A discrete differential evolution algorithm for the permutation flowshop scheduling problem

Abstract

Very recently, Pan et al. [Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation, GECCO07, pp. 126–33] presented a new and novel discrete differential evolution algorithm for the permutation flowshop scheduling problem with the makespan criterion. On the other hand, the iterated greedy algorithm is proposed by [Ruiz, R., & Stützle, T. (2007). A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem. European Journal of Operational Research, 177(3), 2033–49] for the permutation flowshop scheduling problem with the makespan criterion. However, both algorithms are not applied to the permutation flowshop scheduling problem with the total flowtime criterion. Based on their excellent performance with the makespan criterion, we extend both algorithms in this paper to the total flowtime objective. Furthermore, we propose a new and novel referenced local search procedure hybridized with both algorithms to further improve the solution quality. The referenced local search exploits the space based on reference positions taken from a reference solution in the hope of finding better positions for jobs when performing insertion operation. Computational results show that both algorithms with the referenced local search are either better or highly competitive to all the existing approaches in the literature for both objectives of makespan and total flowtime. Especially for the total flowtime criterion, their performance is superior to the particle swarm optimization algorithms proposed by [Tasgetiren, M. F., Liang, Y. -C., Sevkli, M., Gencyilmaz, G. (2007). Particle swarm optimization algorithm for makespan and total flowtime minimization in permutation flowshop sequencing problem. European Journal of Operational Research, 177(3), 1930–47] and [Jarboui, B., Ibrahim, S., Siarry, P., Rebai, A. (2007). A combinatorial particle swarm optimisation for solving permutation flowshop problems. Computers & Industrial Engineering, doi:10.1016/j.cie.2007.09.006]. Ultimately, for Taillard’s benchmark suite, four best known solutions for the makespan criterion as well as 40 out of the 90 best known solutions for the total flowtime criterion are further improved by either one of the algorithms presented in this paper.

Introduction

The permutation flowshop scheduling problem (PFSP) is one of the most comprehensively studied scheduling problems. The PFSP determines the order of n jobs over m machines to optimize certain performance measures when all jobs are to be processed on every machine in the same order. By far, two of the most common measures are the minimization of makespan and total flowtime, which do not only increase the utilization of machines but also lead to a high throughput. In the PFSP, solutions are represented by the permutation of n jobs, i.e., π = {π₁, π₂, …, π_n}. Each job is composed of m operations, and every operation is performed by a different machine. Jobs, once initiated, cannot be interrupted (preempted) by another job on each machine and the release times of all jobs are zero. Thus, given the processing time p_jk for the job j on the machine k, the PFSP is to find the best permutation of jobs ${\mathrm{\pi }}^{\ast }=\{{\mathrm{\pi }}_1^{\ast }\text{,}{\mathrm{\pi }}_2^{\ast }\text{,}\dots \text{,}{\mathrm{\pi }}_n^{\ast }\}$ to be processed on each machine subject to the makespan or total flowtime criterion. Let C(π_j, m) denote the completion time of the job π_j on the machine m. Then given the job permutation π, the completion time for the n-job, m-machine problem is calculated as follows:

$$C({\mathrm{\pi }}_1\text{,}1)=p_{{\mathrm{\pi }}_1\text{,}1}$$

$$C({\mathrm{\pi }}_j\text{,}1)=C({\mathrm{\pi }}_{j-1}\text{,}1)+p_{{\mathrm{\pi }}_j\text{,}1}j=2\text{,}\dots \text{,}n$$

$$C({\mathrm{\pi }}_1\text{,}k)=C({\mathrm{\pi }}_1\text{,}k-1)+p_{{\mathrm{\pi }}_1\text{,}k}k=2\text{,}\dots \text{,}m$$

$$C({\mathrm{\pi }}_j\text{,}k)=\mathit{max}\{C({\mathrm{\pi }}_{j-1}\text{,}k)\text{,}C({\mathrm{\pi }}_j\text{,}k-1)+p_{{\mathrm{\pi }}_j\text{,}k}\}j=2\text{,}\dots \text{,}n\text{;}k=2\text{,}\dots \text{,}m\text{.}$$

So, the makespan of a permutation π can be formally defined as the completion time of the last job π_n on the last machine m, i.e., C_max(π) = C(π_n, m). Therefore, the PFSP with thema kespan criterion is to find the optimal permutation π^∗ in the set of all permutations Π such that C_max(π ^∗) ⩽ C(π_n, m) for each permutation π belonging to Π.

