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

Zahra Roohanizadeh; Ezzatallah Baloui Jamkhaneh; Einolah Deiri

doi:10.5391/IJFIS.2023.23.3.318

Original Article

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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

(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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Abstract
1. Introduction
2. Preliminaries
3. Generalized Intuitionistic Fuzzy Distribution
4. Reliability Characteristics
5. Numerical Example
6. Conclusion
Conflict of Interest
Figures
Tables
References

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: (α_1, α₂)-cut set, Generalized intuitionistic fuzzy distribution, Generalized intuitionistic fuzzy number, Generalized intuitionistic fuzzy reliability, Moore and Bilikam.

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. Generalized Intuitionistic Fuzzy Distribution
4. Reliability Characteristics
5. Numerical Example
6. Conclusion
Conflict of Interest
Figures
Tables
References

Reliability analysis is one of the main topics in systems engineering, in which system reliability is defined as the probability of satisfactory system performance under stated conditions for a specified period. The reliability of a system is evaluated based on probability, satisfactory performance, specific conditions, and time metrics. The classical reliability analysis is accomplished based on precise information such as exact data and parameters. However, in real situations, we encounter some components that cannot be entirely quantified, which leads to imprecise records. Therefore, the fuzzy sets (FSs) theory is a feasible solution. For crisp data, Maihulla et al. [1] considered the reliability analysis of the strength of a solar system contained in four parallel subsystems, such as reliability, mean time to failure, availability, and profit function. Each parallel subsystem has two parallel active components, and the system dependability metric follows from the Gumbel-Hougaard copula family. For ambiguous data, Jamkhaneh [1, 2] provided the system reliability by two fuzzy lifetime distributions as exponential and Weibull distributions. Pak et al. [4] demonstrated the Bayesian estimation of the parameter of the Rayleigh distribution along with the reliability function estimation based on the fuzzy lifetime data.

Atanassov [5] introduced the intuitionistic fuzzy sets (IFSs) theory to handle real-life data with uncertainty to belongingness and non-belongingness to a specific set. The uncertainty and imprecise measurements are accrued owing to several factors such as machine errors, human errors, personal opinions, and estimation errors. The classical lifetime distributions consider crisp parameters, but we may confront vague lifetime data. The imprecise parameters of lifetime data have been illustrated based on the intuitionistic fuzzy numbers (IFNs), which handle parameter uncertainty. Hence, the classic and fuzzy lifetime distributions are developed into intuitionistic fuzzy lifetime distributions.

In [6, 7], the authors provided the reliability of fuzzy systems by intuitionistic fuzzy lifetime distribution. In addition, Shabani and Jamkhaneh [8] introduced a new generalized intuitionistic fuzzy number (GIFN_B) based on the generalized IFS. The authors of [9, 10] demonstrated the system reliability based on the generalized intuitionistic fuzzy (GIF) exponential and Rayleigh lifetime distributions, respectively.

The IFS can be applied in several fields, such as linear programming (LP), shortest path problem (SPP), data envelopment analysis (DEA), reliability redundancy allocation problem (RRAP), and artificial intelligence (AI).

The LP model represents a situation involving some parameters assigned by experts and decision-makers who frequently do not know the precise values of the parameters. In addition, in most optimization problems, there are parameters with imprecise values. In these applications, fuzzy numbers are applied for the LP model parameters. Accordingly, many researchers have shown significant attention to fully fuzzy LP subjects, for instance, [11–13]. Malik et al. [14] considered an approach based on mixed circumscriptions and unlimited variables to solve the fully intuitionistic fuzzy LP problems. The parameters and decision variables are represented by IFNs.

The SPP is the main combinatorial optimization problem in the graph theory environments, which can be promoted to FSs to handle ambiguity in the graphs, studied by [15–17]. The non-parametric DEA procedure is an appropriate tool in productivity measurement and evaluates the relative efficiency in the decision-making unit. The fuzzy DEA technique is recommended to cover the vagueness of the input and output data (for further reading, refer to [18–20]).

In reliability and system engineering, RRAP is a principal concept that concentrates on finding the best strategy to increase reliability. Among available methods to enhance the reliability of a specific system, the reliability optimization of components and structural redundancy has attracted considerable attention. Devi et al. [21] provided a complete and multilateral bibliography from previous research on RRAP in the last two decades. To cover the uncertainty in the reliability component, the RRAP in a fuzzy environment was considered by Taghiyeh et al. [22] based on the modification of fuzzy parametric programming.

AI research includes several concepts, such as multi-agent systems, machine learning, deep learning, neural networks, and FSs. Kahraman et al. [23] performed a literature review of recent developments in FSs and their combinations with other AI techniques.

The concepts of IFS have been developed to the Pythagorean FS (PFS) by Yager [24], such that the square sum of membership and non-membership functions is no more than one. The IFS and PFS are contributed in various domains to consider uncertainty in observation, parameters, and random variables, including decision-making, information measures, and reliability analysis (see [25–31]).

