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Reliability analysis of general systems with bi-uncertain variables

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Abstract

In this paper, the lifetimes of system components are assumed to have independent and nonidentical uncertainty distributions with uncertain parameters. The reliability functions and mean time to failure of the general systems are investigated according to the uncertainty theory. Basic models of the general systems with bi-uncertain variables are established and analyzed, including series, parallel and series–parallel systems. The explicit expressions of reliability function and mean time to failure of each model are presented. Some numerical examples are given to illustrate the applications of the developed models and perform a comparison for the models with uncertain and bi-uncertain variables.

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Abbreviations

\({\text{ M }}\) :

Uncertain measure

\({\vee }\) :

Maximum operator

\({\wedge }\) :

Minimum operator

\({{\xi }_{i}}\) :

Lifetime of component i in series system, \(i = 1,2, \ldots ,n\)

\({{\xi }_{j}}\) :

Lifetime of component j in parallel system, \(j = 1, 2, \ldots , m\)

\({{\xi }_{ij}}\) :

Lifetime of component j for subsystem \({{A}_{i}}\), \(i = 1,2, \ldots , n, j = 1, 2, \ldots ,{m_i}\)

\({k_i}\) :

Number of uncertain parameters contained in component i

\({k_j}\) :

Number of uncertain parameters contained in component j

\({k_{ij}}\) :

Number of uncertain parameters contained in component j for subsystem \({A_i}\)

\({R_i}^*(\; \cdot \; ;t)\) :

Uncertain reliability variable of component i in series system

\({R_j}^*(\; \cdot \; ;t)\) :

Uncertain reliability variable of component j in parallel system

\({{R_{ij}}^*(\; \cdot \; ;t)}\) :

Uncertain reliability variable of component j of subsystem \({A_i}\)

\({R_{{A}_{i}}}^*(\; \cdot \; ;t)\) :

Uncertain reliability variable of subsystem \({A_i}\)

\({\Phi }_{i}(\; \cdot \; ;t)\) :

Uncertainty distribution of component lifetime \({{\xi }_{i}}\) in series system

\({\Phi }_{j}(\; \cdot \; ;t)\) :

Uncertainty distribution of component lifetime \({{\xi }_{j}}\) in parallel system

\({\Phi }_{ij}(\; \cdot \; ;t)\) :

Uncertainty distribution of component lifetime \({{\xi }_{ij}}\) in series–parallel system

\(\Upsilon _{i{{g}_{i}}}^{-1}(\alpha )\) :

Inverse uncertainty distribution of uncertain variable \({{a}_{i{{g}_{i}}}}\)

\(\Upsilon _{j{{g}_{j}}}^{-1}(\alpha )\) :

Inverse uncertainty distribution of uncertain variable \({{a}_{j{{g}_{j}}}}\)

\(\Upsilon _{ij{{g}_{ij}}}^{-1}(\alpha )\) :

Inverse uncertainty distribution of uncertain variable \({{a}_{ij{{g}_{ij}}}}\)

\({{\Psi }^{ - 1}(\alpha )}\) :

Inverse uncertainty distribution of uncertain reliability variable

\({\text{ Z }}\left( {a,b,c} \right) \) :

Zigzag uncertain variable

\({\text{ N }}\left( {e,\sigma } \right) \) :

Normal uncertain variable

\(\text{ LOGN }(e,\sigma )\) :

Lognormal uncertain variable

\({R^*(\; \cdot \; ;t)}\) :

Uncertain reliability variables of system

\(R\left( t \right) \) :

Reliability function of system at time t

\(\mathrm{MTTF }\) :

Mean time to failure

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Acknowledgements

This work is supported in part by the National Natural Science Foundation of China (No. 11601469), the Natural Science Foundation of Hebei Province (No. A2018 203088) and the Science Research Project of Education Department of Hebei Province (No. ZD2017079), People’s Republic of China.

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Authors and Affiliations

  1. School of Science, Yanshan University, Qinhuangdao, 066004, Hebei, People’s Republic of China

    Zhaocai Liu, Linmin Hu & Sijia Liu

  2. College of Mathematical Science, Tianjin Normal University, Tianjin, 300387, People’s Republic of China

    Yuyu Wang

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  1. Zhaocai Liu
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  2. Linmin Hu
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Correspondence to Linmin Hu.

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Liu, Z., Hu, L., Liu, S. et al. Reliability analysis of general systems with bi-uncertain variables. Soft Comput 24, 6975–6986 (2020). https://doi.org/10.1007/s00500-019-04331-6

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