# Base stock policies for periodic review inventory systems

#### Abstract

We study two stochastic inventory systems. The first part deals with an (*s, S*) system system, and the second part with an (* R, T*) system. In a (*s, S*) system, the inventory position of a product is reviewed regularly (every *T* periods), and, if it is found to be below a threshold *s*, an order is placed to bring the item's inventory position to the level *S*. In a (*R, T*) system, an order is placed every *T* periods to bring the inventory up to the base stock *R*; it is a special case of an (*s, S*) system, with *s* set at minus infinity.^ In the first part, we revisit the classical multi-period inventory model with fixed setup cost for both finite-horizon and infinite-horizon settings with the criterion of minimizing total discounted cost. Although efficient algorithms have been found to compute the optimal *s* and * S*, which minimize the long-run average cost, the case of minimizing total discounted cost is seldom considered. In our work, we first offer new bounds for the reorder point *s _{t}* and order-up-to level

*S*. Then, based on these new bounds, we analyze the worst-case behavior (the numbers of local minima) of the cost function with generalized phase type distributed demand: our analysis shows that the relationship among cost parameters (unit purchase, holding and shortage penalty cost) is one of the key factors which determine the number of local minima for the finite-horizon model whether the unmet demand in each period is backordered or lost. Similar results are also found for the infinite-horizon case.^ In the second part, a new control is developed to the (

_{t}*R, T*) inventory system that allow partial backorders and partial lost sales. In each period, demand is accepted till the inventory position reaches zero. For the single retailer model, we show that the steady state distribution of the inventory position has a simple recursive solution. Moreover, we find regularity conditions under which the cost function is unimodal in

*R*, guaranteeing that the optimal solution can be found easily. Similar results are also found for more complex systems.^

#### Degree

Ph.D.

#### Advisors

Maqbool Dada, Purdue University.

#### Subject Area

Business Administration, Management|Engineering, Industrial|Operations Research

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