Purpose. In order to study the impact of grey delivery time uncertainty on customer satisfaction and delivery costs, a vehicle routing problem with grey delivery time windows and multiobjective constraints is defined. Method. The paper first defines the uncertainty of the delivery vehicle’s arrival time to the customer as grey uncertainty and then whitens the grey time windows; at the same time, the customer’s hard time windows is expanded into a soft time windows to measure customer satisfaction when the vehicle arrives. Experiment. In order to verify the validity of the established model, numerical experiments are carried out in two groups based on the Solomon example, and the solution is solved based on the improved quantum evolution algorithm. Analysis. Distribution cost fluctuations and customer satisfaction fluctuations with grey time windows are relatively small; under different satisfaction threshold conditions, the distribution cost is increased gently with the satisfaction threshold. Conclusion. The grey delivery time windows have certain advantages in solving the random travel time vehicle routing problem.

中文翻译：

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灰色时间窗和多目标约束的车辆路径问题建模与求解

目的。为了研究灰色交货时间不确定性对客户满意度和交货成本的影响，定义了具有灰色交货时间窗口和多目标约束的车辆路径问题。方法。本文首先将送货车辆到达客户的不确定性定义为灰色不确定性，然后将灰色时间窗口变白。同时，将客户的困难时间窗口扩展为软时间窗口，以衡量车辆到达时的客户满意度。实验。为了验证所建​​立模型的有效性，在所罗门算例的基础上进行了两组数值实验，并基于改进的量子演化算法进行了求解。分析。带有灰色时间窗口的分销成本波动和客户满意度波动相对较小；在不同的满意度阈值条件下，配送成本随着满意度阈值的增加而逐渐增加。结论。灰色的交货时间窗口在解决随机行驶时间车辆路径问题方面具有某些优势。