﻿ 常见寿命分布组件初始贮存方案评估及优化
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1. 海军工程大学 兵器工程学院, 武汉 430033;
2. 海军工程大学 舰船与海洋学院, 武汉 430033;
3. 海军工程大学 舰船综合电力国防科技重点实验室, 武汉 430033

Evaluation and optimization for initial storage project of common life distribution components
XU Li1, ZHANG Ning1, LI Hua1, RUAN Minzhi2, ZHOU Liang3
1. College of Weaponry Engineering, Naval University of Engineering, Wuhan 430033, China;
2. College of Naval Architecture&Marine Engineering, Naval University of Engineering, Wuhan 430033, China;
3. National Key Laboratory for Vessel Integrated Power System Technology, Naval University of Engineering, Wuhan 430033, China
Abstract: To address the evaluation and optimization of initial storage project for the equipment of long-term storage for one use, the reliability models of different storage life distribution units are analyzed, and the evaluation model of intact quantity is developed for the components and equipment. The cost of parts is selected as the optimization objective, and the probability of reaching the standard is set as constraint. The optimization model for initial storage project is developed and the optimization algorithm by marginal effect value is proposed. A given example shows that the evaluation model of initial storage project established in this paper has high accuracy, and the optimized storage project can get the minimum part purchase cost and satisfies the probability index of reaching the standard at the same time. The optimization algorithm proposed by this paper can effectively improve computational efficiency while guaranteeing computational accuracy. The proposed models and optimization method have a strong reference value for equipment support staff to develop rational support programs.
Keywords: common life distribution    component    initial storage    project evaluation    marginal effect value    optimization

1 问题描述及模型假设 1.1 贮存过程描述

 图 1 装备组织结构 Fig. 1 Framework of equipment

 图 2 组件可用数量随时间变化 Fig. 2 Variation of number of intact component with time
1.2 模型假设

2 初始贮存方案评估模型

2.1 常见寿命分布组件可靠度 2.1.1 指数型组件可靠度

 $R(t)=\mathrm{e}^{\frac{-t}{\mu}}$ （1）
2.1.2 对数正态型组件可靠度

 $R(t)=1-\frac{1}{\sigma \sqrt{2 \pi}} \int_{0}^{t} \frac{1}{x} \exp \left[-\frac{1}{2}\left(\frac{\ln x-\mu}{\sigma}\right)^{2}\right] \mathrm{d} x$ （2）
2.1.3 威布尔型组件可靠度

 $R(t)=\mathrm{e}^{-\left(\frac{t}{\alpha}\right)^{b}}$ （3）
2.2 初始贮存方案评估模型

 $P_{i}\left(s_{i} | n \geqslant M\right)=1-\sum\limits_{m=0}^{M-1} \mathrm{C}_{s_{i}}^{m} R_{i}(t)^{m}\left(1-R_{i}(t)\right)^{s_{i}-m}$ （4）

 $P(\boldsymbol{s} | n \geqslant M)=\prod\limits_{i=1}^{I} P_{i}\left(s_{i} | n_{i} \geqslant M\right)$ （5）
3 初始贮存方案优化模型与算法设计

 $\left\{\begin{array}{ll} \min & \sum\limits_{i} c_{i} s_{i} \\ \mathrm{s.t} & P(\boldsymbol{s} | n \geqslant M) \geqslant P_{0} \end{array}\right.$ （6）

4 算法复杂度分析

1) 穷举法。即穷举所有可能的组件配置方案，选取所有可能方案中满足式(6)的最优方案。若各组件的配置数量上限为Nc，则配置方案的组合数量为NcJ，其中J表示组件的种类，在计算过程中，其时间复杂度T1=O(NcJ)。

2) 智能优化算法。若采用遗传算法为主程序生成组件配置方案。设嵌套法需迭代iter次，每代产生种群大小为num，则其时间复杂度T2=O(iter·num·(log2num+1))，其中，O(iter·num·log2num)为遗传算法本身的时间复杂度。

