Stochastic analysis of a continuous review perishable inventory system with positive lead time and Poisson demand

Abstract

This contribution is a study of the stochastic process associated with the inventory level of a continuous review perishable inventory system of ( S, s) type. The demands to the inventory system are governed by a homogeneous Poisson process. The model considered in the paper admits positive lead time. The inventory is reviewed continuously. The usable age of a batch of arriving items is a known constant. Explicit expression for the stationary distribution of the stochastic process, representing the level inventory is derived for the lost sales case, under a specified aging phenomena of a batch of items. The stationary distribution of the inventory level process is used to obtain a closed form expression for the total expected cost per unit time (in the stationary case) in operating the ( S, s) policy, which may be used as an objective to determine the optinal reorder level. The cost expression is comprehensive and considers all the relevant costs like the fixed ordering cost, proportional unit (ordering) cost, cost of shortage, and storage, cost due to items that perish in a batch. Also, certain first passage time distributions associated with the inventory level stochastic process, which are of special interest (in terms of the inventory system) are derived. Several known special cases of our results are deduced. A simple numerical example is provided.