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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(3): 318-335

Published online September 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.3.318

© The Korean Institute of Intelligent Systems

(α1, α2)-Cut Sets of Reliability Measures in Moore and Bilikam Lifetime Family Using the Generalized Intuitionistic Fuzzy Numbers

Zahra Roohanizadeh, Ezzatallah Baloui Jamkhaneh , and Einolah Deiri

Department of Statistics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

Correspondence to :
Ezzatallah Baloui Jamkhaneh (e_baloui2008@yahoo.com)

Received: September 8, 2022; Revised: June 20, 2023; Accepted: August 4, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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The parameters of lifetime distribution are frequently measured with some imprecision. However, classical lifetime analyses are based on precise measurement assumptions and cannot handle parameter imprecision. Accordingly, to accommodate the imprecision, the generalized intuitionistic fuzzy reliability analysis is preferred over classical reliability analysis. In reliability analysis, generalized intuitionistic fuzzy parameters provide a flexible model and elucidate the uncertainty and vagueness demanded in the reliability analysis. This study generalizes the parameters and reliability characteristics of the Moore and Bilikam family to cover the fuzziness of the lifetime parameters based on the generalized intuitionistic fuzzy numbers. The Moore and Bilikam family includes several lifetime distributions, such that the resulting reliability measures are more comprehensive than other lifetime distributions. The generalized intuitionistic fuzzy reliability functions and their α1-cut and α2-cut sets are provided, such as the reliability, conditional reliability, and hazard rate functions with generalized intuitionistic fuzzy parameters. We also evaluate the bands with upper and lower bounds in reliability measures than the curve. Based on a numerical example, the generalized intuitionistic fuzzy reliability measures are provided based on the Weibull distribution of the Moore and Bilikam family.

Keywords: (α1, α2)-cut set, Generalized intuitionistic fuzzy distribution, Generalized intuitionistic fuzzy number, Generalized intuitionistic fuzzy reliability, Moore and Bilikam.

Conflict of Interest

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No potential conflict of interest relevant to this article was reported.

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(3): 318-335

Published online September 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.3.318

Copyright © The Korean Institute of Intelligent Systems.

(α1, α2)-Cut Sets of Reliability Measures in Moore and Bilikam Lifetime Family Using the Generalized Intuitionistic Fuzzy Numbers

Zahra Roohanizadeh, Ezzatallah Baloui Jamkhaneh , and Einolah Deiri

Department of Statistics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

Correspondence to:Ezzatallah Baloui Jamkhaneh (e_baloui2008@yahoo.com)

Received: September 8, 2022; Revised: June 20, 2023; Accepted: August 4, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The parameters of lifetime distribution are frequently measured with some imprecision. However, classical lifetime analyses are based on precise measurement assumptions and cannot handle parameter imprecision. Accordingly, to accommodate the imprecision, the generalized intuitionistic fuzzy reliability analysis is preferred over classical reliability analysis. In reliability analysis, generalized intuitionistic fuzzy parameters provide a flexible model and elucidate the uncertainty and vagueness demanded in the reliability analysis. This study generalizes the parameters and reliability characteristics of the Moore and Bilikam family to cover the fuzziness of the lifetime parameters based on the generalized intuitionistic fuzzy numbers. The Moore and Bilikam family includes several lifetime distributions, such that the resulting reliability measures are more comprehensive than other lifetime distributions. The generalized intuitionistic fuzzy reliability functions and their α1-cut and α2-cut sets are provided, such as the reliability, conditional reliability, and hazard rate functions with generalized intuitionistic fuzzy parameters. We also evaluate the bands with upper and lower bounds in reliability measures than the curve. Based on a numerical example, the generalized intuitionistic fuzzy reliability measures are provided based on the Weibull distribution of the Moore and Bilikam family.

Keywords: (&alpha,1, ,&alpha,2)-cut set, Generalized intuitionistic fuzzy distribution, Generalized intuitionistic fuzzy number, Generalized intuitionistic fuzzy reliability, Moore and Bilikam.

Fig 1.

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Figure 1.

