Optimal Inventory Policies When the Demand Distribution is Not Known

This paper analyzes the stochastic inventory control problem when the demand distribution is not known. In contrast to previous Bayesian inventory models, this paper adopts a non-parametric Bayesian approach in which the firm's prior information is characterized by a Dirichlet process prior. This provides considerable freedom in the specification of prior information about demand and it permits the accommodation of fixed order costs. As information on the demand distribution accumulates, optimal history-dependent (s,S) rules are shown to converge to an (s,S) rule that is optimal when the underlying demand distribution is known.

[1]  K. Hinderer,et al.  Foundations of Non-stationary Dynamic Programming with Discrete Time Parameter , 1970 .

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

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

[4]  M. Degroot.  Optimal Statistical Decisions , 1970 .

[5]  Prajit K. Dutta,et al.  Parametric continuity in dynamic programming problems , 1994 .

[6]  K. Arrow,et al.  Optimal Inventory Policy. , 1951 .

[7]  Nancy L. Stokey,et al.  Recursive methods in economic dynamics , 1989 .

[8]  D. Iglehart.  The Dynamic Inventory Problem with Unknown Demand Distribution , 1964 .

[9]  J. Fabius.  Asymptotic behavior of bayes' estimates , 1963 .

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

[11]  M. Rothschild,et al.  Searching for the Lowest Price When the Distribution of Prices Is Unknown , 1974, Journal of Political Economy.

[12]  S. Edward Boylan.  Stability Theorems for Solutions to the Optimal Inventory Equation , 1969 .

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

[14]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

[15]  Gabriel Talmain.  Search from an unkown distribution an explicit solution , 1992 .

[16]  Patrick Suppes,et al.  MATHEMATICAL METHODS IN THE SOCIAL SCIENCES, 1959 , 1960 .

[17]  Katy S. Azoury,et al.  A Comparison of the Optimal Ordering Levels of Bayesian and Non-Bayesian Inventory Models , 1984 .

[18]  T. Ferguson.  Prior Distributions on Spaces of Probability Measures , 1974 .

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

[20]  J. Kiefer,et al.  The Inventory Problem: II. Case of Unknown Distributions of Demand , 1952 .

[21]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[22]  D. Blackwell,et al.  Ferguson Distributions Via Polya Urn Schemes , 1973 .

[23]  Manfred SchÄl,et al.  Conditions for optimality in dynamic programming and for the limit of n-stage optimal policies to be optimal , 1975 .

[24]  Donald A. Berry,et al.  Bayesian nonparametric bandits , 1985 .

[25]  S. Karlin.  Dynamic Inventory Policy with Varying Stochastic Demands , 1960 .