Single Vehicle Routing Problems With a Predefined Customer Order, Unified Load and Stochastic Discrete Demands

Abstract

We consider the problem of finding the optimal routing of a single vehicle that delivers K different products to N customers that are served according to a particular order. It is assumed that the demands of the customers for each product are discrete random variables, and the total demand of each customer for all products cannot exceed the vehicle capacity. The joint probability mass function of the demands of each customer is known. It is assumed that all products are stored together in the vehicle's single compartment. The policy that serves all customers with the minimum total expected cost is found by implementing a suitable dynamic programming algorithm. We prove that this policy has a specific threshold-type structure. Furthermore, we study a corresponding infinite-time horizon problem in which the service of the customers is not completed when the last customer has been serviced but it continues periodically with the same customer order. The demands of each customer for the products have the same distributions at different periods. The discounted cost optimal policy and the average-cost optimal policy have the same structure as the optimal policy in the finite-horizon problem. Numerical results are given that illustrate the structural results.