A multi-objective tabu search for a single-machine scheduling problem with sequence-dependent setup times

European Journal of Operational Research 175 (2006) 318–337

www.elsevier.com/locate/ejor

Discrete Optimization

A multi-objective tabu search for a single-machinescheduling problem with sequence-dependent setup times

F. Fred Choobineh *, Esmail Mohebbi, Hansen Khoo

Department of Industrial and Management Systems Engineering, University of Nebraska-Lincoln,

P.O. Box 880518, 175 Nebraska Hall, Lincoln, NE 68588-0518, United States

Received 9 July 2002; accepted 14 April 2005Available online 3 August 2005

Abstract

An m-objective tabu search algorithm for sequencing of n jobs on a single machine with sequence-dependent setuptimes is proposed. The algorithm produces a solution set that is reflective of the objectives� weights and close to the bestobserved values of the objectives. We also formulate a mixed integer linear program to obtain the optimal solution of athree-objective problem. Numerical examples are used to study the behavior of the proposed m-objective tabu searchalgorithm and compare its solutions with those of the mixed integer linear program.� 2005 Elsevier B.V. All rights reserved.

Keywords: Scheduling; Sequence-dependent setup; Multi-objective; Tabu search

1. Introduction

In many real-life scheduling problems, the decision maker is faced with situations in which the appro-priateness of a schedule is measured against multiple objectives. These objectives are often conflictingand no single schedule would simultaneously optimize all objectives. Therefore, in the absence of a globallyoptimum solution, a compromise solution must be sought from among a number of solutions identifiedwithin the scope of the preferences set by the decision maker. Traditional mathematical programming ap-proaches for solving multi-objective optimization problems are computationally intractable for practical

0377-2217/$ - see front matter � 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.ejor.2005.04.038

* Corresponding author.E-mail address: [email protected] (F.F. Choobineh).

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F.F. Choobineh et al. / European Journal of Operational Research 175 (2006) 318–337 319

problems, and thus the use of (meta) heuristic methods such as genetic algorithms, simulated annealing,and tabu search have become popular in the past decade (see Jones et al. (2002) for a review of papersin this area).

A machine scheduling problem is a combinatorial optimization problem, and the most common perfor-mance measures (objectives) are functions of the jobs� completion times. Examples of such objectives to beminimized include the makespan (i.e., the completion time of the last job to leave the system), the (dis-counted) total weighted completion time, the maximum lateness, the total weighted tardiness, and theweighted number of tardy jobs (Pinedo, 2002). The first two objectives are focused on improving resourceutilization and productivity, while the others are mainly perceived as measures of conformity with duedates. Thus, minimizing makespan, number of tardy jobs and total tardiness at the same time, for example,could be a reasonable set of objectives for a machine scheduling problem.

We consider an m-objective non-preemptive scheduling of n jobs on a single machine with sequence-dependent setup times. Sequence-dependent setups are commonly observed in various industrial settingsincluding printing, textile, pharmaceutical, chemical and metallurgical industries. Gagne et al. (2001)describe an industrial application involving holding furnaces that require alloy-dependent draining andcleaning operations between two consecutive castings of metals, and discuss broader applications of se-quence-dependent setups in the printing, textile, pharmaceutical, chemical and metallurgical industries.Das et al. (1995), Franca et al. (1996) and Gravel et al. (2000) document other examples of sequence-depen-dent setups in the plastic and aluminum casting industries.

For the single objective of minimizing the makespan, the problem is known to be a strong NP-hardproblem (Pinedo, 2002). Hence, we conclude that its m-objective problem must also be an NP-hard problemand consequently, devising a heuristic to solve this problem is highly desirable. As such, we propose a multi-objective tabu search algorithm for identifying a bounded solution space. Our algorithm is novel because ituses an independent tabu list for each objective, it tracks the best identified value for each objective (solu-tion to the single objective case), and it creates a bounded solution space whose boundaries are inverselyproportional to the weights assigned to the objectives. That is, bounds for important objectives receivinglarger weights will be closer to their best identified values, whereas bounds for less important objectivesreceiving smaller weights will be further away from their best identified values. In addition, by specifyingthe constant of proportionality for each objective, the decision maker has an opportunity to control thesize of the bounded solution space. The bounded solution space contains a mix of solutions, some of whichare on the Pareto frontier and some may be dominated, thereby offering more alternative solutions to thedecision maker. Furthermore, the proposed algorithm provides a general framework for solving large-scalemulti-objective combinatorial problems.

The rest of this paper is organized as follows. In Section 2 we review the relevant literature. An optimi-zation model for the three-objective single-machine non-preemptive sequencing problem with sequence-dependent setup times is presented in Section 3, followed by a detailed description of the tabu searchalgorithm in Section 4. Section 5 is devoted to our numerical results, and some concluding remarks are pre-sented in Section 6.

