Minimizing makespan in a two-machine no-wait flow shop with batch processing machines

Given a set of jobs and two batch processing machines (BPMs) arranged in a flow shop environment, the objective is to batch the jobs and sequence the batches such that the makespan is minimized. The job sizes, ready times, and processing times on the two BPMs are known. The batch processing machines can process a batch of jobs as long as the total size of all the jobs assigned to a batch does not exceed its capacity. Once the jobs are batched, the processing time of the batch on the first machine is equal to the longest processing job in the batch; processing time of the batch on the second machine is equal to the sum of processing times of all the jobs in the batch. The batches cannot wait between two machines (i.e., no-wait). The problem under study is NP-hard. We propose a mathematical formulation and present a particle swarm optimization (PSO) algorithm. The solution quality and run time of PSO is compared with a commercial solver used to solve the mathematical formulation. Experimental study clearly highlights the advantages, in terms of solution quality and run time, of using PSO to solve large-scale problems.