As for the total flowtime criterion, let F(π_j) represent the flowtime of the job π_j. Clearly F(π_j) is equivalent to the completion time C(π_j, m) of the job π_j on the last machine m since the release times of all jobs are zero. The total flowtime TFT(π) of a permutation π can be computed by summing flowtimes or completion times of all jobs. Then the total flowtime of a permutation π is defined as $\mathit{TFT}(\mathrm{\pi })={\sum }_{j=1}^{n}F({\mathrm{\pi }}_j)={\sum }_{j=1}^{n}C({\mathrm{\pi }}_j\text{,}m)$. Therefore, the PFSP with the total flowtime criterion is to find the optimal permutation π^∗ in the set of all permutations Π such that TFT(π^∗) ⩽ TFT(π) for each permutation π belonging to Π.

The computational complexity of the PFSP with makespan and total flowtime minimization objectives have been proved to be NP-complete by Rinnooy Kan, 1976, Garey et al., 1976, respectively. Although some of exact methods have been reported in the literature (Bansal, 1977, Ignall and Schrage, 1965, Stafford, 1988), they are limited to solve small and/or moderate size problems due to the fact of explosively growing computational expense with the increase of the problem size. Therefore, efforts have been dedicated to finding high-quality solutions in a reasonable computational time by heuristic optimization techniques instead of finding an optimal solution. Past studies have focused on developing heuristics for the makespan minimization problem such as Palmer, 1965, Campbell et al., 1970, Dannenbring, 1977, Nawaz et al., 1983, Taillard, 1990, Framinan et al., 2002. On the topic of flowtime minimization, heuristics are presented by Gupta, 1972, Gelders and Sambandam, 1978, Ho and Chang, 1991, Rajendran and Chaudhuri, 1991, Rajendran, 1993, Ho, 1995, Rajendran and Ziegler, 1997, Wang et al., 1997, Woo and Yim, 1998, Liu and Reeves, 2001, Allahverdi and Aldowaisan, 2002, Framinan and Leisten, 2003, Framinan et al., 2005. To attain a better solution quality, modern metaheuristics have been recently presented for the PFSP with makespan/total flowtime minimization such as ant colony optimization (ACO) in (Stützle, 1998b, Rajendran and Ziegler, 2004), differential evolution (DE) in (Andreas and Omirou, 2006, Onwubolu and Davendra, 2006, Pan et al., 2007, Tasgetiren et al., 2004b, Tasgetiren et al., 2007b, Tasgetiren et al., 2007c), genetic algorithm (GA) in (Chen et al., 1995, Murata et al., 1996, Reeves, 1995, Reeves and Yamada, 1998, Ruiz et al., 2006), iterated greedy algorithm (IG) in (Ruiz & Stützle, 2007), iterated local search (ILS) in (Stützle, 1998a), particle swarm optimization (PSO) in (Lian et al., 2006, Liao et al., 2007; Pan et al., 2006a, Pan et al., 2006b; Pan et al., 2008, Rameshkumar et al., 2005, Tasgetiren et al., 2004a, Tasgetiren et al., 2007a), simulated annealing (SA) in (Ogbu and Smith, 1990, Osman and Potts, 1989), and tabu search (TS) in (Grabowski and Wodecki, 2004, Nowicki and Smutnicki, 1996, Reeves, 1993, Watson et al., 2002, Widmer and Hertz, 1989). Recent review of flowshop heuristics and metaheuristics can be found in Ruiz and Maroto (2005).

The remaining paper is organized as follows. Section 2 introduces the discrete differential evolution algorithm whereas Section 3 presents the iterated greedy algorithm. The details of the referenced local search procedure proposed for the permutation flowshop problem on hand is provided in Section 4. Section 5 discusses the computational results over benchmark problems in both objectives – makespan and total flowtime. Finally, Section 6 summarizes the concluding remarks.