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Applied Mathematics and Computation 183 (2006) 1292–1300

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A bicriteria flowshop scheduling problem with setup times

Tamer Eren a,*, Ertan Guner b

a Department of Industrial Engineering, Faculty of Engineering, Kırıkkale University, 71450 Kırıkkale, Turkeyb Department of Industrial Engineering, Faculty of Engineering and Architecture, Gazi University, 06570 Maltepe, Ankara, Turkey

Abstract

Most of research in production scheduling is concerned with the minimization of a single criterion. However, schedulingproblems often involve more than one aspect and therefore require bicriteria analysis. In this study, bicriteria two-machineflowshop scheduling problem with setup times is considered. The objective function of the problem is minimization of theweighted sum of total completion time and total tardiness. An integer programming model is developed for the problemwhich belongs to NP-hard class. Only small size problems with up to 20 jobs can be solved by the proposed integer pro-gramming model. Heuristic methods are also used to solve large size problems. These heuristics are four tabu search basedheuristics and random search method. According to computational results the tabu search based methods are effective infinding problem solutions with up to 1000 jobs.� 2006 Elsevier Inc. All rights reserved.

Keywords: Flowshop scheduling; Bicriteria; Setup times; Integer programming; Heuristic methods

1. Introduction

Setup times are defined to be the work to prepare the resource (machine), process, or bench for tasks (prod-ucts). In manufacturing environments this includes obtaining tools, positioning work in process material,returning and adjusting tools, cleaning-up, setting the required jigs and fixtures, and inspecting material.The majority of scheduling research on flowshops considers setup times negligible or as part of the processingtimes. While this might be justified for some applications, many other situations call for explicit separate setupconsideration; see [1–7]. An important implication of separate setup times is that the setup on a subsequentmachine may be performed while the job is still in process on the immediately preceding machine. A recentsurvey of scheduling research involving setup times is given by Allahverdi et al. [8] and Cheng et al. [9].

Another area of research in the scheduling literature involves the bicriteria problem. The majority ofresearch on scheduling problems addresses only a single criterion while the majority of real-life problemsrequire the decision maker to consider more than a single criterion before arriving at a decision. The sched-uling literature reveals that the research on bicriteria is mainly focused on the single-machine problem as a

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* Corresponding author.E-mail addresses: [email protected] (T. Eren), [email protected] (E. Guner).

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T. Eren, E. Guner / Applied Mathematics and Computation 183 (2006) 1292–1300 1293

result of the difficulty of the multiple machines problem. This paper addresses a two-machine flowshop sched-uling problem with a bicriteria objective function [10,11].

The bicriteria scheduling problems are generally divided into three classes. In the first class, one of the bicri-teria is considered as the objective to be optimized while the other is considered as a constraint. In the secondclass of problems, both criteria are considered equally important and the problem involves finding efficientschedules. In the third class of problems both criteria are weighted differently and an objective function asthe sum of weighted functions is defined [10,11]. The problem considered in this paper belongs to this class.

To increase system performance of a two-machine flowshop, the lowering of work in process as much aspossible is important. The scheduling criterion of total flow time minimization can effectively reduce workin process. Due-date conformance is one of the performance measures most frequently encountered in prac-tical scheduling problems. The total tardiness criterion, in particular, has been a standard way of quantifyingthis conformance. For instance, if several components are to be assembled into a product, then there is nobenefit gained from finishing some components early and the tardiness penalty is imposed proportionallyon the delay over all jobs [10,11].

In this paper, we consider a bicriteria scheduling problem with separate setup times on two machine flow-shop. The objective function of the problem is minimization of the weighted sum of total completion time andtotal tardiness. The considered problem is denoted as F 2=sji=a

PC þ b

PT . This bicriteria problem is an NP-

hard problem since the simpler single criterion variations of the problem that are F 2=sji=P

C and F 2=sji=P

Tare already NP-hard problems [12,13].

The rest of the study is organized as follows. In Section 2 and Section 3, the problem and the proposed amathematical programming model are described. A random search method and tabu search based methodsthat are used to solve large size problems are presented in the Section 4. The experimental results are givenin the Section 5. Finally, Section 6 provides conclusions and evaluations of the study.

