Optimal delivery of two similar products to N ordered customers with product preferences

Abstract

We study a mathematical model for a specific vehicle routing problem in which a vehicle starts its route from a depot loaded with items of two similar but not identical products. The vehicle must deliver the products to N customers according to a predefined sequence. It is assumed that each customer prefers either product 1 or product 2 with known probabilities and the quantity that each customer demands is a random variable with known distribution. The actual preference and demand of each customer are revealed upon the vehicle's arrival at customer's site. The demand of each customer cannot exceed the vehicle capacity and the vehicle is allowed during its route to return to the depot to restock with quantities of both products. The travel costs between consecutive customers and the travel costs between the customers and the depot are known. If there is shortage for the desired product it is permitted to deliver the other product at a reduced price. The optimal routing strategy is found by implementing a suitable stochastic dynamic programming algorithm. It is possible to prove that the optimal routing strategy has a specific threshold-type structure. Furthermore, if we consider the same problem without the assumption that the customers are ordered, numerical experiments indicate that the optimal routing strategy can be computed for N≤8.