Optimal periodic replenishment policies for spectrally positive L\'evy demand processes

TL;DR

This paper analyzes an inventory control problem with spectrally positive Lévy demand, demonstrating the optimality of a periodic barrier policy and providing explicit solutions and numerical insights.

Contribution

It introduces a novel optimal replenishment policy framework for Lévy demand processes with replenishments at exponential times, using scale functions for explicit solutions.

Findings

Optimal periodic barrier replenishment policy identified

Explicit expressions for value functions derived

Numerical results illustrate policy effectiveness

Abstract

We consider a version of the stochastic inventory control problem for a spectrally positive L\'evy demand process, in which the inventory can only be replenished at independent exponential times. We show the optimality of a periodic barrier replenishment policy that restocks any shortage below a certain threshold at each replenishment opportunity. The optimal policies and value functions are concisely written in terms of the scale functions. Numerical results are also provided.