Journal of the Operational Research Society (1998) 49, 77±85 #1998 Operational Research Society Ltd. All rights reserved. 0160-5682/98 $12.00 Minimising makespan in the two-machine ¯ow-shop with release times R Tadei 1 , JND Gupta 2 , F Della Croce 1 and M Cortesi 1 1 D.A.I., Politecnico di Torino, Italy; and 2 Ball State University, USA This paper considers the two-machine ¯ow-shop problem with the objective of minimising the makespan subject to different release times. In view of the strongly NP-hard nature of this problem, ®ve lower bounds and two new dominance criteria are proposed together with a decomposition procedure that reduces the problem size by setting jobs at the beginning of the sequence. Several branch and bound procedures are described by applying different lower bounds and branching schemes. A detailed computational campaign has been performed on different kinds of instances testing problems with size up to 200 jobs. Keywords: branch and bound; ¯ow-shop scheduling; release times Introduction Traditionally, it is often assumed that all jobs to be scheduled in a shop are simultaneously available at the beginning of the planning period. However, in many practical situations, all jobs are not available at the same time and each job has its own release time that may be determined by the requirements of the customer orders or the output of a Material Requirement Planning (MRP) system. A study of scheduling problems where jobs become available at different times is both useful and necessary. The simplest multiple machine problem involving differ- ent release times of jobs is the two-stage ¯owshop schedul- ing problem. A study of this problem, while useful in its own right, may also provide some approximate solution procedures for the multi-stage ¯owshops with different release times. The two-stage ¯ow-shop problem considered here may be described as follows: a set N f1; 2; ... ; ng of n jobs is to be processed, without preemption, on two machines where each job is processed on machine 1 ®rst and machine 2 next. The processing times of job i 2 N , which becomes available at time r i , called its release time, at machines 1 and 2 are a i and b i respectively. It is desired to ®nd the order (schedule) in which each job should be processed on each of the two machines so as to minimise the total time required to process all n jobs on both machines, called maximum completion time or makespan. For this problem, it is suf®cient to consider only permutation schedules, namely job orderings where identical sequence of jobs is used on both machines. 1,2 With this simpli®cation, the completion times of job p j at the ®rst and second machine in the schedule p p1, p2, ... ; pn, C 1;p j and C 2;p j , are given by the following expressions: C 1;p j max 1 4i 4j r pi P j si a ps 1 C 2;p j max 1 4u 4j C 1;pu P j iu b pi 2 Combining (1) and (2) above, the completion times of a partial schedule s s1, s2; ... ;s j at machine 1 and 2, C 1 s and C 2 s respectively, are expressed as: C 1 s max 1 4i 4j r si P j si a ss 3 C 2 s max 1 4u 4v 4j r su P v iu a si P j iv b si 4 Using the three ®eld notation described by Lawler et al, 3 this problem can be represented as a F 2jr i jC max problem where F 2 indicates that it is a two-stage ¯owshop, r i represents that job i has a release date r i which could be non-zero, and C max indicates that the objective is the minimisation of maximum completion time or makespan. If all jobs are available at time zero, namely, r i 0 for all i 2 N , Johnson's 2 constructive On log n algorithm to ®nd an optimal schedule for this F 2kC max problem can be described as follows: ®rst arrange the jobs with a i 4 b i in Correspondence: Dr F Della Croce, D.A.I., Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy. E-mail: dellacroce@polito.it