A hybrid genetic local search algorithm for the permutation flowshop scheduling problem

Abstract

Traditionally, the permutation flowshop scheduling problem (PFSP) was with the criterion of minimizing makespan. The permutation flowshop scheduling problem to minimize the total flowtime has attracted more attention from researchers in recent years. In this paper, a hybrid genetic local search algorithm is proposed to solve this problem with each of both criteria. The proposed algorithm hybridizes the genetic algorithm and a novel local search scheme that combines two local search methods: the Insertion Search (IS) and the Insertion Search with Cut-and-Repair (ISCR). It employs the genetic algorithm to do the global search and two local search methods to do the local search. Two local search methods play different roles in the search process. The Insertion Search is responsible for searching a small neighborhood while the Insertion Search with Cut-and-Repair is responsible for searching a large neighborhood. Furthermore, the orthogonal-array-based crossover operator is designed to enhance the GA’s capability of intensification. The experimental results show the advantage of combining the two local search methods. The performance of the proposed hybrid genetic algorithm is very competitive. For the PFSP with the total flowtime criterion, it improved 66 out of the 90 current best solutions reported in the literature in short-term search and it also improved all the 20 current best solutions reported in the literature in long-term search. For the PFSP with the makespan criterion, the proposed algorithm also outperforms the other three methods recently reported in the literature.

Introduction

The permutation flowshop scheduling problem (PFSP) is an important scheduling problem and has been extensively studied since it was proposed in 1954 by Johnson (1954). Most of the research on this problem was to minimize makespan. In recent years, the flowshop scheduling problem to minimize total flowtime begins to attract more attention from researchers. The PFSP with this criterion is considered to be more relevant and meaningful for today’s dynamic production environment (Liu and Reeves, 2001) because it tends to stabilize use of resources and minimize the work-in-process inventory. The PFSP with the total flowtime (makespan) criterion is formally defined in the following: n jobs {J₁, J₂, … , J_n} are to be processed on a series of m machines {M₁,M₂, … , M_m} sequentially. The processing time of job J_i on machine M_j is given as T_ij. At any time, each job can be processed on at most one machine and each machine can process at most one job. Also, once a job is processed on a machine, it cannot be terminated before completion. The sequence in which the jobs are to be processed is the same for each machine. And the objective is to find a permutation of jobs such that the total flowtime (makespan) is minimized. Let π = {π₁, π₂, … , π_n} be a permutation of jobs. The completion time of job $J_{{\pi }_i}$ on machine M_j is denoted by C(π_i, j) and is defined as follows:

$$C({\pi }_i\text{,}j)=\max \left\{C({\pi }_{i-1}\text{,}j)\text{,}C({\pi }_i\text{,}j-1)\right\}+T_{{\pi }_i\text{,}j}\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{;}j=1\text{,}2\text{,}\dots \text{,}m$$

where C(π₀, j) = 0 and C(π_i, 0) = 0 for relevant i and j.

The flowtime (or completion time) of job $J_{{\pi }_i}$ is defined to be C(π_i, m), that is, the completion time of job $J_{{\pi }_i}$ on the last machine M_m. Then the total flowtime (or total completion time) of permutation π is defined as $C_{\text{sum}}(\pi )={\sum }_{i=1}^{n}C({\pi }_i\text{,}m)$, that is, the sum of the completion times of all jobs. So, the PFSP with the total flowtime criterion is to find a permutation π^∗ in the set of all permutations Π such that

$$C_{\text{sum}}({\pi }^{\ast })⩽C_{\text{sum}}(\pi )\forall \pi \in \prod \text{.}$$

Makespan of a permutation πis defined as C_max(π) = C(π_n, m), that is, the completion time of the last job $J_{{\pi }_n}$ on the last machine M_m. And the PFSP with the makespan criterion is to find a permutation π^∗ in the set of all permutations Π such that

$$C_{\max }({\pi }^{\ast })=C_{\max }(\pi )\forall \pi \in \prod \text{.}$$

Either to minimize the total flowtime or to minimize makespan, the PFSP was proved NP-hard by Garey et al. (1976). Therefore, instead of trying to find the optimal solution, many efforts had been devoted to design heuristic or metaheuristic methods to find high-quality solutions with a reasonable computation time.

