A NOTE ON A PERIODIC REVIEW INVENTORY MODEL WITH UNCERTAIN DEMAND IN A RANDOM ENVIRONMENT

This paper is on the analysis of a single product, periodic review inventory model, where the distributions of demands vary with the state of the environment variable. The state of the environment is assumed to follow a discrete-time Markov chain. The optimal inventory policy to minimize the total discounted expected cost is derived via dynamic programming. For the finite-horizon model, we show that an environmental-dependent base-stock policy is optimal, and derive some characteristics of the optimal policy. Under additional conditions, we further derive the monotonicity of the optimal policy. 1 Introduction The inventory control has long focused on managing certain specific types of probability in the demand for the products. But, on the other hand, consumer's liking becomes variously in the real-life. The demand is fluctuated by the economic climate, weather condition, trend of public opinion, and so forth. Mere including a purely random component in the demand process will be impossible to express such situations. So, in this paper, under the assumption that the environmental process follows a discrete- time Markov chain, we model a single product inventory system of which the distributions of demands depend on environmental fluctuations, and discuss the management policy. We further investigate the effect of the environmental fluctuations on the optimal policy. The main advantage of the Markov chain approach is that it provides a national and flexible framework for formulating various changes described above. The effect of a randomly changing environment in inventory model received only limited attention in the earlier paper. Kalymon(12) studies a multiple-period inventory model in which the unit cost of the product is determined by a Markov process, and the distribution of demand in each period depends on the current cost. Feldman(8) models the demand en- vironment as a continuous-time Markov chain. The demand is modulated by a compound Poisson process where the parameters are determined by the state of the environment. But he studies only the steady-state distribution of the inventory position. Song and Zipkin(18) present a continuous-review inventory model where the demand process is a Markov mod- ulated Poisson process, and they derive some basic characteristics of the optimal policy and algorithms for computing the optimal policy. In recent articles, ¨ Ozekici and Parlar(15) develop an infinite-horizon periodic-review inventory model with unreliable suppliers where the demand, supply and cost parameters are influenced by a random environment. Cheng and Sethi(2) analyze the joint promotion-inventory management problem for a single item in the context of Markov decision processes. The purpose of this paper is to show that the environmental-dependent base-stock policy is optimal, and that the optimal policy have the monotonicity for review periods by analyz- ing finite-horizon periodic-review inventory model where the demand distribution depend