Abstract In this paper, we presented an e-accurate approach to conduct a continuous optimization on the pollution routing problem (PRP). First, we developed an e-accurate inner polyhedral approximation method for the nonlinear relation between the travel time and travel speed. The approximation error was controlled within the limit of a given parameter e, which could be as low as 0.01% in our experiments. Second, we developed two e-accurate methods for the nonlinear fuel consumption rate (FCR) function of a fossil fuel-powered vehicle while ensuring the approximation error to be within the same parameter e. Based on these linearization methods, we proposed an e-accurate mathematical linear programming model for the continuous PRP (e-CPRP for short), in which decision variables such as driving speeds, travel times, arrival/departure/waiting times, vehicle loads, and FCRs were all optimized concurrently on their continuous domains. A theoretical analysis is provided to confirm that the solutions of e-CPRP are feasible and controlled within the predefined limit. The proposed e-CPRP model is rigorously tested on well-known benchmark PRP instances in the literature, and has solved PRP instances optimally with up to 25 customers within reasonable CPU times. New optimal solutions of many PRP instances were reported for the first time in the experiments.

中文翻译：

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连续污染路径问题

摘要 在本文中，我们提出了一种对污染路径问题（PRP）进行持续优化的 e-accurate 方法。首先，我们为旅行时间和旅行速度之间的非线性关系开发了一种精确的内多面体近似方法。近似误差控制在给定参数 e 的限制内，在我们的实验中可以低至 0.01%。其次，我们为化石燃料动力汽车的非线性燃料消耗率 (FCR) 函数开发了两种 e-accuracy 方法，同时确保近似误差在相同的参数 e 内。基于这些线性化方法，我们提出了连续PRP（简称e-CPRP）的e-accurate数学线性规划模型，其中决策变量如行驶速度、行驶时间、到达/离开/等待时间、车辆负载和 FCR 都在其连续域上同时优化。提供了理论分析以确认 e-CPRP 的解决方案是可行的并且控制在预定的限制内。所提出的 e-CPRP 模型在文献中著名的基准 PRP 实例上经过严格测试，并在合理的 CPU 时间内以最佳方式解决了多达 25 个客户的 PRP 实例。实验中首次报告了许多 PRP 实例的新最优解。并在合理的 CPU 时间内以最佳方式解决了多达 25 个客户的 PRP 实例。实验中首次报告了许多 PRP 实例的新最优解。并在合理的 CPU 时间内以最佳方式解决了多达 25 个客户的 PRP 实例。实验中首次报告了许多 PRP 实例的新最优解。