Improved MILP formulation equipped with valid inequalities for scheduling a batch processing machine with non-identical job sizes

Abstract

We consider the problem of minimizing makespan on a single batch processing machine with a limited capacity. The machine can simultaneously process a group (batch) of jobs, each of them having different sizes and processing times. The processing time of a batch is the longest processing time of all jobs in the batch. We analyze the strength of the LP-relaxations of two mixed integer linear programming (MILP) formulations: a straightforward formulation and an advanced formulation. We prove that while the LP-relaxation of the straightforward formulation can perform arbitrarily bad in providing an LP bound on the optimal makespan, the LP-relaxation of the advanced formulation have some merit in providing a relatively good bound of makespan. With a detailed analysis of the problem, we prove a number of properties of the optimal solution and derive their respective valid inequalities (VI). These VIs are exploited in a novel MILP formulation for the problem. Extensive computations reveal that, with the help of the VIs, the proposed MILP formulation can solve large instances of the problem.