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

• Published in: Operations research Vol. 57; no. 2; pp. 314 - 326
• Main Authors: Archibald, T. W, Black, D. P, Glazebrook, K. D
• Format: Journal Article
• Language: English
• Published: Hanover, MD INFORMS 01.03.2009; Institute for Operations Research and the Management Sciences
• Subjects: Analysis; applications; Applied sciences; Carrying costs; Cost estimates; Decision theory. Utility theory; Delivery costs; Delivery vehicles; Dynamic programming; dynamic programming/optimal control; Exact sciences and technology; Heuristic; Heuristic programming; Heuristics; Inventories; Inventory control; Inventory control, production control. Distribution; inventory/production; Lagrangian function; Mathematical functions; Minimization of cost; Operational research and scientific management; Operational research. Management science; Optimal policy; Perceptual localization; Poisson distribution; policies; probability; Route optimization; Routing; stochastic model applications; Stochastic models; Studies; United States; Stochastic model; Probabilistic approach; Refeeding; Decision making; Priority; Routing; Modeling; Exact solution; Delivery (good); Direct problem; Production management; Lagrange multiplier; Real function; Replenishment; Optimal control; Heuristic method; Greedy algorithm; Restless bandit; Inventory control; Relaxation method; Dynamic programming; dynamic programming/optimal control: applications; Localization; models; inventory/production: policies; probability: stochastic model applications
• ISSN: 0030-364X; 1526-5463
• DOI: 10.1287/opre.1070.0505

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.