A Branch-and-Cut Algorithm for the Inventory Routing Problem with Product Substitution

Abstract

The inventory routing problem arises in vendor managed systems, in which a supplier is responsible for replenishing inventories of products at a set of retailers and manages the logistics over a finite planning horizon with a fleet of capacitated vehicles. We consider this problem in the context of the distribution of two different quality products with one-way product substitution, where the high-quality product may be used to meet the demand for the low-quality product. The routes of vehicles and the quantities of products sent to each retailer in each period are determined in such a way that no stockouts occur and the total cost associated with inventory holding, substitution and transportation is minimized. In this study, we derive a mixed integer linear programming formulation for the problem, strengthen this formulation with valid inequalities and develop a branch-and-cut algorithm as an exact solution method. We conduct experiments using benchmark and randomly generated instances to analyze the effectiveness of our solution method and investigate the relationship between product substitution decisions and system costs under different demand, supply, vehicle capacity and substitution cost settings.