Optimization of Economic Order Quantity and Profits under Markovian Demand

Abstract

Most manufacturers, wholesalers and retailers face a situation of stock depletion over time. Replenishment is usually made using the Economic Order Quantity (EOQ) model. The model assumes deterministic demand of a single item; often at a constant rate whose total inventory costs (ordering and holding) per unit time are minimized. In this paper, a new approach is developed to optimize the economic order quantity (EOQ) of a single item, finite horizon, and periodic review inventory problem with stochastic demand at optimum profits. In the given model, sales price and inventory replenishment periods are uniformly fixed over the planning horizon. Adopting a Markov decision process approach, the states of a Markov chain represent possible sates of demand for the inventory item. The ordering cost, holding cost, shortage cost and sales price are combined with demand and inventory positions to generate profits for the EOQ decision problem. The objective is to determine in each period of the planning horizon an optimal economic order quantity so that the long run profits are maximized for a given state of demand. Using weekly equal intervals, the decisions of how much to order are made using dynamic programming over a finite period planning horizon. A numerical example demonstrates the existence of an optimal state-dependent economic order quantity as well as the corresponding profits of item.