2. Problem description

There are n jobs ready to be processed at time zero. Each job has two operations. The first operation isperformed by the first machine followed by the second operation that is performed by the second machine.Let pji, sji and dj denote the processing time of ith machine, the setup time of ith machine, and the due dateof the job j, respectively. A job once started on any machine must be completed on it without interruption (i.e.,no preemption is allowed). The machine may not process more than one operation at a time. Let Cj and Tj

denote the completion time and tardiness of the job j respectively.Pn

j¼1Cj andPn

j¼1T j ¼Pnj¼1 maxfCj � dj; 0g total completion time and total tardiness, respectively. The considered problem is

denoted as F 2=sji=aPn

j¼1Cj þ bPn

j¼1T j.

3. The proposed integer programming model

The proposed model has n2 + 12n variables and 13n constraints, where n is the number of jobs. Assump-tions, parameters and variables used in the model are given as follows:

3.1. Assumptions

• All jobs are available for processing at time zero.• Each job must be completed if it is started.• Each job has two operations.• To make a job on second machine, it must be completed on the first machine.• Processing times are independent of the schedule.• Jobs can wait the next machine to become idle.• Machines may be idle.• Setup times are known and they are not included to processing times.• Machines never breakdown and are available throughout the scheduling period.• No machine may process more than one operation at the same time.

1294 T. Eren, E. Guner / Applied Mathematics and Computation 183 (2006) 1292–1300

3.2. Parameters

j the number of jobs, j = 1,2, . . . ,n.i the number of machines, i = 1,2.pji the processing time of job j on the ith machine, i = 1,2, j = 1,2, . . . ,n.sji the setup time of job j on the ith machine, i = 1,2, j = 1,2, . . . ,n.dj the due date of job j, j = 1,2, . . . ,n.a the weight for the total completion time, 0 < a < 1.b the weight for the total tardiness, a + b = 1.

3.3. Decision variables

Zjk: if job j is scheduled at the kth rank to be processed, Zjk = 1, otherwise Zjk = 0, j = 1,2, . . . ,n,k = 1,2, . . . ,n.

Xk: the idle time on the second machine between the starting of the kth ranked job and the completion ofthe (k � 1)th ranked job, k = 1,2, . . . ,n.

Yk: the time between its completion at the first machine and its begin processing at the second machine forthe kth ranked job, k = 1,2, . . . ,n.

Zk: the idle time on the second machine between the completion time of the k � 1 ranked job on the secondmachine and the starting time of the kth ranked job on the first time,

Zk ¼ maxfSk � Ck�1;2; 0g; k ¼ 1; 2; . . . ; n:

Sk: the starting time for the kth ranked job at the first machine, k = 1,2, . . . ,n.Tk: the tardiness of the kth ranked job

T k P Ck � d�k ; k ¼ 1; 2; . . . ; n: ð1Þ

3.4. Auxiliary variables

p[ki]: the processing time of the kth ranked job at the ith machine

p½ki� ¼Xk

j¼1

Zjkpji; i ¼ 1; 2; k ¼ 1; 2; . . . ; n; ð2Þ

s[ki]: the setup time of the kth ranked job at the ith machine

s½ki� ¼Xk

j¼1

Zjksji; i ¼ 1; 2; k ¼ 1; 2; . . . ; n; ð3Þ

Ck: the completion time of the kth ranked job at the second machine

Ck ¼Xk

j¼1

X j þXk

j¼1

p½j2�; k ¼ 1; 2; . . . ; n: ð4Þ

d�k : the due date of the kth ranked job

d�k ¼Xn

j¼1

Zjkdj; k ¼ 1; 2; . . . ; n: ð5Þ

3.5. Proposed integer programming model

Objective function: Min Z ¼ aP

C þ bP

T


Constraints:

Xn

j¼1

Zjk ¼ 1; k ¼ 1; 2; . . . ; n; ð6Þ

Xn

k¼1

Zjk ¼ 1; j ¼ 1; 2; . . . ; n; ð7Þ

Sk P Sk�1 þ s½k�1;1� þ p½k�1;1�; k ¼ 2; 3; . . . ; n; ð8ÞC12 ¼ Z1 þ s½1�2 þ X 1 þ p½12�;

Ck2 ¼ Ck�1;2 þ Zk þ s½k;2� þ X k þ p½k2�; k ¼ 2; 3; . . . ; n; ð9ÞZ1 ¼ S1 þ s½1;1� þ p½1;1� þ Y 1 � s½1;2� � X 1;

Zk ¼ Sk þ s½k;1� þ p½k1� þ Y k � Ck�1;2 � s½k;2� � X k k ¼ 2; 3; . . . ; n; ð10ÞX 1 ¼ S1 þ s½1;1� þ p½11� þ Y 1 � Z1 � s½1;2�;

X k ¼ Sk þ s½k;1� þ p½k1� þ Y k � Ck�1;2 � Zk � s½k;2�; k ¼ 2; 3; . . . ; n: ð11Þ

In (1)–(5) equations all variables are positive and integer.Constraint (6) specifies that only one job be scheduled at the kth job priority. Constraint (7) defines that

each job be scheduled only once. Constraint (8) represents that the begin processing time of the kth rankedjob be greater than or equal to the previous jobs completion time at the first machine. Constraints (9) canbe explained the idle time on the second machine to process the kth ranked job (Xk) equals the starting timefor the kth ranked job on the first machine (Sk) plus its processing time on the first machine (pk1) plus the timebetween its completion on the first machine and begin processing time on the second machine (Yk) minus thecompletion time for the (k � 1)th ranked job at the second machine (Ck�1). Constraints (10) can be explainedthe idle time on the second machine before its setup operation of kth ranked job, Constraint (11) can beexplained the same way as Constraints (10). All variables should be greater than or equal to zero and Zjk

is a binary integer.

4. Heuristic methods

Most of the scheduling problems are NP-hard problems. That’s why only the small size problems of thisclass can be solved optimally by enumeration methods such as the branch-and-bound methods, the dynamicprogramming and the integer programming models. However, the real applications in industry require dealingwith the large size problems. Meta-heuristic approaches such as genetic algorithms, simulated annealing, tabusearch, etc. are perhaps the most powerful optimization methods and are suitable for scheduling problems. Inthis study, we used tabu search based heuristic methods and a random search method as the solutionprocedures.

The considered bicriteria flowshop scheduling with separate setup times problem can be solved optimallyfor small size problems with up to 20 jobs by the proposed integer programming model. Also, five heuristicmethods are used to solve large size problems. These heuristics are four tabu search based methods and a ran-dom search method. These methods are explained in the following sections.

4.1. Tabu search heuristic

Tabu search has been used widely in combinatorial optimization. The basic idea is to slightly alter a known(current) solution in a certain manner (called neighborhood structure) and take the best alteration as the newcurrent solution. Such altered solutions are called neighbors of the current solution. An operation that yields aneighbor is called a move. To avoid being trapped at a local optima, the best neighbor that is worse than thecurrent solution is allowed to become the new current solution. To avoid cycling, certain moves are marked astabu. A tabu move may be allowed if an aspiration criterion is satisfied. This procedure continues until a cri-terion is met [14].

Table 1Parameters of tabu search

Parameters Tabu I Tabu II Tabu III Tabu IV

Initial solutions Random Johnson [15] EDD NEH [16]Tabu list length 2

ffiffiffinp

2ffiffiffinp

2ffiffiffinp

2ffiffiffinp

Neighborhood search strategy: API API API APIStopping criterion n Iteration no

improvementn Iteration noimprovement

n Iteration noimprovement

n Iteration noimprovement


In this study, an experimental design was made for defining suitable tabu search parameters and 405 prob-lems were solved. According to experimental results the chosen tabu search parameters in the problem areshown in Table 1. In addition, according to the used initial solutions, the tabu search methods separate fourgroups. For instance, the NEH algorithm is modified our objective function (MNEH) and then the solutionvalue of MNEH is taken as an initial solution at Tabu IV.

