Multiple and nonuniform time grids in discrete-time MIP models for chemical production scheduling

Abstract

The modeling of time plays a key role in the formulation of mixed-integer programming (MIP) models for scheduling, production planning, and operational supply chain planning problems. It affects the size of the model, the computational requirements, and the quality of the solution. While the development of smaller continuous-time scheduling models, based on multiple time grids, has received considerable attention, no truly different modeling methods are available for discrete-time models. In this paper, we challenge the long-standing belief that employing a discrete modeling of time requires a common uniform grid. First, we show that multiple grids can actually be employed in discrete-time models. Second, we show that not only unit-specific but also task-specific and material-specific grids can be generated. Third, we present methods to systematically formulate discrete-time multi-grid models that allow different tasks, units, or materials to have their own time grid. We present two different algorithms to find the grid. The first algorithm determines the largest grid spacing that will not eliminate the optimal solution. The second algorithm allows the user to adjust the level of approximation; more approximate grids may have worse solutions, but many fewer binary variables. Importantly, we show that the proposed models have exactly the same types of constraints as models relying on a single uniform grid, which means that the proposed models are tight and that known solution methods can be employed. The proposed methods lead to substantial reductions in the size of the formulations and thus the computational requirements. In addition, they can yield better solutions than formulations that use approximations. We show how to select the different time grids, state the formulation, and present computational results.