Pergamon 0360-8352(94)00026-3 Computers ind. Engng Vol. 28, No. I, pp. 63-70, 1995 Copyright© 1995 ElsevierScienceLtd Printed in Great Britain. All rights reserved 0360-8352/95 $9.50 +0.00 A TWO-MACHINE FLOWSHOP SEQUENCING PROBLEM WITH LIMITED WAITING TIME CONSTRAINTS DAR-LI YANG and MAW-SHENG CHERN Department of Industrial Engineering, National Tsing Hua University, Hsinchu 30043, Taiwan, R.O.C. (Received for publication 11 May 1994) Abstract--We consider a two-machineflowshopsequencing problem with limited waiting time constraints. This means that for each job the waiting time between two machines cannot be greater than a given upper bound. The objective is to minimize the makespan. There are efficient algorithms for the special cases of infinite waiting time and zero waiting time. The two-machine flowshop sequencing problem with limited waiting time constraints is shown to be NP-hard. A branch-and-bound algorithm is proposed, and computational experiments are provided. INTRODUCTION In this paper, we consider a two-machine flowshop sequencing problem with limited waiting time constraints. We assume that there are n jobs {Ji, J2 .... , J~} and each of the n jobs is processed on 2 machines {M~, M2}. Job Ji consists of 2 operations, O U, i = 1..... n and j = 1,2. The processing time of operation O U on machine Mj is Po. The waiting time of job Ji is the time that elapses between the completion of J~ on MI and the start of processing on M2. We assume that the waiting time of job Ji cannot be greater than a given ui, i = 1..... n. The objective is to minimize the makespan. For example, there is a problem with Pu = 1, Pl2 = 6, P21 = 2, P22 = 3, P31 = 5, P32 = l, u~ = 2, u2 = 1 and u3 = 3. Figure l(a) and (b) illustrate an infeasible schedule and a feasible schedule respectively. In this paper, we relax the no-wait constraint [1, 2] that job must be processed continuously without waiting time between consecutive machines. We consider a sequencing model which is similar to, but somewhat more flexible than the no-wait model. The proposed problem exists in some practical applications such as food production [3], chemical production and steel production. We assume that each machine can process only one operation at a time and different operations of the same job cannot be processed simultaneously, Preemption is not allowed. The permutation schedules constitute an important subclass of schedules, which specify that the processing order of jobs is the same on each machine. We note that this is not the most general model, but it is the most commonly examined one. Our objective is to find the permutation schedule minimizing the makespan. In the proposed problem, if u~ = ~ for all i, then this problem is a conventional flowshop sequencing problem [4]. If u~ = 0 for all i, then this problem is a flowshop sequencing problem with no in-process waiting [2]. There are polynomial time algorithms for solving the above mentioned special cases [5, 6]. However, the two-machine flowshop sequencing problem with limited waiting time constraints is shown to be NP-hard. A branch-and-bound algorithm is proposed and the computational experiments are provided. The m-machine case is also discussed. PROBLEM COMPLEXITY Although there are polynomial time algorithms for the special cases of u~ = ~ for all i [5] and u~ = 0 for all i [6], we will show that the two-machine flowshop sequencing problem with limited waiting time constraints is NP-hard. 63