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Optimal inventory control with shrinkage and observed sales. (English) Zbl 1302.90009

Summary: We study a discrete-time periodic-review inventory system where the unmet demand is lost, and the excess inventory is subject to shrinkage. We first derive the state evolution and then introduce unnormalized conditional probabilities to transform the nonlinear state evolution into a linear one. We then prove the existence and uniqueness of the solution for the Bellman equation in the case of unbounded costs and show that the solution yields the value function.

MSC:

90B05 Inventory, storage, reservoirs
60K10 Applications of renewal theory (reliability, demand theory, etc.)
93E20 Optimal stochastic control
93C41 Control/observation systems with incomplete information
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References:

[1] Bensoussan A., Inventory Control with a Cash Register: Sales Recorded but Not Demand or Shrinkage (2012)
[2] DOI: 10.1137/070688663 · Zbl 1170.90003 · doi:10.1137/070688663
[3] DOI: 10.1137/040620321 · Zbl 1190.90015 · doi:10.1137/040620321
[4] Bensoussan A., The Oxford Handbook of Nonlinear Filtering pp 500– (2011)
[5] DOI: 10.1287/msom.1070.0203 · doi:10.1287/msom.1070.0203
[6] DOI: 10.1016/0022-0396(67)90023-X · Zbl 0158.16801 · doi:10.1016/0022-0396(67)90023-X
[7] Sethi S., Decis. Line 41 pp 16– (2010)
[8] DOI: 10.1007/BF00536382 · Zbl 0164.19201 · doi:10.1007/BF00536382
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