2. A review of past research

The published literature pertinent to this work can be divided into two main categories. The first cate-gory consists of single-machine sequencing procedures with multi-objective and/or sequence-dependent set-up times that are primarily based on traditional operational research global search approaches such asbranch and bound, goal programming, dynamic programming, and trade-off curves. In this regard,Emmons (1975), Chand and Schneeberger (1986, 1988), Chen and Bulfin (1994), Lin and Lee (1995) andKoksalan et al. (1998) propose solution methodologies for various single-machine scheduling problems

320 F.F. Choobineh et al. / European Journal of Operational Research 175 (2006) 318–337

with two objectives with no sequence-dependent setup times. Picard and Queyranne (1978), Barnes andVanston (1981), and White and Wilson (1977) focus on achieving a single objective while taking into ac-count the setup times in sequencing jobs on one machine. Cheng et al. (2000) provide a review of flow-shopscheduling research with setup times.

The second category comprises studies that are based on meta-heuristic methods such as genetic algo-rithms, simulation annealing, and tabu search in solving multi-objective and/or sequence-dependent setuptime scheduling problems. Due to their flexibility, these evolutionary search heuristic approaches have beenutilized more successfully than traditional approaches in solving complex multi-objective scheduling prob-lems. In this category, Glover et al. (1995) presented a tabu search procedure for the problem considered byBarnes and Vanston (1981), which involved minimizing the sum of setup and linear delay penalty costs forthe single-machine scheduling problem with sequence-dependent setups. Rubin and Ragatz (1995) pro-posed a genetic algorithm for the problem of sequencing n jobs on one machine with sequence-dependentsetups such that the total tardiness of all jobs is minimized. Murata et al. (1996) presented a genetic algo-rithm for a flow-shop scheduling problem with m machines and three objectives: minimizing the makespan,total tardiness and total flow time. They used a variety of random-weight combinations for combining mul-tiple objectives into a scalar fitness function to produce the Pareto optimal front. Related work can befound in Horn et al. (1994). Murata et al. was further extended in Ishibuchi and Murata (1998) and Ishibu-chi and Yosshida (2002). Marett and Wright (1996) presented a comparative study between the perfor-mances of simulated annealing and tabu search in solving multi-objective combinatorial problems. Morerecently, Choi and Choi (2002) presented a local search algorithm for a job-shop scheduling problem withalternative operations and sequence-dependent setups in order to minimize the makespan of the schedule.Sarker et al. (2002) analyzed the Pareto-based approaches in multi-objective optimization problems andproposed an evolutionary algorithm which discards the dominated alternatives in each generation. Theresulting number of generated offspring is the number of non-dominated individuals multiplied by a fixedratio. Rabadai et al. (2002) presented a simulated annealing algorithm for scheduling n jobs with a largecommon due date and no preemption on a single machine, with sequence-dependent setup times such thatthe total tardiness and earliness of jobs is minimized. Weng and Sedani (2002) considered a similar early/tardy scheduling problem and presented a tabu search algorithm for the case where jobs have different duedates. Their algorithm, however, did not allow for sequence-dependent setup times. Other applications ofmeta-heuristics in solving multi-objective combinatorial optimization problems in the literature includeSrinivas and Deb (1994), Czyzak and Jaszkiewics (1998), Ulungu et al. (1999), Knowles and Corne(2000) and Jaszkiewics (2002). To the best of our knowledge, none of the existing studies in the literatureconsider sequence-dependent setup times with due-date related objectives within the context of a multi-objective scheduling problem such as the one considered in this work.

3. A mixed integer linear programming formulation

Resource utilization and due-date conformance are two of the most frequently encountered performancemeasures in shop scheduling problems. In a single-machine scheduling problem, resource utilization is com-monly measured in terms of makespan. Due-date related measures are direct indicators of customer servicelevel. In practice, the number of tardy jobs (i.e., jobs completed after their due dates) is typically regarded asa good surrogate measure for the service quality provided to the customers. However, using this measurealone to quantify the level of customer satisfaction can be misleading as it does not take into account themagnitude of lateness associated with each completed job. Hence, a time-dependent measure such as totaltardiness appears to provide a more accurate depiction of due-date conformance and customer service level.As such, let us consider the sequencing problem of n jobs J1, J2, . . ., Jn on one machine with the followingthree objectives (m = 3):


1. Minimizing the completion time of the last job to leave the machine (makespan).2. Minimizing the number of jobs whose completion time is passed their due date (number of tardy jobs).3. Minimizing the sum of the tardiness of all the jobs (total tardiness).

Our main assumptions are:

• All jobs are present at time t = 0.• Each job is processed only once on the machine without preemption.• Job processing times and due dates are known at time t = 0.• Setup time associated with the processing of each job is dependent on its immediately preceding job in

the sequence but is independent of the sequence position.

We next introduce some notations and proceed by presenting a multi-objective mixed integer linear pro-gramming (MILP) formulation of the scheduling problem described above.

Parameters:

pj processing time for job j (pj > 0) j = 1, 2, . . ., n

dj due date for job j (dj > 0) j = 1, 2, . . ., n

s0j setup time of job j in the first sequence position (s0j P 0) j = 1, 2, . . ., n

sij setup time of switching from job i to j (sij P 0) i and j = 1, 2, . . ., n, i 5 j

M a very large number

Decision variables (for i, j, and k = 1, . . ., n):

xjk ¼1 if job j is assigned to kth position in the sequence;

0 otherwise;

(

xijk ¼1 if job j is assigned to kth position in the sequence and it is preceded by job i;

0 otherwise;

(

UðkÞ ¼1 if the job in the kth position is tardy;

0 otherwise.

