Reliable path planning model and algorithm in transportation networks with heterogeneous stochastic travel time in road links
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摘要: 为应对交通网络路段随机行程时间异质性分布情况下，出行者路径选择决策对行程时间可靠性和不可靠性方面的关注，以均值-超量行程时间为优化准则，构建了一种新的可靠路径规划模型；为刻画交通网络中不同路段随机行程时间的异质性分布，以行程时间前四阶矩为输入，解析估计了均值-超量行程时间，以免于现有研究均一化的分布假设；利用均值-超量行程时间的理论特性，提出了以Dijkstra最短路算法和K-最短路算法为基础的精确和近似两阶段算法；以Monte Carlo模拟方法为基准，验证了基于前四阶矩的均值-超量行程时间解析估计方法的精确性；通过求解算例网络中所有OD对的最短均值-超量路径，分析了近似两阶段算法的精确性和计算效率。研究结果表明：现有研究中常用的路段随机行程时间正态分布假设会忽略偏度和峰度系数对可靠路径选择结果的影响，而均值-超量行程时间解析估计方法所得近似值与真值的相对误差不超过0.13%；当K-最短路算法中K为10时，近似两阶段算法求解的全网络OD对最短均值-超量路径中，有0.11%的OD对的最小均值-超量与精确两阶段算法求解结果不同，最大相对误差为3.35%；针对由随机选择的5个节点与其他节点组成的OD对，近似两阶段算法平均计算时间与精确两阶段算法平均计算时间的比值范围为0.87%~22.96%，表明近似两阶段算法不仅保证了其求解准确性，还可有效提高计算效率；提出的模型和算法能够接纳网络中不同路段随机行程时间分布具有异质性的现实特征，其路径规划结果能够更好地体现出行者对按时到达可靠性和迟到风险的关注诉求。Abstract: In view of the concerns of travel time reliability and unreliability from travelers in path selection decisions under the heterogeneous distribution of stochastic travel time in road links in a transportation network, a new reliable path planning model was proposed with the mean-excess travel time (METT) as the optimization criterion. To characterize the heterogeneous distributions of stochastic travel times in different road links in the transportation network, the first four order moments of travel time were used as inputs to analytically estimate the METT, thus avoiding the assumption of homogenous distributions in existing studies. The exact and approximate two-stage algorithms based on the Dijkstra and K-shortest path algorithms were developed according to the theoretical properties of METT. The accuracy of the analytical estimation method of METT based on the first four order moments was verified by using the Monte Carlo simulation method as the benchmark. The accuracy and computational efficiency of the approximate two-stage algorithm were analyzed by finding the shortest mean-excess paths of all OD pairs in the test network. Research results indicate that the commonly used normal distribution assumption of stochastic travel time in road links in existing studies ignores the influences of skewness and kurtosis coefficient on the reliable path selection. The relative error between the approximated value from the analytical estimation method and the true value of the METT is not greater than 0.13%. When K is set as 10 in the K-shortest path algorithm, in the shortest mean-excess paths of OD pairs in the whole network solved by the approximate two-stage algorithm, the minimum mean-excess values of 0.11% OD pairs are different from those solved by the exact two-stage algorithm, with a maximum relative error of 3.35%. For the OD pairs composed of randomly selected five nodes and other nodes, the ratio range of average calculation time of the approximate two-stage algorithm to that of the exact two-stage algorithm is 0.87%-22.96%, indicating that the approximate two-stage algorithm can not only ensure the solution accuracy, but also can improve the calculation efficiency. The proposed model and algorithms can capture the realistic characteristics regarding the heterogeneous distributions of stochastic travel times in different road links, and the corresponding path planning results can better reflect travelers' concerns about the reliability of arriving on time and the risk of encountering delay.
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图 3 Anaheim网络中路段行程时间的异质性分布
Figure 3. Heterogeneous distributions of travel time in road links in Anaheim network
图 4 OD对(327, 405)在不同α下的最短均值-超量路径
Figure 4. Shortest mean-excess paths of OD pair (327, 405) under different α
图 6 两种算法下OD对中可选路径数比值直方图
Figure 6. Histogram of ratios of available paths in OD pairs under two algorithms
表 1 偏度和峰度系数对路径METT的影响
Table 1. Influences of skewness and kurtosis coefficient on path METT
路径 不同场景下的路径METT/min 场景1：S=-0.5，M=2.0 场景2：S=2.0，M=-0.5 场景3：S=0，M=0 路径1 14.00 14.98 14.29 路径2 13.78 15.20 14.20 路径3 13.66 15.55 14.23 
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表 2 解析估计结果对比
Table 2. Comparison of analytical estimation results
路径 路段顺序 路径均值/min METT近似值/min METT真值/min 相对误差/% 路径1 1-2-3-6-9 31.8 37.70 37.65 0.13 路径2 1-2-5-6-9 29.9 37.63 37.64 -0.03 路径3 1-2-5-8-9 28.9 38.84 38.80 0.10 路径4 1-4-5-6-9 44.6 54.89 54.87 0.04 路径5 1-4-5-8-9 43.6 55.58 55.55 0.05 路径6 1-4-7-8-9 56.7 70.70 70.64 0.08
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表 3 可靠性概率α的敏感性分析结果
Table 3. Sensitivity analysis results of reliability probability α
α 可选路径集路径数 最短均值-超量路径(路段顺序) METT/min 0.5 4 路径1：327-328-342-354-355-356-372-388-405 113.33 0.6 5 路径1：327-328-342-354-355-356-372-388-405 116.23 0.7 6 路径2：327-341-353-354-355-356-372-388-405 119.44 0.8 9 路径2：327-341-353-354-355-356-372-388-405 123.23 0.9 11 路径2：327-341-353-354-355-356-372-388-405 129.31
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表 4 近似和精确两阶段算法计算效率比较
Table 4. Comparison of calculation efficiencies between approximate and exact two-stage algorithms
效率 算法 节点2 节点3 节点7 节点20 节点331 平均计算时间/s K=5近似两阶段算法 0.16 0.16 0.18 0.18 0.19 K=10近似两阶段算法 0.28 0.28 0.32 0.33 0.31 K=20近似两阶段算法 0.46 0.45 0.58 0.62 0.49 精确两阶段算法 5.80 2.76 36.92 18.24 1.35 平均可选路径数 K=5近似两阶段算法 4.3 4.2 4.5 4.6 4.4 K=10近似两阶段算法 7.6 7.3 8.4 8.6 7.8 K=20近似两阶段算法 12.3 12.0 15.0 16.1 12.5 精确两阶段算法 74.5 40.4 365.4 236.9 19.6
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