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Branch-and-Cut for the Split Delivery Vehicle Routing Problem with Time Windows

整数规划 数学优化 子网 分支和切割 整数(计算机科学) 路径(计算) 布线(电子设计自动化) 剖切面法 计算机科学 匹配(统计) 车辆路径问题 还原(数学) 选择(遗传算法) 数学 统计 程序设计语言 人工智能 几何学 计算机安全 计算机网络
作者
Nicola Bianchessi,Stefan Irnich
出处
期刊:Transportation Science [Institute for Operations Research and the Management Sciences]
卷期号:53 (2): 442-462 被引量:40
标识
DOI:10.1287/trsc.2018.0825
摘要

The split delivery vehicle routing problem with time windows (SDVRPTW) is a notoriously hard combinatorial optimization problem. First, it is hard to find a useful compact mixed-integer programming (MIP) formulation for the SDVRPTW. Standard modeling approaches either suffer from inherent symmetries (mixed-integer programs with a vehicle index) or cannot exactly capture all aspects of feasibility. Because of the possibility to visit customers more than once, the standard mechanisms to propagate load and time along the routes fail. Second, the lack of useful formulations has rendered any direct MIP-based approach impossible. Up to now, the most effective exact algorithms for the SDVRPTW have been branch-and-price-and-cut approaches using path-based formulations. In this paper, we propose a new and tailored branch-and-cut algorithm to solve the SDVRPTW. It is based on a new, relaxed compact model, in which some integer solutions are infeasible for the SDVRPTW. We use known and introduce some new classes of valid inequalities to cut off such infeasible solutions. One new class is path-matching constraints that generalize infeasible-path constraints. However, even with the valid inequalities, some integer solutions to the new compact formulation remain to be tested for feasibility. For a given integer solution, we build a generally sparse subnetwork of the original instance. On this subnetwork, all time-window-feasible routes can be enumerated, and a path-based residual problem then solved to decide on the selection of routes, the delivery quantities, and thereby the overall feasibility. All infeasible solutions need to be cut off. For this reason, we derive some strengthened feasibility cuts exploiting the fact that solutions often decompose into clusters. Computational experiments show that the new approach is able to prove optimality for several previously unsolved instances from the literature. The e-companion is available at https://doi.org/10.1287/trsc.2018.0825 .
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