A Pickup-and-Delivery Routing Problem with Stochastic Demands

The One-Commodity Pickup-and-Delivery Travelling Salesman Problem (1-PDTSP) is defined on a directed graph as follows. A node of the graph represents a depot and the other nodes represent customers. Every customer has a positive or negative demand of a product. The product is the same for all customers. Customers with positive demand are called pickups and customers with negative demand are called deliveries. A tour is a route for a vehicle which departures from the depot and visits each customer exactly once before returning back to the depot. The vehicle has a given capacity and leaves the depot with some initial load. Given the vehicle capacity and the initial load, a tour is said to be feasible if it completely satisfies the demand of each customer without violating the capacity limits of the vehicle. Given a cost associated with each arc of the graph, an optimal solution is a feasible tour with minimum total cost. The 1-PDTSP is the combinatorial optimization problem of finding such an optimal solution. For low capacities, the problem may be infeasible. If the problem is feasible then the 1-PDTSP also includes finding the appropriated initial load of the vehicle that makes an optimal tour feasible. The initial load may be different from zero and the vehicle capacity. Although the capacity of the vehicle is typically an input of the 1-PDTSP, one could also address the different problem of finding the minimum capacity such that the 1-PDTSP is feasible. While the 1-PDTSP was first introduced in [1], the latter problem has not been approached in the literature (as far as we know). For the 1-PDTSP, instead, optimal and near-optimal algorithms have been proposed. This paper considers a stochastic variant of the 1-PDTSP where we relax the hypothesis that the demand of each customer is deterministic and known in advance. Instead, we assume that the demand of each customer is a random variable with a discrete probability distribution. The particular situation where each random value is replaced by one of each realizations leads to what is called scenario. Therefore, unlike in the 1-PDTSP where only one scenario defining the demands of the customers is possible, in this paper we consider that many scenarios are possible, each one with a known probability. With this stochastic input, the meaning of “feasible tour” and “optimal solution” has to be adapted. Indeed, a tour can turn out to be feasible or unfeasible depending on the scenario. A tour is feasible for a given scenario and a given capacity when it is possible to find an initial load such that all customer requests will be satisfied following this tour. This corresponds to the deterministic situation. Assume first that the capacity has to be decided before the scenario is given and the initial load can be decided when the scenario is given. A tour and a vehicle capacity are said to be adaptable when for each scenario one can find an initial load such that the tour is feasible with the given capacity. Assume now that the vehicle capacity and the initial load have to be decided before the scenario is given. A tour is said to be survivable if there exists a vehicle capacity and an initial load that make the tour feasible for all scenarios. Thus, the deterministic concept of feasibility is replaced by adaptability or survivability, depending on the moment the information on the scenario is made available. In both cases, vehicle capacity has to be decided in advance, which is nothing else than deciding which vehicle to purchase and is typically done under incomplete information. When the information on the scenario may be obtained in advance (e.g., on a daily basis) the concept of adaptability applies. If not, the survivable case is needed. Clearly adaptability is achieved with a smaller capacity than survivability. In this paper we study the question