# Statistical Fuzzy Reliability Assessment of a Blended System

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Reliability Evaluation Using Fuzzy Set Theory

#### 2.2. Reliability Evaluation Using IFS

#### 2.3. Reliability Evaluation Using the UGF Method

## 3. Essential Definitions

#### 3.1. Fuzzy Sets

#### 3.2. Intuitionistic Fuzzy Sets

#### 3.2.1. Operations on IFS

- $\overline{\wp}=({\upsilon}_{\wp},{\mu}_{\wp}).$
- ${\wp}_{\alpha}\cup {\wp}_{\beta}=(\mathrm{max}\{{\mu}_{\alpha},{\mu}_{\beta}\},\hspace{0.17em}\mathrm{min}\{{\upsilon}_{\alpha},{\upsilon}_{\beta}\}).$
- ${\wp}_{\alpha}\cap {\wp}_{\beta}=(\mathrm{min}\{{\mu}_{\alpha},{\mu}_{\beta}\},\hspace{0.17em}\mathrm{max}\{{\upsilon}_{\alpha},{\upsilon}_{\beta}\}).$
- ${\wp}_{\alpha}\oplus {\wp}_{\beta}=({\mu}_{\alpha}+{\mu}_{\beta}-{\mu}_{\alpha}{\mu}_{\beta},{\upsilon}_{\alpha}{\upsilon}_{\beta}).$
- ${\wp}_{\alpha}\otimes {\wp}_{\beta}=({\mu}_{\alpha}{\mu}_{\beta},{\upsilon}_{\alpha}+{\upsilon}_{\beta}-{\upsilon}_{\alpha}{\upsilon}_{\beta}).$
- $\kappa \wp =(1-{(1-{\mu}_{\wp})}^{\kappa},{\upsilon}_{\wp}^{\kappa}),\hspace{0.17em}\kappa >0.$
- ${\wp}^{\kappa}=({\mu}_{\wp}^{\kappa},1-{(1-{\upsilon}_{\wp})}^{\kappa}),\hspace{0.17em}\kappa >0.$
- The subtraction operator for two IFS was defined by Lei and Xu [2]. The subtraction operator for two IFNs is defined as follows:$${\wp}_{\alpha}\ominus {\wp}_{\beta}=\{\begin{array}{l}\left(\frac{{\mu}_{\alpha}-{\mu}_{\beta}}{1-{\mu}_{\beta}},\frac{{\upsilon}_{\alpha}}{{\upsilon}_{\beta}}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mu}_{\alpha}\ge {\mu}_{\beta}\hspace{0.17em}\mathrm{and}\hspace{0.17em}{\upsilon}_{\alpha}\le {\upsilon}_{\beta}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\upsilon}_{\beta}0\hspace{0.17em}\mathrm{and}\hspace{0.17em}{\upsilon}_{\alpha}{\tau}_{\beta}\le {\tau}_{\alpha}{\upsilon}_{\beta}\\ \\ (0,1)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{otherwise}\end{array}.$$

#### 3.2.2. Score Function and Accuracy Function

- ${\wp}_{\alpha}<{\wp}_{\beta}$ if $\sigma ({\wp}_{\alpha})<\sigma ({\wp}_{\beta})$.
- If $\sigma ({\wp}_{\alpha})=\sigma ({\wp}_{\beta})$ then,
- If $\delta ({\wp}_{\alpha})=\delta ({\wp}_{\beta})$, then ${\wp}_{\alpha}={\wp}_{\beta}$.
- If $\delta ({\wp}_{\alpha})<\delta ({\wp}_{\beta})$, then ${\wp}_{\alpha}<{\wp}_{\beta}$.

