Analysis of Deterministic Control and Its Improvements for an Inventory Problem with Multiproduct Batch Differentiation

We consider a multiperiod planning problem faced by a biopharmaceutical firm that must coordinate the production and allocation of batches of intermediate products to end products for multiple markets. This is a challenging problem to solve optimally, so we derive a theoretical bound on the performance of a deterministic control (DC) in which all random variables are replaced by their expected values and the corresponding deterministic optimization problem is solved. This is a variant of an approach that is widely used in practice. We show that while DC can perform poorly in some instances, simple reoptimizations of DC (i.e., certainty equivalent control (CEC)) improve the performance of DC. That said, the benefit of reoptimization decreases as the magnitude of demand variation increases. To address this, and also to provide guidance for heuristic design, we derive performance bounds for two additional heuristic controls—(1) open-loop feedback control (OLFC), which reoptimizes a sequence of stochastic programs with a complete demand distribution information, and (2) multipoint approximation control (MPAC), which solves a dynamic program using an approximate demand distribution that incorporates slightly more distributional information beyond simple expected values. We show that although OLFC outperforms CEC, its improvement can be limited. On the other hand, with carefully chosen approximate demand distributions, MPAC can significantly outperform OLFC. This suggests that, in our setting, fully capturing decision dynamics coupled with a small amount of additional distributional information beyond expected values is more useful than fully capturing the demand distribution but coupling that with insufficient modeling of decision dynamics.