Optimization Method for Train Formation Plan with Fuzzy Chance-Constrained Programming
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摘要:
为提高铁路网的利用能力和运输效率，提出一种高适用性的列车编组计划优化方法. 首先，在车流径路未知的情况下综合考虑车辆集结与改编时间的随机性，采用模糊机会约束规划方法，将集结时间成本与改编时间成本限制在一定的波动区间，构建不确定性的0-1整数规划模型；以货车集结时间成本、货车改编时间成本和货车运输成本最小为目标函数，通过三角模糊数处理时间不确定性，引入车辆集结与改编时间的波动性约束，并采用粒子群算法进行寻优，获取列车编组计划，构造算例以验证所提方法的有效性. 研究结果表明：列车编组计划经优化后，货车在车站总停留时间为
3914 车小时，占货物运输总成本的54%，相较于铁路网实际货车在站平均停留时间降低13%左右，列车编组计划得到了较好的优化.Abstract:To improve the utilization capacity and transportation efficiency of the railway network, a highly applicable method for optimizing freight train formation plans was proposed. First, under the condition of unknown car flow routing, the stochastic nature of both accumulation time and shunting time was considered. A fuzzy chance-constrained programming method was adopted to limit the cost of accumulation time and shunting time within a certain fluctuation range, leading to the construction of a 0-1 integer programming model under uncertainty. By taking the minimum freight car accumulation time cost, shunting time cost, and transportation cost as the objective function, time uncertainty was addressed using triangular fuzzy numbers. The volatility constraints for accumulation time and shunting time were introduced. The particle swarm optimization algorithm was adopted to obtain the train formation plan. A numerical example was then constructed to validate the effectiveness of the proposed method. The results show that the optimized train formation plan reduces the total detention time of freight cars at stations to 3 914 car-hours, accounting for 54% of the total freight transportation cost. This represents a reduction of about 13% compared to the actual average station detention time of freight cars in the railway network, indicating a significant improvement in the freight train formation plan.
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表 1 货流输送方案表
Table 1. Freight transportation schemes
方案 路径 方案弧 改编节点 1 1→2→3 $ \left\{ {{l_{12}},{l_{23}}} \right\} $ 不改编 2 1→2→3 $ \left\{ {{l_{12}}} \right\} $，$ \left\{ {{l_{23}}} \right\} $ 2 3 1→2→4→3 $ \left\{ {{l_{12}},{l_{24}},{l_{43}}} \right\} $ 不改编 4 1→2→4→3 $ \left\{ {{l_{12}}} \right\} $，$ \left\{ {{l_{24}},{l_{43}}} \right\} $ 2 5 1→2→4→3 $ \left\{ {{l_{12}},{l_{24}}} \right\} $，$ \left\{ {{l_{43}}} \right\} $ 4 6 1→2→4→3 $ \left\{ {{l_{12}}} \right\} $，$ \left\{ {{l_{24}}} \right\} $，$ \left\{ {{l_{43}}} \right\} $ 2，4 
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表 2 参数说明
Table 2. Parameter description