3) 本文算法。由第3节模型优化步骤可知，在算法迭代过程中，每次迭代均需遍历计算每类组件的边际效益值，其时间复杂度为O(J)；而优化方法的迭代次数同样与组件种类密切相关，组件种类越多，迭代次数越多，迭代过程的时间复杂度为O(J)，本文优化算法的时间复杂度为T3=O(J2)。

5 算例分析

 序号 寿命分布类型 参数1 参数2 寿命值 费用/万元 1 指数Exp(μ) μ=13.1 13.1 55.7 2 指数Exp(μ) μ=9.5 9.5 30.8 3 对数正态LN(μ, σ2) μ=2.0 σ=0.68 9.31 32.1 4 对数正态LN(μ, σ2) μ=1.8 σ=0.75 8.01 28.3 5 威布尔W(α, b) α=8.0 b=2.1 7.09 20.2 6 威布尔W(α, b) α=11.1 b=2.6 9.86 53.4

 图 3 不同组件完好数量随时间变化曲线 Fig. 3 Variation of number of different intact component with time

 图 4 指数型组件1达标概率评估曲线 Fig. 4 Curves of evaluation for achieving standard probability of exponential component—Component 1
 图 5 对数正态型组件3达标概率评估曲线 Fig. 5 Curves of evaluation for achieving standard probability of lognormal component—Component 3
 图 6 威布尔型组件5达标概率评估曲线 Fig. 6 Curves of evaluation for achieving standard probability of Weibull component —Component 5
 图 7 装备达标概率评估曲线 Fig. 7 Curves of evaluation for achieving standard probability of equipment

