In this paper we deal with an extension of the crossing postman problem to design routes that have to visit different shapes of dimensional elements rather than edges. This problem models the design of routes of drones or other vehicles that must visit a number of geographical elements to deliver some good or service and then move directly to the next using straight line displacements. We present two families of mathematical programming formulations. The first one is time-dependent and captures a number of characteristics of real applications at the price of using three indexes variables. The second family of formulations is not time-dependent, instead it uses connectivity properties to ensure the proper definition of routes. We compare them on a testbed of instances with different shapes of elements: second order cone (SOC) representable and polyhedral neighborhoods and polygonal chains. The computational results reported in this paper show that our models are useful and our formulations can solve to optimality medium size instances of sizes similar to other combinatorial problems including neighborhoods that have already been studied in the literature. To address larger instances we also present a heuristic algorithm that runs in two phases: clustering and Variable Neighborhood Search. This algorithm performs very well since it provides promising feasible solutions and, in addition, it can be used to initialize the solvers with feasible solutions.

在本文中，我们处理交叉邮递员问题的扩展，以设计必须访问不同形状的维度元素而不是边缘的路径。这个问题模拟了无人机或其他车辆的路线设计，这些车辆必须访问多个地理元素以提供一些商品或服务，然后使用直线位移直接移动到下一个。我们介绍了两个系列的数学规划公式。第一个是时间相关的，以使用三个索引变量为代价来捕捉实际应用程序的许多特征。第二类公式与时间无关，而是使用连通性属性来确保正确定义路径。我们在具有不同元素形状的实例的测试台上比较它们：二阶锥（SOC）可表示和多面体邻域和多边形链。本文报告的计算结果表明，我们的模型是有用的，我们的公式可以解决与其他组合问题（包括文献中已经研究过的邻域）类似的中等大小大小的最优实例。为了解决更大的实例，我们还提出了一种分两个阶段运行的启发式算法：聚类和变量邻域搜索。该算法性能非常好，因为它提供了有希望的可行解，此外，它还可用于用可行解初始化求解器。本文报告的计算结果表明，我们的模型是有用的，我们的公式可以解决与其他组合问题（包括文献中已经研究过的邻域）类似的中等大小大小的最优实例。为了解决更大的实例，我们还提出了一种分两个阶段运行的启发式算法：聚类和变量邻域搜索。该算法性能非常好，因为它提供了有希望的可行解，此外，它还可用于用可行解初始化求解器。本文报告的计算结果表明，我们的模型是有用的，我们的公式可以解决与其他组合问题（包括文献中已经研究过的邻域）类似的中等大小大小的最优实例。为了解决更大的实例，我们还提出了一种分两个阶段运行的启发式算法：聚类和变量邻域搜索。该算法性能非常好，因为它提供了有希望的可行解，此外，它还可用于用可行解初始化求解器。