Indexability and Index Heuristics for a Simple Class of Inventory Routing Problems

General rights Copyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. We utilise and develop Whittle's restless bandit formulation to analyse a simple class of inventory routing problems with direct deliveries. These routing problems arise from the practice of vendor-managed inventory replenishment and concern the optimal replenishment of a collection of inventory holding locations controlled centrally by a decision maker who is able to monitor inventory levels throughout the network. We develop a notion of location indexability from a Lagrangian relaxation of the problem and show that (subject to mild conditions) the locations are indeed indexable. We thus have a collection of location indices in closed form, namely, real-valued functions of the inventory level (one for each location), which measure in a natural way (namely, as a fair charge for replenishment) each location's priority for inclusion in each day's deliveries. We discuss how to use such location indices to construct heuristics for replenishment and assess a greedy index heuristic in a numerical study where it performs strongly. A simpler approximate index analysis is available for the case in which the demand at each location is Poisson. This analysis permits a more explicit characterisation of the range of holding cost rates for which (approximate) location indexability is guaranteed.