基于重试机制与转换失效的k/(M+N): G系统可靠性建模与优化

doi:10.15960/j.cnki.issn.1007-6093.2024.04.004

摘要/Abstract

摘要：

本文建立了具有贮备转换失效、Bernoulli休假和工作故障的$k/(M+N):G$可修重试系统可靠性和优化模型。假设当工作部件发生失效时, 由尚未失效的温贮备部件去替换, 替换操作会以一定概率导致温贮备部件发生失效。在重试空间中, 失效部件遵循随机重试原则。利用Runge-Kutta方法和Carmer法则分别求解了系统的瞬态和稳态状态概率, 得到了系统的瞬态和稳态可靠性指标以及一些其他稳态性能指标。基于所定义的成本元素和系统的稳态性能指标, 构建了单位时间总成本函数最小化模型, 采用遗传-粒子群(GA_PSO)混合算法对该优化设计模型进行了求解。通过数值实验评估了不同系统参数对单位时间总成本函数和稳态性能指标的影响。实验结果验证了所建模型的可靠性。

关键词: 重试, Bernoulli休假, 工作故障, 转换失效, 可靠性, 优化

Abstract:

In this paper, the reliability and optimization model of repairable $k/(M+N):G$ retrial system with standby switching failure, Bernoulli vacation and working breakdown is established. It is assumed that when the working component fails, it is replaced by a warm standby component that has not yet failed. The replacement operation will lead to the failure of the warm standby component with a certain probability. In retrial space, the failed components follow the principle of random retrial. By using Runge-Kutta method and Cramer's rule, the transient and steady-state probabilities of the system are solved respectively, and the transient and steady-state reliability indexes and some other steady-state performance indexes of the system are obtained. Based on the defined cost elements and the steady-state performance indexes of the system, a minimization model of the total cost function per unit time is constructed, and the genetic particle swarm optimization (GA_PSO) hybrid algorithm is used to solve the optimization design model. The effects of different system parameters on the total cost function per unit time and steady-state performance index are evaluated by numerical experiments. The experimental results verify the reliability of the established model.

Key words: retrial, Bernoulli vacation, working breakdown, switching failure, reliability, optimization

中图分类号:

O213.2
O224

李晶, 胡林敏, 李明佳. 基于重试机制与转换失效的k/(M+N): G系统可靠性建模与优化[J]. 运筹学学报（中英文）, 2024, 28(4): 44-56.

Jing LI, Linmin HU, Mingjia LI. Reliability modeling and optimization of k=(M+N): G system based on retrial mechanism and switching failure[J]. Operations Research Transactions, 2024, 28(4): 44-56.

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可靠性指标	$ A\left( {20} \right) $	$ m\left( {20} \right) $	$ A\left( \infty \right) $	$ M $	MUT	MDT
结果	0.993 0	0.004 0	0.991 2	0.004 7	212.518 8	1.890 6

其他各项稳态性能指标	$ {P_I} $	$ {P_B} $	$ {P_V} $	$ E\left( I \right) $	$ E\left( R \right) $	$ S.R. $
结果	0.359 9	0.194 4	0.445 7	1.439 1	1.647 2	0.003 6

$ \left( {\lambda , \alpha } \right) $	$ ({0.1, 0.4}) $	$ ({0.2, 0.4}) $	$ ({0.3, 0.4}) $	$ ({0.2, 0.5}) $	$ ({0.2, 0.6}) $
$ C\left( {\mu _b^ * , \mu _d^ * } \right) $	394.250 6	597.732 2	710.112 8	598.724 0	599.017 9
$ \mu _b^ * $	1.878 9	2.438 2	1.936 8	2.345 8	2.242 9
$ \mu _d^ * $	0.782 5	0.864 6	0.520 1	1.578 8	2.230 7
$ {P_I} $	0.278 5	0.128 0	0.071 9	0.127 1	0.126 3
$ {P_B} $	0.298 9	0.329 1	0.382 6	0.311 0	0.312 8
$ {P_V} $	0.422 6	0.542 9	0.545 5	0.561 9	0.560 9
$ S.R. $	0.003 2	0.002 3	0.000 7	0.002 3	0.002 2
$ E\left( I \right) $	1.772 7	3.770 1	5.016 0	3.777 3	3.784 1
$ E\left( R \right) $	1.329 5	0.400 8	0.066 8	0.396 2	0.392 3

