Belief reliability: a scientific exploration of reliability engineering

doi:10.23919/JSEE.2024.000029

Journal of Systems Engineering and Electronics ›› 2024, Vol. 35 ›› Issue (3): 619-643.doi: 10.23919/JSEE.2024.000029

• SYSTEMS ENGINEERING • Previous Articles

Belief reliability: a scientific exploration of reliability engineering

Qingyuan ZHANG¹^,²(), Xiaoyang LI²^,³(), Tianpei ZU²^,⁴(), Rui KANG¹^,²^,³^,*()

¹ International Innovation Institute, Beihang University, Hangzhou 311115, China
² Science and Technology on Reliability and Environmental Engineering Laboratory, Beijing 100191, China
³ School of Reliability Engineering and Systems Engineering, Beihang University, Beijing 100191, China
⁴ School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China

Received:2023-01-29 Online:2024-06-18 Published:2024-06-19
Contact: Rui KANG E-mail:zhangqingyuan@buaa.edu.cn;leexy@buaa.edu.cn;zutp93@buaa.edu.cn;kangrui@buaa.edu.cn
About author:
ZHANG Qingyuan was born in 1993. He received his Ph.D. degree in systems engineering from Beihang University, Beijing, China, in 2020. He is currently an associate professor with International Innovation Institute, Beijing University. His research interests include belief reliability theory, reliability modeling, uncertainty quantification method in engineering, and the reliability design method. E-mail: zhangqingyuan@buaa.edu.cn

LI Xiaoyang was born in 1980. She received her Ph.D. degree in aerospace systems engineering from Beihang University, Beijing, China, in 2007. She is currently a professor, the assistant dean, and the director of the Department of Systems Engineering with the School of Reliability and Systems Engineering, Beihang University, Beijing, China. Her research interests include belief reliability theory, reliability experiment theory, and systematic medicine centered medical-industrial crossover.E-mail: leexy@buaa.edu.cn

ZU Tianpei was born in 1993. She received her Ph.D. degree in systems engineering from Beihang University, Beijing, China, in 2021. She is currently a post doctor in the School of Aeronautic Science and Engineering, Beihang University. Her research interests include belief reliability theory, multi-information fusion, uncertainty quantification, fatigue life prediction, and maintenance optimization.E-mail: zutp93@buaa.edu.cn

KANG Rui was born in 1966. He received his M.S. degree in electrical engineering from Beihang University, Beijing, China, in 1990. He is currently a professor in the School of Reliability and Systems Engineering and International Innovation Institute Beihang University. His main research interests include belief reliability theory, reliability-centered systems engineering, resilience modeling and evaluation for complex system. E-mail: kangrui@buaa.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (62073009;52775020;72201013), the China Postdoctoral Science Foundation (2022M710314), and the Funding of Science & Technology on Reliability & Environmental Engineering Laboratory (6142004210102).

Abstract

Abstract:

This paper systematically introduces and reviews a scientific exploration of reliability called the belief reliability. Beginning with the origin of reliability engineering, the problems of present theories for reliability engineering are summarized as a query, a dilemma, and a puzzle. Then, through philosophical reflection, we introduce the theoretical solutions given by belief reliability theory, including scientific principles, basic equations, reliability science experiments, and mathematical measures. The basic methods and technologies of belief reliability, namely, belief reliability analysis, function-oriented belief reliability design, belief reliability evaluation, and several newly developed methods and technologies are sequentially elaborated and overviewed. Based on the above investigations, we summarize the significance of belief reliability theory and make some prospects about future research, aiming to promote the development of reliability science and engineering.

Key words: belief reliability, performance margin, reliability experiment, chance measure, uncertainty

Qingyuan ZHANG, Xiaoyang LI, Tianpei ZU, Rui KANG. Belief reliability: a scientific exploration of reliability engineering[J]. Journal of Systems Engineering and Electronics, 2024, 35(3): 619-643.

Figures/Tables 10

Fig 1

Table 1

Fig 2

Fig 3

Table 2

Research topics and representative papers about belief reliability modeling and analysis"

Topic	Representative paper
Methodology	Basic model and analysis procedure [7,13,83,85], adjustment factor model [82], structure reliability index method [86,90]
Belief reliability analysis based on the basic equations	Belief reliability analysis of different products: hydraulic servo actuator [82,85], aircraft lock mechanism [91], gear [92], electrical connector [93], harmonic reducer [90,94], DC power supply [95], filter circuit [73], turbine disk of aircraft engine [96]
Uncertainty propagation	Propagation formula and algorithm [85,97], propagation model in different stages [96]
Belief reliability analysis of different system configurations	Series systems [13,78,81,84,98–101], parallel systems [13,81,84,98–101], series-parallel hybrid systems [81,99–102], cold standby systems [99,101,103–105], warm standby systems [103], k-out-of-n systems [84,100,106–109], bridge systems [85,110], general configurations and algorithms [78,81,100,110]
Fault tree analysis	Minimal cut set theorem [13,84], analysis algorithm [84]
Importance index	Uncertain system [108,111], uncertain random system [112]
Belief reliability analysis of systems with degradation-shock dependency	Uncertain degradation with random shock arrival time and uncertain shock size [113,114], uncertain degradation with uncertain shock [115–118], degradation-shock dependency with a change point [119] and uncertain failure threshold [120], belief reliability analysis with different shock modes [121], belief reliability analysis for systems with failure trigger effect [122]
Network system belief reliability	Connectivity belief reliability of transportation network [89] and crude oil maritime network [123], belief reliability of traffic network [124,125], service reliability analysis of cloud data centers [126], cascading failure modeling for circuit network [127]
Belief reliability analysis of multi-state system	Modified universal generating function technique with uncertain measure [128], multi-state k-out-of-n system [108], uncertain state transition chain method considering degradation [129]

