Mathematical Programming Models and Formulations for Deterministic Production Planning Problems

We study in this lecture the literature on mixed integer programming models and formulations for a specific problem class, namely deterministic production planning problems. The objective is to present the classical optimization approaches used, and the known models, for dealing with such management problems. We describe first production planning models in the general context of manufacturing planning and control systems, and explain in which sense most optimization solution approaches are based on the decomposition of the problem into single-item subproblems. Then we study in detail the reformulations for the core or simplest subproblem in production planning, the single-item uncapacitated lot-sizing problem, and some of its variants. Such reformulations are either obtained by adding variables – to obtain so called extended reformulations – or by adding constraints to the initial formulation. This typically allows one to obtain a linear description of the convex hull of the feasible solutions of the subproblem. Such tight reformulations for the subproblems play an important role in solving the original planning problem to optimality. We then review two important classes of extensions for the production planning models, capacitated models and multi-stage or multi-level models. For each, we describe the classical modeling approaches used. Finally, we conclude by giving our personal view on some new directions to be investigated in modeling production planning problems. These include better models for capacity utilization and setup times, new models to represent the product structure – or recipes – in process industries, and the study of continuous time planning and scheduling models as opposed to the discrete time models studied in this review.