Inventory models with multiple order opportunities

Jian Li, Purdue University


We examine the structure of the optimal ordering policies for several inventory models with multiple order opportunities. The inventory models that are examined in this thesis are: (1) the newsvendor model with a second order opportunity; (2) the stochastic multi-period inventory model with limited order opportunities; and (3) the stochastic multi-period inventory model with two supply modes. For (1), we develop three models (Models I, II, and III) that differ in the timing of the second order. In all three models, the first order is placed for delivery at the beginning of the season. In Model I, the second order quantity is determined at the beginning of the season for delivery at a pre-specified time. In Model II, the second order quantity is determined at some pre-specified time that can be any where during the season. In Model III, both the timing and quantity of the second order are determined dynamically. For Model III, we establish for the first time that under appropriate conditions, the decision to place a second order is characterized by a time-dependent (s,S) policy. A counterexample is presented that suggests that the policy structure under more general conditions would likely be more complex. Model II is a generalization of the model of Fisher et al. (2001). We show that under mild regularity conditions, this problem has sufficient structure to reduce to a sequential search of two programs, each of which has at most one local minimum. For Model I, we establish robust conditions under which the optimization behaves well. For (2), we examine a model under the assumptions that shortages are backordered, demand density functions fall in PF2 family and unit purchase cost is non-decreasing as we get closer to the end of the problem horizon. We show that a time varying (s,S) policy is the optimal decision rule for this model. For (3), the two supply modes differ in their delivery leadtimes. The chapter contributes by showing that there are unique "order up to" levels to determine the order quantities from these two suppliers. We identify conditions when it is optimal to order from just one supplier or from both. In case it is optimal to order from both in a period, we show that at the beginning of the period, if the beginning inventory level is between a certain pair of points, then it is optimal to raise the inventory position to the higher point through a slow order. However, if the beginning inventory position is lower than the lower point, then the inventory level is first raised up to this point through a fast order and then the inventory position is raised up to the higher point through a slow order. If the beginning inventory is higher than the higher point, no order needs to be placed. The optimal policies in this chapter are supported by the property that the cost is unimodal in the beginning inventory position and convex in the beginning inventory level. We need the PF2 density assumption to prove this property.




Chand, Purdue University.

Subject Area

Management|Operations research

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