Capacity representation in sales and operations planning: Aggregation through projection

Software systems supporting sales and operations planning operate on the basis of total production volumes of products. They ensure that production volumes are feasible with respect to the capacity at different production sites without allocating them to individual resources. We consider the problem of finding a representation of the set of production volumes that respects the capacity by means of linear inequalities and that only uses variables corresponding to the total production volumes. For single-stage production systems, we derive a complete description of this type analytically. Since this description has exponential size, we present an algorithmic framework for approximating this set. Adopting a polyhedral perspective, our algorithmic framework can be applied to obtain projections of arbitrary polytopes, including those that represent multi-stage production systems. In a case study from a German semiconductor manufacturer, we demonstrate the superiority of our approach over existing methods with respect to the computation time and quality of approximation. Through extensive numerical experiments, we show that our approach can be applied efficiently to instances with a wide range of values for the production system’s parameters, i.e., the number of production stages, resources at each stage, and alternative process plans.