# An (R, S) Based Heuristic Model for the Stochastic Joint Replenishment Problem

## TL;DR

This paper develops a heuristic model based on an (R, S) policy for the stochastic joint replenishment problem, improving decision-making under demand uncertainty with a static-dynamic policy approach.

## Contribution

It extends a MILP model to approximate the optimal (σ, S) control policy for the JRP, demonstrating its effectiveness through computational experiments.

## Key findings

- The proposed approach outperforms existing methods in computational tests.
- The model effectively balances replenishment timing and order quantities under uncertainty.
- The heuristic provides a practical solution for complex inventory coordination problems.

## Abstract

This paper considers the periodic-review stochastic joint replenishment problem (JRP) under Bookbinder and Tan's static-dynamic uncertainty control policy. According to a static-dynamic uncertainty control rule, the decision maker fixes timing of replenishments once and for all at the beginning of the planning horizon, the inventory position is then raised to a predefined order-up-to-position at the beginning of each replenishment period. In this policy, freezing the replenishment times ameliorates the inherent difficulties pertinent to replenishment coordination of multiple products, whereas dynamic order quantities facilitate dealing with uncertain demands. We adapt and extend an earlier mixed integer linear programming (MILP) model for computing static-dynamic uncertainty policy parameters, and demonstrate that the same can be used to approximate the optimal control rule for the JRP, also known as $(\sigma, \vec{S})$ policy. An extensive computational study illustrates the effectiveness of our approach when compared to alternative approaches in the literature.