We consider the problem of finding the optimal routing of a single vehicle that starts its route from a depot and picks up from and delivers K different products to N customers that are served according to a predefined customer sequence. The vehicle is allowed during its route to return to the depot to unload returned products and restock with new products. The items of all products are of the same size. For each customer the demands for the products that are delivered by the vehicle and the quantity of the products that is returned to the vehicle are discrete random variables with known joint distribution. Under a suitable cost structure, it is shown that the optimal policy that serves all customers has a specific threshold-type structure. We also 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 indefinitely with the same customer order. For each customer, the joint distribution of the quantities that are delivered and the quantity that is picked up is the same at each cycle. 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. 
Finite and infinite-horizon single vehicle routing problems with a predefined customer sequence and pickup and delivery