#### 3.3. Interval-Valued Intuitionistic Fuzzy Sets

#### 3.3.1. Interval-Valued Intuitionistic Fuzzy Numbers

#### 3.3.2. Operations on IVIFN

- ${\ell}_{1}\cap {\ell}_{2}=\left(\left[\mathrm{min}\left({\alpha}_{1},{\alpha}_{2}\right),\hspace{0.17em}\mathrm{min}\left({\beta}_{1},{\beta}_{2}\right)\right],\hspace{0.17em}\left[\mathrm{max}\left({\varpi}_{1},{\varpi}_{2}\right),\hspace{0.17em}\mathrm{max}\left({\tau}_{1},{\tau}_{2}\right)\right]\right).$
- ${\ell}_{1}\cup {\ell}_{2}=\left(\left[\mathrm{max}\left({\alpha}_{1},{\alpha}_{2}\right),\hspace{0.17em}\mathrm{max}\left({\beta}_{1},{\beta}_{2}\right)\right],\hspace{0.17em}\left[\mathrm{min}\left({\varpi}_{1},{\varpi}_{2}\right),\hspace{0.17em}\mathrm{min}\left({\tau}_{1},{\tau}_{2}\right)\right]\right).$
- ${\ell}_{1}\otimes {\ell}_{2}=\left(\left[{\alpha}_{1}{\alpha}_{2},{\beta}_{1}{\beta}_{2}\right],\hspace{0.17em}\left[{\varpi}_{1}+{\varpi}_{2}-{\varpi}_{1}{\varpi}_{2},\hspace{0.17em}{\tau}_{1}+{\tau}_{2}-{\tau}_{1}{\tau}_{2}\right]\right).$
- ${\ell}_{1}\oplus {\ell}_{2}=\left(\left[{\alpha}_{1}+{\alpha}_{2}-{\alpha}_{1}{\alpha}_{2},\hspace{0.17em}{\beta}_{1}+{\beta}_{2}-{\beta}_{1}{\beta}_{2}\right],\hspace{0.17em}\left[{\varpi}_{1}{\varpi}_{2},\hspace{0.17em}{\tau}_{1}{\tau}_{2}\right]\right).$
- $\kappa {\ell}_{1}=([1-{(1-{\alpha}_{1})}^{\kappa},1-{(1-{\beta}_{1})}^{\kappa}],\hspace{0.17em}[{\varpi}_{1}^{\kappa},{\tau}_{1}^{\kappa}])$, where $\kappa >0$.
- ${\ell}_{1}^{\kappa}=([{\alpha}_{1}^{\kappa},{\beta}_{1}^{\kappa}],\hspace{0.17em}[1-{\left(1-{\varpi}_{1}\right)}^{\kappa},1-{\left(1-\tau \right)}^{\kappa}])$, where $\kappa >0$.
- Zhao et al. [41] gave another important operation of subtraction in IVIFN defined below:$${\ell}_{1}\hspace{0.17em}\ominus \hspace{0.17em}{\ell}_{2}=\{\begin{array}{l}\left(\left[\frac{{\alpha}_{1}-{\alpha}_{2}}{1-{\alpha}_{2}},\frac{{\beta}_{1}-{\beta}_{2}}{1-{\beta}_{2}}\right],\hspace{0.17em}\left[\frac{{\varpi}_{1}}{{\varpi}_{2}},\frac{{\tau}_{1}}{{\tau}_{2}}\right]\right),\hspace{0.17em}if\hspace{0.17em}{\alpha}_{1}\ge {\alpha}_{2},\hspace{0.17em}{\beta}_{1}\ge {\beta}_{2},\hspace{0.17em}{\varpi}_{1}\le {\varpi}_{2},{\tau}_{1}\le {\tau}_{2}\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}{\varpi}_{2},{q}_{2}0\hspace{0.17em},\hspace{0.17em}{\varpi}_{1}(1-{\alpha}_{2})\le {\varpi}_{2}(1-{\alpha}_{1})\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}{\tau}_{1}(1-{\beta}_{2})\le {\tau}_{2}(1-{\beta}_{1})\\ \left([0,0],\hspace{0.17em}[1,1]\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{otherwise}\end{array}.$$

#### 3.3.3. Score and Accuracy Function of IVIFNs

- ${\ell}_{1}<{\ell}_{2}$ if $\tilde{\sigma}({\ell}_{1})<\tilde{\sigma}({\ell}_{2})$.
- If $\tilde{\sigma}({\ell}_{1})=\tilde{\sigma}({\ell}_{2})$, then,
- ${\ell}_{1}={\ell}_{2}$ if $\tilde{\delta}({\ell}_{1})=\tilde{\delta}({\ell}_{2})$.
- ${\ell}_{1}<{\ell}_{2}$ if $\tilde{\delta}({\ell}_{1})<\tilde{\delta}({\ell}_{2})$.

#### 3.4. Universal Generating Function

#### 3.4.1. Algorithm for Evaluation of Reliability of a k-out-of-n System

- Attain ${U}_{r}(z)={U}_{r-1}(z)\underset{+}{\otimes}{u}_{r}(z)$.
- If ${U}_{r}(z)$ has an expression containing ${z}^{k}$, then it should be removed from ${U}_{r}(z)$ and added to R.

