arXiv 2207.045202022-07-10math.OCcs.AI

车辆路径问题中的局部区域路径

Local Area Routes for Vehicle Routing Problems

University of California, Irvine（加州大学欧文分校）；Laminaar Optimization Research Group（Laminaar优化研究组）

Udayan Mandal, Amelia Regan, Julian Yarkony

摘要：

我们考虑一种方法来提高效率的列生成（CG）方法解决车辆路径问题。我们引入局部区域（LA）的路由松弛，一种替代/补充常用的NG路由松弛和递减状态空间松弛（DSSR）内的CG配方。LA路由是ng-routes的子集和基本路由的超集。通常，CG的定价阶段必须使用计算上昂贵的过程来产生基本路由，这些基本路由是没有重复客户的路由。非基本路线至少访问一个客户一次以上，形成一个循环。LA路线以允许有效定价的方式放松了作为基本路线的约束。LA路线最好理解为NG路线松弛。Ng-routes是允许在空间中具有非局部循环的路由;这意味着循环中至少有一个中间客户（称为断路器）必须考虑循环中的起始客户在空间上远离。LA路由使用一组对应于从路由开始到结束排序的路由上的客户的特殊索引来描述。LA路由松弛通过另外强制断路器必须位于特殊索引处来进一步限制超出ng-routes的允许循环的集合，其中特殊索引的集合被递归地定义如下。路由中的第一个特殊索引位于索引1处，这意味着它与路由中的第一个客户相关联。第k个特殊索引对应于第k-1个特殊索引之后的第一个客户，其不被认为是位于第k-1个特殊索引处的客户的邻居（被认为在空间上远离）。我们表明，LA路线松弛可以显着提高计算速度的定价相比，标准的DSSR。

英文摘要：

We consider an approach for improving the efficiency of column generation (CG) methods for solving vehicle routing problems. We introduce Local Area (LA) route relaxations, an alternative/complement to the commonly used ng-route relaxations and Decremental State Space Relaxations (DSSR) inside of CG formulations. LA routes are a subset of ng-routes and a super-set of elementary routes. Normally, the pricing stage of CG must produce elementary routes, which are routes without repeated customers, using processes which can be computationally expensive. Non-elementary routes visit at least one customer more than once, creating a cycle. LA routes relax the constraint of being an elementary route in such a manner as to permit efficient pricing. LA routes are best understood in terms of ng-route relaxations. Ng-routes are routes which are permitted to have non-localized cycles in space; this means that at least one intermediate customer (called a breaker) in the cycle must consider the starting customer in the cycle to be spatially far away. LA routes are described using a set of special indexes corresponding to customers on the route ordered from the start to the end of the route. LA route relaxations further restrict the set of permitted cycles beyond that of ng-routes by additionally enforcing that the breaker must be a located at a special index where the set of special indexes is defined recursively as follows. The first special index in the route is at index 1 meaning that it is associated with the first customer in the route. The k'th special index corresponds to the first customer after the k-1'th special index, that is not considered to be a neighbor of (considered spatially far from) the customer located at the k-1'th special index. We demonstrate that LA route relaxations can significantly improve the computational speed of pricing when compared to the standard DSSR.

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英文摘要：

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