In this paper, we study an extension of the orienteering problem where travel times are random and time-dependent and service times are random. Additionally, the service at each selected customer is subject to a soft time window; that is, violation of the window is allowed but subject to a penalty that increases in the delay. A solution is a tour determined before the vehicle departs from the depot. The objective is to maximize the sum of the collected prizes net of the expected penalty. The randomness of the travel and service times is modeled by a set of scenarios based on historical data that can be collected from public geographical information services. We present an exact solution method for the problem based on a branch-and-bound algorithm enhanced by a local search procedure at the nodes. A numerical experiment demonstrates the merits of the proposed solution approach. This study is the first to consider an orienteering problem with stochastic travel times and soft time windows, which are more relevant than hard time windows in stochastic settings.

在本文中，我们研究了定向运动问题的扩展，其中旅行时间是随机且与时间相关的，而服务时间是随机的。此外，每个选定客户的服务都受到软时间窗口的限制；也就是说，违反窗口是允许的，但会受到随着延迟而增加的处罚。解决方案是在车辆离开停车场之前确定行程。目标是使所收集的奖金扣除预期处罚后的总和最大化。出行和服务时间的随机性是通过基于可以从公共地理信息服务收集的历史数据的一组场景来建模的。我们提出了一种基于分支定界算法的问题的精确解决方法，该算法通过节点处的本地搜索过程增强。数值实验证明了所提出的解决方法的优点。这项研究首次考虑了具有随机行进时间和软时间窗的定向运动问题，这比随机设置中的硬时间窗更相关。