(

C(k) completion time of the job in the kth sequence positionS(k) setup time of the job in the kth sequence positionP(k) processing time of the job in the kth sequence positionD(k) due date of the job in the kth sequence positionT(k) tardiness of the job in the kth sequence position

The model:

minimize Z1 ¼ CðnÞ ð1Þ

minimize Z2 ¼Xn

k¼1

UðkÞ ð2Þ

minimize Z3 ¼Xn

k¼1

T ðkÞ ð3Þ


subject toXn

i¼1

xik ¼ 1 k ¼ 1; . . . ; n; ð4Þ

Xn

k¼1

xjk ¼ 1 j ¼ 1; . . . ; n; ð5Þ

Xn

i

Xn

j

xijk ¼ 1 k ¼ 2; . . . ; n; ð6Þ

xjk þ xik�1 � 1 6 xijk i and j ¼ 1; . . . ; n; ði 6¼ jÞ and k ¼ 2; . . . ; n; ð7Þ

Sð1Þ ¼Xn

i¼1

s0ixi1; ð8Þ

SðkÞ ¼Xn

i¼1

Xn

j¼1

sijxijk k ¼ 2; . . . ; n; ð9Þ

PðkÞ ¼Xn

j¼1

xjkpj k ¼ 1; . . . ; n; ð10Þ

DðkÞ ¼Xn

j¼1

xjkdj k ¼ 1; . . . ; n; ð11Þ

CðkÞ ¼ Cðk � 1Þ þ SðkÞ þ PðkÞ k ¼ 1; . . . ; n; ð12Þ� CðkÞ þ dðkÞ 6 Mð1� UðkÞÞ k ¼ 1; . . . ; n; ð13ÞCðkÞ � dðkÞ 6 MUðkÞ k ¼ 1; . . . ; n; ð14ÞT ðkÞP CðkÞ � dðkÞ k ¼ 1; . . . ; n; ð15ÞUðkÞ ¼ 0 or 1 k ¼ 1; . . . ; n; ð16Þxik ¼ 0 or 1 k ¼ 1; . . . ; n; ð17Þxjik ¼ 0 or 1 k ¼ 1; . . . ; n; ð18ÞT ðkÞP 0 k ¼ 1; . . . ; n. ð19Þ

We note the similarity between the above formulation and that of the classical traveling salesman prob-lem (TSP), where the setup time incurred in switching from one job to another can be interpreted as thetraveling distance between two cities. Observe that this formulation contains a total of n3 + n2 + 6n vari-ables and n(n � 1)2 + 10n � 1 constraints. Eqs. (1)–(3) represent the objective functions, namely minimiz-ing the makespan, minimizing the number of tardy jobs and minimizing the total tardiness, respectively. Eq.(4) ensures that only one job is assigned to a sequence position; Eq. (5) assures that only one sequence posi-tion is allocated to a job. Eqs. (6) and (7) assure that only one job, job j, is positioned after job i when job i

is assigned to the (k � 1)th sequence position, and prevent the MILP from generating an invalid solutionthat (in TSP terminology) involves sub-tours. Eqs. (8) and (9) determine the setup times for the first se-quence position (initial setup) and subsequence positions. The processing time and the due date of thekth sequence position are acquired by Eqs. (10) and (11), respectively. The completion time of the job atthe kth sequence position is obtained by Eq. (12). The inequalities displayed in Eqs. (13) and (14) are for-mulated to determine whether the job at the kth sequence position is late under the if–else condition. Sim-ilarly, the relation in (15) is set to obtain the tardiness of the job at the kth sequence position. Eqs. (16)–(18)impose the binary constraint. Finally, the condition set in Eq. (19) assures non-negativity of the tardiness.

Clearly, unless there is a unique sequence that simultaneously minimizes all three objectives, optimizingwith respect to one objective at a time is at the expense of deviating from the optimal solution of the others.Therefore, it is advisable to seek a compromise solution that minimizes the overall relative deviationfrom individual optimal objective values by considering all objectives concurrently, a task that becomes


excessively difficult as the number of objectives grows. One common approach in dealing with such situa-tions is to establish a weighted (composite) objective function based on the significance of individual objec-tives, or equivalently, the criticality of deviating from the optimal value of each individual objective. Forexample, if the decision maker assigns a series of relative weights W1, W2, and W3 to Z1, Z2, and Z3, respec-tively, the composite objective function can be readily established as the weighted normalized total sum ofdeviations from optimal objective values in the form

Minimize Z ¼ W 1

Z1 � Z�1Z�1

þ W 2

Z2 � Z�2Z�2

þ W 3

Z3 � Z�3Z�3

; ð20Þ

where Z�i is the optimal value of the MILP when Zi, i = 1, 2, 3, is its only objective. Clearly, if W1 > W2, forexample, the value of the Z1 associated with the compromise solution sought in this way will be closer to Z�1than that of Z2 to Z�2, reflecting the fact that the decision maker is more amenable to compromise objective2 in return for achieving objective 1.

In the next section, we propose a tabu search procedure that not only alleviates the computational com-plexity of solving the optimization model for large-scale problems, but also provides the decision makerwith a bounded set of alternative solutions for a more convenient trade-off analysis.