符号 含义 $ P $ 运输需求$ p $的集合，$ {p_{ij}} $为节点$ i $去向节点$ j $的运输需求 $ L_r $ 路径$ r $的基础弧集合 $ R_p $ 运输需求$ p $的可选径路集合；$ R_{pl} = \left\{{r \in R_p \left| {l \in L_r } \right.} \right\} $，为需求p的基础弧集合 $ {w}_i^+ $ 以节点$ i $为起点的方案弧集合 $ {w}_i^{-} $ 以节点$ i $为终点的方案弧集合 $ {c_{pr}} $ 运输需求$ p $通过路径$ r $运输时在始发站的集结参数，为随机变量 $ {m_r} $ 路径$ r $的列车平均编成辆数 $ {m_p} $ 运输需求$ p $内的车辆数 $ {t_r} $ 货车在路径$ r $上的走行时间 $ {t_{{\mathrm{run}}i}} $ 货车无改编通过节点$ i $的车小时节省，为随机变量 $ {c_1} $ 货车单位时间集结成本 $ {c_2} $ 货车单位时间改编成本 $ {c_3} $ 货车单位时间运输成本 $ V_i$ 节点$ i $的改编能力 $ U_l$ 区间$ l $的通过能力 $ w_{l^*l} $ 0-1变量，当$ l^* $包含$ l $时，其值为1；否则为0 $ o_p $ 运输需求$ p $的起点 $ d_p $ 运输需求$ p $的终点
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表 3 成都局各技术站相关参数
Table 3. Parameters related to technical stations of Chengdu Railway Bureau
车站 无改编通过节省小时/h 改编能力/列 1 (2.8,3.0,3.2) 149 2 (2.4,2.6,2.8) 55 3 (2.5,2.7,2.9) 44 4 (3.3,3.5,3.7) 39 5 (2.0,2.2,2.4) 50 6 (1.6,1.8,2.0) 68 7 (2.7,2.9,3.1) 53 8 (2.9,3.1,3.3) 168 9 (1.8,2.0,2.2) 109
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表 4 部分集结参数
Table 4. Accumulation parameters
方向 集结参数 方向 集结参数 1-2 （7.2,7.8,8.2） 4-7 （7.9,8.3,8.7） 1-4 （7.4,7.8,8.2） 5-6 （8.1,8.5,8.9） 1-5 （7.7,8.1,8.5） 5-7 （9.7,10.1,10.5） 1-6 （7.5,7.9,8.3） 5-8 （9.0,9.4,9.8） 2-3 （8.7,9.1,9.5） 5-9 （8.1,8.5,8.9） 2-6 （8.9,9.3,9.7） 6-8 （7.5,7.9,8.3） 3-6 （10.1,10.5,10.9） 7-9 （7.6,8.0,8.4） 4-5 （9.3,9.7,10.1） 8-9 （9.2,9.6,10.0）
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表 5 算法参数取值
Table 5. Parameter values for algorithm
符号 数值 $ N_{\mathrm{in}} $ $ \left| L \right| $ $ N_{\mathrm{out}} $ $ \left| N \right| \times \left| N \right| $ $ {N_{{\mathrm{in,stop}}}} $ $ N'/3 $ $ {N^{{\mathrm{out,stop}}}} $ $ N''/3 $ $ n $ $ \left| N \right| \times 3 $ $ v_{\max} $ 6 $ {w_{\max }} $ 0.9 $ {w_{\min }} $ 0.1 $ {\alpha _1} $、$ {\alpha _2} $ 2 $ f_{\mathrm{best}} $ 10000000 $ k $ 0.4 $ \alpha $ 1 $ s $ 0.5 $ A $ 10000 $ B $ 5000 $ \varOmega $ 2
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表 6 部分求解结果
Table 6. Solution results
发站 途径站 到站 编组内容 车数/辆 1 2 1-2,4-2,5-2,7-2,9-2 146 1 2 3 1-3,4-3 78 1 4 1-4,2-4,3-4 49 1 5 1-5,2-5 74 1 6 1-6 57 1 4 7 1-7,2-7,4-7 103 1 5 8 1-8,5-8,7-8 54 1 5 9 1-9,2-9,5-9 89 2 1 2-1,3-1 146
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表 7 运输成本
Table 7. Transport costs
运输成本 费用值/车小时 运输总成本 7165.8 列车集结时间成本 2035.0 货车改编时间成本 1879.0 货物途中运输时间成本 3251.8
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表 8 各技术站的负荷情况
Table 8. Load conditions of technical stations
车站 改编车数/辆 改编能力/辆 改编能力利用率/% 1 67 149 44.90 2 5 55 9.09 3 0 44 0 4 9 39 23.07 5 4 50 8.00 6 7 68 10.20 7 0 53 0 8 41 168 24.40 9 16 109 14.60
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表 9 部分线路区间的负荷情况
Table 9. Load conditions of railway line sections
区间 区间通过
能力/列列车平均编成
辆数/辆通过车
数/辆区间占
用率/%1-2 30 65 1569 80.46 1-4 34 65 1530 69.23 1-5 26 60 1435 91.99 1-6 23 65 1381 92.37 2-3 25 55 1042 75.78 2-6 21 60 641 50.87 3-6 33 55 589 32.45 4-5 28 60 483 28.75 4-7 30 65 754 38.67 5-6 24 65 681 43.65 5-7 24 60 948 65.83 5-8 22 60 354 26.82 5-9 26 65 1280 75.74 6-8 24 50 681 56.75 7-9 33 55 1637 90.19
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