In fuzzy science, Senapati and Yager [32] introduced the q-rung orthopair FSs, called Fermatean FSs (FFSs), followed by an extensive comparison with PFS and IFS. Moreover, they derived a set of operations, score and accuracy functions, and Euclidean distance of FFSs, (also see [33, 34]).

The multi-criteria decision-making (MCDM) designates the best option from a set of available candidates by evaluating some criteria. Senapati and Yager [35] assigned the technique for order preference based on the similarity to the ideal solution (TOPSIS) with Fermatean fuzzy information to consider uncertain information in the MCDM process. The aggregation of fuzzy information is a new area of IFS, where the aggregation operators of FFS were introduced by [35]. They introduced several weighted aggregated operators over the FFSs, including the weighted average, weighted geometric, weighted power average, and weighted power geometric operators. In addition, they provided an MCDM strategy based on the four operators with Fermatean fuzzy information.

Mishra et al. [36] provided the complex proportional assessment (COPRAS) strategy for MCDM in the FFS fields. They introduced a Fermatean fuzzy Archimedean copula-based Maclaurin symmetric mean operator and integrated it with the Fermatean fuzzy-COPRAS approach. Rong et al. [37] proposed the cubic FFS by integrating cubic FS and FFS. They provided the score, accuracy functions, comparison laws, and generalized distance measures of cubic FFS. Senapati et al. [38] described several intuitionistic fuzzy aggregation operators based on the Aczel-Alsina operations, including weighted averaging, ordered weighted averaging, and hybrid averaging operators. They also investigated MCDM issues based on novel operators (see also [39, 40]).

The Moore and Bilikam (MB) family includes several important lifetime models, such as exponential, Rayleigh, Weibull, Gompertz, Pareto, and two-parameter exponential distributions. The MB family has attracted the attention of several researchers from different branches. For example, Jamkhaneh et al. [41] concentrated on the exponentiated lifetime distribution and developed the Gompertz distribution as a particular case of the MB family. They provided a two-parameter inverse Gompertz distribution along with a reliability analysis. The applicability of the new distribution was verfied by the relief times of 20 patients receiving an analgesic. Considering a series-parallel system, Danjuman et al. [42] provided several system characteristics, including reliability, availability, maintainability, dependability, mean time between failures, and mean time to failure. In each subsystem, the failure and repair rates follow the exponential distribution of the MB family.

The classical reliability analysis is used when sufficient knowledge of data and parameters is available. However, fuzzy environments guide our understanding of system reliability by allowing parameter uncertainties. Hence, we consider the well-known MB family with imprecise shape and scale parameters based on GIFNs and improve the reliability analysis using GIF. Subsequently, we evaluate the reliability of systems using the MB family, where the lifetime parameters are taken as GIFN_B with linear and nonlinear membership and non-membership functions. The reliability of the generalized intuitionistic fuzzy MB (GIFMB) family is developed based on the α₁-cut, α₂-cut and (α₁, α₂)-cut of GIFN_Bs. Regarding the intuitionistic fuzzy environment, the reliability characteristics of the Weibull distribution are provided as numerical examples.

This paper is organized as follows: The preliminaries and basic concepts of GIFN_Bs are represented in Section 2. The GIFMB family and GIF reliability characteristics are provided in Sections 3 and 4, respectively. The numerical example of Weibull distribution with GIFN parameters and several GIF reliability characteristics are discussed in Section 5. Finally, Section 6 offers concluding remarks.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. Generalized Intuitionistic Fuzzy Distribution
4. Reliability Characteristics
5. Numerical Example
6. Conclusion
Conflict of Interest
Figures
Tables
References

This section briefly reviews the MB family and definitions and terminologies related to GIFN_B used throughout the paper.

Definition 2.1

The probability density function (PDF) and cumulative distribution function (CDF) of the MB family are demonstrated as

f (x,θ,β)=βθg′(x)gβ-1(x) exp (-gβ(x)θ), x, θ,β≥0,F (x,θ,β)=1-exp (-gβ(x)θ), x,θ,β≥0,

where g(x) is a real-valued, strictly increasing function of x with g(0⁺) = 0, g(∞) = ∞. g′(x) denotes the derivative of g(x) with respect to x.

Definition 2.2 ([43])

Consider the non-empty set X, the GIFN_BA in X is defined as

A={⟨x,μA(x),νA(x)⟩:x∈X},

where μ_A : X → [0, 1] and ν_A : X → [0, 1], denote the degree of membership and non-membership functions of x in A, respectively, and 0≤μAδ(x)+νAδ(x)≤1, for each x ∈ X and δ = n or 1n, n = 1, 2, …, ℕ.

