﻿ 基于竞争博弈的多目标可靠性优化设计方法<sup>*</sup>
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1. 北京航空航天大学可靠性与系统工程学院, 北京 100083;
2. 北京航空航天大学可靠性与环境工程技术重点实验室, 北京 100083

Multi-objective reliability design optimization approach based on competition game
FENG Jiazhen1,2, ZHANG Jianguo1,2, QIU Jiwei1,2
1. School of Reliability and Systems Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China;
2. Science and Technology on Reliability and Environmental Engineering Laboratory, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2017-05-31; Accepted: 2017-08-01; Published online: 2017-09-22 15:07
Foundation item: National Key R&D Program of China (2013CB733000); National Natural Science Foundation of China(51675026, 71671009)
Corresponding author. ZHANG Jianguo, E-mail: zjg@buaa.edu.cn
Abstract: Aimed at the subjectivity of the selection of target weights, multi-objective reliability design optimization approach based on competition game is proposed. In this approach, every design objective is treated as the corresponding game player, and the random design variable set is decomposed into multiple strategy sets that are allocated to the corresponding player through the random design variable set mapping (RDVSM) technology. Then, combined with the performance measurement analysis method, every player takes its payoff as the single objective function for the reliability design optimization in its own strategy set, and the optimal results of all players form a group of strategies in this game round. After multi-round games, the equilibrium solution of the game is acquired. The study of a pressure vessel case and a gear reducer case shows that the proposed approach avoids the selection of target weights, and the design results have high objectivity.
Key words: multi-objective     reliability design optimization     competition game     random design variable set mapping (RDVSM)     game equilibrium solution

1 基础理论 1.1 可靠性分析的性能测量方法

 (1)

 (2)

1.2 博弈理论

 (3)

2 MRDO的竞争博弈方法

2.1 基于RDVSM的博弈方策略集分解

1) 影响因子集合构建

 (4)

2) 基于模糊聚类的策略集分解

① 构建模糊相似矩阵R

 (5)

② 构建模糊等价矩阵

③ 模糊聚类。

2.2 竞争博弈的求解算法

 (6)

3 案例分析 3.1 压力容器MRDO案例

1) 压力容器MRDO数学模型

 图 1 压力容器示意图[16] Fig. 1 Schematic diagram of pressure vessel[16]

 变量 均值 变异系数 均值范围 r/mm 595.8611 0.05 [2.54, 914.4] l/mm 999.7745 0.05 [2.54, 3556] t/mm 62.4510 0.05 [12.7, 152.4]

 (7)
 (8)

2) 基于RDVSM的策略集分解

3) 竞争博弈求解

4) 优化结果对比分析

 变量 非劣解1 非劣解2 非劣解3 μtN/mm 107.554 0 108.746 8 106.403 1 μlN/mm 1 059.278 0 1 509.354 4 1 601.394 3 μrN/mm 795.133 5 815.710 1 827.923 2 μwN/kg 12 393.860 9 15 135.898 9 15 523.401 2 μvN/m3 4.207 6 5.425 9 5.822 7

 变量 权重组合1(ω1=0.01，ω2= 0.99) 权重组合2(ω1=0.1，ω2= 0.9) 权重组合3(ω1= ω2= 0.5) 权重组合4(ω1=0.9，ω2=0.1) 权重组合5(ω1=0.99，ω2= 0.01) μtW/mm 105.092 8 105.099 4 105.096 6 83.953 1 12.7 μlW/mm 1 719.527 4 1 713.351 9 1 721.302 9 2 093.012 8 962.011 8 μrW/mm 840.228 2 841.113 6 837.490 1 672.022 5 71.356 5 μwW/kg 16 187.338 1 16 184.143 9 16 121.364 6 10 387.539 0 54.265 2 μvW/m3 6.295 3 6.297 5 6.250 2 4.238 7 0.016 9

3.2 减速器MRDO案例

1) 减速器MRDO数学模型

 (9)
 (10)
 图 2 减速器传动原理示意图[17] Fig. 2 Schematic diagram of drive principle of reducer[17]

 cm 变量 均值 标准差 均值范围 齿面宽度x1 3.58 0.05 [2.6, 3.6] 齿轮模数x2 0.72 0.01 [0.7, 0.8] 轴1轴承间距x4 7.48 0.05 [7.3, 8.3] 轴2轴承间距x5 7.83 0.05 [7.3, 8.3] 轴1直径x6 3.37 0.05 [2.9, 3.9] 轴2直径x7 5.26 0.05 [5.0, 5.5] 注：齿轮1齿数x3(取整数)均值范围为[17, 28]。

2) 减速器的博弈均衡解

3) 优化结果对比分析

 变量 非劣解1 非劣解2 非劣解3 μx1N/cm 3.579 8 3.588 4 3.582 3 μx2N/cm 0.701 7 0.700 9 0.701 1 μx3N 17 17 17 μx4N/cm 8.042 3 8.043 8 8.039 6 μx5N/cm 7.927 6 7.962 3 8.046 0 μx6N/cm 3.537 9 3.456 1 3.509 1 μx7N/cm 5.379 2 5.390 5 5.395 9 μf1N/cm3 3 194.976 0 3 195.355 4 3 210.576 2 μf2N/MPa 93.504 2 100.295 7 95.823 8 μf3N/MPa 80.688 1 80.183 7 79.944 4

 变量 权重组合1(ω1=0.02, ω2=ω3=0.49) 权重组合2(ω1=0.2, ω2=ω3=0.4) 权重组合3(ω1=ω2=ω3=1/3) 权重组合4(ω1=0.8, ω2=ω3=0.1) 权重组合5(ω1=0.98, ω2=ω3=0.01) μx1W/cm 3.50 3.50 3.50 3.50 3.50 μx2W/cm 0.7 0.7 0.7 0.7 0.7 μx3W 17 17 17 17 17 μx4W/cm 7.986 8 7.750 1 7.750 6 7.475 1 7.476 4 μx5W/cm 7.950 1 7.952 0 8.112 9 7.995 9 7.995 3 μx6W/cm 3.9 3.9 3.9 3.434 2 3.434 0 μx7W/cm 5.5 5.5 5.369 0 5.369 2 5.369 2 μf1W/cm3 3 313.345 3 3 310.516 3 3 226.071 3 3 077.533 3 3 077.463 2 μf2W/MPa 69.813 3 69.783 6 69.783 7 102.155 3 102.177 7 μf3W/MPa 75.490 7 75.493 9 81.153 4 81.143 2 81.143 2

4 结论

1) 基于竞争博弈提出了一种解决MRDO问题的方法。通过RDVSM将随机设计变量集转换成策略集，所有博弈方均以自身收益为目标在各自策略集中进行SRDO，由优化结果构成一轮博弈的策略组合，然后经过多轮博弈获得均衡解。2个案例设计结果表明了方法的可行性，所求博弈均衡解具有较高的客观性。

2) 在利用PMA对MRDO模型中的可靠性约束进行分析评估时，只考虑了单一的随机不确定性，工程实际当中有多种类型不确定性共存，应考虑在此基础上对基于竞争博弈的MRDO求解方法做进一步的研究和改进。

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#### 文章信息

FENG Jiazhen, ZHANG Jianguo, QIU Jiwei

Multi-objective reliability design optimization approach based on competition game

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(4): 887-894
http://dx.doi.org/10.13700/j.bh.1001-5965.2017.0367