An Envelope Theorem for Bayesian Dynamic Program and Its Application to an Inventory Problem

A generalized envelope theorem is established for a Bayesian dynamic program problem. An application of the theorem is given in a Bayesian inventory management problem with unobserved lost sales. Specifically, we show that the optimal inventory level with unobserved lost sales is greater than the optimal inventory level with observed lost sales. We prove this result under the continuous demand distribution, which complements the existing work in the literature. We further comment that the results can be easily extended to the Markov-modulated demand process.

[1]  Dimitri P. Bertsekas,et al.  Stochastic optimal control : the discrete time case , 2007 .

[2]  H. Scarf.  Some remarks on bayes solutions to the inventory problem , 1960 .

[3]  H. Scarf.  Bayes Solutions of the Statistical Inventory Problem , 1959 .

[4]  Evan L. Porteus,et al.  Stalking Information: Bayesian Inventory Management with Unobserved Lost Sales , 1999 .

[5]  W. Lovejoy.  Myopic policies for some inventory models with uncertain demand distributions , 1990 .

[6]  Suresh P. Sethi,et al.  A Multiperiod Newsvendor Problem with Partially Observed Demand , 2007, Math. Oper. Res..

[7]  Jing-Sheng Song,et al.  Optimal Policies for Multiechelon Inventory Problems with Markov-Modulated Demand , 2001, Oper. Res..

[8]  M. Dada,et al.  Dynamic learning, pricing, and ordering by a censored newsvendor , 2007 .

[9]  Jing-Sheng Song,et al.  Inventory Control with Unobservable Lost Sales and Bayesian Updates , 2008 .

[10]  C. R. Sox,et al.  Adaptive Inventory Control for Nonstationary Demand and Partial Information , 2002, Manag. Sci..

[11]  Woonghee Tim Huh,et al.  A Nonparametric Asymptotic Analysis of Inventory Planning with Censored Demand , 2009, Math. Oper. Res..

[12]  William S. Lovejoy,et al.  Suboptimal Policies, with Bounds, for Parameter Adaptive Decision Processes , 1993, Oper. Res..

[13]  Li Chen,et al.  Dynamic Inventory Management with Learning About the Demand Distribution and Substitution Probability , 2008, Manuf. Serv. Oper. Manag..

[14]  J. Arthur.  Stochastic Models in Operations Research, Volume II. Stochastic Optimization (Daniel P. Heyman and Matthew J. Sobel) , 1985 .

[15]  R. L. Winkler,et al.  Learning, Experimentation, and the Optimal Output Decisions of a Competitive Firm , 1982 .

[16]  Katy S. Azoury.  Bayes Solution to Dynamic Inventory Models Under Unknown Demand Distribution , 1985 .

[17]  Martin L. Puterman,et al.  The Censored Newsvendor and the Optimal Acquisition of Information , 2002, Oper. Res..

[18]  Ananth V. Iyer,et al.  Improved Fashion Buying with Bayesian Updates , 1997, Oper. Res..

[19]  Retsef Levi,et al.  Adaptive Data-Driven Inventory Control Policies Based on Kaplan-Meier Estimator for Censored Demand , 2010 .

[20]  Li Chen,et al.  Bounds and Heuristics for Optimal Bayesian Inventory Control with Unobserved Lost Sales , 2010, Oper. Res..

[21]  A. Mas-Colell,et al.  Microeconomic Theory , 1995 .

[22]  Suresh P. Sethi,et al.  A Note on 'The Censored Newsvendor and the Optimal Acquisition of Information' , 2009, Oper. Res..

[23]  S. Tokdar,et al.  Newsvendor Problem with Newsvendor Demand Distributions , 2010 .

[24]  Jing-Sheng Song,et al.  Analysis of Perishable-Inventory Systems with Censored Demand Data , 2008, Oper. Res..