Definition 2.3 ([8])

An especial class of the GIFN_BA is defined as

μA(x)={(x-ab-a)1δ,a≤x≤b,1,a≤x≤c,(d-xd-c)1δ,c≤x≤d,0,o.w,νA(x)={(b-xb-a1)1δ,a1≤x≤b,0,b≤x≤c,(x-cd1-c)1δ,c≤x≤d1,1,o.w,

where a₁ ≤ a ≤ b ≤ c ≤ d ≤ d₁ and 0≤μAδ(x)+νAδ(x)≤1, for each x ∈ X. The GIFN_BA is denoted as A = (a₁, a, b, c, d, d₁, δ).

Notation 2.1

If δ = 1, the GIFN_B reduces to a trapezoidal intuitionistic fuzzy number and, for a₁ = a and d = d₁, it reduces to the trapezoidal fuzzy number. In addition, in Definition 2.3, if relation a ≤ a₁ ≤ b ≤ c ≤ d₁ ≤ d is established, the GIFN_BA is denoted as A = (a, a₁, b, c, d₁, d, δ).

Definition 2.4 ([44])

Let α₁, α₂ ∈ [0, 1] be fixed numbers such that 0≤α1δ+α2δ≤1. A set of (α₁, α₂)-cut generated by a GIFN_BA is defined by

A[α1,α2,δ]={⟨x,μA(x)≥α1,νA(x)≤α2⟩:x∈X}.

The α₁-cut set GIFN_BA is a crisp subset of ℝ, which is defined as

Aμ[α1,δ] ={⟨x,μA(x)≥α1⟩:x∈X} =[AμL[α1], AμU[α1]], 0≤α1≤1,AμL[α1]=a+(b-a) α1δ,AμU[α1]=d-(d-c) α1δ.

Similarly, a α₂-cut set GIFN_BA is a crisp subset of ℝ, and we have

Aν[α2,δ] ={⟨x,νA(x)≥α2⟩:x∈X} =[AνL[α2], AνU[α2]], 0≤α2≤1,AνL[α2]=b (1-α2δ)+a1 α2δ,AνU[α2]=c(1-α2δ)+d1 α2δ.

Therefore the (α₁, α₂)-cut set of the GIFN_B is represented as follows:

A [α1,α2,δ] ={x,x∈[AμL[α1], AμU[α1]]∩[AνL[α2], AνU[α2]]}.

The GIFN_B based on the α₁-cut and α₂-cut sets is given by

A (α1,α2,δ) =(Aμ[α1,δ], Aν[α2,δ]).

Definition 2.5

Let [a, b] and [c, d] be two α-cut sets. We define the following relations and operations on α-cut sets as follows:

1. [a, b] ≼ [c, d] ⇔ a ≤ c and b ≤ d,
2. k ⊗ [a, b] = [ka, kb], k > 0; and k ⊗ [a, b] = [kb, ka], k < 0,
3. k⊕[a, b] = [k+a, k+b]; and k⊖[a, b] = [k−b, k−a],
4. [a, b] ⊕ [c, d] = [a + c, b + d].

Definition 2.6

We define relations and operations on the GIFN_B as follows:

1. A(α₁, α₂, δ) ⊕ B(α₁, α₂, δ) = (A_μ[α₁, δ] ⊕ B_μ[α₁, δ], A_ν[α₂, δ] ⊕ B_ν[α₂, δ]),
2. k ⊗ A(α₁, α₂, δ) ⊕ b = (k ⊗ A_μ[α₁, δ] ⊕ b, k ⊗ A_ν[α₂, δ] ⊕ b),
3. b ⊝ A(α₁, α₂, δ) = (b ⊝ A_μ[α₁, δ], b ⊝ A_ν[α₂, δ]),
4. A(α₁, α₂, δ) ≼ B(α₁, α₂, δ), if and only if A_μ[α₁, δ] ≼ B_μ[α₁, δ] and A_ν[α₂, δ] ≼ B_ν[α₂, δ],
5. A(α₁, α₂, δ) = B(α₁, α₂, δ), if and only if A_μ[α₁, δ] = B_μ[α₁, δ] and A_ν[α₂, δ] = B_ν[α₂, δ],

where A(α₁, α₂, δ) and B(α₁, α₂, δ) are two GIFN_Bs.

Throughout the paper, we consider the following relations 0 ≤ α₁, α₂ ≤ 1, 0≤α1δ+α2δ≤1 and (i, j) = (1, μ), (2, ν).

3. Generalized Intuitionistic Fuzzy Distribution

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Abstract
1. Introduction
2. Preliminaries
3. Generalized Intuitionistic Fuzzy Distribution
4. Reliability Characteristics
5. Numerical Example
6. Conclusion
Conflict of Interest
Figures
Tables
References

Consider X as a continuous random variable from the PDF f(x, θ̃, β̃), where θ̃ and β̃ are GIFN_B. The set of α₁-cut of membership and α₂-cut set of non-membership functions of corresponding CDF (GIF unreliability function [GIFUF]) is defined as

Fj(x)[αi,δ]={F (x,θ,β)∣θ∈θj[αi,δ],β∈βj[αi,δ]}=F[FjL(x)[αi], FjU (x)[αi]],

where F (x, θ, β) is the crisp CDF,

FjL[αi]=infθ∈θj[αi,δ]β∈βj[αi,δ] F(x,θ,β),FjU[αi]=supθ∈θj[αi,δ]β∈βj[αi,δ] F(x,θ,β).

