On the otherhand, when the short-term memory TS uses acandidate list strategy, it is reasonable to dropscreening and explore all moves around the eliteswith ‘‘No CLS’’.

3.2.2. Parameter calibration

In the second stage of experimentation, the 40-job set is used as well, and the promising strategiesare further fine-tuned for both sets with the aimof establishing some empirical rules to set theparameter values with respect to problem size.The resulting empirical dynamic tenure pattern isas follows: When ‘‘No CLS’’ is used in the short-term memory TS phase, the dynamic tenure attainsa form of (15)–(n)–(2.25n) for small–medium–largetenures, where n is the number of jobs. If a CLSis used, then the dynamic tenure pattern becomes(15)–(0.5n)–(1.5n). The tenure values obtained inthis manner are quite robust, i.e. varying the S orL tenure values by ±5 around these values doesnot cause a change in the solutions for 40-job

Table 4Final results for 50-job problem set for selected TS strategies and fin

50-Job problem set

Method TS1

Initial solution heuristic R&M

Candidate list strategy

Job NONEMove NONE

Tabu classification ARC+

Dynamic tenure

S 15M 50L 110Cycle string SMLM

Intensification

Maximum elite count 3Elite window 300Stopping condition 1500Tenure multiplier 0.5Job TARDYMove NONE

ARPD 0.001NO 1Average CPU (entire search) 170.712Average CPU (time elapsed up to best) 16.91

problems, and for 50-job problems the numberof non-optimal solutions may rise by up to three,the ARPD may increase by up to 0.003. The tenurevalues computed from these empirical patterns areshown in Table 3, and for the 100-job problem setthey are directly used without further fine-tuning.The tenure multiplier of the intensification phaseis another parameter to fine-tune, the rangessearched are [0.3, 0.6] for no CLS and [1.0, 1.7]for a CLS during intensification. As the problem

e-tuned parameters

TS2 TS3 TS4

R&M+ R&M+ R&M+

TARDY TARDY TARDYNONE A-DIV B-DIV

ARC+ ARC+ ARC+

15 15 1525 25 2575 75 75SMLM SMLM SMLM

3 3 3300 300 3001500 1500 15001.4 1.4 1.4NONE NONE NONENONE NONE NONE

0.002 0.004 0.0033 4 342.712 44.576 41.5364.61 8.42 8.68

U. Bilge et al. / European Journal of Operational Research 176 (2007) 1423–1435 1431

size gets larger, selecting a slightly lower value oftenure multiplier seems more suitable to achievethe intensifying effect. A further decision made atthis stage is to eliminate TARDY-A-OR candidatelist strategy since it is outperformed by the others.

The results for the entire problem set that arepresented in the next sub-section are obtained withthese best performing strategy and parametersettings.

3.3. Results

The final results for the 50-job problem set arepresented in Table 4. ‘‘No CLS’’ in the short-termmemory TS phase and intensifying with theTARDY candidates list strategy provides thebest results, optimally solving 124 of the 125instances with the percent deviation of the singleinstance from optimal being only 0.08%. TARDY,TARDY-A-DIV and TARDY-B-DIV prove to befast and efficient in that they succeed in solving allthe problem instances with only three, four and

Table 5Final results for 40-job problem set for selected TS strategies and fin

40-Job problem set

Method TS1

Initial solution heuristic R&M

Candidate list strategy

Job NONEMove NONE

Tabu classification ARC+

Dynamic tenure

S 15M 40L 90Cycle string SMLM

Intensification

Maximum elite count 3Elite window 300Stopping condition 1500Tenure multiplier 0.4Job TARDYMove NONE

ARPD 0NO 0Average CPU (entire search) 98.46Average CPU (time elapsed up to best) 8.32

three non-optimal solutions, respectively as seenin Table 4. When the search behaviors areobserved via the graphical display of WinMeta, itis seen that the ‘‘DIVERSE’’ strategies follow dif-ferent search paths resulting in a diverse set of elitesolutions as intended.

