Stochastic single vehicle routing problem with ordered customers and partial fulfilment 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 order. The vehicle during its route can return to the depot for replenishment. It is assumed a stochastic demand for each customer. The actual demand of each customer becomes known upon the vehicle's arrival at 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. 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 smaller or equal to nine.