Membership and non-membership functions of GIFP for (a) δ = 1, and (b) δ = 2.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 318-335https://doi.org/10.5391/IJFIS.2023.23.3.318

Fig 2.

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Figure 2.

GIFR bands for α1 = 0 and α2 = 1.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 318-335https://doi.org/10.5391/IJFIS.2023.23.3.318

Fig 3.

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Figure 3.

Membership and non-membership functions of GIFR for δ = 2.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 318-335https://doi.org/10.5391/IJFIS.2023.23.3.318

Fig 4.

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Figure 4.

Reliability bands S(t) [0.3, 0.1, δ] for δ = 0.5, 1 and 2.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 318-335https://doi.org/10.5391/IJFIS.2023.23.3.318

Fig 5.

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Figure 5.

Reliability bands S(t) [α1, α2, 1] for (α1, α2) = (0, 1), (1, 0), (0.8, 0.2).

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 318-335https://doi.org/10.5391/IJFIS.2023.23.3.318

Fig 6.

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Figure 6.

GIFCR bands for α1 = 0 and α2 = 1.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 318-335https://doi.org/10.5391/IJFIS.2023.23.3.318

Fig 7.

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Figure 7.

Membership and non-membership functions of GIFCR for δ = 2.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 318-335https://doi.org/10.5391/IJFIS.2023.23.3.318

Fig 8.

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Figure 8.

Membership and non-membership functions of GIFH for t = δ = 2.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 318-335https://doi.org/10.5391/IJFIS.2023.23.3.318

Fig 9.

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Figure 9.

The GIFH bands for α1 = 0 and α2 = 1.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 318-335https://doi.org/10.5391/IJFIS.2023.23.3.318

Fig 10.

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Figure 10.

GIFUF bands for α1 = 0 and α2 = 1.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 318-335https://doi.org/10.5391/IJFIS.2023.23.3.318

Table 1 . Different cut sets of GIFP for δ = 1, 2.

(α1, α2)δ = 1δ = 2
Pμ[α1]Pν[α2]P[α1, α2]Pμ[α1]Pν[α2]P[α1, α2]
(0, 1)[0.1472, 0.2255][0.1231, 0.2706][0.1472, 0.2255][0.1472, 0.2256][0.1231, 0.2706][0.1472, 0.2256]
(0.3, 0.8)[0.1537, 0.2162][0.1313, 0.2536][0.1537, 0.2162][0.1491, 0.2227][0.1382, 0.2408][0.1491, 0.2227]
(0.4, 0.7)[0.1559, 0.2132][0.1355, 0.2455][0.1559, 0.2132][0.1506, 0.2205][0.1450, 0.2294][0.1506, 0.2205]
(0.5, 0.5)[0.1581, 0.2102][0.1445, 0.2301][0.1581, 0.2102][0.1526, 0.2177][0.1567, 0.2123][0.1567, 0.2123]
(0.7, 0.4)[0.1627, 0.2043][0.1493, 0.2228][0.1627, 0.2043][0.1579, 0.2105][0.1613, 0.2062][0.1613, 0.2062]
(1, 0)[0.1698, 0.1958][0.1698, 0.1958][0.1698, 0.1958][0.1698, 0.1958][0.1698, 0.1958][0.1698, 0.1958]

Table 2 . Different cut sets of GIFR for δ = 2.

(α1, α2)Sμ (0.2) [α1, 2]Sν (0.2) [α2, 2]S (0.2) [α1, α2, 2]
(0, 1)[0.7379, 0.8105][0.6960, 0.8357][0.7379, 0.8105]
(0.3, 0.8)[0.7404, 0.8086][0.7229, 0.8202][0.7404, 0.8086]
(0.4, 0.7)[0.7424, 0.8072][0.7334, 0.8134][0.7424, 0.8072]
(0.5, 0.5)[0.7449, 0.8053][0.7492, 0.8017][0.7492, 0.8017]
(0.7, 0.4)[0.7514, 0.8002][0.7549, 0.7972][0.7549, 0.7972]
(1, 0)[0.7647, 0.7888][0.7647, 0.7888][0.7647, 0.7888]

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