## Inventory control in a fluctuating demand environment.(English)Zbl 0798.90035

An inventory-control model is considered. The order lead time is either fixed or stochastic. There are linear costs for holding inventory and for backorders. It is assumed that the demand rate varies with an underlying state-of-the-world variable. The states of the world are described by a Markov chain with a countable state space. When the world is in state $$i$$, demand follows a Poisson process with rate $$\lambda^ i$$.
The inventory problem is formulated as a dynamic programming problem. There are two state variables: the world and inventory position. The order decision depends on the current state of the both variables. An order policy which minimizes the expected total discounted cost over an infinite horizon is described. If the order cost is linear in the quantity ordered then an optimal policy is characterized by a single number for every value of world. If there is also a fixed cost to place in order then a world-dependent $$(s,S)$$-type policy is optimal. Some algorithms to compute an optimal policy and some extensions of the model are presented.

### MSC:

 90B05 Inventory, storage, reservoirs 90C39 Dynamic programming
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