A branch-and-cut-and-price approach for the pickup and delivery problem with shuttle routes
Section snippets
Benefits of transfers and motivation for the work
One reason to consider the transfer of requests is the potential savings that can be achieved by this practice. This is illustrated by the example given in Fig. 1. We consider 9 requests: requests 1, 2 and 3 share the pickup point p123, requests 4, 5 and 6 share the pickup point p456, requests 7, 8 and 9 share the pickup point p789. Requests 1, 2, 4, 5, 7 and 8 must be delivered to d124578 while requests 3, 6, and 9 must be delivered to d369. The nodes o and o′ represent the starting depot and
Related works and position of the problem
Although the PDP has been intensively studied in the last 15 years, the PDPT has been the subject of very few works. Cortés et al. (2010) present a mathematical formulation of the PDPT that is solved using a branch-and-cut algorithm. Instances with 6 requests and 2 vehicles are solved to optimality. Kerivin et al. (2008) consider a PDP where every request can be split as well as transferred from one vehicle to another at every node of the problem. This problem has no time window and is solved
The pickup and delivery problem with shuttle routes
This section formally presents three mathematical models for the PDPS. The PDPS was motivated by applications in the field of transportation of people with disabilities (Lehuédé, Pavageau, & Péton, 2009). This concerns people who require daily trips to and from their home to schools or vocational rehabilitation centers. The centers or local authorities have to pay for the increasing transportation costs. In practice, dozens of people share the same delivery point so the number of pickup points
A branch-and-cut-and-price algorithm
The branch-and-cut-and-price algorithm consists in using column generation to compute the linear relaxation of each node of the branch-and-cut tree. This method has already been successfully applied to solve the related Pickup and Delivery Problem with Time Windows (Ropke and Cordeau, 2009, Baldacci et al., 2011). For each node, as long as valid inequalities strengthening the linear relaxation are found, they are added. This section describes algorithms for solving the pricing sub-problem
Computational experiments
The algorithm presented in this paper was developed in C++ and run on an i3-530 Ubuntu 10.04 computer. We used IBM Ilog Cplex 12.2 to solve all linear relaxations. The main goal of this section is to evaluate the performance of the algorithm described in the preceding section. We first assess the performance of arc-based and path-based formulations on a set of small-sized instances. Next, we measure the efficiency of various combinations of valid inequalities. Models SP1 and SP2 are then
Conclusion
Consolidation of requests at transfer points is likely to improve the efficiency of complex door-to-door transportation systems. In this paper, we proposed a graph model for transfer points, an arc-based and two path-based models for the Pickup and Delivery Problem with Shuttle routes (PDPS). We described a branch-and-cut-and-price method to solve the PDPS. The branch-and-price algorithm raises two sub-problems: an elementary shortest path problem with resource constraints solved by a labeling
Acknowledgments
The authors thank the doctoral school ED STIM for funding the research stay of Renaud Masson at the DTU. They are grateful to the Conseil Général de Loire-Atlantique and the ADAPEI 44 for their authorization to use their data. The work of Stefan Ropke was sponsored by the Danish Agency for Science, Technology and Innovation (Project “Intelligent Freight Transport Systems”).
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