A Periodic Review Inventory Model in a Random Environment

We consider a single product, periodic review inventory model with variable cqacity, random yield, and uncertain demand in a random environment. All model parameters and distributions depend on environmental fluctuations. It is assuumed that the environmental process follows a discrete-time Markov chain. The optimd inventory policy to minimize the total discounted expected cost is derived via dynamic programming. For the finite-horizon model, we show that the objective function is quasi-convex and that the structure of the optimal policy is characterized by a single environmental-dependent critical number for the initial inventory level at each period. Expressions for solving the critical number and the optimal planned ordering are obtained. We further show that the solution for the finite-horizon model converges to that of the infinite-horizon model.