4. A multi-objective tabu search solution methodology

Introduced and elucidated by Glover (1986, 1989, 1990) tabu search is a meta-heuristic neighborhoodsearch methodology. The basic idea of this methodology is to search for the next candidate solution fromamong a carefully constructed neighborhood of the current trial solution by allowing for the possibility ofthe new solution being worse than the existing one. Tabu search attempts to overcome the issue of local opti-mality by maintaining a list of prohibited moves, called a tabu list, at each iteration of the search procedure.More specifically, tabu search is able to explore the search space beyond the local optimum by remembering alist of recent moves that are not to be repeated in taking the next step toward generating a new solution. Thetabu restriction, however, can be overridden provided that a forbidden (tabu) move meets aspiration criteria.Aspiration criteria are specifically designed to authorize a tabu move so that a better solution that has not beenvisited yet is not overlooked. A commonly used aspiration criterion in tabu search is that if at any iteration ofthe search process a tabu move can generate a solution than is better than the best solution found so far, thetabu restriction is revoked. We adopt the same criterion in our proposed methodology.

With a few exceptions (e.g., Tan et al., 2003), tabu search has been mainly applied to single-objectiveoptimization problems. When applying tabu search to multiple-objective optimization problems, all objec-tives may be combined into a normalized weighted linear function. However, this approach is not desirablein most multi-objective decision making situations in industrial settings since the decision maker would nor-mally like to consider several good quality solutions before making a final decision. This consideration isespecially helpful in evaluating trade-offs among conflicting objectives, some of which may have not beenspecifically modeled. One possible approach for identifying several solutions is to determine the set of thePareto optimal solutions (Murata et al., 1996, among others). Another approach is to identify a set of solu-tions which are within certain percentages of the best-known individual objective values. The latter ap-proach provides the decision maker with a set of good quality solutions, some of which may be on thePareto front. This approach is particularly desirable when the Pareto optimal front is hard to establish.Clearly, the idea of establishing an admissible range for candidate solutions with respect to each objectiveis closely tied to the degree of decision maker�s flexibility in deviating from its best-known value. Our pro-posed tabu search algorithm utilizes the idea of relative weights to form the bounded solution space de-scribed above. More specifically, our flexibility in admitting feasible solutions with respect to anobjective is inversely proportional to the weight assigned to that objective; the higher the weight of anobjective, the smaller the range from its best-known value within which candidate solutions are admissible.


Next we present a detailed description of the parameters and modules of the algorithm, followed by apseudocode of the proposed m-objective tabu search algorithm.

4.1. Input parameters

The following input parameters are required:m the number of objectives,n the number of jobs,Wobj relative weight of objective obj, obj = 1, . . ., m,Max_iteration the maximum number of iterations allowed in the search process,Tabu_size the size of the tabu list, i.e., the maximum number of immediate past moves that are considered

to be tabu at every iteration,Move_distance the maximum allowable distance between two positions in a given sequence when conduct-

ing a ‘‘swap’’ transaction at any iteration. Note that for a given sequence P = {J[1], J[2], . . ., J[n]},where J[i] denotes the job scheduled in position [i] ([i] = 1, 2, . . ., n) of the sequence, and the distancebetween any two positions [i] and [j] ([i] < [j]) is [j] � [i] � 1.

4.2. Admissible range for the bounded solution space

Recall that the bounded solution space consists of all solutions whose objective values fall within certainpercentages of the best-known values of desired objectives. Let Dobj denote the percentage corresponding tothe admissible range for objective obj (obj = 1, . . ., m). Then, when applying the proposed multi-objectivetabu search, for a solution to be included in the bounded solution space at an iteration, its value with re-spect to each and every objective obj, say Zobj, must satisfy

Zobj 6 Rangeobj ¼ Best known valueobjð1þ DobjÞ if obj is in a minimization format;

Zobj P Rangeobj ¼ Best known valueobjð1� DobjÞ if obj is in a maximization format;

(

where Best_known_valueobj, obj = 1, . . ., m, is the best value of objective obj known up to that iteration.Clearly, the bound imposed by the Rangeobj tightens as new values for Best_known_valueobj are foundthroughout the search process, thereby systematically reducing the number of solutions in the solutionspace. A simple procedure for determining Dobjs is as follows:

Step 1: Calculate the ratio Rj ¼ W objPm

obj¼1W obj

for j = obj = 1, . . ., m.

Step 2: Sort Rjs from the smallest to the largest.Step 3: Determine Robj by assigning the smallest Rj to the objective with the largest weight, the next small-

est Rj to the objective with the next largest weight and so on.Step 4: Determine Dobj by multiplying Robj by the scale factor aobj, 0 < aobj 6 1, obj = 1, . . ., m.

When aobjs are equal to 1, the values of Dobjs are primarily influenced by the ordinal ranking of the objec-tive weights (Wobjs). For example, in case of four objectives (m = 4) with W1 = 200, W2 = 400, W3 = 100,and W4 = 300, the above procedure yields D1 = 0.3, D2 = 0.1, D3 = 0.4, and D4 = 0.2. In order to explicitlyaccount for the cardinal values of the objective weights, we can, in the above example, set the values of aobj

to (minobj{Wobj})/Wobj which results in D1 = 0.15, D2 = 0.025, D3 = 0.4, and D4 = 0.666. Other means ofsetting aobjs may include various quantitative measures of relative differences in objective weights, the deci-sion maker�s preference, or a combination thereof.


