We introduce the location-or-routing problem (LoRP), which integrates the facility location and the vehicle routing problems by uncovering a new connection from the customer coverage perspective. In the LoRP, open facilities cover the customers in their neighborhood and the uncovered customers are transported to open facilities by capacitated vehicles. Each facility has a maximum coverage range and each vehicle route is constrained by a maximum length. In this setting, a customer can be covered either by ‘location’ or by ‘routing’, hence the problem name. We discuss several application areas of LoRP and present its relation to the location and routing problems. We develop a set covering model and a branch-and-price algorithm as an exact solution methodology. The results show that the facility coverage range is an important determinant of the number and location of open facilities. We find that the vehicle routes play a decreasing role on the total cost as facility range increases. Furthermore, trade-off analyses on random graphs show that the total cost decreases almost linearly by increasing facility coverage range. We investigate the reasons behind this observation using arguments from asymptotic analysis and find that it is a common property when the customers are uniformly distributed.

我们介绍了位置或路由问题（LoRP），该问题通过从客户覆盖范围的角度发现新的连接来集成设施位置和车辆路线问题。在LoRP中，开放式设施覆盖了附近的客户，而未发现的客户则被载有车辆的车辆运送到开放式设施中。每个设施都具有最大覆盖范围，并且每个车辆路线都受到最大长度的限制。在这种情况下，可以通过“位置”或“路由”来覆盖客户，因此可以用问题名称来覆盖。我们讨论了LoRP的几个应用领域，并提出了它与位置和路由问题的关系。我们开发了一套覆盖模型和一种分支价格算法作为精确的解决方法。结果表明，设施覆盖范围是开放设施数量和位置的重要决定因素。我们发现，随着设施范围的增加，车辆路线在总成本中的作用逐渐降低。此外，对随机图的权衡分析表明，总成本几乎会随着设施覆盖范围的增加而线性降低。我们使用来自渐近分析的参数来调查此观察的原因，并发现当客户均匀分布时，这是一个共同的属性。