The average CPU times required by each strat-egy are presented in two different formats. The firstis the duration of the entire search. The secondmeasure is the time elapsed from the beginning ofthe search until the best solution is found in thatsearch, denoted as ‘‘Time elapsed up to best’’. AllCPU measurements are in seconds. The results pre-sented in Table 4 show that although ‘‘No CLS’’yields higher quality solutions, the total computa-tional time it requires is three times the total com-putational time required by the alternativemethods. The TARDY-A-DIV and TARDY-B-DIV strategies have almost equal CPU, and theiractual CPU requirements is half that required by‘‘No CLS’’. The best CPU performance is givenby the TARDY candidate list strategy, and it

e-tuned parameters

TS2 TS3 TS4

R&M+ R&M+ R&M+

TARDY TARDY TARDYNONE A-DIV B-DIV

ARC+ ARC+ ARC+

15 15 1520 20 2060 60 60SMLM SMLM SMLM

3 3 3300 300 3001500 1500 15001.6 1.5 1.5NONE NONE NONENONE NONE NONE

0 0 0.0340 0 128.46 81.01 27.122.74 10.86 3.54

1432 U. Bilge et al. / European Journal of Operational Research 176 (2007) 1423–1435

succeeds in optimally solving 122 out of 125instances in 4.61 seconds per instance on the aver-age. These strategies are named as TS1, TS2, TS3and TS4, as seen in the first row of Table 4, for easeof reference.

It is seen in Table 5 that, all of the probleminstances in the 40-job problem set can be solvedto optimality by TS1, TS2 and TS3, while TS4can solve 124 of the 125 instances. The best timeperformance is obtained with TS2, where the bestsolution is reached in an average time of 2.74seconds.

For the 100-job set (Table 6), R&M+ becomesthe preferred the initial solution heuristic also forTS1. This is expected, since the search space forthe 100-job set is much larger compared to the40-job set and it is better to initiate the search atthe best possible starting point.

In the 100-job problem set, the best performanceis achieved by TS4 which succeeds in obtaining 108of the best-known solutions with an ARPD of only0.007. This is achieved in an average computation

Table 6Final results for 100-job problem set for selected TS strategies and fi

100-Job problem set

Method TS1

Initial solution heuristic R&M+

Candidate list strategy

Job NONEMove NONE

Tabu classification ARC+

Dynamic tenure

S 15M 100L 225Cycle string SMLM

Intensification

Maximum elite count 3Elite window 300Stopping condition 1500Tenure multiplier 0.5Job TARDYMove NONE

ARPD 0.014NO 18Average CPU (entire search) 341.26Average CPU (time elapsed up to best) 76.18

time of 2 minutes. However, even the worst per-forming strategy, TS1 in this case, yields an ARPDof only 0.014 and this is accomplished at an averagetime requirement of 76 seconds. These robustresults justify the empirical parameter selectionrules based on problem size.

Crauwels et al. (1998) solved 123 of the 40-joband 118 of the 50-job problem instances optimallywith their TS algorithm, as opposed to 125 and124, respectively in our case. For the 100 job prob-lem set they obtained 103 of the best solutionsknown then, but an improved set of results areused here (we obtained 108 of the best solutionsin the updated benchmark). The ARPD for the100-job instances is reported as 0.04 in the afore-mentioned study, and keeping in mind that thisis calculated relative to an inferior set of best solu-tions, it is significantly worse than the 0.007 in ourcase. Furthermore, the multi-start TS algorithmthey present is not deterministic and the statisticsreported is based on the best solution obtainedat the end of a single run of five random restarts.

ne-tuned parameters

TS2 TS3 TS4

R&M+ R&M+ R&M+

TARDY TARDY TARDYNONE A-DIV B-DIV

ARC+ ARC+ ARC+

15 15 1550 50 50150 150 150SMLM SMLM SMLM

3 3 3300 300 3001500 1500 15001.2 1.3 1.4NONE NONE NONENONE NONE NONE

0.010 0.007 0.00719 21 17271.39 277.48 283.4682.88 125.47 127.59

U. Bilge et al. / European Journal of Operational Research 176 (2007) 1423–1435 1433

Although it is not possible to make an exact com-parison on the basis of time performance (theyused an HP 9000-G50 computer), the TS algo-rithm presented here dominates this previous TSalgorithm on the basis of solution quality.

The iterated dynasearch algorithm of Congramet al. (2002) succeeded to solve the 40-job and 50-job sets optimally and yielded an updated bestsolution set for the 100-job set. Both their timeperformance and ARPD are reported in a cor-rected form relative to the findings of Crauwelset al. (1998); for instance, whenever they found asolution which is better than the best solutionknown to Crauwels et al. its relative percent devi-ation is negative and actually reduces their ARPD.For this reason, it is difficult to make an exactcomparison of the average performance of the iter-ated dynasearch algorithm with respect to our TSfor the 100-job case. Even if we assume theextreme case of all best solutions being found inall ten replications they performed, the average rel-ative percent deviation of our TS from this methodwould be only 0.007.

