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1. 军械工程学院 电子与光学工程系, 石家庄 050003;
2. 洛阳电子设备试验中心, 洛阳 471000

Condition-based change decision for multi-state system based on opportunistic policy
PAN Gang1,2, SHANG Chaoxuan1, CAI Jinyan1, LIANG Yuying1, MENG Yafeng1
1. Department of Electronic and Optic Engineering, Ordnance Engineering College, Shijiazhuang 050003, China;
2. Luoyang Electronics Equipment Test Center, Luoyang 471000
Received: 2016-03-02; Accepted: 2016-07-01; Published online: 2016-08-29
Foundation item: National Natural Science Foundation of China (61372039, 61271153)
Abstract: A condition-based change and maintenance decision method based on opportunistic policy was proposed for multi-state system with economic correlation and performance correlation among components. Markov model was utilized to characterize the degradation process of components and the universal generating function was used to analyze the reliability indicator of system. Based upon opportunistic policy and from the perspective of "component change", a new condition-based change and maintenance decision method is proposed to ensure the maximum economic benefit within the system's limited service period. The change and maintenance decision for a radar power amplifying system is analyzed. The results indicate that the total maintenance frequency is reduced to a certain extent in consideration of the economic correlation of components. And the proposed method enables an enhancement on the battlefield support capacity of equipment system, and shows strong versatility and engineering application value.
Key words: opportunistic policy     Markov model     universal generating function     condition-based change     radar power amplifying system

1 Markov模型描述

 (1)

k=k′, 则有pk, k(t, Δt)=Pr{X(tt)=kX(t)=k′}, 它表示Δt时刻内部件驻留在状态k的概率，且有。由此可定义[16-17]

 (2)
 (3)

 图 1 部件的Markov模型 Fig. 1 Component Markov model

 (4)

2 运算法则 2.1 部件状态划分规则

1) 当部件i单独构成分系统l，且分系统l可以与其他分系统近似的认为是串联结构时，则有：，其中：ki=1, 2, …, Mixi, ki为部件i在状态ki时的性能参数；xi, max为部件i的最大性能参数。

2) 当n′个部件i构成分系统l，且分系统l的性能输出为n′个部件i的性能输出和，同时分系统l可以与其他分系统近似的认为是串联结构时，则有：

2.2 通用生成函数定义

 (5)

2.3 通用生成函数运算法则

 (6)

1) 当gksgi, kigi′, ki的和时，定义δ1运算符：

2) 当gksgi, kigi′, ki的乘积时，定义δ2运算符：

3) 当gksgi, kigi′, ki的最小值时，定义δ3运算符：

3 基于通用生成函数的多态系统可靠性指标分析

3.1 多态系统可靠性分析

 (7)

 (8)

3.2 多态系统平均瞬态性能分析

 (9)
4 更换决策模型

4.1 基于部件状态的更换维修模型

4.1.1 模型假设

1) 假定系统可以由n个多态部件以串并联结构方式连接。

2) 假定构成系统的多态部件的状态转移过程可以用Markov模型进行描述。

3) 假定对系统内某一部件进行更换维修时，对其他部件的性能没有影响。

4) 系统在服役期间只有“工作”和“维修”2种状态，且对部件的任意次维修更换均需对系统进行停机。

5) 系统在运行过程中，可通过在线测试手段，实现对部件状态的准确监测。

6) 部件之间的维修费用具有一定经济相关性，即系统每次停机维修都会有较高的固定维修费用(如系统拆装、维修设备调试等)。

4.1.2 更换维修模型

1) 若部件i跳转至更换阈值状态θi及以下时，将立即更换，从维修效果角度来讲为“修复如新”。

2) 为了减少系统停机维修更换次数，增强系统的任务执行能力，引入机会维修决策的观念。若部件i跳转至更换阈值状态θi及以下状态时，且在经过ΔT时间后，部件j将跳转至更换阈值状态θj，则在更换部件i的同时，将部件j一并更换，其中j=1, 2, …, n，且ji

3) 对部件进行机会维修更换可节省对其维修时所需的一些固定费用，降低系统的运行风险。

4.2 更换决策目标

 (10)

