ORIGINAL ARTICLE An improved mixed integer linear formulation and lower bounds for minimizing makespan on a flow shop with batch processing machines Ali Husseinzadeh Kashan & Behrooz Karimi Received: 24 September 2007 / Accepted: 3 January 2008 / Published online: 2 February 2008 # Springer-Verlag London Limited 2008 Abstract This paper considers a flow shop scheduling problem with batch processing machines. Each batch processing machine has a limited capacity and can process a group of jobs, each of them having a different known capacity requirement, simultaneously. Job processing time on each machine is known and arbitrary. The processing time of a batch on each machine is the longest processing time of all jobs in the batch. We improve the only existing mixed integer linear formulation (MILF) of the problem through significant reduction in size complexity of the model. Results justify that the improved MILF is clearly more efficient in reducing the required time for obtaining optimal makespan of small-size problems, in comparison with the existing MILF . Motivated by relaxing variety of the problem assumptions, several valid lower bounds on the optimal makespan are also proposed that can furtheraccel- erate obtaining optimal solution through proposed MILF . Robustness evaluation of each bound under the different problem settings is reported through computations. Keywords Batch-processing machine . Flow shop . Lower bound . Mixed integer linear formulation . Scheduling 1 Introduction The formal description to the scheduling problem consid- ered in this paper is as follows. There is a set of N jobs simultaneously available for processing in a flow shop environment with M operating stages and an unlimited buffer capacity between consecutive stages. Each stage comprises a batch processing machine in which several jobs can be processed simultaneously in a batch. The set of jobs are grouped into batches and these batches are processed, in a fixed process flow, on all the machines. Thus the problem considered here is a permutation flow shop problem, i.e., all batches are processed on each machine in the same order. Each job j has size s j with processing time p jm on machine m. Each machine m has a capacity equal to S m and can process a batch as long as the total size of the batch is not more than its capacity. Also jobs cannot be split across the batches. At stage m, the processing time of batch b (B b ), which is denoted by P bm , is given by the longest processing time among all jobs in that batch. All jobs in a batch have a common completion time that is equal to the completion time of the batch. Once a batch is formed, adding or removing a job from it, is not permitted and in each stage once processing of a batch is initiated, it cannot be interrupted. A scheduling problem of this type is called parallel-batching scheduling problem. On the other side, there is serial-batching scheduling problem where, jobs may be batched if they share the same setup on a machine and only one job is processed at a time [23]. A practical application of the problem can be found in electronics industries, wherein the environmental stress screening (EES) chambers are used. ESS is a screening process to detect defects that could not be detected by visual inspection or in-circuit testing. ESS has been evolved from burn-in techniques and would require the entire product to be exposed to tests at conditions specified by the customer. The chambers employed in an ESS are of varying capacities and would depend on the number of assemblies to be tested and also on the test conditions. The Int J Adv Manuf Technol (2009) 40:582–594 DOI 10.1007/s00170-008-1377-9 A. H. Kashan : B. Karimi (*) Department of Industrial Engineering, Amirkabir University of Technology, P.O. Box 15875-4413, Tehran, Iran e-mail: B.Karimi@aut.ac.ir A. H. Kashan e-mail: A.Kashani@aut.ac.ir