4. Conclusions

This study aims to develop a robust TS algo-rithm tailored for the single machine totalweighted tardiness (SMTWT) problem. The TSmethod devised is completely deterministic andattains the best performance by employing a can-didate list strategy based on problem context, adynamic tenure structure and an intensificationphase around elite solutions after the short-termmemory TS is completed.

The results indicate that rather than employinga single tabu tenure value for the entire duration ofTS, it is better to systematically vary the tenure toovercome some difficulties presented by the topol-ogy of the search space and to induce a balancebetween intensification and diversification. Theinitial solution is also a dominant effect in TS,and if the problem structure is not considered forappropriate selection of initial solution, TS deteri-orates in its quality, as reflected in the preliminaryexperimentation phase. R&M+ heuristic performswell in this respect.

Candidate list strategies are critical in determin-ing the efficiency of the TS strategy, and this studymakes use of various candidate list strategies thatincorporate the problem specific information inthe screening mechanism. The tabu classificationused is compatible with the candidate list strategiesin that it functions in accord with what the CLSaims to accomplish in terms of broken arcs in amove. The neighborhood used contains all insertand swap moves and the candidate list strategiesbring clever and efficient screening that enable afast search over this large neighborhood. Thishowever, like any screening mechanism, posesthe risk of skipping some good neighbors. There-fore, intensification is performed around elite solu-tions that are identified during the short-termmemory TS phase. This intensification phaseemploys a different neighborhood screeningapproach than the one in the initial phase so thatdifferent search paths are followed around the elitesolutions when the search restarts. Hence, the can-didate list strategy employed in the short-termmemory TS phase is complemented with the candi-date list strategy employed in the intensificationphase. As such, the overall strategy becomes toemploy one of two methods: The first method isto search intensely in the short-term memory TSphase with ‘‘No CLS’’ and then to intensifyaround the selected elites with TARDY Candi-dates List Strategy in a fast way. The secondapproach is to first perform a fast and efficientsearch in the short-term memory TS phase withTARDY, TARDY-A-DIV or TARDY-B-DIVcandidate list strategy and then to restart thesearch around the elites with an intense search atti-tude, namely, with no candidate list strategy. TheCPU time required for the first approach (TS1)is high as compared to the cases employing the sec-ond approach while quality of results is similar.Therefore, TS2, TS3 and TS4 which employ CLSscome forth as effective solution techniques forSMTWT.

As a result of these observations, the algorithmpresented in Fig. 2 is suggested as a robust TSalgorithm for the SMTWT Problem.

The proposed TS approach yields very highquality results for the set of benchmark problemsobtained from the literature. The results show that

1. Generate an initial solution by means of a heuristic. Suggested strategy:

• Use R&D+ heuristic.

2. Employ short-term TS search.Suggested strategy:

• Apply an arc-based tabu classification (i.e. ARC+).

• Apply TARDY candidate list strategy to screen the neighborhood of all insert and swap moves.

• For larger size problems apply also a move type CLS with diversification effect (i.e. B-DIV) together with TARDY.

• Use dynamic tenure with small (S), medium (M) and large (L) tenure values along a repeated pattern of SMLS, each for a duration of (2x M) iterations.

• Select tenure values as S=15, M=0.5n, and L=1.5n (for a CLS strategy as described above) where n is the number of jobs.

3. Apply intensification.Suggested strategy:

• Use three elite solutions identified during the short-term memory TS phase to restart the search.

• Drop CLS at intensification phase (for a short-term CLS strategy as described above).

• Adapt the short-term tenure values with a multiplier around [1.4,1.6] for a CLS strategy as described above.

Fig. 2. A robust TS algorithm for SMWTP.

1434 U. Bilge et al. / European Journal of Operational Research 176 (2007) 1423–1435

the set of 40-job SMTWT problems is solved tooptimality via the TS approach. As for the 50-job and 100-job problem sets, 99.2% and 86.4%of the best-known values reported in the literatureare reached, respectively, with very insignificantdeviations for the rest. These results are betterthan those for other state-of-the-art TS algorithmsfor SMTWT. The performance of proposed TSalgorithm is also quite competitive with respectto the iterated dynasearch algorithm of Congramet al. (2002). The look-ahead capability of the lat-ter approach resulting from allowing severalmoves in a single iteration draws the attention toincorporation of compound moves into TS neigh-borhood as well. Compound moves, an oftenneglected issue in TS literature (Glover andLaguna, 1997), may result into very large neigh-

borhoods, but then again good candidate list strat-egies may help speeding up the search.

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