 序号 组件1 组件2 组件3 组件4 组件5 组件6 增加组件的序号 P(n≥M) 费用/万元 贮存数量 边际效益值 贮存数量 边际效益值 贮存数量 边际效益值 贮存数量 边际效益值 贮存数量 边际效益值 贮存数量 边际效益值 1 11 7.1×10-7 11 1.7×10-6 11 5.6×10-7 11 1.2×10-6 11 1.2×10-6 11 1.2×10-7 2 1.8×10-5 2 425.5 2 11 2.8×10-6 12 2.8×10-6 11 2.2×10-6 11 4.8×10-6 11 4.6×10-6 11 4.7×10-7 4 7.0×10-5 2 456.3 3 11 8.3×10-6 12 8.1×10-6 11 6.5×10-6 12 5.0×10-6 11 1.3×10-5 11 1.4×10-6 5 0.000 21 2 484.6 4 11 1.9×10-5 12 1.9×10-5 11 1.5×10-5 12 1.2×10-5 12 9.6×10-6 11 3.2×10-6 1 0.000 48 2 504.8 5 12 2.4×10-5 12 6.1×10-5 11 4.9×10-5 12 3.8×10-5 12 3.1×10-5 11 1.0×10-5 2 0.001 55 2 560.5 6 12 5.2×10-5 13 7.2×10-5 11 0.000 11 12 8.4×10-5 12 6.9×10-5 11 2.3×10-5 3 0.003 43 2 591.3 7 12 0.000 11 13 0.000 14 12 6.0×10-5 12 0.000 17 12 0.000 14 11 4.7×10-5 4 0.006 92 2 623.4 8 12 0.000 18 13 0.000 24 12 0.000 10 13 0.000 13 12 0.000 24 11 7.9×10-5 2 0.011 70 2 651.7 9 12 0.000 29 14 0.000 23 12 0.000 17 13 0.000 21 12 0.000 39 11 0.000 13 5 0.019 24 2 682.5 10 12 0.000 41 14 0.000 33 12 0.000 24 13 0.000 30 13 0.000 20 11 0.000 18 1 0.027 09 2 702.7 11 13 0.000 37 14 0.000 60 12 0.000 44 13 0.000 55 13 0.000 37 11 0.000 34 2 0.050 14 2 758.4 12 13 0.000 50 15 0.000 49 12 0.000 60 13 0.000 75 13 0.000 51 11 0.000 46 4 0.068 75 2 789.2 13 13 0.000 66 15 0.000 64 12 0.000 78 14 0.000 46 13 0.000 67 11 0.000 61 3 0.090 02 2 817.5 14 13 0.000 84 15 0.000 82 13 0.000 31 14 0.000 59 13 0.000 86 11 0.000 78 5 0.115 21 2 849.6 15 13 0.000 97 15 0.000 95 13 0.000 36 14 0.000 68 14 0.000 36 11 0.000 89 1 0.132 53 2 869.8 16 14 0.000 69 15 0.001 33 13 0.000 51 14 0.000 95 14 0.000 51 11 0.001 26 2 0.186 56 2 925.5 17 14 0.000 85 16 0.000 96 13 0.000 62 14 0.001 16 14 0.000 62 11 0.001 54 6 0.227 59 2 956.3 18 14 0.001 15 16 0.001 31 13 0.000 85 14 0.001 58 14 0.000 84 12 0.000 30 4 0.309 62 3 009.7 19 14 0.001 32 16 0.001 50 13 0.000 97 15 0.000 83 14 0.000 96 12 0.000 35 2 0.354 39 3 038 20 14 0.001 49 17 0.000 99 13 0.001 09 15 0.000 94 14 0.001 09 12 0.000 39 1 0.400 53 3 068.8 21 15 0.000 92 17 0.001 19 13 0.001 32 15 0.001 13 14 0.001 31 12 0.000 47 3 0.483 62 3 124.5 22 15 0.001 00 17 0.001 29 14 0.000 43 15 0.001 23 14 0.001 43 12 0.000 51 5 0.526 00 3 156.6 23 15 0.001 05 17 0.001 36 14 0.000 45 15 0.001 30 15 0.000 51 12 0.000 54 2 0.554 80 3 176.8 24 15 0.001 13 18 0.000 83 14 0.000 48 15 0.001 40 15 0.000 55 12 0.000 58 4 0.596 83 3 207.6 25 15 0.001 21 18 0.000 89 14 0.000 52 16 0.000 65 15 0.000 59 12 0.000 62 1 0.636 38 3 235.9 26 16 0.000 66 18 0.000 98 14 0.000 57 16 0.000 72 15 0.000 65 12 0.000 69 2 0.703 51 3 291.6 27 16 0.000 69 19 0.000 56 14 0.000 60 16 0.000 76 15 0.000 68 12 0.000 76 4 0.733 70 3 322.4 28 16 0.000 71 19 0.000 58 14 0.000 61 17 0.000 32 15 0.000 70 12 0.000 74 6 0.754 96 3 350.7 29 16 0.000 74 19 0.000 61 14 0.000 65 17 0.000 34 15 0.000 73 13 0.000 10 1 0.794 28 3 404.1 30 17 0.000 37 19 0.000 64 14 0.000 68 17 0.000 35 15 0.000 77 13 0.000 11 5 0.835 65 3 459.8 31 17 0.000 38 19 0.000 65 14 0.000 69 17 0.000 36 16 0.000 25 13 0.000 11 3 0.851 23 3 480 32 17 0.000 39 19 0.000 67 15 0.000 19 17 0.000 37 16 0.000 25 13 0.000 12 2 0.873 43 3 512.1 33 17 0.000 39 20 0.000 36 15 0.000 20 17 0.000 38 16 0.000 26 13 0.000 12 1 0.893 98 3 542.9 34 18 20 15 17 16 13 0.915 97 3 598.6

 图 8 穷举法及本文算法计算的贮存方案效费曲线对比 Fig. 8 Comparison of curves of effectiveness with cost for storage scheme of exhaustion method and algorithm of this paper

 方法 最优方案 P(n≥M) 费用/万元 运行时间/s 穷举法 [18, 20, 15, 17, 16, 13] 0.916 0 3 598.6 1 154 智能优化算法 [18, 20, 15, 17, 16, 13] 0.916 0 3 598.6 293 本文算法 [18, 20, 15, 17, 16, 13] 0.916 0 3 598.6 0.119
6 结论