Finally, it can be concluded that

F˜(x)=F(x)(α1,α2,δ)=(Fμ(x)[α1,δ], Fν(x)[α2,δ]),

and a set of (α₁, α₂)-cut GIF distribution function is given by

F(x)[α1,α2,δ]={w,w∈Fμ(x)[α1,δ]∩ Fν(x)[α2,δ]},

where F_j(x) [α_i, δ] is a two-variate function with regards to α_i, i = 1, 2 and x. Consider the number x₀, the GIF probability (GIFP) of X ≤ x₀ is represented by F̃ (x₀). In this method, for every given α₁₀ and α₂₀, shapes of F_μ(x) [α₁₀, δ] and F_ν(x) [α₂₀, δ], behave as bands with upper and lower bounds.

The distribution bands satisfy the following properties

1. F_j (0) [α_i₀, δ] = [0, 0],
2. F_j (∞) [α_i₀, δ] = [1, 1],
3. F_j (x₁) [α_i₀, δ] ≼ F_j (x₂) [α_i₀, δ] ⇔ x₁ ≤ x₂, i.e., bands of F_j(x) [α_i₀, δ] are non-decreasing.

Based on the definition of GIFUF, we define a set of α₁-cut of membership and α₂-cut set of non-membership functions of the GIFP of C as follows:

Pj(C)[αi,δ]={P(C,θ,β)∣θ∈θj[αi,δ],β∈βj[αi,δ]}=[PjL(C)[αi], PjU(C)[αi]],

where P (C, θ, β) = ∫_Cf (x, θ, β) dx, PjL[αi]=infθ∈θj[αi,δ]β∈βj[αi,δ]P (C, θ, β), and PjU[αi]=supθ∈θj[αi,δ]β∈βj[αi,δ]P(C,θ,β). Finally, we have

P˜ (C)=P(C) (α1,α2,δ)=(Pμ(C) [α1,δ], Pν(C) [α2,δ]).

The set of (α₁, α₂)-cut GIFP of C is represented as

P (C)[α1,α2,δ]={w,w∈Pμ(C) [α1,δ]∩ Pν(C) [α2,δ]}.

[0, 0] ≼ P_μ (C) [α₁, δ], P_ν (C) ≼ [1, 1], and

[0,0]=P (∅) [α1,α2,δ]≼P (C) [α1,α2,δ]≼P (S) [α1,α2,δ]=[1,1],

where ∅︀ and S, are the empty and universal sets, respectively.

Consider the lifetime random variable X from the MB family with GIF lifetime parameters θ̃ = (a₁₁, a₁, b₁, c₁, d₁, d₁₁, δ) and β̃ = (a₂₂, a₂, b₂, c₂, d₂, d₂₂, δ), say MB(θ̃, β̃). Since F (x,θ,β)=1-exp (-g(x)βθ) is a monotonically decreasing function of θ, for g(x) ≥ 1, the function F (x, θ, β) is increasing and otherwise decreasing with respect to β, and

Fμ(x) [α1,δ]= {[1-exp {-g(x)a2+(b2-a2)α1δd1-(d1-c1) α1δ}, 1-exp {-g(x)d2-(d2-c2)α1δa1+(b1-a1) α1δ}], g(x≥1) {[1-exp {-g(x)d2-(d2-c2)α1δd1-(d1-c1) α1δ}, 1-exp {-g(x)a2+(b2-a2)α1δa1+(b1-a1) α1δ}], g(x)<1,Fν(x) [α2,δ]= {[1-exp {-g(x)b2+(1-α2δ)+a22α2δc1 (1-α2δ)+d11α2δ}, 1-exp {-g(x)c2(1-α2δ)+d22α2δb1 (1-α2δ)+a11α2δ}], g(x)≥1[1-exp {-g(x)c2(1-α2δ)+d22α2δc1 (1-α2δ)+d11α2δ}, 1-exp {-g(x)b2(1-α2δ)+a22α2δb1 (1-α2δ)+a11α2δ}], g(x)<1.

Corollary 3.1

P (C^c) (α₁, α₂, δ) = 1 ⊝ P(C) (α₁, α₂, δ), where C^c means complement set of C.

