## Branch-and-cut-and-price for the pickup and delivery problem with time windows (2007)

Citations: | 4 - 3 self |

### BibTeX

@MISC{Ropke07branch-and-cut-and-pricefor,

author = {Stefan Ropke and Jean-François Cordeau},

title = {Branch-and-cut-and-price for the pickup and delivery problem with time windows},

year = {2007}

}

### OpenURL

### Abstract

In the pickup and delivery problem with time windows (PDPTW), vehicle routes must be designed to satisfy a set of transportation requests, each involving a pickup and a delivery location, under capacity, time window, and precedence constraints. This paper introduces a new branch-and-cut-and-price algorithm in which lower bounds are computed by solving through column generation the linear programming relaxation of a set partitioning formulation. Two pricing subproblems are considered in the column generation algorithm: an elementary and a non-elementary shortest path problem. Valid inequalities are added dynamically to strengthen the relaxations. Some of the previously proposed inequalities for the PDPTW are also shown to be implied by the set partitioning formulation. Computational experiments indicate that the proposed algorithm outperforms a recent branch-and-cut algorithm.

### Citations

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Citation Context ...those introduced by Ropke et al. (2007) (data set one) and the instances proposed by Li and Lim (2001) (data set two). The instances proposed by Li and Lim originate from the Solomon VRPTW instances (=-=Solomon, 1987-=-). For these instances we minimize the total traveled distance in our objective. The Li and Lim instances are divided into two series. Those in the first series have a short planning horizon while tho... |

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Citation Context ...vely. We refer to the paper by Ropke et al. (2007) for examples and figures. 4.3 Rounded capacity inequalities Rounded capacity inequalities which are often used in the context of the VRP (see, e.g., =-=Naddef and Rinaldi, 2002-=-) can also be used for the PDPTW. For any node subset S ⊆ P ∪D, let κ(S) be a lower bound on the number of times that vehicles must enter the set. The following inequality is then clearly valid: x(δ +... |

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Citation Context ...xact methods has been somewhat limited. Two main approaches have been used to solve the PDPTW exactly: branch-and-price and branch-and-cut. Branch-and-price methods (see, e.g., Barnhart et al., 1998; =-=Desaulniers et al., 1998-=-) use a branch-and-bound scheme in which lower bounds are computed by column generation. The first branch-and-price algorithm for the PDPTW was proposed by Dumas et al. (1991) who considered a set par... |

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Citation Context ...roblems for the PDPTW in the literature. In the first application of column generation to the PDPTW (Dumas et al., 1991) a non-elementary shortest path problem was solved while later implementations (=-=Sol, 1994-=-; Sigurd et al., 2004) have used an elementary shortest path problem. Both shortest path problems are N P-hard. Little is known about how the relaxations obtained by solving these two subproblems diff... |

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Citation Context ...ice algorithm for the PDPTW. It is well known that set partitioning formulations of vehicle routing problems tend to provide stronger lower bounds than formulations based on arc (flow) variables (see =-=Bramel and Simchi-Levi, 2002-=-). Two different shortest path problems have been considered as pricing subproblems for the PDPTW in the literature. In the first application of column generation to the PDPTW (Dumas et al., 1991) a n... |

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Citation Context ... the problem, work on exact methods has been somewhat limited. Two main approaches have been used to solve the PDPTW exactly: branch-and-price and branch-and-cut. Branch-and-price methods (see, e.g., =-=Barnhart et al., 1998-=-; Desaulniers et al., 1998) use a branch-and-bound scheme in which lower bounds are computed by column generation. The first branch-and-price algorithm for the PDPTW was proposed by Dumas et al. (1991... |

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Citation Context ...in the vehicle. The resulting problem is called the Dial-a-Ride Problem (DARP). The VRP and VRPTW are well known combinatorial optimization problems which have received a lot of attention (see, e.g., =-=Toth and Vigo, 2002-=-). Since it generalizes the VRPTW, the PDPTW is clearly N P-hard. Over the last few decades, several heuristics have been proposed for the PDPTW. However, because of the difficulty of the problem, wor... |

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