In general, the scale factor aobjs can be used not only to account for the cardinal value of the weight butalso to control the size of the bounded solution space. Small scale factors result in smaller bounded solutionspace and thereby a smaller number of solutions. It should also be noted that aobjs can be adjusted dynam-ically to populate the bounded solution space at initial stages of the algorithm (i.e., looser admissible ranges),and to further reduce the number of solutions admitted (i.e., tighter admissible ranges) in a systematicmanner as the number of iterations increases. We note in passing that a more elaborate means of settingadmissible ranges may include using the ideal and Nadir vectors of objectives; however, as indicated inEhrgott et al. (2003), the problem of computing the Nadir values for multi-objective problems in general canbe very difficult. Therefore, devising heuristics such as the procedure described above appears more appealing.

4.3. Starting solution

The starting solution space comprises m randomly generated sequences P1,P2, . . .,Pm, each serving asthe starting solution for an objective obj (obj = 1, . . ., m).

4.4. Neighborhood generation: Swap move

A pairwise position exchange (swap) move with a maximum allowable move distance specified byparameter Move_distance is used for generating neighborhood solutions. In other words, a candidate solu-tion in the neighborhood of a current solution is generated by replacing a job in a marked position of thecurrent sequence (say [i]) with another job in a position [j] ([j] > [i]) such that [j] � [i] 6 Move_distance + 1.For example, given P = {1, 2, 3, 4, 5} with Move_distance = 2, the neighborhood solutions resultingfrom ‘‘allowable’’ swap moves are: {2, 1, 3, 4, 5}, {3, 2, 1, 4, 5}, {4, 2, 3, 1, 5}, {1, 3, 2, 4, 5}, {1, 4, 3, 2, 5},{1, 5, 3, 4, 2}, {1, 2, 4, 3, 5}, {1, 2, 5, 4, 3}, and {1, 2, 3, 5, 4}. In general, for a schedule with n jobs andMove_distance = r (r 6 n � 2), this swap procedure results in a neighborhood consisting of (2n �r � 2)(r + 1)/2 possible solutions. The move distance restriction, however, is revoked for n consecutive iter-ations once an improvement is made to an objective value. An alternative strategy calls for revoking therestriction on Move_distance at iterations where no better solution is found and imposing the limitationwhen a better solution becomes available. In this regard, we note that the notion of Move_distance canbe interpreted as a measure of degree to which the current solution is changed as the result of a swap move.Barnes and Laguna (1993) point out that a large Move_distance tends to produce a greater change, andconsequently, may result in undesired changes in terms of the objective values. Therefore, these movesare less likely to be chosen for generating the next solution. On the other hand, imposing a rigid constraintwith regard to the Move_distance in generating new solutions can limit the exploration ability of the searchprocess. We believe the proposed strategy has the advantage of saving computation time, while allowing thesearch process to branch out and explore new neighborhoods at the same time. We also note that the heu-ristic can be readily modified to allow for the ‘‘insertion’’ (or ‘‘shift’’) move, instead of the proposed ‘‘swap’’move, for generating neighborhood solutions. A comparison of the effectiveness of these moves has beencarried out by Fink and Voß (2003). They showed that that the insertion move performed better thanthe swap move when tabu search and simulated annealing were used for solving several cases of continuousflow-shop scheduling problems.

4.5. Tabu lists

Due to the fact that there are m-objectives involved in the problem, our proposed methodology main-tains and updates m independent tabu lists—each corresponding to a distinct objective—throughout thesearch process. More specifically, starting from each randomly generated initial sequence Pobj,obj = 1, . . ., m, the neighborhood search process is conducted with respect to objective obj and the


corresponding tabu list is updated accordingly after each iteration of the search process is complete. Cou-pled with the formation process of the bounded solution space described above, this implies that we con-duct the m tabu search process in parallel to establish a repository of quality near-optimal solutions.

4.6. The proposed tabu search algorithm

In addition to those notations introduced above, the following notations are also used in describing thealgorithm:

½i�½j�Pobj the sequence obtained after exchanging (swapping) jobs scheduled in positions [i] and [j]

([i], [j] = 1, 2, . . ., n, [i] 5 [j]) of any given sequencePobj {J[1], J[2], . . ., J[n]}, obj = 1, . . ., m

Tabu_timeobj [i][j] the iteration at which the swap move [i] M [j] currently present in the tabu list obj(obj = 1, . . ., m) was last executed

Improve_timeobj the iteration at which Best_known_valueobj was last improvedRecorded_valueobj the recorded value of objective obj at each iteration of the neighborhood search processP�obj the best known schedule at each iteration of the search process starting from the initial schedule

Pobj, obj = 1, 2, . . ., m

The m-objective tabu search algorithm will be presented for the situation when all objectives are to beminimized. However, the algorithm could easily be modified to accommodate maximization of some orall objectives.