Proof

Based on GIFP definition, it can be concluded that

Pj (Cc)[αi,δ]={1-P (C,θ,β)∣θ∈θj [αi,δ],β∈βj [αi,δ]}=[PjL (Cc) [αi], PjU (Cc) [αi]]=[infθ∈θj[αi,δ]β∈βj[αi,δ] (1-P (C,θ,β)), supθ∈θj[αi,δ]β∈βj[αi,δ] (1-P (C,θ,β))]=[1-supθ∈θj[αi,δ]β∈βj[αi,δ] P (C,θ,β),1- infθ∈θj[αi,δ]β∈βj[αi,δ] P (C,θ,β)]=1⊝[PjL (C) [αi],PjU (C) [αi]],

by Definition 2.6; thus, the proof is completed.

Corollary 3.2

If C₁ ⊂ C₂, then P(C₁)(α₁, α₂, δ) ≼ P(C₂)(α₁, α₂, δ).

Proof

Since P (C₁, θ, β) ≤ P(C₂, θ, β), we have

Pj (C1)[αi,δ]=[infθ∈θj[αi,δ]β∈βj[αi,δ] P (C1,θ,β), supθ∈θj[αi,δ]β∈βj[αi,δ]P (C1,θ,β)]≼[infθ∈θj[αi,δ]β∈βj[αi,δ] P (C2,θ,β), supθ∈θj[αi,δ]β∈βj[αi,δ]P (C2,θ,β)]=Pj (C2) [αi,δ],

regarding Definition 2.6; thus, the proof is accomplished.

4. Reliability Characteristics

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1. Introduction
2. Preliminaries
3. Generalized Intuitionistic Fuzzy Distribution
4. Reliability Characteristics
5. Numerical Example
6. Conclusion
Conflict of Interest
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Consider X as a random variable with density function f(x, θ̃, β̃) where θ̃ and β̃ are GIFN_B. The GIF reliability characteristic (GIFRC) is denoted by ψ̃(t). A set of α₁-cut of membership and α₂-cut set of non-membership functions GIFRC of lifetime component is defined by

ψj(t)[αi,δ]={ψ (t,θ,β)∣θ∈θj[αi,δ],β∈βj[αi,δ]}=[ψjL(t)[αi], ψjU (t)[αi]],

where

ψjL(t)[αi]=infθ∈θj[αi,δ]β∈βj[αi,δ] ψ(t,θ,β), ψjU(t)[αi]=supθ∈θj[αi,δ]β∈βj[αi,δ] ψ(t,θ,β).

The function ψ(t, θ, β) can be considered as the reliability, conditional reliability, hazard rate, cumulative risk, and reverse hazard functions.

Finally, as ψ(t) (α₁, α₂, δ) = (ψ_μ(t) [α₁, δ], ψ_ν(t) [α₂, δ]) and (α₁, α₂)-cut set of GIFRC is given by

ψ(t)[α1,α2,δ]={w,w∈ψμ (t) [α1,δ]∩ψν(t) [α2,δ]},

where ψ_j(t) [α_i, δ], i = 1, 2 is a two-variate function in terms of α_i and t. For t₀, ψ̃ (t₀) is a GIFN_B. In this method, for every especial α₁₀ and α₂₀, shapes of ψ_μ(t) [α₁₀, δ], ψ_ν(t) [α₂₀, δ] and ψ(t) [α₁₀, α₂₀, δ] are same as bands with upper and lower bounds.

4.1 Generalized Intuitionistic Fuzzy Reliability

The GIF reliability (GIFR) function, denoted by S̃(t), is the GIF probability of a unit surviving for more than time t. The cut sets of GIFR function are indicated as

Sj(t)[αi,δ]={S (t,θ,β)∣θ∈θj [αi,δ],β∈βj [αi,δ]}=[SjL(t)[αi], SjU (t)[αi]],

where S(t,θ,β)=∫t∞f(x,θ,β) dx,SjL[αi]=infθ∈θj[αi,δ]β∈βj[αi,δ] S(t,θ,β) and SjU(t)[αi]=supθ∈θj[αi,δ]β∈βj[αi,δ] S(t,θ,β).

Finally, the GIFR is shown as S(t)(α₁, α₂, δ) = (S_μ(t)[α₁, δ], S_ν(t)[α₂, δ]) and the set of (α₁, α₂)-cut GIFR is represented by

S(t)[α1,α2,δ]={w,w∈Sμ(t) [α1,δ]∩Sν(t) [α2,δ]}.

S_j(t)[α_i, δ], i = 1, 2 are two-variable functions in terms of α_i and t. For t₀, S̃(t₀) is a GIFN_B. In this method, for every especial α₁₀ and α₂₀, shapes of S_μ(t)[α₁₀, δ] and S_ν(t)[α₂₀, δ] are the same as bands with upper and lower bounds.

The reliability bands have the following properties

1. S_j (0) [α_i₀, δ] = [1, 1]; i.e,. no one starts off dead,
2. S_j (∞) [α_i₀, δ] = [0, 0],; i.e., everyone dies ultimately,
3. S_j (t₁) [α_i₀, δ] ≽ S_j (t₂) [α_i₀, δ] ⇔ t₁ ≤ t₂ ; i.e. bands of S_j(t) [α_i₀, δ] decline monotonically.

