A very fast tabu search algorithm for the permutation flow shop problem with makespan criterion

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Abstract

This paper deals with a classic flow-shop scheduling problem with makespan criterion. Some new properties of the problem associated with the blocks have been presented and discussed. These properties allow us to propose a new very fast local search procedure based on a tabu search approach. Computational experiments (up to 500 jobs and 20 machines) are given and compared with the results yielded by the best algorithms discussed in the literature. These results show that the algorithm proposed solves the flow-shop instances with high accuracy in a very short time. The presented properties and ideas can be applied in any local search procedures.

Scope and purpose The permutation flow shop scheduling problem with makespan criterion is relatively simply formulated, but it is unfortunately a hard combinatorial problem. For this reason many various algorithms have been proposed and tested to solve the problem in a short time. This paper deals with the new properties and techniques, which allow one to construct a very efficiency algorithm based on a tabu search approach. In the algorithm, we propose to use the lower bounds on the makespans instead of computing the makespans explicitly, for selecting the best solution. Besides a dynamic tabu list is used that assists additionally to avoid trapped at a local optimum. Finally, the appropriate perturbations are introduced that guide the search to the more promising areas of solution space, where ”good solutions” can be found. All these components make that proposed algorithm solves the large-size flow shop instances with high accuracy in very short time.

Introduction

The paper deals with the classic permutation flow-shop problem, which can be considered as a permanent indicator of the practical applicability of the scheduling theory. This problem is relatively simply formulated and successfully applied in industry, has a finite but large number of solutions, it is unfortunately NP-hard. For this reason many various algorithms have been proposed and tested to solve the problem in a short time.

Skipping the review of a large number of papers dealing with this problem, we only mention the most recent and representative ones [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. This paper deals with properties and techniques, which allow us to solve flow-shop instances up to 500 jobs and 20 machines with high accuracy in a very short time. In order to reduce the computational effort, we propose to employ some ideas presented in the paper [11] for the job-shop problem.

The paper is organized as follows. In Section 2, the notations and basic definitions are introduced. Section 3 presents the new properties of the problem, moves and neighbourhood structure, methods to evaluate the moves, search strategy, dynamic tabu list, perturbations, and algorithm based on tabu search approach. Computational results are shown in Section 4 and compared with those taken from the literature. Section 5 gives our conclusions and remarks.

Section snippets

Problem formulation and preliminaries

The permutation flow-shop problem can be formulated as follows.

Problem

Each of n jobs from the set J={1,2,…,n} has to be processed on m machines 1,2,…,m in that order. Job j, jJ, consists of a sequence of m operations Oj1,Oj2,…,Ojm; operation Ojk corresponds to the processing of job j on machine k during an uninterrupted processing time pjk. We want to find a schedule such that the processing order is the same on each machine and the maximum completion time is minimal.

Each schedule of jobs can be

Tabu search (TS) algorithm

We present a tabu search algorithm, which is capable of handling any sequencing problem with the described structure and is currently one of the most effective methods of the local search techniques designed to find a near-optimal solution of many combinatorial optimization problems (see Glover [15] and [16]). Briefly speaking, the basic idea of this method consists in starting from an initial basic permutation of n jobs and searching through its neighbourhood, a set of permutations related to

Algorithm TSGW

In the algorithm, the asterisk (*) refers to the best values found, the zero superscript (o) refers to initial value, and its lack denotes the current values. The algorithm starts from a given initial permutation πo (which can be found by any algorithm). The algorithm stops when Maxiter iterations have been performed.

  • INITIALISATION.

    Set ππo, C≔Cmaxo), π≔πo, T≔∅, iter≔0, retp≔0.

  • SEARCHING.

    Set iteriter+1, modify (if it is appropriate) LengthT of the tabu list according to the method described earlier, and for

Computational results

Algorithm TSGW was coded in C++, run on IBM RS6000/43P/200MHz and tested on the benchmark problems taken from the literature. The results obtained by our algorithm were then compared with results from the literature. So far the best approximation algorithms for a permutation flow-shop problem with Cmax criterion were presented in papers by Ishubuchi et al. [2], Grabowski and Pempera [1], Nowicki and Smutnicki [3], Ogbu and Smith [4], Osman and Potts [5], Reeves and Yamada [6], Taillard [7], [8]

Conclusions

In this paper we have presented and discussed some new properties of blocks in the flow-shop problem. These properties allow us to propose a new very fast algorithm based on the tabu search approach. In order to decrease the computational effort for the search in TS, we propose calculation of the lower bounds on the makespans instead of computing makespans explicitly for selecting the best solution. These lower bounds are used for evaluations of the moves for selecting the “best” one. Also, we

Acknowledgements

This research has been supported by KBN Grant 4 T11A 016 24. The authors are due to the referees for their valuable comments and suggestions.

Józef Grabowski is a Full Professor at Wroclaw University of Technology, Poland. He did his Ph.D.(hab) in the area of scheduling from the same institute. His research interests are in operations management and operations research. He has regularly published papers in this and other international journals.

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Józef Grabowski is a Full Professor at Wroclaw University of Technology, Poland. He did his Ph.D.(hab) in the area of scheduling from the same institute. His research interests are in operations management and operations research. He has regularly published papers in this and other international journals.

Mieczyslaw Wodecki received his Ph.D. from University of Wroclaw, Institute of Computer Science. His research interests are in scheduling and optimization. He has published several papers in academic and international journals.

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