Ant-colony algorithms for permutation flowshop scheduling to minimize makespan/total flowtime of jobs

Abstract

The problem of scheduling in permutation flowshops is considered with the objective of minimizing the makespan, followed by the consideration of minimization of total flowtime of jobs. Two ant-colony optimization algorithms are proposed and analyzed for solving the permutation flowshop scheduling problem. The first algorithm extends the ideas of the ant-colony algorithm by Stuetzle [Proceedings of the 6th European Congress on Intelligent Techniques and Soft Computing (EUFIT ’98), vol. 3, Verlag Mainz, Aachen, Germany, 1998, p. 1560], called max–min ant system (MMAS), by incorporating the summation rule suggested by Merkle and Middendorf [Proceedings of the EvoWorkshops 2000, Lecture Notes in Computer Science No. 1803, Springer-Verlag, Berlin, 2000, p. 287] and a newly proposed local search technique. The second ant-colony algorithm is newly developed. The proposed ant-colony algorithms have been applied to 90 benchmark problems taken from Taillard [European Journal of Operational Research 64 (1993) 278]. First, a comparison of the solutions yielded by the MMAS and the two ant-colony algorithms developed in this paper, with the heuristic solutions given by Taillard [European Journal of Operational Research 64 (1993) 278] is undertaken with respect to the minimization of makespan. The comparison shows that the two proposed ant-colony algorithms perform better, on an average, than the MMAS. Subsequently, by considering the objective of minimizing the total flowtime of jobs, a comparison of solutions yielded by the proposed ant-colony algorithms with the best heuristic solutions known for the benchmark problems, as published in an extensive study by Liu and Reeves [European Journal of Operational Research 132 (2001) 439], is carried out. The comparison shows that the proposed ant-colony algorithms are clearly superior to the heuristics analyzed by Liu and Reeves. For 83 out of 90 problems considered, better solutions have been found by the two proposed ant-colony algorithms, as compared to the solutions reported by Liu and Reeves.

Introduction

Many exact and heuristic algorithms have been proposed over the years for solving the static permutation flowshop scheduling problem with the objectives of minimizing makespan and total flowtime of jobs, considered either separately or simultaneously (e.g. Johnson, 1954; Ignall and Schrage, 1965; Campbell et al., 1970; Gelders and Sambandam, 1978; Miyazaki et al., 1978; Miyazaki and Nishiyama, 1980; Nawaz et al., 1983; Rajendran, 1993, Rajendran, 1995; Ho, 1995; Wang et al., 1997; Woo and Yim, 1998; Liu and Reeves, 2001). In addition, metaheuristics such as genetic algorithms, simulated annealing and tabu search are used to solve flowshop scheduling problems (e.g. Widmer and Hertz, 1989; Ishibuchi et al., 1995; Nowicki and Smutnicki, 1996; Ben-Daya and Al-Fawzan, 1998).

In recent times, attempts are being made to solve combinatorial optimization problems by making use of ant-colony optimization (ACO) algorithms (or simply, ant-colony or ACO algorithms). The pioneering work has been done by Dorigo (1992), and an introduction to the ACO algorithms has been dealt with in Dorigo et al. (1996). Attempts have been made recently to solve scheduling problems using ACO algorithms (e.g. Stuetzle, 1998 dealing with permutation flowshop scheduling problem with the objective of minimizing the makespan, and Merkle and Middendorf, 2000 considering the single-machine scheduling problem with the objective of minimizing the sum of weighted tardiness of jobs).

In this paper, we investigate the problem of scheduling in permutation flowshops by using ACO algorithms, first with the consideration of minimizing the makespan, followed by the consideration of minimizing the sum of total flowtime of jobs. Two ant-colony algorithms are proposed in this paper. The first algorithm incorporates the concept of summation rule (presented by Merkle and Middendorf, 2000), a modification with respect to the selection of the job to be appended to the partial ant-sequence, and a new local search technique in the max–min ant system (MMAS) which is an ant-colony algorithm developed by Stuetzle (1998). The second proposed ant-colony algorithm is based on new ideas that are developed in the current work. By considering the objective of minimizing the makespan, we evaluate the ACO algorithms in comparison with the upper bound values of makespan presented by Taillard (1993) for the 90 benchmark permutation flowshop scheduling problems taken from Taillard (1993). Next, we consider the objective of minimizing the total flowtime of jobs and evaluate the solutions yielded by the two proposed ACO algorithms relative to the best heuristic solutions reported by Liu and Reeves (2001) for the same 90 benchmark permutation flowshop problems.