Tightened Formulation and Resolution of Energy-Efficient Job-Shop Scheduling

Job shops are an important production environment for low-volume high-variety manufacturing. When there are urgent orders, the speeds of certain machines can be adjusted with a high energy and wear and tear cost. Scheduling in such an environment is to achieve on-time deliveries and low energy costs. The problem is, however, complicated because part processing time depends on machine speeds, and machines need to be modeled individually to capture energy costs. This paper is to obtain near-optimal solutions efficiently. The problem is formulated as a Mixed-Integer Linear Programming (MILP) form to make effective use of available MILP methods. This is done by modeling machines in groups for simplicity while approximating energy costs, and by linking part processing status and machine speed variables. Nevertheless, the resulting problem is still complicated. The formulation is therefore transformed by extending our previous tightening approach for machines with constant speeds. The idea is that if constraints can be transformed to directly delineate the convex hull, then the problem can be solved by linear programming methods. To solve the problem efficiently, our advanced decomposition and coordination method is used. Numerical results show that near-optimal solutions are obtained, demonstrating significant benefits of our approach on on-time deliveries and energy costs.