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2009The Pickup and Delivery Problem with Transfers Formulation and a Branch-and-Cut Solution Method.pdf

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2009The Pickup and Delivery Problem with Transfers Formulation and a Branch-and-Cut Solution Method

THE PICKUP AND DELIVERY PROBLEM WITH TRANSFERS: FORMULATION AND A BRANCH-AND-CUT SOLUTION METHOD CRISTIA?N E. CORTE?S, MARTI?N MATAMALA, AND CLAUDIO CONTARDO Abstract. In this paper, a strict formulation of a generalization of the classical pick up and de- livery problem is presented. Here, we add the flexibility of providing the option for passengers to transfer from one vehicle to another at specific locations. As part of the mathematical formula- tion, we include transfer nodes where vehicles may interact interchanging passengers. Additional variables to keep track of customers along their route are considered. The formulation has been proven to work correctly, and by means of a simple example instance, we conclude that there exist some configurations in which a scheme allowing transfers results in better quality optimal solutions. Finally, a solution method based on Benders decomposition is addressed. We compare the computational effort of this application with a straight branch and bound strategy; we also provide insights to develop more efficient set partitioning formulations and associated algorithms for solving real-size problems. 1. Introduction The pickup and delivery problem (PDP), as defined in the specialized literature, has been one of the most studied network logistic problems in the last decade [Desrosiers et al., 1995, Savelsbergh and Sol, 1995, Desrochers et al., 1988]. The PDP can be conceptually described as finding the optimal way of assigning a set of transportation requests to a fleet of vehicles (initially located at several depots), by minimizing a specific purpose objective function, subject to a variety of constraints. The objective function may include components such as operational costs, number of vehicles, customer’s level of service, and so on. A transportation request consists of picking up a certain number of passengers from a predeter- mined pickup location and dropping them off at a predetermined delivery location. In the version