 (11)

 (12)
4.3 更换决策变量

4.4 基于机会维修策略的部件更换决策分析

1) 根据求解部件i的平均正常工作时间，并将部件i的平均正常工时间Ti按照由大到小的顺序进行排序，可得Tz=Θ({T1, T2, …, Ti, …, Tn})，1≤inΘ(·) 为排序函数。

2) 根据，求得部件i可能的最大维修次数numi

3) 令Ti, zi={Ti, 2Ti, …, numiTi}表示部件i在有限的服役时间内可能出现的更换时刻，其中0≤zi≤numi

4) 从时间坐标轴上，根据构成系统部件可能出现的更换时刻进行逐次维修，假定在t时刻对部件i进行第一次更换，然而引入机会维修概念后，若在tT时刻后部件j的状态退化到θj及以下，那么在对部件i更换的同时，对部件j进行第一次机会维修更换，则对部件j第一次维修的时刻变为Tj, 2=Ti, 1+Tj, 1，同时对部件j的更换时刻集合Tj, zj进行更新，变为Tj, zj={Tj, Ti+Tj+Tmj, …, (numi-1)Ti+Tj+Tmj}，依次逐渐对各部件的维修时刻进行更新，直至到达系统的最高服役年限，其中j∈{1, 2, …, n}且jiTmj为对部件j维修更换所需的时间，0≤ΔT≤min{T1, T2, …, Tn}。

5 算例分析

 图 2 某型雷达功率放大系统 Fig. 2 A certain type of radar power amplifying system

 部件编号 cr/(万元·a-1) cm/(万元·a-1) 1 5 2 2 3 1 3 4 2 4 4.4 2.2 5 4.8 2.4 6 5 2.6 7 3 1.5

5.1 部件的状态和概率分析

1) 当分系统3的状态性能处于完好工作状态gsub3=100%时，分系统1部件此时所需的状态性能最小为gsub1=75%。

2) 当分系统1的状态性能处于完好工作状态gsub1=100%时，分系统3所需的状态性能最小为gsub3=75%。

3) 当分系统1和分系统3均处于不完全工作状态且2个分系统的性能相同时，此时2个分系统所需的最低性能为

4) 为了增大系统状态与最小性能需求之间的区分度，将系统的状态性能在最小性能需求区域附近尽量将其细化，当部件具有5个状态时，结合本节1)~3) 分析，将分系统1的状态性能分别定义为{0, 81%, 87%, 93.5%, 100%}，同理将分系统3的状态定义为{0, 81%, 87%, 100%}，因分系统3的性能为各部件性能之和，将其内各部件状态定义为{0, 20.25%, 21.75%, 25%}。

 部件编号 性能水平/% 状态转移率/a-1 更换率/a-1 1 g51=100 λ5, 41=0.12, λ5, 31=0.135λ5, 21=0.145, λ5, 11=0.25 μ4, 51=180 μ3, 51=150 μ2, 51=120 μ1, 51=100 g41=93.5 λ4, 31=0.30, λ4, 21=0.35 λ4, 11=0.40 g31=87 λ3, 21=0.50, λ3, 11=0.65 g21=81 λ2, 11=0.85 g11=0 2 g22=100 λ2, 12=0.145 μ1, 22=100 g12=0 3 g43=25 λ4, 33=0.105, λ4, 23=0.15λ4, 13=0.20 μ3, 43=200 μ2, 43=150 μ1, 43=100 g33=21.75 λ3, 23=0.30, λ3, 13=0.40 g23=20.25 λ2, 13=0.50 g13=0 4 g43=25 λ4, 34=0.115, λ4, 24=0.15λ4, 14=0.20 μ3, 44=200 μ2, 44=150 μ1, 44=100 g33=21.75 λ3, 24=0.30, λ3, 14=0.40 g23=20.25 λ2, 14=0.50 g13=0 5 g43=25 λ4, 35=0.125, λ4, 25=0.15λ4, 15=0.20 μ3, 45=200 μ2, 45=150 μ1, 45=100 g33=21.75 λ3, 25=0.30, λ3, 15=0.40 g23=20.25 λ2, 15=0.50 g13=0 6 g43=25 λ4, 36=0.135, λ4, 26=0.15λ4, 16=0.20 μ3, 46=200 μ2, 46=150 μ1, 46=100 g33=21.75 λ3, 26=0.30, λ3, 16=0.40 g23=20.25 λ2, 16=0.50 g13=0 7 g27=100 λ2, 17=0.165 μ1, 27=100 g17=0

