Parallel Simulated Annealing with Genetic Enhancement for flowshop problem with C_sum

Abstract

In this paper, parallelisable Simulated Annealing with Genetic Enhancement (SAwGE) algorithm is presented and applied to Permutation Flowshop Scheduling Problem with total flowtime criterion. This problem is proved to be $\mathcal{NP}$-complete in a strong sense for more than one machine. SAwGE is based on a Clustering Algorithm for Simulated Annealing (SA), but introduces a new mechanism for dynamic SA parameters adjustment, based on genetic algorithms. Computational experiments, based on 120 benchmark datasets by Taillard, show that SAwGE outperforms other heuristics and metaheuristics presented recently in literature. Moreover SAwGE obtains 118 best solutions, including 81 newly discovered ones.

Introduction

Permutation Flowshop Scheduling Problem (PFSP) has been studied since 1954, when it was first defined by Johnson (1954). The goal of PFSP is to order n jobs to be processed on m machines. Processing time for each job, on each machine, is known and fixed. All jobs are available when processing starts, and they must be processed uninterrupted, on each machine in the same order. Sequence of processing jobs on each machine is assumed to be the same, therefore solution to PFSP is a permutation of jobs entering first machine, which optimises some criterion. The most common one is the makespan criterion (C_max), but this paper addresses the total flowtime criterion (C_sum), which became quite popular recently. Garey, Johnson, and Sethi (1976) proved that PFSP with C_max is $\mathcal{NP}$-complete in the strong sense when m ⩾ 3, and with C_sum when m ⩾ 2.

Attempts to solve PFSP with C_sum criterion with exact methods are mainly based on Branch and Bound algorithm, including those by Ignall and Schrage, 1965, Bansal, 1977, and more recently by Chung, Flynn, and Kirca (2002). Due to their computation time these algorithms can be only applied to relatively small problems.

Simple heuristics used in constructive methods, allows them to obtain quite good results in a short time. This group of algorithms include those by Rajendran and Ziegler, 1997, Wang et al., 1997 and by Woo and Yim (1998). Liu and Reeves (2001) provided a comparison of these heuristics and also proposed a new one, referred to as LR. More constructive methods has been proposed since then, including those by Allahverdi and Aldowaisan, 2002, Framinan et al., 2005, and most recently by Li, Wang, and Wu (2009).

Also metaheuristics has been successfully applied to solve PFSP with C_sum criterion, including Genetic Local Search (GLS) by Yamada and Reeves (1998), Ant Colony Optimisation (ACO) by Rajendran and Ziegler (2004), and Particle Swarm Optimisation: PSO by Tasgetiren, Liang, Sevkli, and Gencyilmaz (2007) and H-CPSO by Jarboui, Ibrahim, Siarry, and Rebai (2008). The most recent algorithms include Hybrid Genetic Local Search (HGLS) by Tseng and Lin (2008), Hybrid Genetic Algorithm (HGA) by Zhang, Li, and Wang (2009), Iterated Local Search (ILS) by Dong, Huang, and Chen (2009), and Estimation of Distribution Algorithm with Variable Neighbourhood Search (VNS and EDA–VNS) by Jarboui, Eddaly, and Siarry (2009).

Due to $\mathcal{NP}$-completeness of PFSP, thus huge demand for computational power, algorithms designed for parallel architectures have been developed for both makespan and total flowtime criteria. Vallada and Ruiz (2009) provide very detailed review of parallel algorithms for PFSP, and propose a new cooperative metaheuristic for makespan and total tardiness criteria. Parallel version of exact method for those two criteria, as well as for multi-objective optimisation, was proposed by Lemesre, Dhaenens, and Talbi (2007). Parallel algorithms for PFSP with total flowtime criterion include parallel Simulated Annealing by Wodecki and Bożejko (2002), as well as parallel Tabu Search, Genetic Algorithm and Scatter Search by Bożejko and Wodecki, 2002, Bożejko and Wodecki, 2004, Bożejko and Wodecki, 2008. Furthermore, Bożejko (2009) provide detailed analysis of theoretical and empirical speedups obtained from parallelisation of objective function evaluation, for both C_max and C_sum criteria.

Simulated Annealing (SA) was proposed by Kirkpatrick, Gelatt, and Vecchi (1983), and applied to the Travelling Salesman Problem (TSP). The main disadvantage is that quality of the results is very susceptible to SA configuration (start temperature, cooling scheme, etc.).

Ram, Sreenivas, and Subramaniam (1996) proposed Clustering Algorithm for Simulated Annealing — a parallel approach following the island model — and applied it to solve Job Shop Scheduling problem and TSP. To reduce influence of SA configuration on quality of results obtained by such parallel algorithm, it could be adjusted dynamically during runtime. In this paper, a Simple Genetic Algorithm approach is considered, and parallel Simulated Annealing with Genetic Enhancement (SAwGE) algorithm is proposed, and applied to PFSP with C_sum criterion.

This paper is organised as follows: PFSP with C_sum criterion is formulated in Section 2, SAwGE algorithm is described in details in Section 3. Section 4 contains computational experiments results, their analysis and discussion. Finally, conclusions are given in Section 5.