## 4. Model Description

## 5. Computation of Reliability Function

## 6. Proposed Methodologies

#### IFN-Based Approach for Fuzzy Reliability Evaluation

## 7. Evaluation of IFR and IVIFR

#### 7.1. IFR Computation

^{2}) × (6(0.60, 0.30)

^{5}− 15(0.60, 0.30)

^{4}+ 10(0.60, 0.30)

^{3}).

#### 7.2. IVIFR Computation

^{2}) × (6([0.50, 0.60], [0.20, 0.30])

^{5}− 15([0.50, 0.60], [0.20, 0.30])

^{4}+ 10([0.50, 0.60], [0.20, 0.30])

^{3}),

## 8. Results and Discussion

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] - Lei, Q.; Xu, Z. Derivative and differential operations of intuitionistic fuzzy numbers. Int. J. Intell. Syst.
**2015**, 30, 468–498. [Google Scholar] [CrossRef] - Traneva, V.; Tranev, S. Intuitionistic Fuzzy Analysis of Variance of Movie Ticket Sales. In Intelligent and Fuzzy Techniques: Smart and Innovative Solutions. INFUS 2020. Advances in Intelligent Systems and Computing; Kahraman, C., Cevik Onar, S., Oztaysi, B., Sari, I., Cebi, S., Tolga, A., Eds.; Springer: Cham, Switzerland, 2021; Volume 1197, pp. 340–363. [Google Scholar]
- Lian, K.; Wang, T.; Wang, B.; Wang, M.; Huang, W.; Yang, J. The Research on Relative Knowledge Distances and Their Cognitive Features. Int. J. Cogn. Comput. Eng.
**2023**, 4, 135–148. [Google Scholar] [CrossRef] - Bai, W.; Zhang, C.; Zhai, Y.; Sangaiah, A.K. Incomplete intuitionistic fuzzy behavioral group decision-making based on multigranulation probabilistic rough sets and MULTIMOORA for water quality inspection. J. Intell. Fuzzy Syst.
**2023**, 44, 4537–4556. [Google Scholar] [CrossRef] - Utkin, L.V.; Gurov, S.V. A general formal approach for fuzzy reliability analysis in the possibility context. Fuzzy Sets Syst.
**1996**, 83, 203–213. [Google Scholar] [CrossRef] - Bing, L.; Meilin, Z.; Kai, X. A practical engineering method for fuzzy reliability analysis of mechanical structures. Reliab. Eng. Syst. Saf.
**2000**, 67, 311–315. [Google Scholar] [CrossRef] - Dong, Y.G.; Chen, X.Z.; Cho, H.D.; Kwon, J.W. Simulation of fuzzy reliability indexes. KSME Int. J.
**2003**, 17, 492–500. [Google Scholar] [CrossRef] - Kumar, A.; Yadav, S.P.; Kumar, S. Fuzzy reliability of a marine power plant using interval valued vague sets. Int. J. Appl. Sci. Eng.
**2006**, 4, 71–82. [Google Scholar] - Abdelgawad, M.; Fayek, A.R. Fuzzy reliability analyzer: Quantitative assessment of risk events in the construction industry using fuzzy fault-tree analysis. J. Constr. Eng. Manag.
**2011**, 137, 294–302. [Google Scholar] [CrossRef] - Chandna, R.; Ram, M. Fuzzy reliability modeling in the system failure rates merit context. Int. J. Syst. Assur. Eng. Manag.
**2014**, 5, 245–251. [Google Scholar] [CrossRef] - Chaube, S.; Singh, S.B. Fuzzy Reliability Theory Based on Membership Function. Int. J. Math. Eng. Manag. Sci.
**2016**, 1, 34–40. [Google Scholar] [CrossRef] - Wang, L.; Xiong, C.; Wang, X.; Liu, G.; Shi, Q. Sequential optimization and fuzzy reliability analysis for multidisciplinary systems. Struct. Multidiscip. Optim.
**2019**, 60, 1079–1095. [Google Scholar] [CrossRef] - Yang, J.; Xing, L.; Wang, Y.; He, L. Combinatorial Reliability Evaluation of Multi-State System with Epistemic Uncertainty. Int. J. Math. Eng. Manag. Sci.
**2022**, 7, 312–324. [Google Scholar] [CrossRef] - Atanassov, K.T. Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia deposed in Central Sci. Tech. Libr. Bulg. Acad. Sci.
**1983**, 1697, 84. [Google Scholar] - Mahapatra, G.S.; Roy, T.K. Reliability evaluation using triangular intuitionistic fuzzy numbers arithmetic operations. World Acad. Sci. Eng. Technol.
**2009**, 50, 574–581. [Google Scholar] - Kumar, M.; Yadav, S.P.; Kumar, S. A new approach for analysing the fuzzy system reliability using intuitionistic fuzzy number. Int. J. Ind. Syst. Eng.
**2011**, 8, 135–156. [Google Scholar] [CrossRef] - Garg, H.; Rani, M.; Sharma, S.P.; Vishwakarma, Y. Intuitionistic fuzzy optimization technique for solving multi-objective reliability optimization problems in interval environment. Expert Syst. Appl.
**2014**, 41, 3157–3167. [Google Scholar] [CrossRef] - Song, Y.; Wang, X.; Zhu, J.; Lei, L. Sensor dynamic reliability evaluation based on evidence theory and intuitionistic fuzzy sets. Appl. Intell.
**2018**, 48, 3950–3962. [Google Scholar] [CrossRef] - Kumar, A.; Singh, S.B.; Ram, M. Reliability appraisal for consecutive-k-out-of-n: F system of non-identical components with intuitionistic fuzzy set. Int. J. Oper. Res.
**2019**, 36, 362–374. [Google Scholar] [CrossRef] - Akbari, M.G.; Hesamian, G. Time-dependent intuitionistic fuzzy system reliability analysis. Soft Comput.
**2020**, 24, 14441–14448. [Google Scholar] [CrossRef] - Kumar, A.; Ram, M.; Goyal, N.; Bisht, S.; Kumar, S.; Pant, R.P. Analysis of Fuzzy Reliability of the System Using Intuitionistic Fuzzy Set. In Intelligent Communication, Control and Devices. Advances in Intelligent Systems and Computing; Choudhury, S., Gowri, R., Sena Paul, B., Do, D.T., Eds.; Springer: Singapore, 2021; Volume 1341, pp. 371–378. [Google Scholar]
- Ushakov, I.A. A universal generating function. Sov. J. Comput. Syst. Sci.
