An improved formulation for the inventory routing problem with time-varying demands

The Inventory Routing Problem (IRP) is a broad class of complex routing problems where the quantities of delivered products must also be determined. In this paper, we consider the classic IRP where a single supplier must determine when to visit its customers, how much to deliver and how to combine the customer visits in each period into routes. We propose a branch-and-cut algorithm based on a new mathematical formulation for the IRP, improving the average lower bound obtained from algorithms based on the branch-and-cut methodology. The new formulation substitutes parts of the original formulation with a convex combination of extreme points. We call these extreme points customer schedules and for each customer they contain information about delivery periods and corresponding delivered quantities. We show that this algorithm outperforms a state-of-the-art branch-and-cut algorithm on instances with time-varying demands. The customer schedule-based algorithm obtains better lower bounds, which improves the average optimality gap by 29% and 15% on two new sets of instances with time-varying demands.