Partially Observed Inventory Systems: The Case of Rain Checks
SIAM Journal of Control and Optimization, 47(5), 2008, 2490-2519
27 Pages Posted: 30 Jan 2008 Last revised: 24 Jun 2020
Date Written: February 8, 2008
Abstract
In many inventory control contexts, inventory levels are only partially (i.e., not fully) observed. This may be due to non-observation of demand, spoilage, misplacement, or theft of inventory. We study a periodic review inventory system where the unmet demand is backordered. When inventory level is nonnegative, the inventory manager does not know the exact inventory level. Otherwise, inventory shortages occur and the inventory manager issues rain checks to customers. The shortages are fully observable via the rain checks. The inventory manager determines the order quantity based on the partial information on the inventory level. The objective is to minimize the expected total discounted cost over an infinite horizon. The dynamic programming formulation of this problem has an infinite dimensional state space. We use the methodology of the unnormalized probability to establish the existence of an optimal feedback policy when the periodic cost has linear growth. Moreover, uniqueness and continuity of the solution to dynamic programming equations are proved when the discount factor is sufficiently small.
Keywords: Partially Observed Systems, inventroy control, Rain Checks, incomplete information, lost sales, optimal control, incomplete information, feedback policy, dynamic programming, unnormalized probability, infinite dimensional space
JEL Classification: C61, M11, D81, D83
Suggested Citation: Suggested Citation
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