5.2 基于部件状态的功率放大系统更换决策

5.3 更换决策下功率放大系统可靠性分析

 图 3 系统可靠度与时间的关系 Fig. 3 Relationship between system reliability and time

5.4 更换决策下功率放大系统平均性能分析

 图 4 系统平均瞬态性能与时间的关系 Fig. 4 Relationship between system mean instantaneous performance and time

6 结论

1) 采用本文所提基于机会维修的多态部件更换决策研究方法，通过尽可能多的将部件级维修组合起来，在保证系统处于较高可靠度的前提下降低了维修费用。

2) 充分结合机会维修和部件状态性能维修的优势，制定更加高效合理的维修决策方案，提高了雷达功率放大系统的任务执行能力。

 [1] 任淑红.民航发动机性能可靠性评估与在翼寿命预测方法研究[D].南京:南京航空航天大学, 2010:88-90. REN S H.Research on methods of performance reliability assessments and life on wing prediction for civil aero engine[D].Nanjing:Nanjing University of Aeronautics and Astronautics, 2010:88-90(in Chinese). [2] 程志君, 杨征, 谭林. 基于机会策略的复杂系统视情维修决策模型[J]. 机械工程学报, 2012, 48 (6): 168–174. CHENG Z J, YANG Z, TAN L. Condition-based maintenance model of deteriorating complex system based on opportunistic policy[J]. Chinese Journal of Mechanical Engineering, 2012, 48 (6): 168–174. DOI:10.3901/JME.2012.06.168 (in Chinese) [3] 蔡景, 左洪福, 王华伟. 多部件系统的预防性维修优化模型研究[J]. 系统工程理论与实践, 2007, 27 (2): 133–138. CAI J, ZUO H F, WANG H W. A study on preventive maintenance optimization model for multi-unit system[J]. Systems Engineering-Theory & Practice, 2007, 27 (2): 133–138. (in Chinese) [4] LEVITIN G, LISNIANSKI A. Joint redundancy and maintenance optimization for multistate series-parallel systems[J]. Reliability Engineering & System Safety, 1999, 64 (1): 33–42. [5] 刘宇.多状态复杂系统可靠性建模及维修决策[D].成都:电子科技大学, 2011:69-79. LIU Y.Multi-state complex system reliability modeling and maintenance decision[D].Chengdu:University of Electronic Science and Technology of China, 2011:69-79(in Chinese). [6] 成国庆, 李玲, 唐应辉. 多态退化串联可修系统的最优维修更换策略[J]. 系统工程理论与实践, 2012, 32 (5): 1118–1123. CHENG G Q, LI L, TANG Y H. Optimal replacement policy for a deteriorating series repairable system with multi-state[J]. Systems Engineering-Theory & Practice, 2012, 32 (5): 1118–1123. (in Chinese) [7] 黄傲林, 李庆民, 黎铁冰, 等. 劣化系统周期预防性维修策略的优化[J]. 系统工程与电子技术, 2014, 36 (6): 1103–1107. HUANG A L, LI Q M, LI T B, et al. Optimization of periodic preventive maintenance policies for deteriorating repairable system[J]. Systems Engineering and Electronics, 2014, 36 (6): 1103–1107. (in Chinese) [8] 孙志礼, 王健, 印明昂, 等. 可修系统预防性维修时间的确定[J]. 东北大学学报(自然科学版), 2014, 35 (1): 84–87. SUN Z L, WANG J, YIN M A, et al. Determination about preventive maintenance time of the repairable system[J]. Journal of Northeastern University (Natural Science), 2014, 35 (1): 84–87. (in Chinese) [9] 狄鹏, 黎放, 杨元. 基于机会维修策略的预防性维修优化模型研究[J]. 工程设计学报, 2012, 19 (4): 263–267. DI P, LI F, YANG Y. Optimal preventive maintenance model based on opportunistic maintenance policy[J]. Chinese Journal of Engineering Design, 2012, 19 (4): 263–267. (in Chinese) [10] ZHOU Y, ZHANG Z, LIN T R, et al. Maintenance optimisation of a multi-state series-parallel system considering economic dependence and state-dependent inspection intervals[J]. Reliability Engineering & System Safety, 2013, 111 : 248–259. [11] CUONG D D, MING J Z, MAYANK P. Selective maintenance for multi-state series-parallel systems under economic dependence[J]. Reliability Engineering & System Safety, 2014, 121 : 240–249. [12] 葛小凯, 胡剑波, 张博锋. 