# Advanced Problems in Inventory Theory

Project: Research

## Description

Inventory Control is one of the first and most important domains of application in Operations Research. Although static models like EOQ (Economic Order Quantity) and Newsvendor problem are very old, the dynamic models were considered only in the late fifties and sixties, when Dynamic Programming techniques became available. H. Scarf introduced in 1960 the concept of s,S policy and since then considerable developments have followed both in Applied Mathematics and Operations Research. Motivated by Inventory Control, the Author introduced with J.L. Lions the concept of Quasi- Variational Inequalities to solve Impulse Control problems. Since then, the mathematical developments have followed a route which provides abstract results like existence and uniqueness of the solution of the Q.V.I. and existence of an optimal feedback rule. Numerical Analysis has also progressed considerably. Although essential, these results and methods do not sufficiently meet the need of practitioners to use simple decision rules, even though the complexity has increased considerably. The overwhelming success of s,S policy stems from the fact that it is very intuitive for practitioners. There is a strong need to put the mathematical effort on addressing more realistic and complex problems than the standard ones, still trying to design simple decision rules which can be optimal or quasi-optimal. The advantage of relying on general mathematical results and numerical analysis approaches is that we can obtain a benchmark to test the quality of the decision rules which are designed. In the numerous works in which one tries to extend the Base Stock and s,S policies, this is generally not done, thus the optimality is not proven, or in case of approximations the quality of the approximation is not established.We have chosen three areas of research as the main ones. The first one concerns optimization of average costs, which we want to study with the powerful techniques of Ergodic Theory and Ergodic Control. We want to study how the transients converge towards the steady state. The full justification of the results needs advanced methods of Ergodic Theory. The second area concerns more sophisticated models for the demand. Basic models consider demand external and a sequence of independent variables. Markov demand has been considered only recently. The third area concerns inventory control with partial information, in which the A. has been active in recent years in the U.S.

## Detail(s)

Project number 9041857 GRF Finished 1/01/12 → 27/06/16