Scheduling a two-stage hybrid flow shop with parallel machines at the first stage J.N.D. Gupta Department of Management, Ball State University, Muncie, IN 47306, USA A.M.A. Hariri Department of Statistics, King Abdul-Aziz University, P.O. Box 9028, Jeddah 21413, Saudi Arabia C.N. Potts Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, UK This paper considers a non-preemptive two-stage hybrid flow shop problem in which the first stage contains several identical machines, and the second stage contains a single machine. Each job is to be processed on one of the first-stage machines, and then on the second-stage machine. The objective is to find a schedule which minimizes the maximum completion time or makespan. The problem is NP-hard in the strong sense, even when there are two machines at the first stage. Several lower bounds are derived and are tested in a branch and bound algorithm. Also, constructive heuristics are presented, and a descent algorithm is proposed. Extensive computational tests with up to 250 jobs, and up to 10 machines in the first stage, indicate that some of the heuristics consistently generate optimal or near-optimal solutions. Keywords: Flexible flow shop, identical parallel machines, branch and bound, heuristics. 1. Introduction Consider a two-stage hybrid flow shop scheduling problem where each of n jobs is to be processed first at stage 1 and then at stage 2. In the general version of this problem, there are several identical parallel machines at each stage. A non-preemptive schedule is required for which no interruption in the processing of an operation is allowed. Each job i (i = 1,…, n) has integer processing times a i and b i at the two stages. The objective is to find a schedule which minimizes the maximum completion time or makespan. © J.C. Baltzer AG, Science Publishers Annals of Operations Research 69(1997)171 – 191 171