Achieving High Individual Service Levels Without Safety Stock? Optimal Rationing Policy of Pooled Resources

Resource pooling is a fundamental concept that has many applications in operations management for reducing and hedging uncertainty. An important problem in resource pooling is to decide (1) the capacity level of pooled resources in anticipation of random demand of multiple customers and (2) how the capacity should be allocated to fulfill customer demands after demand realization. In this paper, we present a general framework to study this two-stage problem when customers require individual and possibly different service levels. Our modeling framework generalizes and unifies many existing models in the literature and includes second-stage allocation costs. We propose a simple randomized rationing policy for any fixed feasible capacity level. Our main result is the optimality of this policy for very general service level constraints, including type I and type II constraints and beyond. The result follows from a semi-infinite linear programming formulation of the problem and its dual. As a corollary, we also prove the optimality of index policies for a large class of problems when the set of feasible fulfilled demands is a polymatroid.