Let the lifetime random variable X has MB(θ̃, β̃); the cut sets of GIFR function are as follows:

Sj(t)[αi,δ]={exp (-g(t)βθ)∣θ∈θj [αi,δ],β∈βj [αi,δ]}.

Subsequently, the α₁-cut set and α₂-cut set of GIFR are

Sμ(t) [α1,δ]= {[exp {-g(t)d2-(d2-c2)α1δa1+(b1-a1) α1δ}, exp {-g(t)a2+(b2-a2)α1δd1-(d1-c1) α1δ}], g(t)≥1,[exp {-g(t)a2+(b2-a2)α1δa1+(b1-a1) α1δ}, exp {-g(t)d2+(d2-c2)α1δd1-(d1-c1) α1δ}], g(t)<1,Sν(t) [α2,δ]= {[exp {-g(t)c2(1-α2δ)+d22α2δb1(1-α2δ)+a11α2δ}, exp {-g(t)b2(1-α2δ)+a22α2δc1(1-α2δ)+d11α2δ}], g(t)≥1, {[exp {-g(t)b2(1-α2δ)+a22α2δb1(1-α2δ)+a11α2δ}, exp {-g(t)c2(1-α2δ)+d22α2δc1(1-α2δ)+d11α2δ}], g(t)<1.

Corollary 4.1

For every δ, we have

S(t) [1,0,δ]= {[exp {-g(t)c2b1}, exp{-g(t)b2c1}], g(t)≥1,[exp {-g(t)b2b1}, exp{-g(t)c2c1}], g(t)<1,S(t) [0,1,δ]= {[exp {-g(t)d2a1}, exp{-g(t)a2d1}], g(t)≥1,[exp {-g(t)a2b1}, exp{-g(t)d2d1}], g(t)<1.

4.2 Generalized Intuitionistic Fuzzy Conditional Reliability

The GIF conditional reliability (GIFCR) function of the lifetime component is denoted by S̃ (t| τ). The set of α₁-cut of membership and α₂-cut set of non-membership functions of S̃ (t| τ) are given by

Sj(t∣τ) [αi,δ]={S (t,θ,β∣τ)∣θ∈θj [αi,δ],β∈βj [αi,δ]}=[SjL (t∣τ) [αi],SjU (t∣τ) [αi]],

where S(t,θ,β∣τ)=S(t+τ,θ,β)S(τ,θ,β),SjL[t∣τ][αi]=infθ∈θj[αi,δ]β∈βj[αi,δ] S(t,θ,β∣τ) and SjU(t∣τ) [αi]=supθ∈θj[αi,δ]β∈βj[αi,δ] S(t,θ,β∣τ). Hence, GIFCR can be given as S (t| τ )(α₁, α₂, δ) = (S_μ( t| τ ) [α₁, δ], S_ν( t| τ )[α₂, δ]) and a set of (α₁, α₂)-cut GIFCR function is described as

S(t∣τ) [α1,α2,δ]={w,w∈Sμ(t∣τ) [α1,δ]∩Sν(t∣τ) [α2,δ]}.

Consider the random variable X from MB(θ̃, β̃); cut sets of the GIFCR function are given as

Sj(t∣τ)[αi,δ]={exp {-g(t+τ)β-g (τ)βθ}∣θ∈θj [αi,δ],β∈βj [αi,δ]}.

The α₁-cut set and α₂-cut set of GIFCR are computed as follows:

Sμ(t∣τ) [α1,δ]= [exp {-g(t+τ)d2-(d2-c2)α1δ-g (τ)d2-(d2-c2)α1δa1+(b1-a1) α1δ}, exp {-g(t+τ)a2+(b2-a2)α1δ-g (τ)a2+(b2-a2)α1δd1-(d1-c1) α1δ}],Sν(t∣τ) [α2,δ]= [exp {-g(t+τ)c2(1-α2δ)+d22α2δ-g(τ)c2(1-α2δ)+d22α2δb1 (1-α2δ)+a11α2δ}, exp {-g(t+τ)b2(1-α2δ)+a22α2δ-g(τ)b2(1-α2δ)+a22α2δc1(1-α2δ)+d11α2δ}].

Corollary 4.2

For every δ, we have

S (t∣τ) [1,0,δ]= [exp {-g (t+τ)c2-g (τ)c2b1}, exp{-g (t+τ)b2-g (τ)b2c1}],S(t∣τ) [0,1,δ]= [exp {-g (t+τ)d2-g (τ)d2a1}, exp{-g (t+τ)a2-g (τ)a2d1}].

