A bicriteria flowshop scheduling problem with setup times

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  • problems often involve more than one aspect and therefore require bicriteria analysis. In this study, bicriteria two-machine

    ucts). In manufacturing environments this includes obtaining tools, positioning work in process material,

    uling literature reveals that the research on bicriteria is mainly focused on the single-machine problem as a

    * Corresponding author.E-mail addresses: teren@kku.edu.tr (T. Eren), erguner@gazi.edu.tr (E. Guner).

    Applied Mathematics and Computation 183 (2006) 12921300

    www.elsevier.com/locate/amc0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved.returning and adjusting tools, cleaning-up, setting the required jigs and xtures, and inspecting material.The majority of scheduling research on owshops considers setup times negligible or as part of the processingtimes. While this might be justied for some applications, many other situations call for explicit separate setupconsideration; see [17]. 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-owshop 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 eective innding 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 dened to be the work to prepare the resource (machine), process, or bench for tasks (prod-A bicriteria owshop scheduling problem with setup times

    Tamer Eren a,*, Ertan Guner b

    a Department of Industrial Engineering, Faculty of Engineering, Krkkale University, 71450 Krkkale, 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, schedulingdoi:10.1016/j.amc.2006.05.160

  • T. Eren, E. Guner / Applied Mathematics and Computation 183 (2006) 12921300 1293result of the diculty of the multiple machines problem. This paper addresses a two-machine owshop sched-uling problem with a bicriteria objective function [10,11].

    The bicriteria scheduling problems are generally divided into three classes. In the rst 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 nding ecientschedules. In the third class of problems both criteria are weighted dierently and an objective function asthe sum of weighted functions is dened [10,11]. The problem considered in this paper belongs to this class.

    To increase system performance of a two-machine owshop, the lowering of work in process as much aspossible is important. The scheduling criterion of total ow time minimization can eectively 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 nobenet gained from nishing 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 ow-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 bP T . 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 rst operation isperformed by the rst 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 Tjdenote the completion time and tardiness of the job j respectively.

    Pnj1Cj and

    Pnj1T j Pn

    j1 maxfCj dj; 0g total completion time and total tardiness, respectively. The considered problem isdenoted as F 2=sji=a

    Pnj1Cj b

    Pnj1T 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 rst 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.

  • j the number of jobs, j = 1,2, . . . ,n.i the number of machines, i = 1,2.

    k k k1;2

    T k P Ck dk ; k 1; 2; . . . ; n: 1

    1294 T. Eren, E. Guner / Applied Mathematics and Computation 183 (2006) 12921300dk : the due date of the kth ranked job

    dk Xn

    j1Zjkdj; k 1; 2; . . . ; n: 5

    3.5. Proposed integer programming model

    P Ppki j1

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

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

    ski Xk

    j1Zjksji; i 1; 2; k 1; 2; . . . ; n; 3

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

    Ck Xk

    j1X j

    Xk

    j1pj2; k 1; 2; . . . ; n: 4. Auxiliary variables

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

    Xk3.4Sk: the starting time for the kth ranked job at the rst machine, k = 1,2, . . . ,n.Tk: the tardiness of the kth ranked job

    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 rst 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 rst time,

    Z maxfS C ; 0g; k 1; 2; . . . ; n:3.2. ParametersObjective function: Min Z a C b T

  • withsearcthis sproce

    Thfor smmethdom

    4.1. T

    Ta(currecurreneighcurre

    T. Eren, E. Guner / Applied Mathematics and Computation 183 (2006) 12921300 1295tabu. A tabu move may be allowed if an aspiration criterion is satised. This procedure continues until a cri-

    terionthe large size problems. Meta-heuristic approaches such as genetic algorithms, simulated annealing, tabuh, etc. are perhaps the most powerful optimization methods and are suitable for scheduling problems. Intudy, we used tabu search based heuristic methods and a random search method as the solutiondures.e considered bicriteria owshop scheduling with separate setup times problem can be solved optimallyall size problems with up to 20 jobs by the proposed integer programming model. Also, ve heuristic

    ods are used to solve large size problems. These heuristics are four tabu search based methods and a ran-search method. These methods are explained in the following sections.

    abu search heuristic

    bu search has been used widely in combinatorial optimization. The basic idea is to slightly alter a knownnt) solution in a certain manner (called neighborhood structure) and take the best alteration as the newnt solution. Such altered solutions are called neighbors of the current solution. An operation that yields abor is called a move. To avoid being trapped at a local optima, the best neighbor that is worse than thent solution is allowed to become the new current solution. To avoid cycling, certain moves are marked asConstraints:

    Xn

    j1Zjk 1; k 1; 2; . . . ; n; 6

    Xn

    k1Zjk 1; j 1; 2; . . . ; n; 7

    Sk P Sk1 sk1;1 pk1;1; k 2; 3; . . . ; n; 8C12 Z1 s12 X 1 p12;Ck2 Ck1;2 Zk sk;2 X k pk2; k 2; 3; . . . ; n; 9Z1 S1 s1;1 p1;1 Y 1 s1;2 X 1;Zk Sk sk;1 pk1 Y k Ck1;2 sk;2 X k k 2; 3; . . . ; n; 10X 1 S1 s1;1 p11 Y 1 Z1 s1;2;X k Sk sk;1 pk1 Y k Ck1;2 Zk sk;2; k 2; 3; . . . ; n: 11

    In (1)(5) equations all variables are positive and integer.Constraint (6) species that only one job be scheduled at the kth job priority. Constraint (7) denes 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 rst 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 rst machine (Sk) plus its processing time on the rst machine (pk1) plus the timebetween its completion on the rst machine and begin processing time on the second machine (Yk) minus thecompletion time for the (k 1)th ranked job at the second machine (Ck1). 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 Zjkis a binary integer.

    4. Heuristic methods

    Most of the scheduling problems are NP-hard problems. Thats 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 dealingis met [14].

  • In this study, an experimental design was made for dening 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 modied 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 specic number of solution point (a sample size) randomly fromthe solution space. Random search evaluates the selected points (sequences) about their objective function val-

    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

    n

    p2n

    p2n

    p2n

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

    improvementn Iteration noimprovement

    n Iteration noimprovement

    n Iteration noimprovement

    Table 3

    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 sjk

    Pnj1pj un 1pj

    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

    1296 T. Eren, E. Guner / Applied Mathematics and Computation 183 (2006) 12921300The 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 61.64

    14 114.87 293.92 134.18 118.6116 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

  • ues and identies 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 prespecied numberof iterations. There are two parameters in random search method: The rst is the specication 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 20number of job (n)

    number of job (n)

    CPU

    tim

    e (sn

    )CP

    U ti

    me

    (sn)

    (sn)

    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

    randomtabu-Itabu-II...

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