1) 针对“长期贮存，一次使用”的装备，开展组件初始贮存方案评估及优化问题研究。分析了不同寿命分布贮存单元的可靠度模型，建立了贮存期内组件及装备完好数量评估模型；该模型能够在组件贮存方案确定之后，评估贮存期内各组件以及装备完好数量满足某一规定数值的概率。

2) 以装备数量达标概率指标为约束条件，组件购置费用为优化目标，建立了组件初始贮存方案优化模型，提出了基于边际效益值的方案优化算法。该优化算法能够在组件购置费用最低的情况下选出符合约束条件的方案，具有较高的计算效率，适合计算多种类、大批量的组件初始贮存方案。提出的模型和优化算法可扩展至2层以上的装备结构，为装备保障人员制定合理的装备组件贮存方案提供决策支持。

 [1] 田智文, 刘文宝, 王敬贤. 宇航元器件长期贮存及寿命评价方法研究[J]. 质量与可靠性, 2015(3): 32-34. TIAN Z W, LIU W B, WANG J X. Study on long-term storage and life assessment method of aerospace components[J]. Quality and Reliability, 2015(3): 32-34. (in Chinese) Cited By in Cnki | Click to display the text [2] 刘震宇, 马小兵, 赵宇. 非恒定温度场弹上性能退化型部件贮存可靠性评估方法[J]. 航空学报, 2012, 33(9): 1671-1678. LIU Z Y, MA X B, ZHAO Y. Storage reliability assessment for missile component with degradation failure mode in a temperature varying environment[J]. Acta Aeronautica et Astronautica Sinica, 2012, 33(9): 1671-1678. (in Chinese) [3] OKAN Y, BAYINDIER K, GOKHAN O O. Reliability assessment of solid-propellant rocket motors under storage and transportation loads[J]. Journal of Spacecraft and Rockets, 2017, 54(6): 1356-1366. Click to display the text [4] 罗巍, 张春华, 谭源源, 等. 系统贮存可靠度近似置信下限的Boots trap评估方法[J]. 宇航学报, 2009, 30(4): 1725-1730. LUO W, ZHANG C H, TAN Y Y, et al. Bootstrap estimate method of approximate confidence lower limits of system storage reliability[J]. Journal of Astronautics, 2009, 30(4): 1725-1730. (in Chinese) Cited By in Cnki (8) | Click to display the text [5] CHEN Y X, ZHANG Q, CAI Z Y, et al. Storage reliability assessment model based on competition failure of multi-components in missile[J]. Journal of Systems Engineering & Electronic, 2017, 28(3): 606-616. Click to display the text [6] LUO W, ZHANG C H, CHEN X, et al. Accelerated reliability demonstration under competing failure modes[J]. Reliability Engineering and System Safety, 2015, 136: 75-84. Click to display the text [7] 谭源源, 张春华, 陈循. 竞争失效场合步进应力加速试验统计分析[J]. 航空学报, 2011, 32(3): 429-437. TAN Y Y, ZHANG C H, CHEN X. Analysis of step stress accelerated testing with competing failure modes[J]. Acta Aeronautica et Astronautica Sinica, 2011, 32(3): 429-437. (in Chinese) Cited By in Cnki (10) | Click to display the text [8] 张仕念, 颜诗源, 张国彬, 等. 基于能执行任务率的导弹武器装备贮存寿命综合评估方法[J]. 系统工程理论与实践, 2015, 35(2): 513-520. ZHANG S N, YAN S Y, ZHANG G B, et al. Storage life synthesis evaluation method of guided missile weapon based on mission capable rate[J]. Systems Engineering Theory & Practice, 2015, 35(2): 513-520. (in Chinese) Cited By in Cnki | Click to display the text [9] ITO K, NAKAGAWA T, NISHI K. Extended optimal inspection policies for a system in storage[J]. Mathematical and Computer Modeling, 1995, 22(10): 83-87. Click to display the text [10] ITO K, NAKAGAWA T. Optimal inspection policies for a storage system with degradation at periodic tests[J]. Mathematical and Computer Modeling, 2000, 31(10): 191-195. Click to display the text [11] 蔡静, 赵明. 指数型寿命分布的贮存-使用模型可靠性评估方法[J]. 