考虑依赖性的多部件系统状态维修优化仿真建模[J]. 航空学报, 2013, 34 (8): 1854–1863. GE X K, HU J B, ZHANG B F. Simulation modeling for condition based maintenance optimization of multi-component systems with system with dependencies[J]. Acta Aeronautica et Astronautica Sinica, 2013, 34 (8): 1854–1863. (in Chinese) [13] 张卓琦, 吴甦, 李斌锋. 考虑故障相关的两部件系统机会维修策略[J]. 清华大学学报(自然科学版), 2012, 52 (1): 122–127. ZhANG Z Q, WU S, LI B F. Opportunistic maintenance policy for a two-unit system with failure interactions[J]. Journal of Tsinghua University (Science and Technology), 2012, 52 (1): 122–127. (in Chinese) [14] 赵洪山, 鄢盛腾, 张小田. 风电机组确定性机会更换维修策略的研究[J]. 太阳能学报, 2014, 35 (4): 568–575. ZHAO H S, YAN S T, ZHANG X T. Deterministic opportunistic replacement maintenance strategy for wind turbine[J]. Acta Energiae Solaris Sinica, 2014, 35 (4): 568–575. (in Chinese) [15] 赵洪山, 张健平, 高夺, 等. 风电机组的状态-机会维修策略[J]. 中国电机工程学报, 2015, 35 (15): 3851–3858. ZHAO H S, ZHANG J P, GAO D, et al. A condition based opportunistic maintenance strategy for wind turbine[J]. Proceedings of the CSEE, 2015, 35 (15): 3851–3858. (in Chinese) [16] LIANIANSKI A, LEVITIN G. Multi-state system reliability:Assessment, optimization and applications[M]. Toh Tuck Link: World Scientific, 2003: 92-94. [17] LIANIANSKI A, FRENKEL I, DING Y. Multi-state system reliability analysis and optimization for engineers and industrial managers[M]. London: Springer, 2010: 40-45. [18] 潘刚, 尚朝轩, 梁玉英, 等. 考虑认知不确定的雷达功率放大系统可靠性评估[J]. 北京航空航天大学学报, 2016, 42 (6): 1185–1194. PAN G, SHANG C X, LIANG Y Y, et al. Reliability evaluation of radar power amplification system considering epistemic uncertainty[J]. Journal of Beijing University of Aeronautics and Astronautics, 2016, 42 (6): 1185–1194. (in Chinese) [19] 章恩泽, 陈庆伟. 不确定可靠性优化问题的多目标粒子群优化算法[J]. 控制与决策, 2015, 30 (9): 1701–1705. ZHANG E Z, CHEN Q W. Multi-objective particle swarm optimization for uncertain reliability optimization problems[J]. Control and Decision, 2015, 30 (9): 1701–1705. (in Chinese) [20] 李春洋.基于多态系统理论的可靠性分析与优化设计方法研究[D].长沙:国防科学技术大学, 2010:92-114.LI C Y. Research on reliability analysis and optimization based on the multi-state system theory[D].Changsha:National University of Defense Technology, 2010:92-114(in Chinese).

#### 文章信息

PAN Gang, SHANG Chaoxuan, CAI Jinyan, LIANG Yuying, MENG Yafeng

Condition-based change decision for multi-state system based on opportunistic policy

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(2): 319-327
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0155