Procedure Main

Start procedureInput Max_iteration, Tabu_size, Move_distance, W1, W2, . . ., Wm

Calculate D1,D2, . . .,Dm

Generate m random sequences, P1,P2, . . .,Pm

For (obj = 1 to m) do{Best_known_valueobj, Rangeobj 1}

EndforExecute Reset Tabu Array

For (Iteration = 1 to Max_iteration) do{
For obj = 1 to m do{Recorded_valueobj 1Generate Neighborhood for Pobj
For (each and every ½i�½j�Pobj neighbor of Pobj) do{Calculate Z1, Z2, . . ., Zm values

If (Z1 6 Range1, Z2 6 Range2, . . ., and Zm 6 Rangem) then{Deposit ½i�½j�Pobj into the Bounded Solution Space}EndifIf (Move Status [i] M [j] is not TABU, or Zobj < Best_known_valueobj then{

If (Zobj < Recorded_valueobj) then{Recorded_value Zobj

P�obj ½i�½j�Pobj}}}

EndifEndif

Endfor


Execute Update Tabu Array [i] M [j]Pobj P�obj

If (Recorded_valueobj < Best_known_valueobj) then{Best_known_valueobj Recorded_valueobj

Improve_timeobj Iteration}}Endif

EndforIf (Best_known_valueobj is changed for any obj, obj = 1, 2, . . ., m) then{Range_obj [(1 + Dobj)(Best_known_valueobj)] for obj = 1, 2, . . ., m

Execute Update Bounded Solution Space}}Endif

EndforStop procedure

The main procedure calls five subprograms (shown in bold). The pseudocodes for these subprograms aregiven in Appendix.

5. Numerical experiments

Extensive numerical experiments were conducted to observe the behavior of the proposed algorithm.The results of two sets of experiments are presented here. The parameter values (i.e., processing times,due dates, and setup times) for the scheduling problems used in each experiment were generated randomlyusing the following probabilistic patterns:

Processing time ðpj; j ¼ 1; . . . ; nÞ � Uniform½1; a�;Setup time ðsij; i ¼ 0; 1; . . . ; n; j ¼ 1; 2; . . . ; n; i 6¼ jÞ � Uniform½1; b�;

Due date ðdjÞ �Xn

j¼1

pj=2þUniform �Xn

j¼1

pj=4;Xn

j¼1

pj=4

" #.

5.1. Convergence rate of the bounded solution space

The first set of experiments is aimed at depicting the convergence rate of the bounded solution space. Weconsidered a two-objective (m = 2) scheduling problem aimed at minimizing the makespan (Z1) and thetotal tardiness (Z3) with n = 10 jobs, W1 = 120, W3 = 80, a = 15 and b = 5. Note that the MILP formu-lation of this problem can be readily devised by eliminating Z3 and U(k) and the corresponding relations(2), (13), (14) and (16) from the model presented in Section 3. The tabu algorithm was run for Max_iter-

ation = 5000 iterations with Tabu_size = 10, and Move_distance = 5, and scale factors a1 = a3 = 0.2. For asequence P = {J[1], J[2], . . ., J[n]}, the values of Z1 and Z3 were calculated using Eqs. (1) and (3), respectively.To illustrate convergence of the process, the job sequences found in the bounded solution space were re-corded every time the set was updated throughout the run and the associated two-objective values wereplotted. Fig. 1 contains individual plots of the bounded solution sets in the 2D space of objective Z1

and Z3 at iterations 1, 5, 66, and 5000, as well as an overall display of the transitions of the solution spaceboundaries at various iterations of our proposed procedure.

Note that the Best_known_valueobj, obj = 1, 3 for the stated iterations are [87, 199], [79, 146], [77, 138],and [77, 138], respectively. The minimum values of individual objective functions obtained from solvingthe MILP problem are Z�1 ¼ 77 and Z�3 ¼ 138.

Fig. 1. Convergence of the bounded solution space with two objectives.



Next, we considered the three-objective (m = 3) scheduling problem of Section 3. The relative objectiveweights for the three-objective problem were W1 = 200, W2 = 100 and W3 = 100, a1 = a2 = a3 = 0.4, and

Fig. 2. Convergence of the bounded solutions space with three objectives.


the value of Max_iteration was set to 50,000. All other parameter values were kept unchanged. Fig. 2 dis-plays the 3D plots of the bounded solution space at iterations 1, 10, 27, 120, 155, and 50,000 with corre-sponding Best_known_valueobj, obj = 1, 2, 3, as [89, 8, 246], [78, 5, 144], [77, 5, 144], [77, 5, 138], [77, 4, 138],and [77, 4, 138], respectively. The minimum values of individual objective functions obtained from solvingthe MILP problem are therefore Z�1 ¼ 77; Z�2 ¼ 4, and Z�3 ¼ 138.