4.2. Random search

Random search is a method that selects a specific number of solution point (a sample size) randomly fromthe solution space. Random search evaluates the selected points (sequences) about their objective function val-

Table 3The average CPU times of the integer programming model (n 6 20)

n Weights (a,b) Setup times (sji)

�U[0,9] �U[0,24] �U[0,49] �U[0,99]

10 (0.25,0.75) 5.09 4.00 3.20 6.7012 32.76 56.50 39.12 52.2414 122.74 229.92 128.97 128.2716 338.87 370.08 302.16 293.5418 959.95 801.16 683.54 537.9220 3429.02 2799.86 2252.47 1834.85

10 (0.50,0.50) 6.73 4.56 4.53 10.5512 86.40 56.71 53.87 61.6414 114.87 293.92 134.18 118.6116 255.88 359.71 229.04 248.0818 1188.07 1027.87 849.60 431.9720 3281.21 3089.58 2974.03 2210.35

10 (0.75,0.25) 3.66 3.35 3.73 9.0712 35.82 74.17 40.22 61.7514 118.28 222.79 114.18 94.0616 442.14 336.49 235.35 198.6318 957.51 940.27 703.15 548.1420 2635.10 2354.78 2434.45 2446.56

Table 2The experimental set for the small size problems

Parameters Alternatives Values

Number of job (n) 6 10, 12, 14, 16, 18, 20Processing time (pji) 1 �U[1,100]Setup time (sji) 4 �U[0,9] �U[0,24] �U[0,49] �U[0,99]Due date (dj) 1 dj ¼ �sjk þ

Pnj¼1pj þ uðn� 1Þ�pj

Weights (a,b) 3 (0.25,0.75); (0.50,0.50); (0.75,0.25)Number of solved problems 10

Total problem 6 · 1 · 4 · 1 · 3 · 10 = 720


ues and identifies the best sequence in the sample. The best sequence is stored at the memory and then thisprocess is repeated. When the new best sequence is better than the previous one, the previous sequence isupdated with the new sequence otherwise it does not change. The search is stopped at a prespecified numberof iterations. There are two parameters in random search method: The first is the specification of the samplesize and the other is the stopping rule. Stopping can be realized either at a certain number of iterations or whenthere is no improvement a given number of consecutive iterations [17,18].

In experimental study we took (n � 1) as the sample size and no improvement for n repetitions as the stop-ping rule for the problem.

0.00

0.05

0.10

0.15

0.20

10 12 14 16 18 20

number of job (n)

number of job (n)

number of job (n)

CPU

tim

e (s

n)C

PU ti

me

(sn)

CPU

tim

e (s

n)

random

tabu-I

tabu-II

Tabu-III

Tabu-IV

0.00

0.05

0.10

0.15

0.20

10 12 14 16 18 20

random

tabu-I

tabu-II

Tabu-III

Tabu-IV

0.00

0.05

0.10

0.15

0.20

10 12 14 16 18 20

random

tabu-I

tabu-II

Tabu-III

Tabu-IV

b

c

Fig. 1. Heuristic performance for small number of jobs: (a) a = 0.25, b = 0.75; (b) a = 0.50, b = 0.50; (c) a = 0.75, b = 0.25.


5. Experimental results

The integer programming model was used to find the optimal solutions of the considered problem usingHyper LINDO/PC 6.01 software package. Tabu search based heuristic methods used in this paper were codedon C++ Builder 5. The experimental design is similar to Rajendran ve Ziegler [19]’s and processing times onmachines were generated from a uniform distribution in the ranges [1, 100]. The setup times are randomly gen-erated from another uniform distribution on the integers between [0, 9], [0,24], [0,49] and [0, 99]. Due dates (dj)are defined as follows:

0.00

0.10

0.20

0.30

0.40

100 200 300 400 500 600 700 800 900 1000

random

tabu-I

tabu-II

Tabu-III

Tabu-IV

0.00

0.10

0.20

0.30

0.40

100 200 300 400 500 600 700 800 900 1000

random

tabu-I

tabu-II

Tabu-III

Tabu-IV

0.00

0.10

0.20

0.30

0.40

100 200 300 400 500 600 700 800 900 1000

random

tabu-I

tabu-II

Tabu-III

Tabu-IV

b

c

number of job (n)

number of job (n)

number of job (n)

CPU

tim

e (s

n)C

PU ti

me

(sn)

CPU

tim

e (s

n)

Fig. 2. Heuristic performance for a large number of jobs: (a) a = 0.25, b = 0.75; (b) a = 0.50, b = 0.50; (c) a = 0.75, b = 0.25.

