Conic programming models for production planning with clearing functions: Formulations and duality

Concave clearing functions that model the expected throughput of a production resource as a function of its planned workload have yielded promising results when used in production planning models. The most common of these models take the form of linear programs (LPs) obtained by piecewise linearization of the clearing function constraints, leading to large formulations and inaccurate estimates of dual prices for resources. We show that several clearing function forms considered in the literature to date can be reformulated as conic programs (CPs), for which efficient solution methods and an elegant duality theory analogous to that for linear programming exist. We derive expressions for the optimal values of the dual variables and present computational experiments showing that the dual prices obtained from the CP formulation are more accurate than those obtained from the piecewise linearized LP models. In addition, the CP solution outperforms the LP solutions with respect to nervousness.