Heuristics for two-machine flowshop scheduling with setup times and an availability constraint

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Abstract

This paper studies the two-machine flowshop scheduling problem with anticipatory setup times and an availability constraint imposed on only one of the machines where interrupted jobs can resume their operations. We present two heuristics and show that their worst-case error bounds are no larger than 23.

Introduction

Machine scheduling problems with availability constraints motivated by preventive maintenance have received increasing attention from researchers. The studies in the literature on this topic mainly deal with three situations, namely resumable, nonresumable, and semiresumable. If a job cannot be finished before the unavailable period of a machine and the job can continue after the machine becomes available again, it is called resumable. On the other hand, if the job has to restart rather than continue, the situation is called nonresumable. If the unfinished job will have to partially restart after the machine becomes available again, the situation is called semiresumable. The recent research results on this subject can be found in the review papers by Lee et al. [1], Sanlaville and Schmidt [2], and Schmidt [3].

The two-machine flowshop scheduling problem with availability constraints was first studied by Lee [4]. Under the resumable assumption, he proved that the problem is NP-hard when an availability constraint is imposed on only one machine and proposed a pseudo-polynomial dynamic programming algorithm to solve the problem optimally. He also developed two heuristics. The first heuristic is for solving the problem where the availability constraint is imposed on machine 1, which has a worst-case error bound of 12. The second heuristic is for solving the problem where the availability constraint is imposed on machine 2, which has a worst-case error bound of 13. Lee [5] further studied the semiresumable case and developed a pseudo-polynomial dynamic programming algorithm and heuristics. For the resumable case, Cheng and Wang [6] developed an improved heuristic when the availability constraint is imposed on the first machine, and the heuristic has a worst-case error bound of 13. Breit [7] presented an improved heuristic for the problem with an availability constraint only on the second machine and showed that the heuristic has a worst-case error bound of 14. Cheng and Wang [8] considered a special case of the problem where the availability constraint is imposed on each machine, and the two availability constraints are consecutive. They developed a heuristic and showed that it has a worst-case error bound of 23 for the nonresumable situation. In addition, the two-machine flowshop scheduling problem with availability constraints has also been studied under the no-wait processing environment by Cheng and Liu [9], [10]. For the general flowshop scheduling problem with availability constraints, Aggoune [11] proposed a heuristic based on a genetic algorithm and a tabu search.

In all the above-mentioned flowshop scheduling models, setup times are not considered; in other words, setup times are assumed to be included in processing times. However, in many industrial settings, it is necessary to treat setup times as separated from processing times (see, for example [12], [13]). In this paper we consider the two-machine flowshop scheduling problem with anticipatory setup times, where the availability constraint is imposed on only one machine. The setup times are anticipatory, i.e., the setup for the second operation of any job on machine 2 can start before the completion of its first operation on machine 1 whenever there is some idle time on machine 2. We assume that the processing order of jobs is the same on each machine. That is, we confine ourselves to finding solutions that are permutation schedules for the problem. We also assume that all the jobs and their setups are resumable. The objective is to minimize the makespan. It is evident from Lee [4] that our problem is NP-hard. In the next section, we introduce the notation and some preliminaries. In Sections 3 and 4, we study the cases where the availability constraint is imposed on machines 1 and 2, respectively. Some concluding remarks are given in the last section.

Section snippets

Notation and preliminaries

For the problem under consideration, we introduce the following notation to be used throughout this paper.

    S={J1,,Jn}:

    a set of n jobs;

    M1,M2:

    machine 1 and machine 2;

    Δl=tl-sl:

    the length of the unavailable interval on Ml, where Ml is unavailable from time sl to tl, 0sltl, l=1,2;

    si1,si2:

    setup times of Ji on M1 and M2, respectively, where si1>0,si2>0;

    ai,bi:

    processing times of Ji on M1 and M2, respectively, where ai>0,bi>0;

    π=[Jπ(1),,Jπ(n)]:

    a permutation schedule, where Jπ(i) is the ith job in π;

    π*:

The unavailable interval is on M1

In this section we develop a heuristic for the problem F2/setup, r-a(M1)/Cmax and evaluate its worst-case error bound. The basic ideas of our heuristic are to combine a few simple heuristic rules and then improve the schedules by re-arranging the order of some special jobs with large setup times or large processing times on M2 in different situations.

Heuristic H1:

  • (1)

    Find jobs Jp and Jq such that sp2+bpsq2+bqmax{si2+bi|JiS{Jp,Jq}}.

  • (2)

    Sequence the jobs by YHA. Let the corresponding schedule be π1

The unavailable interval is on M2

In this section we provide a heuristic for the problem F2/setup, r-a(M2)/Cmax and analyze its worst-case error bound.

Heuristic H2:

  • (1)

    Find two jobs Jp and Jq such that sp2+bpmaxsi2+bi|JiS{Jp}and sq1+aqmax{si1+ai|JiS{Jq}}.

  • (2)

    Sequence the jobs by YHA. Let the corresponding schedule be π1 and the corresponding makespan be Cmax(π1).

  • (3)

    Sequence the jobs in nonincreasing order of (si2+bi)/(si1+ai). Let the schedule be π2 and the corresponding makespan be Cmax(π2).

  • (4)

    Sequence job Jq in the last position, and

Conclusions

In this paper we studied the two-machine flowshop scheduling problem with anticipatory setup times and a resumable availability constraint imposed on only one of the machines. Since the problem is NP-hard, we developed two polynomial-time heuristics and analyzed their worst-case error bounds.

Acknowledgements

This research was supported in part by The Hong Kong Polytechnic University under a grant from the Area of Strategic Development in China Business Services.

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