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A Branch-and-Cut Algorithm for a Vendor-Managed Inventory-Routing Problem

Department of Quantitative Methods, University of Brescia, Brescia, Italy
Department of Quantitative Methods, University of Brescia, Brescia, Italy
Canada Research Chair in Distribution Management, HEC Montréal and Center for Research on Transportation, Montréal, Canada
Department of Quantitative Methods, University of Brescia, Brescia, Italy
archetti{at}eco.unibs.it
bertazzi{at}eco.unibs.it
gilbert{at}crt.umontreal.ca
speranza{at}eco.unibs.it

We consider a distribution problem in which a product has to be shipped from a supplier to several retailers over a given time horizon. Each retailer defines a maximum inventory level. The supplier monitors the inventory of each retailer and determines its replenishment policy, guaranteeing that no stockout occurs at the retailer (vendor-managed inventory policy). Every time a retailer is visited, the quantity delivered by the supplier is such that the maximum inventory level is reached (deterministic order-up-to level policy). Shipments from the supplier to the retailers are performed by a vehicle of given capacity. The problem is to determine for each discrete time instant the quantity to ship to each retailer and the vehicle route. We present a mixed-integer linear programming model and derive new additional valid inequalities used to strengthen the linear relaxation of the model. We implement a branch-and-cut algorithm to solve the model optimally. We then compare the optimal solution of the problem with the optimal solution of two problems obtained by relaxing in different ways the deterministic order-up-to level policy. Computational results are presented on a set of randomly generated problem instances.

Key Words: supply chain management; vendor-managed inventory (VMI); deterministic order-up-to level; inventory-routing problem; branch and cut
History: Received: October 2005; revised: June 2006; accepted: November 2006.

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