Table 2

Fig 4

Fig 5

Table 3

Table 4

Table 5

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Type	Representative method (mathematical basis)	Existing problem if applied to reliability engineering
Imprecise probability-based method	Bayesian reliability (Bayesian theory)	Due to the inclusion of subjective information, Bayesian probabilities are not probabilities in the sense of frequency, so it is questionable to still follow Kolmogolov’s axiomatic system for subsequent calculations. When available information is scarce, the results of Bayesian reliability analysis are greatly sensitive to the prior knowledge and are often not sufficiently complete to support decision making.
	Evidence reliability (evidence theory)	The basic formula of $ {\text{Bel}} \leqslant \Pr \leqslant {\text{Pl}} $ has not been strictly proved. The results of the interval form will cause interval expansion problems in the system reliability calculation.
	Interval reliability (interval analysis theory)	The results can also bring interval expansion problems. The interval analysis is not self-consistent in mathematics (for two events related to interval analysis with $\Lambda \subset \Delta $, we may find $ \text{π} \{ \Lambda \} \gt \text{π} \{ \Delta \} $, where $ \text{π} $ is a likelihood measure; one can refer to [40] for an example).
New mathematical measure-based method	Posbist reliability (fuzzy theory)	We can derive results that are not self-consistent, i.e., the sum of reliability and unreliability is not equal to 1.

Element of metadata	Meaning of being “same”	Meaning of being “different”
${\boldsymbol{X}}$	The products corresponding to the collected metadata are from the same population, i.e., the types and design values of ${\boldsymbol{X}}$ are same for each product	The products corresponding to the collected metadata are from the different populations, i.e., the types and design values of ${\boldsymbol{X}}$ for each product may be different, but similar, for example, the digital simulation prototype, improved version, etc
${\boldsymbol{Y}}$	The type and size of the stress that the products bear during working are all the same	The type of the stress that the products bear during working is same, but the size is different, for example, different stress level in tests, stress in both tests and actual use, etc
${\boldsymbol{t}}$	—	The degradation time or lifetime/failure time obtained are usually different due to the uncertainty of products
$P,{P_{{\mathrm{th}}}}$	The products corresponding to the collected metadata are designed for same function with same performance parameters and requirements	(i) The products corresponding to the collected metadata are designed for same function with different performance parameters and requirements (ii) The products corresponding to the collected metadata are designed for different functions

Method symbol	Method or model
1	Graduation formula method [131], the kth moment method based on maximal entropy principle [132,145], order statistics method with Bernard approximation [146]
2	Metadata with information of degradation	(i) Uncertain process-based models [75,147,148] $ \begin{gathered} P(t) = (e(s) + {\sigma _s}{\xi _s}) \cdot {t^\beta } + \sigma C({t^\beta }), \\ C({t^\beta }) \sim {\mathrm{N}}(0,{t^\beta }),\;e(s) \sim {\mathrm{N}}({\mu _e}(s),{\sigma _e}(s)),{\sigma _s}{\xi _s} \sim {\mathrm{N}}(0,{\sigma _s}s) \\ \end{gathered} $ (ii) Time variant uncertainty distribution model [149] $ {\Phi _p}(x\|t) = {\left( {1 + \exp \left( {\dfrac{{\text{π} (e(t) - x)}}{{\sqrt 3 \sigma (t)}}} \right)} \right)^{ - 1}} $ (iii) Uncertain differential equation-based model [150] $\begin{gathered} {\mathrm{d}}P(t) = {\beta _0}\gamma \exp ({\beta _1}\phi (s)){t^{\gamma - 1}}{\mathrm{d}}t + \sigma \gamma \exp ({\beta _1}\phi (s)){t^{\gamma - 1}}{\mathrm{d}}C(t), \\ C(t)\sim {\mathrm{N}}(0,t),\;\;\;{\kern 1pt} P(0) = 0 \\ \end{gathered} $ (iv) Performance and health status margin degradation framework for belief reliability evaluation [151]
2	Metadata with information of lifetime/failure time	Data fusion method: consistent belief degree method (data equivalence method and constant coefficient of variation method) [79], maximal cross entropy method [152]
3	Similarity fusion method [152]

Field	Representative paper
Belief reliability centered maintenance and supportability optimization	(i) Maintenance optimization model: maintenance indexes and analysis for repairable systems [153], optimization of the level of repair considering spare stocks [154] (ii) Spare parts optimization: variety optimization [155,156], quantity optimization [45,157], depot location optimization [158–160]
Risk analysis	Uncertainty representation and propagation in risk analysis [161–163], aviation risk assessment [164,165], transportation risk assessment and avoidance [89]
Prognostics and health management	Failure prognostics with scarce data [166], remaining useful life prediction with degradation data [167]
Software belief reliability assessment	Software belief reliability growth model using uncertain differential equation with perfect [168] or imperfect [169] debug processes
Others	Belief reliability analysis of supply chain [170], soil slopes design [171], assembly line analysis and optimization [172], path selection in maritime transportation [173]

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Belief reliability: a scientific exploration of reliability engineering

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