Int. J. Production Economics 105 (2007) 407–424 Metaheuristic approaches to the hybrid flow shop scheduling problem with a cost-related criterion Adam Janiak a , Erhan Kozan b , Maciej Lichtenstein a,Ã , Ceyda Og˘uz c,1 a Institute of Engineering Cybernetics, Wroclaw University of Technology, Janiszewskiego 11/17, Wroclaw, Poland b Department of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld. 4001, Australia c Department of Logistics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong SAR Received 2 February 2004; accepted 2 May 2004 Available online 12 October 2005 Abstract In the paper, the flow-shop scheduling problem with parallel machines at each stage (machine center) is studied. For each job its release and due date as well as a processing time for its each operation are given. The scheduling criterion consists of three parts: the total weighted earliness, the total weighted tardiness and the total weighted waiting time. The criterion takes into account the costs of storing semi-manufactured products in the course of production and ready-made products as well as penalties for not meeting the deadlines stated in the conditions of the contract with customer. To solve the problem, three constructive algorithms and three metaheuristics (based one Tabu Search and Simulated Annealing techniques) are developed and experimentally analyzed. All the proposed algorithms operate on the notion of so-called operation processing order, i.e. the order of operations on each machine. We show that the problem of schedule construction on the base of a given operation processing order can be reduced to the linear programming task. We also propose some approximation algorithm for schedule construction and show the conditions of its optimality. r 2005 Elsevier B.V. All rights reserved. Keywords: Scheduling; Flow shop; Tabu search; Simulated annealing 1. Introduction A variety of production processes can be modeled as a flow-shop scheduling problem. In the classical flow- shop problem, a set of jobs flow through multiple stages in the same order, where each stage consists of only one machine. However, being not sufficient for many production processes in technologically developing manufacturing environments, this classical flow-shop model was extended almost two decades after Johnson’s seminal work (Johnson, 1954) on classical two-stage flow-shop scheduling problem. In this extended model, each stage consists of a number of identical parallel machines, and can be viewed as a machine center. This type of flow-shop environment is usually referred to as a hybrid flow-shop or flow-shop with multiple machines (processors) in the literature (Hoogeveen et al., 1996; Janiak and Lichtenstein, 2001; Lee and Vairaktarakis, 1994; Riane et al., 1998). ARTICLE IN PRESS www.elsevier.com/locate/ijpe 0925-5273/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2004.05.027 Ã Corresponding author. Tel.: +48 71 3202906; fax: +48 71 3212677. E-mail address: maciej.lichtenstein@pwr.wroc.pl (M. Lichtenstein). 1 Currently with the Department of Industrial Engineering, Koc - University, Turkey.