Journal of Heuristics, 5, 5–28 (1999) c  1999 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. A3/2 Algorithm for Two-Machine Open Shop with Route-Dependent Processing Times V. A. STRUSEVICH V.Strusevich@greenwich.ac.uk University of Greenwich, London, U.K. A. J. A. VAN DE WAART Erasmus University, Rotterdam, The Netherlands R. DEKKER Erasmus University, Rotterdam, The Netherlands Abstract. This paper considers the problem of minimizing the schedule length of a two-machine shop in which not only can a job be assigned any of the two possible routes, but also the processing times depend on the chosen route. This problem is known to be NP-hard. We describe a simple approximation algorithm that guarantees a worst-case performance ratio of 2. We also present some modifications to this algorithm that improve its performance and guarantee a worst-case performance ratio of 3/2. Keywords: approximation, open shop scheduling, heuristics, worst-case analysis 1. Introduction Many practical situations can be described using multi-stage (or shop) scheduling models. In these models each job has to be processed on a number of sequential machines. For an overview of the results in this area see (Lawler et al., 1993) and (Tanaev et al., 1994). However, in classical scheduling theory, even if a choice of the route for a job is allowed, it is normally assumed that the processing times of all operations remain unchanged. This may not be the case in practice. For instance, the authors have observed an example at engineering works where cylinders for marine diesel engines are manufactured on special turning lathes and the processing times depend upon the order in which the lathes are used. Despite their practical relevance, shop scheduling models with route-dependent process- ing times have not received much attention. To obtain more insight, this paper analyzes the simplest problem of this type called the two-machine open shop scheduling problem with route-dependent processing times to minimize the makespan. This problem is known to be NP-hard (see (Adiri and Amit, 1983)), this is why we concentrate on heuristic algorithms and worst-case analysis of their performance. This paper is organized as follows. Section 2 contains the definition of the problem and some notation. Section 3 considers a simple heuristic algorithm that guarantees a worst-case ratio bound of 2. A schedule found by that heuristic is to be transformed by two algorithms presented in Section 4. These algorithms require O (n 2 ) time each and they eventually generate a schedule which is at most 3/2 times worse than an optimal one. Analysis of these algorithms is given in Sections 5 through 7. Final remarks are contained in Section 8.