**1986**, 24, 118–129. [Google Scholar] - Levitin, G.; Lisnianski, A. Importance and sensitivity analysis of multi-state systems using the universal generating function method. Reliab. Eng. Syst. Saf.
**1999**, 65, 271–282. [Google Scholar] [CrossRef] - Ding, Y.; Lisnianski, A. Fuzzy universal generating functions for multi-state system reliability assessment. Fuzzy Sets Syst.
**2008**, 159, 307–324. [Google Scholar] [CrossRef] - An, Z.W.; Huang, H.Z.; Liu, Y. A discrete stress–strength interference model based on universal generating function. Reliab. Eng. Syst. Saf.
**2008**, 93, 1485–1490. [Google Scholar] [CrossRef] - Li, Y.F.; Zio, E. A multi-state model for the reliability assessment of a distributed generation system via universal generating function. Reliab. Eng. Syst. Saf.
**2012**, 106, 28–36. [Google Scholar] [CrossRef] - Mi, J.; Li, Y.F.; Liu, Y.; Yang, Y.J.; Huang, H.Z. Belief universal generating function analysis of multi-state systems under epistemic uncertainty and common cause failures. IEEE Trans. Reliab.
**2015**, 64, 1300–1309. [Google Scholar] [CrossRef] - Meena, K.S.; Vasanthi, T. Reliability analysis of mobile ad hoc networks using universal generating function. Qual. Reliab. Eng. Int.
**2016**, 32, 111–122. [Google Scholar] [CrossRef] - Jaiswal, N.; Negi, S.; Singh, S.B. Reliability analysis of non-repairable weighted k-out-of-n system using belief universal generating function. Int. J. Ind. Syst. Eng.
**2018**, 28, 300–318. [Google Scholar] [CrossRef] - Kumar, A.; Ram, M. Computation Interval-Valued Reliability of Sliding Window System. Int. J. Math. Eng. Manag. Sci.
**2019**, 4, 108–115. [Google Scholar] [CrossRef] - Liu, X.; Yao, W.; Zheng, X.; Xu, Y. Reliability Analysis of Complex Multi-State System Based on Universal Generating Function and Bayesian Network. arXiv
**2022**, arXiv:2208.04130. [Google Scholar] - Li, J.; Lu, Y.; Liu, X.; Jiang, X. Reliability analysis of cold-standby phased-mission system based on GO-FLOW methodology and the universal generating function. Reliab. Eng. Syst. Saf.
**2023**, 233, 109125. [Google Scholar] [CrossRef] - Xu, Z.; Yager, R.R. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen. Syst.
**2006**, 35, 417–433. [Google Scholar] [CrossRef] - Xu, Z. Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst.
**2007**, 15, 1179–1187. [Google Scholar] - Gou, X.; Xu, Z. Exponential operations for intuitionistic fuzzy numbers and interval numbers in multi-attribute decision making. Fuzzy Optim. Decis. Mak.
**2017**, 16, 183–204. [Google Scholar] [CrossRef] - Chen, S.M.; Tan, J.M. Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst.
**1994**, 67, 163–172. [Google Scholar] [CrossRef] - Hong, D.H.; Choi, C.H. Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst.
**2000**, 114, 103–113. [Google Scholar] [CrossRef] - Atanassov, K.T. Interval Valued Intuitionistic Fuzzy Sets. In Intuitionistic Fuzzy Sets. Studies in Fuzziness and Soft Computing; Physica: Heidelberg, Germany, 1990; Volume 35, pp. 139–177. [Google Scholar]
- Xu, Z.; Chen, J. On Geometric Aggregation Over Interval-Valued intuitionistic Fuzzy Information. In Proceedings of the Fourth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2007), Haikou, China, 24–27 August 2007; Volume 2, pp. 466–471. [Google Scholar]
- Zhao, H.; Xu, Z.; Yao, Z. Interval-valued intuitionistic fuzzy derivative and differential operations. Int. J. Comput. Intell. Syst.
**2016**, 9, 36–56. [Google Scholar] [CrossRef] - Levitin, G. The Universal Generating Function in Reliability Analysis and Optimization; Springer: London, UK, 2005; Volume 6. [Google Scholar]
- Romeu, J.L. Understanding series and parallel systems reliability. Sel. Top. Assur. Relat. Technol. (START) Dep. Def. Reliab. Anal. Cent. (DoD RAC)
**2004**, 11, 1–8. Available online: https://web.cortland.edu/matresearch/SerieslParallelSTART.pdf (accessed on 4 April 2023).

