Optimization of the Transportation Network of Hazardous Materials Considering Bounded Rationality and Equity
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摘要: 针对含有风险控制的危险品运输网络优化问题，讨论了运输商的有限理性路径选择行为对运输风险的影响。基于鲁棒优化的方法构建了双层规划模型，通过增加各路段最大风险值的上界约束来实现不同路段之间运输风险分布的公平性，上层规划表示政府部门通过关闭部分路段来最小化最大运输网络总风险、路段最大风险的上界值以及路段关闭总数；下层规划表示有限理性的运输商在考虑感知偏差的情形下选择总成本最小的运输路径。考虑到传统启发式算法容易陷入局部最优解，通过重新定义上下层问题，设计了割平面算法求解该模型，并给出了算例分析。结果表明：虽然有限理性运输商的总成本增加了3.5%，但危险品运输网络的最大总风险下降了约8.4%；通过改变政府部门对各目标的关注度，可以影响有限理性运输商的路径选择行为，使得方差系数和基尼系数分别下降了约36.1%和26.2%，实现了不同路段之间风险分布的公平性目标；在实施车辆限行策略的情形下，针对有限理性运输商的感知偏差进行灵敏度分析，发现运输网络最大总风险的最小值不会改变，但会对路段关闭总数产生影响。在考虑运输商为有限理性决策者的情形下，可为政府部门设计更加符合实际情况的危险品运输网络，从而有效降低运输风险。Abstract: For the optimization of the transportation network of hazardous materials (hazmat) with risk control, the effects of route selection for hazmat carriers considering bounded rationality on transportation risk is studied. A bi-level programming model is developed based on a robust optimization method to achieve risk equity by increasing the upper bound constraint on the maximum link risk. In which, the upper level aims to minimize the maximum total risk of the transportation network, the upper bound value of maximum link risk, and the total number of link closures by closing quite a few links. The lower lever indicates that the hazmat carriers considering bounded rationality chose the route with minimum total cost considering perceptual errors. For the traditional heuristic algorithms easily fall into the local optimal solutions, a cutting plane algorithm is proposed to solve the model by redefining the problems of upper and lower levels, and finally a numerical example is given. The results show that, the total cost of hazmat carriers considering bounded rationality increases by 3.5%, but the maximum total risk of the transportation network of hazmat decreases by 8.4%. By changing the focus of government departments on each objective, boundedly rational route choice behaviors of hazmat carriers can be influenced. The variance coefficient and the Gini coefficient decrease by 36.1% and 26.2%, respectively, which results in achieving the goal of risk equity between different links. In a case of vehicle restriction strategy, a sensitivity analysis is carried out on the perceptual errors of hazmat carriers considering bounded rationality. It shows that the minimum value of the maximum total risk of the transportation network would not change, but has impacts on the total number of link closures. In the case that hazmat carriers are bounded rational decision makers, a more realistic transportation network for hazmat can be designed for government departments, thus effectively reducing transportation risks.
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图 4 公平性约束有无情况下不同λ取值时的方差系数和基尼系数
Figure 4. The coefficient of variance and Gini coefficient at different values of λ with and without fairness constraint
图 5 不同权重取值时，k取值对优化前后最大总风险的最小值和路段关闭数量的影响
Figure 5. Effect of k on the minimum value of the maximum total risk and the number of link closures before and after optimization at different weights
表 1 符号说明
Table 1. Thesymbol table
符号 说明 i, j, p, q 运输网络G中的节点，i, j, p, q ∈ N (i, j), (p, q) 运输网络G中的路段，(i, j)∈ A, (p, q)∈ A n 运输网络的总路段数 S 危险品的所有流向 s 第s种流向，由运输的起点和终点决定，且s ∈ S Os 第s种流向的起点 Ds 第s种流向的终点 Rij, Rpq 各路段的运输风险值 R 运输网络的总风险值 R 路段最大总风险的均值，即R = maxR/n γ 方差系数 ω 基尼系数 θ 各路段上最大风险的上界值 ρij 路段(i，j)沿线的人口密度 ds 第s种流向所需车辆数，辆 α 关闭的总路段数量对于运输风险的转换系数 ζs 运输商对于第s种流向上产生的运输成本的感知偏差集合 λ 最大总风险在上层规划中的权重λ ∈ [0, 1] yij 0-1决策变量，当政府决定开放路段(i，j)时取值为1，否则，取值为0 aijs 对于第s种流向，当运输商选择路段(i，j)时取值为1，否则，取值为0 εs 运输商对于第s种流向上产生的运输成本的感知偏差 
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表 2 路段运输成本和沿线人口密度
Table 2. Travel costof each link and population density along the line
路段 运输成本（cij) 人口密度ρij /（人/km2） 1-5 50 100 4-5 70 150 5-9 35 112 4-9 85 238 1-12 38 115 5-6 40 198 12-6 55 130 6-7 36 110 7-11 40 129 8-2 32 115 6-10 56 137 9-10 58 148 10-11 45 123 9-13 60 232 13-3 86 211 11-3 42 156 12-8 38 178 7-8 46 108 11-2 30 103
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表 3 危险品运输网络的优化结果（λ = 0, 0.2, 0.4, 0.6）
Table 3. Optimization results of hazmat transportation network (λ = 0, 0.2, 0.4, 0.6)
路段(i, j) 决策变量yij 路段(i, j) 决策变量yij 1-12 1 12-8 1 1-5 1 12-6 1 4-5 1 6-7 1 4-9 1 6-10 1 5-9 1 10-11 1 5-6 0 7-8 1 9-10 1 7-11 1 9-13 1 8-2 1 13-3 0 11-2 1 11-3 1 — —
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表 4 危险品运输网络的优化结果（λ = 0.8）
Table 4. Optimization results of hazmat transportation network (λ = 0.8)
路段(i, j) 决策变量yij 路段(i, j) 决策变量 yij 1-12 1 12-8 1 1-5 1 12-6 1 4-5 1 6-7 1 4-9 1 6-10 1 5-9 0 10-11 1 5-6 0 7-8 1 9-10 1 7-11 1 9-13 1 8-2 1 13-3 1 11-2 1 11-3 1
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表 5 危险品运输网络的优化结果（λ = 1.0）
Table 5. Optimization results of hazmat transportation network (λ = 1.0)
路段(i, j) 决策变量yij 路段(i, j) 决策变量yij 1-12 1 12-8 1 1-5 1 12-6 0 4-5 0 6-7 1 4-9 1 6-10 1 5-9 1 10-11 1 5-6 0 7-8 1 9-10 1 7-11 1 9-13 0 8-2 1 13-3 1 11-2 1 11-3 1 — —
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表 6 不同权重取值时，优化后各流向的可行路径
Table 6. The feasible paths for each shipment after optimization at different weights
权重 ds 可行路径 总风险 总成本 λ = 0, 0.2, 0.4, 0.6 d1 = 2 1-12-6-7-11-3 1 280 422 1-5-9-10-11-3 1 278 460 d2 = 3 4-9-10-11-2 1 836 654 4-5-9-10-11-2 1 908 714 λ = 0.8 d1=2 1-12-6-7-11-3 1 280 422 d2 = 3 4-9-10-11-2 1 836 654 λ = 1.0 d1=2 1-5-9-10-11-3 1 278 460 d2 =3 4-9-10-11-2 1 836 654
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