An Inventory System with Two Supply Modes and L\'evy Demand

Abstract

This study considers a continuous-review inventory model for a single item with two replenishment modes. Replenishments may occur continuously at any time with a higher unit cost, or at discrete times governed by Poisson arrivals with a lower cost. From a practical standpoint, the model represents an inventory system with random deal offerings. Demand is modeled by a spectrally positive L\'evy process (i.e., a L\'evy process with only positive jumps), which greatly generalizes existing studies. Replenishment quantities are continuous and backorders are allowed, while lead times, perishability, and lost sales are excluded. Using fluctuation theory for spectrally one-sided L\'evy processes, the optimality of a hybrid barrier policy incorporating both kinds of replenishments is established, and a semi-explicit expression for the associated value function is computed. Numerical analysis is provided to support the optimality result.