As expected, the solution space generally gets smaller as the number of iterations increases until the min-imum possible values of all individual objectives (i.e., the tightest bounds) have been acquired. We also notethat the algorithm moves fairly rapidly toward capturing those solutions associated with the best possiblevalues of the objectives. The boundaries of the bounded solution space were not improved beyond iteration155 (iteration 66 in case of m = 2), because the minimum possible values of all objectives had already beenfound. The algorithm, however, continued to identify additional solutions that were within the boundariesof the bounded solution space in the remaining iterations before reaching Max_iteration. Table 1 contains alist of 22 sequences and their respective objective values that were in the bounded solution space for the case

Table 1The bounded solution set for a 10-job 3-objective problem after 50,000 iterations (W1 = 200, W2 = 100, W3 = 100)

No. Z1 Z2 Z3 Z Job sequence

1 82 5 166 58.3 8 10 2 7 6 5 3 9 4 12 83 5 152 50.7 10 2 7 6 9 8 5 3 4 13 83 5 164 59.4 6 7 2 4 5 10 8 3 9 14 83 4 165 35.1 6 7 2 4 8 5 10 3 9 15 83 4 165 35.1 2 7 6 4 8 5 10 3 9 16 83 5 166 60.9 2 7 6 9 4 5 10 8 1 37 83 5 166 60.9 2 7 6 10 8 5 3 9 4 18 84 5 154 54.8 2 7 6 9 10 8 3 4 5 19 84 5 156 56.2 2 7 6 9 10 8 3 4 1 5

10 84 5 162 60.6 10 2 7 6 9 8 1 3 4 511 84 5 165 62.7 4 2 7 6 9 8 5 10 1 312 85 5 149 53.8 10 2 7 6 9 8 3 4 5 113 85 5 151 55.2 10 2 7 6 9 8 3 4 1 514 85 5 156 58.8 6 7 2 4 9 10 8 5 1 315 85 5 156 58.8 2 7 6 4 9 10 8 5 1 316 85 5 161 62.4 4 2 7 6 9 8 5 10 3 117 85 5 163 63.9 10 2 7 6 9 8 1 4 5 318 85 5 164 64.6 2 7 6 9 8 10 3 4 5 119 85 5 164 64.6 2 7 6 9 4 5 10 8 3 120 85 5 165 65.3 10 2 7 6 9 8 5 1 4 321 85 5 166 66.1 10 6 2 7 9 8 5 3 4 122 85 5 166 66.1 2 7 6 9 8 10 3 4 1 5

Table 2Various relative weight assignments in experimental problems with n = 15 and n = 20

W1 W2 W3

a 150 100 50b 150 50 100c 100 150 50d 50 150 100e 100 50 150f 50 100 150


of m = 3 at iteration 50,000. Table 1 is sorted in ascending value of the highest weight objective, Z1; thecomposite objective value, Z, is also reported. We note that the fourth sequence in Table 1 is the optimalsequence obtained from solving the MILP with a single composite objective function defined by Eq. (20). Itshould be noted that the fifth sequence is an alternative solution of the MILP that was identified by the tabusearch.

Fig. 3. Average CPU times for the experimental problems with n = 15 and n = 20.


5.2. Impact of parameter variations

The second set of experiments focused on studying the impact of varying scheduling parameters on theperformance of the proposed tabu search. The experiments were conducted for the 3-objective schedulingproblem. We considered two 3-objective (m = 3) problems, one with 15 jobs (problem 1) and the other with20 jobs (problem 2). For each scheduling problem, two sets of processing times (pjs with a = 15), and twosets of due dates (djs) were randomly generated from the probability distribution forms given earlier. Forboth scheduling problems, three different sets of setup times (sijs) were drawn from the uniform distributionU � [1, b] with b = 5, 10, and 15. For both problems, six different sets of objective weights were used, asshown in Table 2.

The algorithm was set to run for Max_iterations = 100,000 iterations with Tabu_size = 20 andMove_distance = 5 on a Pentium III 450 MHz computer. The three scale factors were set to an equal valuea1 = a2 = a3 = 0.3. Each experiment was run for 10 replications. Fig. 3 shows the average CPU time for theexperiments conducted. It is clear from these results that the average CPU time increases as the setup rangedecreases from U � [1, 15] to U � [1, 5] for each relative weight assignment scenario. This is mainly due tothe fact that the number of solutions in the bounded solution space in most cases increases as the setup timechanges from U � [1, 15] to U � [1, 5]. The average number of solutions in the bounded solution space ineach experiment is depicted in Table 3.

Recall that in our proposed algorithm the m boundaries of the bounded solution space at every iterationare defined in terms of certain percentages, Dobjs, of the best-known values of individual objectives. As aresult, the values of Dobjs have an ultimate impact on the number of solutions allowed in the bounded solu-tion space. Clearly, having multiple solutions is essential in finding a compromise solution to any multi-objective decision-making problem. Nevertheless, a large number of solutions in the final boundedsolutions space can overwhelm the decision maker. A simple way to reduce the number of solutions inthe bounded solution space is to decrease the scale factor aobj. Fig. 4 provides an illustration of the outcomeof such adjustments for problem 1 with n = 20, objective weights W1 = 150, W2 = 100, and W3 = 50, andsij � U [1, 5]. The base (100%) values of scale factors were a1 = a2 = a3 = 0.3. Additional scale factors were80% and 60% of the base values. It should be noted that when the scale factor was 40% of its base value, thebounded solution space was empty.

