The Vehicle Routing Problem with Selective Backhauls (VRPSB) aims to minimize the total routing costs minus the total revenue collected at backhaul customers. We explore a VRPSB under uncertain revenues. A deterministic VRPSB is formulated as a mixed-integer programming problem and two robust counterparts are derived. A novel method to estimate the probabilistic bounds of constraint violation is designed. A robust metaheuristic is developed, requiring little time to obtain feasible solutions with average gap of 1.40%. The robust approach studied demonstrates high potential to tackle the problem, requiring similar computing effort and maintaining the same tractability as the deterministic modeling.

带有选择性回程的车辆路由问题（VRPSB）的目的是使总的路由成本减至从回程客户处收集的总收入减到最小。我们在收入不确定的情况下探索VRPSB。确定性VRPSB被公式化为混合整数编程问题，并得出两个健壮的对应物。设计了一种新的估计约束违反概率边界的方法。开发了强大的元启发式方法，只需很少的时间即可获得平均间隙为1.40％的可行解。研究的鲁棒方法证明了解决该问题的巨大潜力，需要类似的计算工作并保持与确定性建模相同的可处理性。