Batching and resource allocation decisions on an m-machine proportionate flowshop

This article considers an m-machine proportionate flowshop scheduling problem where each stage of production consists of a batching operation. Moreover, we assume that the job processing times are controllable through the allocation of a non-renewable resource. The objective consists of minimising the makespan. The scheduler's task consists of (1) allocating jobs to batches; (2) scheduling batches on the m-machine flowshop; (3) allocating resources to batches; and (4) allocating the resources within each batch to jobs. We show that there exists an optimal solution to the problem that consists of sequencing the jobs in ?-shape order based on their workloads. Furthermore, the jobs that are sorted in non-decreasing order of workload are scheduled in batches of equal size, whereas the remaining jobs are allocated to batches of different sizes. We present an time algorithm based on the observation that all jobs scheduled in the first segment of the ?-shape have workloads that are smaller than (or equal to) that of the last job in the sequence.