The multi-trip time-dependent vehicle routing problem with soft time windows and overtime constraints (MT-TDVRPSTW-OT) is considered in this paper. The modified hierarchical multi-objective formulation and the equivalent single-objective formulation are proposed. The iterative multi-trip tour construction and improvement (IMTTCI) procedure and the single-trip tour counterpart procedure with post-processing greedy heuristic (ISTTCI-GH) are proposed to solve the problem. These procedures are based on the ruin and recreate principle, and consider trade-offs among cost components (vehicle usage, number of early/late soft time window occurrences, transport distance, transport time, overtime and early/late soft time window penalty costs) in the search process. From the computational experiment, the IMTTCI procedure outperforms the existing efficient insertion heuristic with 42.09% improvement in number of vehicles and 24.30% improvement in travel time for the constant-speed multi-trip vehicle routing problem with hard time windows and shift time limits, a special case of MT-TDVRPSTW-OT problem, on all problem instances. For the MT-TDVRPSTW-OT problem, the ISTTCI-GH and the IMTTCI outperform the ISTTCI on all problem groups by 43.21% and 69.44% in the number of vehicles (the primary objective), respectively. The IMTTCI outperforms the ISTTCI-GH in number of vehicles by 51.07%, but takes 42.83% longer CPU time than the ISTTCI-GH. The performance of IMTTCI in terms of the primary objective improves with the increase of mean speed as well as the increase of time window width and planning horizon across all customer configuration types, tighter/looser time windows, shorter/longer planning horizon and various time-dependent travel speed profiles. The sensitivity analysis of different problem parameters is performed. The complexity analysis of the proposed procedures shows that the proposed procedures are solvable in polynomial time, and from the computational results the relationships between the CPU time and problem size confirm this. For the MT-VRPSTW-OT, a mixed integer program and the special case of MT-TDVRPSTW-OT, on 12-customer instances, the proposed IMTTCI algorithm yields the average optimality gap of 1.35% with the average CPU time of 0.66 seconds, whereas the GAMS/CPLEX solver yields the average optimality gap of 0.83% with the average CPU time of 256.39 seconds. The proposed IMTTCI algorithm yields only 0.52% greater optimality gap but 388 times faster CPU time than the commercial solver.

中文翻译：

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带有软时间窗和加班限制的多行程时变车辆路径问题

本文考虑了具有软时间窗和加班限制的多行程时间相关车辆路径问题（MT-TDVRPSTW-OT）。提出了改进的分层多目标公式和等效的单目标公式。为解决该问题，提出了迭代多程旅行构建与改进（IMTTCI）程序和具有后处理贪婪启发式的单程旅行对应程序（ISTTCI-GH）。这些程序基于“破坏并重新创建”原理，并考虑成本要素（车辆使用，早/晚软时间窗出现次数，运输距离，运输时间，加班和早/后软时间窗惩罚成本）之间的权衡。在搜索过程中。根据计算实验，IMTTCI程序的性能优于现有的有效插入启发式算法，对于具有硬时间窗和换挡时间限制的恒速多程车辆路径问题（MT-T的特殊情况），车辆数量减少了42.09％，行进时间减少了24.30％ TDVRPSTW-OT问题，在所有问题实例上。对于MT-TDVRPSTW-OT问题，ISTTCI-GH和IMTTCI在所有问题组上的车辆数量（主要目标）分别超过ISTTCI 43.21％和69.44％。IMTTCI的车辆数量比ISTTCI-GH好51.07％，但比ISTTCI-GH的CPU时间长42.83％。在所有客户配置类型中，IMTTCI的主要目标方面的性能会随着平均速度的提高以及时间窗口宽度和计划范围的增加而提高，更紧密/更宽松的时间范围，更短/更长的规划范围以及各种随时间变化的行驶速度曲线。进行不同问题参数的敏感性分析。所提出的过程的复杂性分析表明，所提出的过程在多项式时间内是可解的，并且从计算结果来看，CPU时间与问题大小之间的关系证实了这一点。对于MT-VRPSTW-OT（混合整数程序）和MT-TDVRPSTW-OT的特殊情况，在12个客户实例上，建议的IMTTCI算法产生的平均最优差距为1.35％，平均CPU时间为0.66秒，而GAMS / CPLEX求解器的平均最优差距为0.83％，平均CPU时间为256.39秒。提出的IMTTCI算法仅产生0。