Abstract
This research introduces a new method, called Interval \({L}_{z}\)-transform (ILz), designed to estimate the reliability indices of Multi-State systems (MSS) even when data is uncertain or insufficient. Traditionally, precise values of state probabilities and performance metrics for each component were required, which could be challenging when data is lacking. To address this, the Interval \({L}_{z}\) function is proposed, along with corresponding operators, enabling the calculation of interval-valued reliability indices for MSS. To demonstrate the effectiveness of the proposed method, it is applied to a numerical example of a series–parallel system. In this example, we determine interval-valued reliability indices such as reliability, availability, mean expected performance, and expected profit, considering uncertain values for the performance and failure rates of each multi-state component.
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Abbreviations
- M :
-
The system’s total number of conceivable states
- \(g_{ij}\) :
-
Performance level of the element i in state j
- \(\mathop {\underline {g} }\nolimits_{ij}\) :
-
Element i’s performance level lower bound in state j
- \(\mathop {\overline{g}}\nolimits_{ij}\) :
-
Element i’s performance level upper bound in state j
- \(\mathop \phi \nolimits_{ij}\) :
-
Probability of the element i in state j
- \(\mathop {\underline {\phi } }\nolimits_{ij}\) :
-
Lower bound of probability of component i which is in state j
- \(\mathop {\overline{\phi }}\nolimits_{ij}\) :
-
Upper bound of probability of component i which is in state j
- \(\mathop \lambda \nolimits_{ij}^{k}\) :
-
Failure rate, i.e. rate by which component k is transitioned from one level/state i to j
- \(\mathop \mu \nolimits_{ji}^{k}\) :
-
Repair rate, i.e. rate by which component k is transitioned from level/state j to i
- \(\left[ w \right]\) :
-
Required performance level
- \(\left[ R \right]\) :
-
Interval-valued reliability of the system
References
Barlow RE, Wu AS (1978) Coherent systems with multi-state components. Math Oper Res 3(4):275–281
Bisht V, Singh SB (2021) Reliability estimation of 4× 4 SENs using UGF method. J Reliab Stat Stud. https://doi.org/10.13052/jrss0974-8024.1418
Bisht V, Singh SB (2022) Reliability analysis of 8 × 8 SEN- using UGF method. In: Ram M, Pham H (eds) Reliability and maintainability assessment of industrial systems. Springer, Cham, pp 329–342
Bisht V, Singh SB (2023) Lz-transform approach to evaluate reliability indices of multi-state repairable weighted K-out-of-n systems. Qual Reliab Eng Int 39(3):1043–1057. https://doi.org/10.1002/qre.3279
Ding Y, Zuo MJ, Lisnianski A, Tian Z (2008) Fuzzy multi-state systems: general definitions, and performance assessment. IEEE Trans Reliab 57(4):589–594. https://doi.org/10.1109/TR.2008.2006078
El-Neweihi E, Proschan F, Sethuraman J (1978) Multistate coherent systems. J Appl Probab 15(4):675–688. https://doi.org/10.2307/3213425
Frenkel I, Bolvashenkov I, Khvatskin L, Lisnianski A (2018) The Lz-transform method for the reliability and fault tolerance assessment of norilsk-type ship’s diesel-geared traction drives. Transp Telecommun 19(4):284–293. https://doi.org/10.2478/ttj-2018-0023
Frenkel I, Khvatskin L, Daichman S, Lisnianski A (2013) Assessing water cooling system performance: Lz-transform method. In: 2013 international conference on availability, reliability and security (pp. 737–742). Regensburg, Germany
Hu L, Zhang Z, Su P, Peng R (2017) Fuzzy availability assessment for discrete time multi-state system under minor failures and repairs by using fuzzy Lz-transform. Eksploatacja i Niezawodność. https://doi.org/10.17531/ein.2017.2.5
Jaulin L, Kieffer M, Didrit O, Walter É (2001) Interval analysis. In: Jaulin L, Kieffer M, Didrit O, Walter É (eds) Applied interval analysis. Springer, London, pp 11–43
Jia H, Jin W, Ding Y, Song Y, Yu D (2017). Multi-state time-varying reliability evaluation of smart grid with flexible demand resources utilizing
Khati A, Singh SB (2021) Reliability assessment of replaceable shuffle-exchange network by using interval-valued universal generating function. In: Kumar A, Ram M (eds) The handbook of reliability, maintenance, and system safety through mathematical modeling. Academic Press, pp 419–455
Li W, Zuo MJ (2008) Reliability evaluation of multi-state weighted k-out-of-n systems. Reliab Eng Syst Saf 93(1):160–167. https://doi.org/10.1016/j.ress.2006.11.009
Lisnianski A (2012) L z-Transform for a discrete-state continuous-time Markov process and its applications to multi-state system reliability. In: Lisnianski A, Frenkel I (eds) Recent advances in system reliability. Springer, London, pp 79–95
Lisnianski A, Haim HB (2013) Short-term reliability evaluation for power stations by using Lz-transform. J Modern Power Syst Clean Energy 1(2):110–117. https://doi.org/10.1007/s40565-013-0021-3
Lisnianski A, Levitin G (2003) Multi-state system reliability: assessment, optimization and applications. World scientific, Singapore
Lisnianski A, Frenkel I, Khvatskin L (2017) On sensitivity analysis of aging multi-state system by using Lz-transform. Reliab Eng Syst Saf 166:99–108. https://doi.org/10.1016/j.ress.2016.12.001
Lisnianski A, Frenkel I, Khvatskin L (2021) Modern dynamic reliability analysis for multi-state systems. Springer, Berlin
Negi S, Singh SB (2015) Reliability analysis of non-repairable complex system with weighted subsystems connected in series. Appl Math Comput 262:79–89. https://doi.org/10.1016/j.amc.2015.03.119
Qin J, Li Z (2019) Reliability and sensitivity analysis method for a multistate system with common cause failure. Complexity. https://doi.org/10.1155/2019/6535726
Ushakov IA (1986) A universal generating function. Sov J Comput Syst Sci 24(5):118–129
Yeh WC (2008) A simple universal generating function method for estimating the reliability of general multi-state node networks. IIE Trans 41(1):3–11. https://doi.org/10.1080/07408170802322622
Zhao X, Wu C, Wang S, Wang X (2018) Reliability analysis of multi-state k-out-of-n: G system with common bus performance sharing. Comput Ind Eng 124:359–369. https://doi.org/10.1016/j.cie.2018.07.034
Acknowledgements
The first author gratefully acknowledges the Department of Science and Technology (India) for providing INSPIRE fellowship (DST/INSPIRE/03/2022/005092) and financial support the course of this study.
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Bisht, V., Singh, S.B. Interval valued reliability indices assessment of multi-state system using interval \(L_{z}\)-transform. Int J Syst Assur Eng Manag (2024). https://doi.org/10.1007/s13198-024-02337-4
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DOI: https://doi.org/10.1007/s13198-024-02337-4