Table 3Average number of solutions in the bounded solution space

Setup (1–5) Setup (1–10) Setup (1–15)

Prob 1 Prob 2 Prob 1 Prob 2 Prob 1 Prob 2

n = 15

a 35.50 50.00 1.70 3.00 2.70 0.90b 186.30 214.50 9.90 14.10 6.30 3.90c 1.00 15.00 4.10 0.00 1.00 1.10d 1.00 15.00 6.30 0.00 1.10 1.60e 621.10 476.40 222.90 53.60 59.10 20.10f 49.60 90.00 92.90 10.00 11.60 3.50

n = 20

a 158.90 1037.70 34.70 180.90 1.20 5.00b 565.40 7474.60 139.20 259.00 1.30 14.50c 7.00 39.30 11.50 56.60 0.00 5.80d 6.60 60.60 9.70 62.70 1.20 9.00e 3063.60 16568.00 500.40 2524.50 28.60 61.60f 406.20 1715.20 139.80 527.40 13.80 59.30

Fig. 4. The effect of reducing aobjs for problem 1 with n = 20.



Panel I in Fig. 4 shows the impact of reducing aobj on the average number of solutions in the boundedsolution space for 10 replications of the experiment. As expected, the average number of solution decreasesas aobj is reduced. Panels II–V, respectively, depict the impact of reducing aobj on the values of Z (the com-posite objective), Z1 (makespan), Z2 (number of tardy jobs) and Z3 (total tardiness) for 10 replications ofthe problem. In each panel, there are three adjacent columns per entry that represent the best minimumvalue, the mean of minimum values, and the average of solutions values associated with the correspondingobjective for 10 replications. The average objective values associated with the bounded solution space (thethird column) in cases of composite objective, makespan, and total tardiness decrease as aobj is decreased,but this does not hold true for the number of tardy jobs. This suggests that as we reduce aobj proportionally,we are more likely to exclude those solutions from the bounded solution space that result in poor valueswith respect to one or more objective. In particular, the behavior of the average number of tardy jobsfor the bounded solution space in our experiment can be attributed to the fact that when aobj is large,the bounded space contains more solutions with desirable values (in terms of the number of tardy jobs)but poor values (with respect to makespan and total tardiness) than any other scenario.

Another interesting observation can be made by considering the best minimum values of 10 replicationsfor the composite and individual objectives for various percentages of aobj:

Z ¼ 12.49 Z1 ¼ 181 Z2 ¼ 5 Z3 ¼ 300 for 100% of aobj;

Z ¼ 12.54 Z1 ¼ 179 Z2 ¼ 5 Z3 ¼ 297 for 80% of aobj;

Z ¼ 18.01 Z1 ¼ 179 Z2 ¼ 6 Z3 ¼ 297 for 60% of aobj.

Note that the best Z-value increases as the percentage of aobj decreases from 100% to 60%. However, whilethe best value of Z2 increases accordingly, the best values of Z1 and Z3 both decrease. This implies thatusing a composite function may introduce bias in choosing the right solution, which emphasizes the impor-tance of providing the decision maker with several solutions.

6. Conclusion

We have proposed an m-objective tabu search heuristic for job sequencing on a single machine with se-quence-dependent setup times. The heuristic has three unique features: it uses m-parallel tabu lists, it accountsfor the weight of each objective, and it creates a bounded solution space. The bounded solution space includessolutions within certain percentages of the best solution values with respect to each objective, some of whichmay be on the Pareto optimal front. Furthermore, the heuristic methodology is general enough that it can beapplied to similar multi-optimization problems with a mix of maximizing and minimizing objectives.

Performance of a three-objective heuristic was investigated through numerical experiments, and its sug-gested solutions were compared with the optimal solution obtained from solving a mixed linear integer pro-gram. The results indicate that the proposed heuristic obtains optimal or close-to-optimal solutions inpolynomial time as the problem size increases. On the other hand, under the same problem size, the com-putational time increases as the setup ranges decrease. This is due to the fact that more solutions are oftenqualified to be included in the bounded solution space in a problem with tighter setup ranges than a prob-lem with broader setup ranges.

Acknowledgement

We wish to thank the editor and two anonymous referees for their insightful comments which havehelped us improve the presentation of this paper. This research was supported by the NSF grant EPS-0091900.


Appendix

Procedure Reset Tabu Array
Start ProcedureFor obj = 1, . . ., m {
For [i] = 1, . . ., n {

For [j] = [i] + 1, . . ., n do{tabu_timeobj[i][j] Max_Iteration}}}Endfor

EndforEndforStop procedure

Procedure Neighborhood
Start procedure
If (Iteration – Improve_timeobj < n) then{
For [i] = 1 to n{
For [j] = [i] + 1 to n do {Swap the job in position [i] of Pobj with the one is position [j] ([i] M [j])}}}Endfor

EndforElse {

For [i] = 1 to n {

For [j] = [i] + 1 to [i] + 1 + Move_distance do {Swap the job in position [i] of Pobj with the one is position [j] ([i] M [j])}}}Endfor

EndforEndif

Stop procedure

Procedure Move Status [i] M [j]
Start procedure
If (Tabu_timeobj[i][j] > Iteration�Tabu_size) then {Return TABU}Endif

Stop procedure

Procedure Update Tabu Array [i] M [j]
Start procedureTabu_timeobj[i][j] IterationStop procedure
Procedure Update Bounded Solution Space
Start procedure
For (each solution in the Bounded Solution List) do{
If (Zobj 6 Rangeobj, for obj = 1, . . ., m) then {Re-deposit the solution into the Bounded Solution Space}}


EndifEndfor

Stop procedure

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A multi-objective tabu search for a single-machine scheduling problem with sequence-dependent setup times