TableExperi

Param

NumbProcesSetupDue dWeighNumb

Total p


dj ¼ �sjk þXn

j¼1

pj þ uðn� 1Þ�pj;

where �sjk is mean sequence-dependent setup time and �pj is mean processing time, u is a random number be-tween [0, 1]. The experimental set is given in Table 2. As seen from Table 2, totally 720 problems are solved.

The average CPU times of the integer programming model (in terms of seconds) in relation to the numberof jobs are given in Table 3. three different weight factors are used for the problems.

The optimal solutions of the considered problem can be found up to 20 job sizes. To solve large size prob-lems in a short time and to find the optimal or near optimal solutions, heuristics should be used. For the prob-lem considered, five heuristic methods were used. Heuristic error is calculated as follows.

Error ¼ Heuristic solution�Optimal solution

Optimal solution:

The results of the heuristic methods were compared with the results of the optimal solutions obtained by theinteger model, and the errors of these heuristics are given in Fig. 1. As seen from this Fig. 1, the tabu search-IVgives fairly good results in terms of error and it is quite effective in this bicriteria problem.

The results of the large size problems (100 6 n 6 1000) were also computed using heuristic methods. Sincethe optimal results of these problems have not been known, the results of the heuristics were compared withthe best result in order to define their performances. In this comparison, the error, formulated below, was usedas a performance measure:

Error ¼ Heuristic solution� Best heuristic solution

Best heuristic solution

Fig. 2 presents, the solution qualities of heuristics according to the weight values. The tabu search-IV gives thebest results of the problem for all weight values.

0

10000

20000

30000

40000

100

200 300

400

500

600

700 800 900

1000

number of job (n)

CP

Uti

me

(sn

) random

Tabu-I

Tabu-II

Tabu-III

Tabu-IV

Fig. 3. The average CPU times of heuristics (100 6 n 6 1000).

4mental set for large problems

eters Alternatives Values

er of job 10 100,200, . . . , 1000sing time (pji) 1 �U[1,100]time (sji) 4 �U[0,9] �U[0,24] �U[0,49] �U[0,99]ate (dj) 1 dj ¼ �sjk þ

Pnj¼1pj þ uðn� 1Þ�pj

ts (a,b,c) 3 (0.25,0.75); (0.50,0.50); (0.75,0.25)er of solved problems 10

roblem 10 · 1 · 4 · 1 · 3 · 10 = 1200


The experimental set of the large size problems is given in Table 4. As seen in Table 4, totally 1200 problemswere solved.

The execution times of heuristics for small size problems were so small that they were neglected. However,the average execution times of the heuristics for large size problems can not be neglected as shown in Fig. 3. Itis clear from this Fig. 3 that the longest execution times belong to the random search, Tabu II follows it.

6. Conclusion

In this paper, we consider the bicriteria flowshop scheduling problem with the weighted sum of total com-pletion time and total tardiness the consideration of the separate setup times. The proposed integer program-ming model can solved the problems up to 20 jobs. The base of this model is structured on Su and Chou [20]’sinteger programming model. To solve the large sizes problems up to 1000 jobs, a random search and tabusearch based heuristics methods were used. The performances of heuristics about the solution error were eval-uated with the integer model results for small size problems and each other for large size problems. Totally 720small size problems and 1200 large size problems were solved. According to results, the tabu search IV heu-ristic for all weight values was the more effective than others.

In scheduling the multi-machine or multicriteria cases under the sequence-dependent setup times can bealso interesting issues for future studies.

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http://dx.doi.org/10.1016/j.amc.2005.11.112