Motivated by the challenges of non-emergency patient transportation services in the healthcare industry, this study investigated a multi-trip vehicle routing problem incorporating multi-skilled manpower with downgrading. We aimed to find an optimal plan for vehicle routing and multi-skilled manpower scheduling in tandem with the objective of minimizing the total cost, including travel and staff costs, without violating time windows and lunch break constraints. To address this, two mathematical models were formulated: an arc-flow model and a trip-based set-covering model. In addition, a branch-and-price-and-cut algorithm, based on the set-covering model, was proposed to solve practical-scale instances. To determine the feasibility of the integer solutions, we introduce a feasibility check model. To address the multi-trip characteristics of the proposed problem, a novel two-phase column generation algorithm was introduced to solve the subproblem. This approach differs from traditional one-phase labeling algorithms and involves a tailored labeling algorithm for obtaining non-dominated labels in the first phase and a strategy to identify the trip with the minimum reduced cost for each label in the second phase. Furthermore, novel and efficient staff-based inequalities were developed by improving the k-path inequalities. Extensive numerical experiments were conducted to demonstrate the solution performance of the proposed algorithm and reveal managerial insights for non-emergency ambulance operations. The results demonstrate that our algorithm can successfully solve instances with up to 50 patients to optimality within two hours. Moreover, we demonstrated the value of jointly optimizing vehicle routing and staff planning, which can result in significant cost savings of up to 19.4%.

中文翻译：

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具有时间窗和多技能人力的多行程车辆路径问题的精确算法


受医疗保健行业非紧急患者运输服务挑战的启发，本研究调查了将多技能人力与降级相结合的多次行程车辆路线问题。我们的目标是找到车辆路线和多技能人力调度的最佳计划，以最大限度地降低总成本（包括差旅和员工成本），同时不违反时间窗口和午休限制。为了解决这个问题，制定了两个数学模型：弧流模型和基于行程的集合覆盖模型。此外，还提出了一种基于集合覆盖模型的分支价格切割算法来解决实际规模的实例。为了确定整数解的可行性，我们引入了可行性检查模型。为了解决所提出问题的多行程特征，引入了一种新颖的两阶段列生成算法来解决子问题。这种方法与传统的单阶段标记算法不同，涉及用于在第一阶段获取非支配标签的定制标记算法，以及在第二阶段识别每个标签成本降低最小的行程的策略。此外，通过改进 k 路径不等式，开发了新颖且高效的基于员工的不等式。进行了大量的数值实验，以证明所提出的算法的解决方案性能，并揭示非紧急救护车操作的管理见解。结果表明，我们的算法可以在两小时内成功解决最多 50 名患者的实例并达到最优。 此外，我们还展示了联合优化车辆路线和员工规划的价值，这可以显着节省高达 19.4% 的成本。