$ \left( {\gamma , \delta } \right) $	$ ({2.0, 0.5}) $	$ ({3.0, 0.5}) $	$ ({4.0, 0.5}) $	$ ({3.0, 0.3}) $	$ ({3.0, 0.4}) $
$ C\left( {\mu _b^ * , \mu _d^ * } \right) $	394.250 6	387.141 1	383.428 2	489.931 0	426.880 7
$ \mu _b^ * $	1.878 9	1.855 0	1.839 6	1.820 1	1.875 7
$ \mu _d^ * $	0.782 5	0.780 9	0.784 5	0.693 6	0.754 3
$ {P_I} $	0.278 5	0.268 0	0.262 2	0.149 5	0.216 0
$ {P_B} $	0.298 9	0.305 5	0.309 4	0.260 1	0.284 1
$ {P_V} $	0.422 6	0.426 5	0.428 4	0.590 4	0.499 9
$ S.R. $	0.003 2	0.003 2	0.003 3	0.002 1	0.002 8
$ E\left( I \right) $	1.772 7	1.689 4	1.646 8	2.859 4	2.141 1
$ E\left( R \right) $	1.329 5	1.388 6	1.419 0	0.827 1	1.155 2

$ \Big( {\tilde \lambda , g} \Big) $	$ ({0.05, 0.4}) $	$ ({0.06, 0.4}) $	$ ({0.07, 0.4}) $	$ ({0.05, 0.5}) $	$ ({0.05, 0.6}) $
$ C\left( {\mu _b^ * , \mu _d^ * } \right) $	394.250 6	398.684 0	402.905 9	426.929 5	460.245 9
$ \mu _b^ * $	1.878 9	1.878 0	1.876 4	1.949 1	1.988 5
$ \mu _d^ * $	0.782 5	0.789 7	0.778 2	0.782 1	0.755 6
$ {P_I} $	0.278 5	0.266 7	0.255 7	0.221 4	0.171 9
$ {P_B} $	0.298 9	0.303 8	0.308 7	0.275 0	0.255 9
$ {P_V} $	0.422 6	0.429 5	0.435 6	0.503 6	0.572 2
$ S.R. $	0.003 2	0.003 1	0.003 0	0.002 8	0.002 3
$ E\left( I \right) $	1.772 7	1.833 6	1.892 8	2.141 1	2.521 5
$ E\left( R \right) $	1.329 5	1.279 9	1.231 8	1.115 6	0.905 3

$ \left( {q, \beta } \right) $	$ ({0.02, 2.0}) $	$ ({0.03, 2.0}) $	$ ({0.04, 2.0}) $	$ ({0.03, 2.5}) $	$ ({0.03, 3.0}) $
$ C\left( {\mu _b^ * , \mu _d^ * } \right) $	396.976 7	398.891 1	400.800 0	398.615 1	398.083 4
$ \mu _b^ * $	1.767 0	1.767 6	1.765 4	1.829 6	1.883 8
$ \mu _d^ * $	1.637 9	1.629 6	1.638 9	1.211 2	0.764 0
$ {P_I} $	0.274 3	0.272 1	0.269 7	0.273 3	0.274 2
$ {P_B} $	0.302 2	0.303 2	0.304 2	0.301 5	0.300 5
$ {P_V} $	0.423 5	0.424 8	0.426 0	0.425 2	0.425 3
$ S.R. $	0.006 2	0.009 3	0.012 3	0.009 3	0.009 4
$ E\left( I \right) $	1.800 5	1.822 0	1.843 5	1.817 0	1.814 2
$ E\left( R \right) $	1.308 3	1.293 7	1.279 0	1.298 5	1.301 5