﻿ 轨道养护计划多目标模型及求解
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 同济大学学报(自然科学版)  2018, Vol. 46 Issue (7): 926-933.  DOI: 10.11908/j.issn.0253-374x.2018.07.010 0

### 引用本文

XU Yude, LIU Yiming, SHEN Jianfeng, QIU Junxing. Multi-objective Track Maintenance Plan Model and Solution[J]. Journal of Tongji University (Natural Science), 2018, 46(7): 926-933. DOI: 10.11908/j.issn.0253-374x.2018.07.010.

### 文章历史

1. 同济大学 道路与交通工程教育部重点实验室，上海 201804;
2. 中交投资有限公司，北京 100029

Multi-objective Track Maintenance Plan Model and Solution
XU Yude1, LIU Yiming1, SHEN Jianfeng2, QIU Junxing1
1. Key Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji University, Shanghai 201804, China;
2. CCCC Investment Company Ltd, Beijing 100029, China
Abstract: In order to solve the multi-objective model of track maintenance plan, the fast and elitist non-dominated sorting generic algorithm (NSGA-Ⅱ algorithm) is improved through coding improvement, constraint condition control in the whole process and self-adaption of genetic coefficients. It can maintain the diversity of the population in solving the multi-objective problem of track maintenance to avoid premature convergence, the global optimal solution is obtained by accelerating convergence at the later stage. These methods solve the decision-making problem of large maintenance machineries' optimal operation time and place. Case in Shanghai and Kunming line shows, established planning method has high computational efficiency and the effect of maintenance plan is obvious, the track irregularity of all sections did not exceed the limit of maintenance.
Key words: multi-objective model    coding improvement    constraint condition    self-adaption of genetic coefficients    improved non-dominated sorting generic algorithm(NSGA-Ⅱ algorithm)

(1) 全局解空间较大庞大，年度养护计划中将一年分36旬，1 km线路对应的决策变量为36个，如果是某个铁路局管内的所有线路进行轨道养护计划的编制，那决策变量的数量是惊人的，模型计算的时间和空间复杂度都将增加；

(2) 标准NSGA-Ⅱ算法中没有处理约束条件的算子，约束条件需要通过添加约束算子来对非支配解集进行搜索；

(3) 传统的遗传算法通常采用固定遗传参数的办法，但参数的设置是否恰当将直接影响到算法的搜索效果，过大或者过小的参数将会使算法无法收敛或者后期收敛速度过慢.

1 轨道养护计划多目标模型

1.1 轨道状态最优

 ${\sigma _i}\left( T \right) = {\sigma _0} \cdot {e^{{k_i}t}} + \varepsilon \left( t \right)$ (1)

 $\begin{array}{l} {\rm{Min}}\;{O_1} = {\sum _{u \in U}}{\sum _{j \in J}}({g_{uj}} \cdot {\sigma _{u0}} \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{{\rm{e}}^{[j \cdot {k_{un}} + \left( {36 - j} \right) \cdot {k_{un + 1}}] \cdot T}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{g_{uj}} \cdot {\varepsilon _{nj}} \cdot {{\rm{e}}^{\left( {36 - j} \right) \cdot {k_{un + 1}} \cdot T)}} \cdot {w_{uj}} \end{array}$ (2)

1.2 成本费用最小

 $\begin{array}{l} {\rm{Min}}\;\;{{\rm{O}}_2} = {\sum _{u \in U}}{\sum _{j \in J}}({C_{\rm{t}}} \cdot {w_{uj}} + {C_{\rm{d}}} - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{L_{\rm{b}}} \cdot {P_{\rm{b}}} - {L_{{\rm{cft}}}} \cdot {E_{{\rm{bad}}}}) \end{array}$ (3)

1.3 轨道养护计划约束条件

(1) 养路机械总量约束

 ${W_{uj}} \le 1$ (4)

(2) 定点定时养护约束

 ${W_{{u_1}{j_1}}} = 1$ (5)

(3) 区段上限个数约束

 ${\sum _u}{W_{uj}} \le {A_j}, j \in J$ (6)

(4) 同一区段养护次数约束

 ${\sum _j}{W_{uj}} \le 1$ (7)

(5) 恶化状态上限约束

 $\sum\limits_{j = 1}^{{J_u} - 1} {{W_{uj}} = 1}$ (8)

(6) 养护成本总量约束

 ${\sum _{u \in U}}{\sum _{j \in J}}{C_{uj}} \le {C_{\max }}$ (9)

2 轨道养护计划多目标模型求解 2.1 改进的NSGA-Ⅱ算法

2.1.1 编码设计改进

2.1.2 全过程约束

2.1.3 遗传系数自适应

 $P =\left\{ {\begin{array}{*{20}{l}} { {P_1} - ({P_1} - {P_2}) \cdot \frac{{{R_{{\rm{avg}}}} - {R_{{\rm{ind}}}}}}{{{R_{{\rm{avg}}}}}}, }&{{R_{{\rm{ind}}}} \le {R_{{\rm{avg}}}}}\\ {{P_1}, }&{{R_{{\rm{ind}}}} > {R_{{\rm{avg}}}}} \end{array}} \right.$ (10)

2.2 轨道养护计划编制

 图 1 轨道养护计划的实现流程图 Fig.1 Flow chart of annual maintenance plan for track
3 算例分析

 图 2 养护年度计划的帕累托最优前沿 Fig.2 Pareto optimal frontier of annual maintenance plan

 图 3 养护维修计划1的作业安排 Fig.3 Arrangement of maintenance plan 1

 图 4 不同养护维修计划下各区段的高低不平顺标准差对比图 Fig.4 Comparison of standard deviation of longitudinal irregularity in each segment under different maintenance plans

 图 5 计划1条件下各区段全年的高低不平顺标准差变化图 Fig.5 Variation of standard deviation of longitudinal irregularity in each segment of plan 1

4 结论

(1) 建立的轨道状态-成本费用多目标优化模型能将轨道几何状态最优及成本费用最省两个目标作为一个整体进行考虑.轨道养护计划多目标模型能在确保列车安全运行的同时，减少“过度维修”现象的发生.

(2) 通过采用0-1编码将个体放入三维数组、全过程控制约束条件以及遗传系数自适应这三种方法建立了改进的NSGA-Ⅱ算法，使之求解所建立的轨道养护计划多目标优化模型.改进的NSGA-Ⅱ算法在求解轨道养护计划多目标问题时能更好地保持种群多样性，避免“早熟”，并在后期加速收敛得到全局最优解.

(3) 选取既有沪昆上行线K230+600~K240+600区段对模型和算法进行了验证分析，所得的养护维修计划中：对于初始不平顺较大且高低不平顺标准差恶化较快的区段给予了优先养护维修，而对于初始不平顺较小且高低不平顺标准差恶化较慢的区段，则都安排在了年度计划的后期，或者不予安排养护维修作业；存在养护维修作业的区段，不论养护维修时间被安排在哪一旬，一般都会在高低不平顺标准差在1.3 mm左右时得到养护作业，且其恢复效果都很好，基本上能恢复到0.7 mm以下.

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