Case | $\mathit{\rho}$ | Score Function | Ω | Score Function |
---|---|---|---|---|

I | (0.70, 0.30) | 0.40 | (0.60, 0.30) | 0.30 |

II | (0.82, 0.10) | 0.72 | (0.75, 0.25) | 0.50 |

III | (0.90, 0.05) | 0.85 | (0.83, 0.10) | 0.73 |

Case | Intuitionistic Fuzzy Reliability (IFR) | Score Function |
---|---|---|

I | (0.46698, 0.42795) | 0.03903 |

II | (0.55251, 0.41912) | 0.13338 |

III | (0.79573, 0.17693) | 0.61880 |

Case | $\mathit{\rho}$ | Score Function | Ω | Score Function |
---|---|---|---|---|

I | ([0.60, 0.70], [0.20, 0.30]) | 0.40 | ([0.50, 0.60], [0.20, 0.30]) | 0.30 |

II | ([0.77, 0.82], [0.05, 0.10]) | 0.72 | ([0.55, 0.60], [0.05, 0.10]) | 0.50 |

III | ([0.88, 0.90], [0.01, 0.07]) | 0.85 | ([0.79, 0.86], [0.08, 0.11]) | 0.73 |

Case | Interval-Valued Intuitionistic Fuzzy Reliability (IVIFR) | Score Function |
---|---|---|

I | ([0.32062, 0.46698], [0.28145, 0.42795]) | 0.03910 |

II | ([0.43368, 0.51097], [0.06861, 0.13840]) | 0.36882 |

III | ([0.74669, 0.82461], [0.07631, 0.13277]) | 0.68111 |

Case | IFR | IVIFR |
---|---|---|

I | 0.03903 | 0.03910 |

II | 0.13338 | 0.36882 |

III | 0.61880 | 0.68111 |

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**MDPI and ACS Style**

Chachra, A.; Kumar, A.; Ram, M.; Triantafyllou, I.S.
Statistical Fuzzy Reliability Assessment of a Blended System. *Axioms* **2023**, *12*, 419.
https://doi.org/10.3390/axioms12050419

**AMA Style**

Chachra A, Kumar A, Ram M, Triantafyllou IS.
Statistical Fuzzy Reliability Assessment of a Blended System. *Axioms*. 2023; 12(5):419.
https://doi.org/10.3390/axioms12050419

**Chicago/Turabian Style**

Chachra, Aayushi, Akshay Kumar, Mangey Ram, and Ioannis S. Triantafyllou.
2023. "Statistical Fuzzy Reliability Assessment of a Blended System" *Axioms* 12, no. 5: 419.
https://doi.org/10.3390/axioms12050419