Analysis of an improved branch-and-cut formulation for the Inventory-Routing Problem with Transshipment

The Inventory-Routing Problem with Transshipment (IRPT) is an extension of the Inventory-Routing Problem (IRP), in which not only the supplier can deliver goods to the retailers, but also transshipments between retailers or between the supplier and a retailer are possible. In this paper, we investigate the branch-and-cut (B&C) formulation of the IRPT and propose four different types of improvements for it. We first develop two new sets of valid inequalities for the problem which greatly strengthen the linear relaxation of the problem. We then improve the bounds on the continuous delivery variables and use these improved bounds to tighten the inventory management constraints. Next, we reformulate the routing component of the problem by exploiting the possible presence of direct shipments in the optimal solution. Finally, we prove that some integer and continuous variables can be eliminated out of the mathematical formulation. Experimental results demonstrate that these improvements drastically reduce the computational burden for solving the IRPT to optimality. On a large set of benchmark instances our proposed branch-and-cut algorithm, which integrates these improvements, outperforms the current best results in the literature. In addition, we are able to find the optimal solutions for two instances whose optimal solution was not known until now.