A branch-and-cut-and-price approach for the pickup and delivery problem with shuttle routes

doi:10.1016/j.ejor.2013.08.042

European Journal of Operational Research

Volume 236, Issue 3, 1 August 2014, Pages 849-862

https://doi.org/10.1016/j.ejor.2013.08.042 Get rights and content

Highlights

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We introduce the Pickup and Delivery Problem with Shuttle routes (PDPS).
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We propose one arc based model and two path based model for the PDPS.
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We develop a branch-and-cut-and-price algorithm.
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The method is evaluated on generated and real-world instances with up to 193 requests.
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Instances with up to 87 requests are solved to optimality in less than one hour.

Abstract

The Pickup and Delivery Problem with Shuttle routes (PDPS) is a special case of the Pickup and Delivery Problem with Time Windows (PDPTW) where the trips between the pickup points and the delivery points can be decomposed into two legs. The first leg visits only pickup points and ends at some delivery point. The second leg is a direct trip – called a shuttle – between two delivery points. This optimization problem has practical applications in the transportation of people between a large set of pickup points and a restricted set of delivery points.

This paper proposes three mathematical models for the PDPS and a branch-and-cut-and-price algorithm to solve it. The pricing sub-problem, an Elementary Shortest Path Problem with Resource Constraints (ESPPRC), is solved with a labeling algorithm enhanced with efficient dominance rules. Three families of valid inequalities are used to strengthen the quality of linear relaxations. The method is evaluated on generated and real-world instances containing up to 193 transportation requests. Instances with up to 87 customers are solved to optimality within a computation time of one hour.

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

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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View full text

A branch-and-cut-and-price approach for the pickup and delivery problem with shuttle routes

Highlights

Abstract

Section snippets

Benefits of transfers and motivation for the work

Related works and position of the problem

The pickup and delivery problem with shuttle routes

A branch-and-cut-and-price algorithm

Computational experiments

Conclusion

Acknowledgments

European Journal of Operational Research

European Journal of Operational Research

Computers & Industrial Engineering

Computers & Operations Research

Computers & Industrial Engineering

An exact algorithm for the pickup and delivery problem with time windows

Operations Research

Static pickup and delivery problems: A classification scheme and survey

TOP

Recent models and algorithms for one-to-one pickup and delivery problems

Cutting planes for branch-and-price algorithms

Networks

An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems

Networks

Minimum makespan multi-vehicle dial-a-ride

Shortest path problems with resource constraints

Subset-row inequalities applied to the vehicle routing problem with time windows

Operations Research

The splittable pickup and delivery problem with reloads

European Journal of Industrial Engineering

2-path cuts for the vehicle routing problem with time windows

Transportation Science

Two exact algorithms for the distance constrained vehicle routing problem

Networks