A comparison of solution procedures for two-machine flow shop scheduling with late work criterion Jacek Blazewicz a, * , Erwin Pesch b , Malgorzata Sterna c , Frank Werner d a Institute of Computing Science, Poznan University of Technology, Piotrowo 3A, 60-965 Poznan, Poland b Institute of Information Systems, FB 5-Faculty of Economics, University of Siegen, Hoelderlinstrasse 3, 57068 Siegen, Germany c Institute of Computing Science, Poznan University of Technology, Piotrowo 3A, 60-965 Poznan, Poland d Faculty of Mathematics, Otto-von-Guericke-University, PSF 4120, 39016 Magdeburg, Germany Received 27 May 2004; received in revised form 20 July 2005; accepted 29 September 2005 Abstract In this paper we analyze different solution procedures for the two-machine flow shop scheduling problem with a common due date and weighted late work criterion, i.e. for problem F2jd j ZdjY w , which is known to be binary NP-hard. In computational experiments we compare the practical efficiency of a dynamic programming approach, an enumerative method and a heuristic list scheduling procedure. Test results show that each solution method has its advantages and none of them can be rejected from consideration a priori. q 2005 Elsevier Ltd. All rights reserved. Keywords: Flow shop; Late work; Dynamic programming; Enumerative method; List scheduling 1. Introduction The process of investigating every new optimization problem follows the same well-known scheme (cf. e.g. Blazewicz, Ecker, Pesch, Schmidt, & Weglarz, 2001; Brucker, 2001). If researchers have failed in constructing an optimal polynomial-time method solving the problem under consideration, they attempt to prove the hardness of the problem. For NP-hard problems, exact but exponential time methods can be proposed (including pseudo-polynomial time approaches for binary NP-hard problems) or efficient but approximate ones. We have followed the same route in our research on the two-machine flow shop problem with a common due date and weighted late work criterion, F2jd j Z djY w , which had been shown to be binary NP-hard (Blazewicz, Pesch, Sterna, & Werner, 2005). In the problem considered in this paper, we have to find an optimal schedule for a set of jobs JZ{J 1 ,., J j ,., J n } on two dedicated machines M 1 , M 2 . Each job J j 2J consists of two tasks T 1j and T 2j , executed on machines M 1 , M 2 for p 1j , p 2j time units, respectively. Moreover, to each job J j a weight w j is assigned. Each job has to be performed, without preemptions, first on machine M 1 and then on M 2 . Furthermore, each job can be processed on at most one machine at the same time and each machine can perform at most one task at the same time. We are looking for a schedule that minimizes the total weighted late work in the system (Y w ). The late work (Y j ) for job J j 2J is determined Computers & Industrial Engineering 49 (2005) 611–624 www.elsevier.com/locate/dsw 0360-8352/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2005.09.001 * Corresponding author. Tel.: C48 61 8790 790; fax: C48 61 8771 525. E-mail addresses: Jacek.Blazewicz@cs.put.poznan.pl (J. Blazewicz), pesch@fb5.uni-siegen.de (E. Pesch), Malgorzata.Sterna@cs.put.poznan. pl (M. Sterna), Frank.Werner@mathematik.uni-magdeburg.de (F. Werner).