Optimal control of production time of perishable inventory system with postponed demands

We consider a continuous review production inventory system in which the demands arrive according to a Poisson process. We assume that the demands that occur during stock out periods enter a pool which has finite capacity. Any demand that arrives when the pool is full and the inventory level is zero, is also assumed to be lost. The demands in the pool are selected one by one, if the stock is above the prefixed value, with interval time between any two successive selections is distributed as exponential. In order to replenish stock, a production facility is switched on when the inventory level drops to prefixed value s and items are produced one by one until the stock level reaches a maximum capacity S(> s). The successive production time is assumed to follow an exponential distribution whose parameter is chosen from a given set of positive values. The objective of this paper is to investigate the optimal set of production rates that minimises the total expected cost per unit time. We formulate this problem as a semi-Markov decision problem. The stationary optimal policy is computed using linear programming algorithm and the results are illustrated numerically.