4.3 Generalized Intuitionistic Fuzzy Hazard

The GIF hazard (GIFH) function of the lifetime component is denoted by h̃(t), which means the GIF probability of the system failing at a distance Δt when it was active until t. A set of α₁-cut of membership, α₂-cut set of non-membership functions of GIFH are

hj(t) [αi,δ]={h (t,θ,β)∣θ∈θj [αi,δ],β∈βj [αi,δ]}=[hjL (t) [αi],hjU (t) [αi]],

where

hjL(t)[αi]=infθ∈θj[αi,δ]β∈βj[αi,δ] h(t,θ,β),hjU(t)[αi]=supθ∈θj[αi,δ]β∈βj[αi,δ] h(t,θ,β),

and h (t,θ,β)=f (t,θ,β)S (t,θ,β).

The GIFH can be expressed as h(α₁, α₂, δ) = (h_μ(t)[α₁, δ], h_ν(t)[α₂, δ]) and a set of (α₁, α₂)-cut a GIFH function is defined by

h(t) [α1,α2,δ]={w,w∈hμ (t) [α1,δ]∩hν(t) [α2,δ]}.

h_j(t)[α_i, δ], i = 1, 2 is a two-variate function in terms of α_i and t. In addition, cut sets of the GIF cumulative hazard (GIFCH) function are

Hj(t) [αi,δ]={H (t,θ,β)∣θ∈θj [αi,δ],β∈βj [αi,δ]}=[HjL (t) [αi], HjU (t) [αi]],

where

HjL(t)[αi]=infθ∈θj[αi,δ] H(t,θ,β),HjU(t)[αi]=supθ∈θj[αi,δ] H(t,θ,β),

and H (t,θ,β)=∫0tf (u,θ,β)S (u,θ,β)du.

Based on the random variable X from MB(θ̃, β̃) family, the cut sets of GIFH function are

hj(t) [αi,δ]={βθg′(x)gβ-1(x)∣θ∈θj [αi,δ],β∈βj [αi,δ]}=[hj(t)L [αi], hj(t)U [αi]],

where

hjL(t) [αi]=inf {βθg′(x)gβ-1(x)∣θ∈θj [αi,δ],β∈βj [αi,δ]},hjU(t) [αi]=sup {βθg′(x)gβ-1(x)∣θ∈θj [αi,δ],β∈βj [αi,δ]}.

In addition,

Hj(t) [αi,δ]={gβ(x)θ∣θ∈θj [αi,δ],β∈βj [αi,δ]}=[Hj(t)L [αi], Hj(t)U [αi]],

where

Hμ(t) [α1,δ]={[g(t)a2+(b2-a2)α1δd1-(d1-c1) α1δ,g(t)d2-(d2-c2)α1δa1+(b1-a1) α1δ], g(t)≥1[g(t)d2+(d2-c2)α1δd1-(d1-c1) α1δ,g(t)a2-(b2-a2)α1δa1+(b1-a1) α1δ], g(t)<1,

and

Hν(t) [α2,δ]={[g(t)b2+(1-α2δ)+a22α2δc1 (1-α2δ)+d11α2δ,g(t)c2+(1-α2δ)+d22α2δb1 (1-α2δ)+a11α2δ], g(t)≥1,[g(t)c2(1-α2δ)+d22α2δc1 (1-α2δ)+d11α2δ,g(t)b2(1-α2δ)+a22α2δb1 (1-α2δ)+a11α2δ], g(t)<1.

Corollary 4.3

For every δ, we have

H(t) [1,0,δ]= {[ g(t)b2c1,g(t)c2b1], g(t)≥1,[ g(t)c2c1,g(t)b2b1], g(t)<1,H(t) [0,1,δ]= {[ g(t)a2d1,g(t)d2a1], g(t)≥1,[ g(t)d2d1,g(t)a2a1], g(t)<1.

Corollary 4.4

Consider ψ(t, θ, β) as the unreliability, reliability, conditional reliability, hazard rate, and cumulative hazard functions,

(i) If δ₁ ≤ δ₂, then ψ_μ(t) [α₁, δ₁] ⊂ ψ_μ(t) [α₁, δ₂] and ψ_ν(t) [α₂, δ₂] ⊂ ψ_ν(t) [α₂, δ₁]; therefore, increasing δ leads to wider bandwidth of ψ_μ(t) [α₁, δ] and narrower bandwidth of ψ_ν(t) [α₂, δ].
(ii) If (α1(1),1-α2(1))≼(α1(2),1-α2(2)), then ψμ(t) [α1(2),δ]⊂ψμ(t) [α1(1),δ] and ψν(t) [α2(2),δ]⊂ψν(t) [α2(1),δ]. Hence, as (α₁, 1−α₂) increases, bandwidth of ψ_μ(t)[α₁, δ] and ψ_ν(t) [α₂, δ] reduces.
(iii) Let 1-α1δ=α2δ, a_i₁ = a_i and d_i₁ = d_i, i = 1, 2. Then, we have

ψμ(t) [α1,δ]=ψν(t) [α2,δ]=ψ(t) [α1,α2,δ].

