Parallel Simulated Annealing with Genetic Enhancement for flowshop problem with Csum☆
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 (Cmax), but this paper addresses the total flowtime criterion (Csum), which became quite popular recently. Garey, Johnson, and Sethi (1976) proved that PFSP with Cmax is -complete in the strong sense when m ⩾ 3, and with Csum when m ⩾ 2.
Attempts to solve PFSP with Csum 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 Csum 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 -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 Cmax and Csum 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 Csum criterion.
This paper is organised as follows: PFSP with Csum 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.
Section snippets
Permutation Flowshop Scheduling Problem (PFSP)
In PFSP, a set of n jobs {J1, J2, … , Jn} is to be processed through a set of m machines {M1, M2, … , Mm}. Each job must be processed uninterrupted on each machine in the same order (starting with M1, then M2, and so on, until it is processed on machine Mm). While sequence of processing jobs on each machine is assumed to be the same, the goal of PFSP with Csum criterion, is to find permutation of jobs entering M1, that minimises total flowtime.
Let pi,j denote processing time of job Jj on machine Mi.
Simulated Annealing with Genetic Enhancement
Classic Simulated Annealing, and all modifications leading to Simulated Annealing with Genetic Enhancement (SAwGE) for PFSP with Csum criterion are described in this section.
Computational experiments
SAwGE algorithm is very flexible in many aspects. One of them is that it can be run as a parallel algorithm, using many worker nodes, as well as a sequential algorithm, using just one worker node. In this section, computational experiments on both approaches are presented.
Section 4.1 contains details on testing platform used for experiments. Analysis of parallel SAwGE performance is presented in Section 4.2. Next two subsections contain results of comparison of sequential SAwGE with other
Conclusions
Simulated Annealing with Genetic Enhancement (SAwGE) algorithm for Permutation Flowshop Scheduling Problem with total flowtime criterion is proposed in this paper. SAwGE introduces a dynamic adjustment of parameters of Simulated Annealing, during algorithm’s runtime. This mechanism is based on a Simple Genetic Algorithm.
SAwGE is designed for parallel computing environments, but can act as a sequential algorithm, when run with only one worker node. Parallel SAwGE wastes little time for
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This manuscript was processed by Area Editor Edwin Cheng.