Inventory Control with Modulated Demand and a Partially Observed Modulation Process

TL;DR

This paper studies an inventory control problem with a partially observed modulation process affecting demand, providing conditions for optimal policies and characterizing their structure within the belief space.

Contribution

It introduces an attainability condition ensuring optimal policies depend on the belief state, and characterizes their structure using linear inequalities.

Findings

Optimal base stock level is constant within belief regions when $K=0$.

$s$ and $S$ values are region-constant for $K>0$.

Heuristic bounds are provided when the attainability condition does not hold.

Abstract

We consider a periodic review inventory control problem having an underlying modulation process that affects demand and that is partially observed by the uncensored demand process and a novel additional observation data (AOD) process. We present an attainability condition, AC, that guarantees the existence of an optimal myopic base stock policy if the reorder cost $K=0$ and the existence of an optimal $(s, S)$ policy if $K>0$, where both policies depend on the belief function of the modulation process. Assuming AC holds, we show that (i) when $K=0$, the value of the optimal base stock level is constant within regions of the belief space and that each region can be described by two linear inequalities and (ii) when $K>0$, the values of $s$ and $S$ and upper and lower bounds on these values are constant within regions of the belief space and that these regions can be described by a finite set of linear inequalities. A heuristic and bounds for the $K=0$ case are presented when AC does not hold. Special cases of this inventory control problem include problems considered in the Markov-modulated demand and Bayesian updating literatures.