Multi-Objective Robust Optimisation Method for Stochastic Time-Dependent Vehicle Routing Problem
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摘要: 车辆路径问题 (vehicle routing problem,VRP) 是物流配送的核心问题之一,为了提高物流配送的时效性,在传统VRP模型的基础上,同时考虑了路网交通状态的时变性和随机性,基于最小最大准则,提出了一种带硬时间窗的随机时变车辆路径问题 (stochastic time-dependent vehicle routing problem,STDVRP) 的多目标鲁棒优化模型. 设计了一种非支配排序蚁群算法 (non-dominated sorting ant colony optimisation,NSACO),求解STDVRP多目标优化模型;通过测试算例,对比分析了NSACO算法与改进型非支配排序遗传算法 (non-dominated sorting genetic algorithm II,NSGA-II). 研究结果表明:对于车辆数最小的Pareto边界解,NSACO算法的平均车辆数比NSGA-II算法小3.33%;对于最坏行程时间最小的Pareto边界解,NSACO算法的平均最坏行程时间比NSGA-II算法小17.49%.Abstract: The vehicle routing problem (VRP) is a core issue of distribution logistics. In order to improve the timeliness of deliveries, a multi-objective robust optimisation model based on the minimax criterion was proposed for the stochastic time-dependent vehicle routing problem (STDVRP) with hard time windows, considering both the stochastic and time-varying nature of link travel times. A non-dominated sorting ant colony optimisation (NSACO) algorithm was designed to solve this multi-objective optimisation model for the STDVRP. The NSACO algorithm was compared with the non-dominated sorting genetic algorithm II (NSGA-II) through computational instances. The results show that for the Pareto boundary of the minimised number of vehicles, the average number of vehicles for NSACO is 3.33% less than that of NSGA-II, and for the Pareto boundary of the minimised worst travel time, the average worst travel time for NSACO is 17.49% less than that of NSGA-II.
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表 1 各类路段的行程车速
Table 1. Travel speed of various links
路段类型 时段 1 2 3 4 1 1.00 3.00 1.20 2.80 2 1.80 2.20 1.60 2.40 3 1.60 2.40 1.80 2.20 4 1.40 2.60 1.00 3.00 5 1.20 2.80 1.40 2.60 表 2 STDVRP算例的计算结果
Table 2. Computational results of STDVRP instances
算例 算法 边界解 车辆数/辆 最坏行程时间 最坏等待时间 期望行程时间 期望等待时间 C101 NSGA-II A 18 1 319.93 4 007.66 1 271.38 4 040.33 B 12 1 482.58 778.83 1 282.24 912.98 NSACO A 12 826.78 1 024.67 703.85 1 083.16 B 12 826.78 1 024.67 703.85 1 083.16 C201 NSGA-II A 7 970.18 8 344.81 858.08 8 451.99 B 5 999.46 5 415.92 864.21 5 542.61 NSACO A 5 617.23 5 032.93 534.08 5 097.59 B 4 1 610.53 1 461.69 1 380.19 1 642.52 R101 NSGA-II A 20 1 342.00 1 113.83 1 238.72 1 239.63 B 16 1 375.66 626.21 1 195.39 778.21 NSACO A 18 1 267.96 1 060.74 1 096.56 1 192.29 B 16 1 350.62 653.77 1 183.74 801.94 R201 NSGA-II A 7 1 206.09 1 630.94 1 120.11 1 721.53 B 3 1 385.58 145.18 1 199.96 299.86 NSACO A 7 1 087.86 2 507.04 956.45 2 597.12 B 3 1 387.48 206.84 1 201.50 355.61 RC101 NSGA-II A 16 1 334.95 576.61 1 261.48 666.44 B 14 1 350.06 357.83 1 210.61 470.25 NSACO A 16 1 337.90 638.40 1 140.44 735.28 B 14 1 371.84 306.04 1 218.40 403.19 RC201 NSGA-II A 8 1 349.75 2 591.40 1 213.29 2 740.82 B 4 1 393.53 741.07 1 244.36 864.97 NSACO A 8 1 132.08 3 102.31 969.50 3 201.67 B 4 1 539.07 312.02 1 370.44 438.52 表 3 NSACO算法最优解的相对值
Table 3. Relative values of NSACO algorithm’s optimal solutions
算例 边界解 车辆数
相对值/%最坏行程时
间相对值/%期望行程时
间相对值/%C101 A 66.67 62.64 55.36 B 100.00 55.77 54.89 C201 A 71.43 63.62 62.24 B 80.00 161.14 159.71 R101 A 90.00 94.48 88.52 B 100.00 98.18 99.03 R201 A 100.00 90.20 85.39 B 100.00 100.14 100.13 RC101 A 100.00 100.22 90.40 B 100.00 101.61 100.64 RC201 A 100.00 83.87 79.91 B 100.00 110.44 110.13 平均 A 88.02 82.51 76.97 B 96.67 104.55 104.09 -
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