The Joint Location Inventory ReplenishmentProblem at a Supermarket Chainunder Stochastic Demand

Abstract

We consider a joint location inventory replenishment problem involving a chain of supermarkets at designated locations. Associated with each supermarket is stochastic stationary demand where inventory replenishment periods are uniformly fixed over the supermarkets. Considering inventory positions of the supermarket chain, we formulate a finite state Markov decision process model where states of a Markov chain represent possible states of demand for milk powder product. The unit replenishment cost, shortage cost, demand and inventory positions are used to generate the total inventory cost matrix; representing the long run measure of performance for the Markov decision process problem. The problem is to determine for each supermarket at a specific location an optimal replenishment policy so that the long run inventory costs are minimized for the given state of demand. The decisions of replenishing versus not replenishing at a given location are made using dynamic programming over e finite period planning horizon. We test the model using data from two supermarket locations. The model demonstrates the existence of an optimal state-dependent replenishment policy and inventory costs for two selected supermarket locations.