Heuristic methods for minimizing makespan had been proposed by Palmer, 1965, Campbell et al., 1970, Dannenbring, 1977, Nawaz et al., 1983. These methods are constructive heuristic methods. They in general are very fast but suffer from poor solution quality. The best method among them is NEH proposed by Nawaz et al. (1983). But the mean relative error of the makespan obtained by NEH to the lower bound of the optimal makespan is still above 5%. In recent years various metaheuristic methods had been applied to solve the PFSP with makespan minimization criterion. These include simulated annealing (e.g. Ogub and Simith, 1990, Ishubuchi et al., 1995), tabu search (e.g. Taillard, 1990, Nowicki and Smutnicki, 1996, Grabowski and Wodecki, 2004), genetic algorithm (e.g. Reeves, 1995) and hybrid metaheuristics (e.g. Reeves and Yamada, 1998, Wang and Zheng, 2003). Metaheuristic methods in general obtain much better solution quality than heuristic methods do. And the computation time of metaheuristic methods is much less than that of exact methods.

Heuristic methods for minimizing the total flowtime can be classified into three categories: constructive heuristics (e.g. Wang et al., 1997), improvement heuristics (e.g. Ho, 1995), and composite heuristics that combine different heuristics. Rajendran and Ziegler (1997) presented a constructive heuristic incorporated with an improvement phase which consists of one round of insertion search. Woo and Yim (1998) also proposed a constructive heuristic but they tried to find the best position to insert a selected job into the partial sequence. Liu and Reeves (2001) proposed a constructive heuristic and several composite heuristics that combined the proposed constructive heuristic with local search methods. They reported that composite heuristics had better performance. Allahverdi and Aldowaisan (2002) applied pair-wise exchange to existing heuristics to improve the performance. Framinan et al. (2005) gave a review of the heuristic methods for the total flowtime flowshop scheduling problem. In addition to the review, they also suggested two composite heuristics that outperformed other heuristics. From the trend of designing heuristics for this problem, it is noted that composite heuristics have better performance than constructive heuristics and improvement heuristics. Besides heuristics, recently, several metaheuristics were proposed for the total flowtime flowshop scheduling problem. In general, metaheuristics obtained solutions of better quality than heuristics did, but metaheuristics needed more computation time. Yamada and Reeves (1998) presented a genetic local search algorithm for the total flowtime flowshop scheduling problem. Their algorithm obtained good quality solutions but spent much computation time. Rajendran and Ziegler (2004) proposed two ant colony optimization methods called M-MMAS and PACO for both the total flowtime flowshop scheduling problem and the makespan flowshop scheduling problem. They compared the performance of their methods on the total flowtime flowshop problem with that of Liu and Reeves (2001) and reported that for 83 out of the 90 benchmark problems taken from Taillard (1990), better solutions had been found by their methods. The next year, they proposed still other two ant colony optimization algorithms (Rajendran and Ziegler, 2005) that slightly improved the performance of M-MMAS and PACO. Tasgetiren et al. (2007) suggested a particle swarm optimization method. With this method, 57 out of the 90 best known solutions reported by Liu and Reeves, 2001, Rajendran and Ziegler, 2004 were improved. In this paper, we proposed a hybrid genetic algorithm that hybridizes the genetic algorithm with two local search methods for both the total flowtime flowshop scheduling problem and the makespan flowshop scheduling problem. The experimental results reveal that our hybrid genetic algorithm outperforms M-MMAS, PACO (Rajendran and Ziegler, 2004) and PSO (Tasgetiren et al., 2007) on the total flowtime flowshop scheduling problem. Also, with less computation time compared to that used by Yamada and Reeves (1998), our method obtained better solutions than their method did. Since M-MMAS, PACO and PSO had also been applied to solve the makespan flowshop scheduling problem. The results of our algorithm on the makespan flowshop scheduling problem were also compared with their results, and this comparison reveals that our algorithm outperforms these algorithms in both the solution quality and the computation time.

The remaining parts of the paper are organized as follows. In Section 2, our hybrid genetic algorithm is presented. In Sections 3 Experimental results, 4 Conclusions and future works, experimental results and conclusions are given respectively.