Optimal production and inventory control of multi-class mixed backorder and lost sales demand class models

Manufacturers usually face customers with different service requirements. The challenge they face is how to determine which customers to serve when the inventory supply is limited. We consider the problem of a manufacturer serving two types of customers: those with long term commitment, divided into several backorder classes; and those without commitment consisting of a single lost sales class. We treat the problem within a continuous time integrated production and inventory control framework. We propose a two-step strategy to fully characterize the optimal policy. In a first step, we use the concept of L-natural convexity to partially describe the optimal policy. Based on these results, in a second step, we reformulate the problem and fully characterize the optimal policy. We show that the latter is characterized by state-dependent multidimensional thresholds. Due to the computational complexity of the problem, we propose three heuristic policies: The first uses linear thresholds that mimic the optimal policy. These thresholds are computed through a decomposition of the original problem into a series of single-lost sales, single-backorder class problems. The second heuristic treats all demand classes equally. The third heuristic uses static thresholds to control production and inventory rationing among demand classes. Extensive numerical results show that, for problems with one lost sales and two or three backorder classes, the first heuristic outperforms the other two with a cost deviation less than 1.25%, from the optimal. Furthermore, computing the thresholds of this heuristic is orders of magnitude faster than computing the optimal policy.