济南大学学报:自然科学版, 2013, 27(2): 205-209. CAI J, ZHAO M. A reliability estimate method of storage-usage model based on exponential distribution[J]. Journal of University of Jinan (Sci. & Tech.), 2013, 27(2): 205-209. (in Chinese) Cited By in Cnki (12) | Click to display the text [12] SCARF P A, CAVALCANTE C A V. Hybrid block replacement and inspection policies for a multi-component system with heterogeneous component lives[J]. European Journal of Operational Research, 2010, 206(2): 384-394. Click to display the text [13] CAVALCANTE C A V, SCARF P A, DE ALMEIDA A T. A study of a two phase inspection policy for a preparedness system with a defective state and heterogeneous lifetime[J]. Reliability Engineering & System Safety, 2011, 96(6): 627-635. Click to display the text [14] LAGGOUNE R, CHATEAUNEUF A, AISSANI D. Impact of few failure data on the opportunistic replacement policy for multi-component systems[J]. Reliability Engineering & System Safety, 2010, 95(2): 108-119. Click to display the text [15] TAGHIPOUR S, BANJEVIC D. Optimum inspection interval for a system under periodic and opportunistic inspections[J]. IIE Transactions, 2012, 44(11): 932-948. Click to display the text [16] TAGHIPPOUR S, BANJEVIC D. Optimal inspection of a complex system subject to periodic and opportunistic inspections and preventive replacements[J]. European Journal of Operational Research, 2012, 220(3): 649-660. Click to display the text [17] NAKAGAWA T, MIZUTANI S, CHEN M. A summary of periodic and random inspection policies[J]. Reliability Engineering and System Safety, 2010, 95(8): 906-911. Click to display the text [18] 杨力, 马小兵. 维修-更换串联系统贮存可用度建模及费用分析[J]. 兵工学报, 2015, 36(3): 552-558. YANG L, MA X B. Storage availability modeling and cost analysis for a repair-replacement series system[J]. Acta Armamentarii, 2015, 36(3): 552-558. (in Chinese) Cited By in Cnki | Click to display the text [19] 马小兵, 杨力. 贮存可用度约束下的可修系统寿命评估与优化[J]. 系统工程与电子技术, 2015, 37(3): 572-576. MA X B, YANG L. Life evaluation and optimization for a repairable system under the constraint of storage availability[J]. Systems Engineering and Electronics, 2015, 37(3): 572-576. (in Chinese) Cited By in Cnki | Click to display the text [20] 甘茂治, 康建设, 高崎. 军用装备维修工程学[M]. 2版. 北京: 国防工业出版社, 2010: 42-47. GAN M Z, KANG J S, GAO Q. Military equipment maintenance engineering[M]. 2nd ed. Beijing: National Defense Industry Press, 2010: 42-47. (in Chinese) [21] SHERBROOKE C C. 装备备件最优库存建模:多级技术[M]. 2版. 北京: 电子工业出版社, 2008: 23-27. [31] SHERBROOKE C C. Optimal inventory modeling of systems multi-echelons[M]. Beijing: Publishing House of Electronics Industry, 2008: 23-27.
http://dx.doi.org/10.7527/S1000-6893.2019.23441

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#### 文章信息

XU Li, ZHANG Ning, LI Hua, RUAN Minzhi, ZHOU Liang

Evaluation and optimization for initial storage project of common life distribution components

Acta Aeronautica et Astronautica Sinica, 2020, 41(4): 223441.
http://dx.doi.org/10.7527/S1000-6893.2019.23441