A Counterintuitive Example in Inventory Management

TL;DR

This paper presents a counterintuitive example in inventory management showing that a finite long-term average cost does not necessarily imply tightness of average expected ordering measures, challenging common assumptions.

Contribution

It provides a novel example demonstrating that finite average costs do not guarantee tightness of ordering measures, contradicting intuitive expectations in inventory control theory.

Findings

Finite average cost does not imply tightness of ordering measures.

Counterexample where occupation measures are tight but ordering measures are not.

Challenges assumptions about cost and measure tightness in inventory models.

Abstract

The paper \cite{helm:17} studies an inventory management problem under a long-term average cost criterion using weak convergence methods applied to average expected occupation and average expected ordering measures. Under the natural condition of inf-compactness of the holding cost rate function, the average expected occupation measures are seen to be tight and hence have weak limits. However inf-compactness is not a natural assumption to impose on the ordering cost function. For example, the cost function composed of a fixed cost plus proportional (to the size of the order) cost is not inf-compact. Intuitively, it would seem that imposing a requirement that the long-term average cost be finite ought to imply tightness of the average expected ordering measures; a lack of tightness should mean that the inventory process spends large amounts of time in regions that have arbitrarily large holding costs resulting in an infinite long-term average cost. This paper demonstrates that this intuition is incorrect by identifying a model and an ordering policy for which the resulting inventory process has a finite long-term average cost, tightness of the average expected occupation measures but which lacks tightness of the average expected ordering measures.