Single Vehicle Routing Problem with a Predefined Customer Sequence, Stochastic Demands and Partial Satisfaction of Demands

We consider the problem of finding the optimal routing of a single vehicle that starts its route from a depot and delivers a product to N customers that are served according to a particular sequence. The vehicle during its route can return to the depot for restocking. The demands of the customers are random variables with known distributions. The actual demand of each customer is revealed as soon as the vehicle visits the customer's site. It is permissible to satisfy fully or to satisfy partially or not to satisfy the demand of a customer. The cost structure includes travel costs between consecutive customers, travel costs between the customers and the depot and penalty costs if a customer's demand is not satisfied or if it is satisfied partially. A dynamic programming algorithm is developed for the determination of the optimal routing policy. It is shown that the optimal routing policy has a specific threshold-type structure.