Inventory Policies for Two Products under Poisson Demand: Interaction between Demand Substitution, Limited Storage Capacity and Replenishment Time Uncertainty

TL;DR

This paper models a two-product inventory system with Poisson demand, limited storage, and substitution, analyzing how demand substitution and replenishment uncertainty impact optimal policies and profits.

Contribution

It develops a Markov chain model for such systems and proves submodularity of the profit function, enabling efficient optimization of order quantities.

Findings

Demand substitution increases profit under limited capacity.

Replenishment time uncertainty affects optimal order levels.

Submodularity property improves computational efficiency.

Abstract

We consider a two-product inventory system with independent Poisson demands, limited joint storage capacity and partial demand substitution. Replenishment is performed simultaneously for both products and the replenishment time may be fixed or exponentially distributed. For both cases we develop a Continuous Time Markov Chain model for the inventory levels and derive expressions for the expected profit per unit time. We prove that the profit function is submodular in the order quantities, which allows for a more efficient algorithm to determine the optimal ordering policy. Using computational experiments we assess the effect of substitution and replenishment time uncertainty on the order quantities and the profit as a function of the storage capacity.