Optimal control of service for facilities holding inventory

This paper addresses the problem of optimally controlling service rates for an inventory system of service facilities. We consider a finite capacity system with Poisson arrivals and exponentially distributed leadtimes and service times. For given values of maximum inventory and reorder levels, we determine the service rates to be employed at each instant of time so that the long-run expected cost rate is minimized. The problem is modelled as a semi-Markov decision problem. We establish the existence of a stationary optimal policy and we solve it by employing linear programming. Several instances of a numerical example, which provide insight into the behaviour of the system, are presented.