A Bayesian approach to a dynamic inventory model under an unknown demand distribution

Decision making on stock levels in cases of uncertain demand rate

The theory of uniformly most accurate upper tolerance limits is presented for families of discrete distributions having monotone likelihood ratios. Explicit formulae are derived for binomial, negative-binomial and Poisson distributions. The relationship to systems of uniformly most accurate upper confidence limits is exposed.

Uniformly Most Accurate Upper Tolerance Limits for Monotone Likelihood Ratio Families of Discrete Distributions

This paper compares four major schemes used for forecasting mean demand to be used as input into an inventory model so that ‘ optimum ’ stockage levels can be obtained. The inventory model is the classical order up to S, infinite horizon model with carry-over from period to period and complete back-ordering. Maximum likelihood, exponential smoothing, standard Bayes and adaptive Bayes schemes are used and results, via Monte Carlo simulation, are obtained on the average costs per period for (1) stationary demand, (2) long-term trend and (3) ‘ shock ’ changes in mean demand.

http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0742340&Location=U2&doc=GetTRDoc.pdf

A Comparison of Maximum Likelihood Exponential Smoothing and Bayes Forecasting Procedures in Inventory Modeling

The two inventory echelons under consideration are the depot, D, and k tender ships E1, …, Ek. The tender ships supply the demand for certain parts of operational boats (the customers). The statistical model assumes that the total monthly demands at the k tenders are stationary independent Poisson random variables, with unknown means λ1, …, λk. The stock levels on the tenders, at the heginning of each month, can be adjusted either by ordering more units from the depot, or by shipping bach to the depot an excess stock. There is no traffic of stock between tenders which is not via the depot. The lead time from the depot to the tenders is at most 1 month. The lead time for orders of the depot from the manufacturer is L months. The loss function due to erroneous decision js comprised of linear functions of the extra monthly stocks, and linear functions of shortages at the tenders and at the depot over the N months. A Bayes sequential decision process is set up for the optimal adjustment levels and orders of the two echelons. The Dynamic Programming recursive functions are given for a planning horizon of N months.

A two‐echelon multi‐station inventory model for navy applications

Bayesian determination of optimal stock levels is studied for the case of Poisson distribution of the demand variable, and prior gamma distribution of the expected demand. Bayes sequential procedure is derived, assuming that stock level can be adjusted at the beginning of each period so that a shortage can be immediately replenished and an overstock can be corrected. The Bayes sequential procedure is more difficult to obtain if this assumption is removed. A dynamic programming method for obtaining the general Bayes sequential procedure is outlined. Finally, an empiric Bayes estimation procedure of the optimal Bayesian stock level is presented.

S. Zacks

Papers

Bayes sequential design of stock levels