Definition 4.1

Let F̃(x) be a GIFUF, then

(i) It belongs to the class of distributions with an increasing failure rate (IFR) (decreasing failure rate [DFR]) if S_j(t)[α_i, δ] increases (decreases) with respect to t.
(ii) It belongs to the class of distributions with an increasing failure rate average (IFRA) if 1tHj(t)[αi,δ] increases with respect to t.
(iii) It belongs to the new better than used (NBU) class, if

Fj(t1+t2)[αi,δ]≺_Fj(t1)[αi,δ]⊗Fj(t2)[αi,δ].

4.4 GIFR Function of k out of n Systems

If n-components are connected in k out of n systems with independent and identically distributed lifetime variables, then the α_i-cut (i = 1, 2) of GIFR with GIF distribution is

Sj(t) [αi,δ]={S (t,θ,β)∣θ∈θj [αi,δ],β∈βj [αi,δ]}=[SjL (t) [αi],SjU (t) [αi]],

where

SjL(t)[αi]=infθ∈θj[αi,δ]β∈βj[αi,δ] S(t,θ,β)SjU(t)[αi]=supθ∈θj[αi,δ]β∈βj[αi,δ] S(t,θ,β),

and

S (t,θ,β)=∑r=kn(nr)S(t,θ,β)rF(t,θ,β)n-r.

If n-components are connected serially, cut sets in the GIFMB family are represented by

Sj (t) [αi,δ]={exp(-ng(t)βθ)∣θ∈θj [αi,δ],β∈βj [αi,δ]}.

If n-components are connected parallelly, cut sets with the GIFMB family are given as

Sj (t) [αi,δ]={1-(1-exp(-g(t)βθ))n∣θ∈θj [αi,δ],β∈βj [αi,δ]}.

5. Numerical Example

Go to

Abstract
1. Introduction
2. Preliminaries
3. Generalized Intuitionistic Fuzzy Distribution
4. Reliability Characteristics
5. Numerical Example
6. Conclusion
Conflict of Interest
Figures
Tables
References

Let the lifetime of an electronic component be modeled by the Weibull distribution denoted by (W(θ̃, β̃)), a sub-model of the MB family, with GIF parameters θ̃ = (2, 2.2, 2.3, 2.4, 2.5, 2.7, δ) and β̃ = (0.2, 0.25, 0.3, 0.35, 0.4, 0.45, δ), where δ = 1, 2. The cut sets of GIFP of X ≤ 0.1 is provided as

Pj (X≤0.1) [αi]={1-exp (-0.1βθ)∣θ∈θj [αi,δ],β∈βj [αi,δ]}.

Hence, for δ = 1, 2,

Pμ (X≤0.1) [α1,δ]=[1-exp (-0.10.4-0.05α1δ2.5-0.1α1δ),1-exp{0.10.25+0.05α1δ2.2+0.1α1δ}]Pν (X≤0.1) [α2,δ]=[1-exp (-0.10.35+0.1α2δ2.4+0.3α2δ),1-exp{0.10.3-0.1α2δ2.3-0.3α2δ}].

Based on Figure 1 and Table 1, we infer that the ambiguity in the GIFP is decreased by increasing α₁ and decreasing α₂. In Figure 1, the membership and non-membership functions of GIFP are smoother for δ = 2 than δ = 1. For δ = 1, cut sets of the membership function of the GIFP have less bandwidth than δ = 2, whereas that of the GIFP for δ = 1 have large bandwidth than δ = 2 (see Table 2).

The cut sets of GIFR are given by

Sj(t) [αi,δ]={exp (-tβθ)∣θ∈θj[αj,δ],β ∈βj [αi,δ]},Sμ(t) [α1,δ]={[exp{-t0.4-0.05α1δ2.2+0.1α1δ}, exp{-t0.25+0.05α1δ2.5-0.1α1δ}], t≥1,[exp{-t0.25-0.05α1δ2.2+0.1α1δ}, exp{-t0.04-0.05α1δ2.5-0.1α1δ}] t<1,Sν(t) [α2,δ]={[exp{-t0.35+0.1α2δ2.3-0.3α2δ}, exp{-t0.3-0.1α2δ2.4+0.3α2δ}], t≥,[exp{-t-0.03-0.1α2δ2.3-0.3α2δ}, exp{-t-0.35+0.1α2δ2.4+0.3α2δ}], t<1.

The GIFR bands for α₁ = 0 and α₂ = 1 are computed as follows:

Sμ(t) [0,δ]={[exp {-t0.42.2}, exp {-t0.252.5}], t≥1,[exp {-t0.252.2}, exp {-t0.42.5}], t<1,Sν(t) [1,δ]={[exp {-t0.452}, exp {-t0.22.7}], t≥1,[exp {-t0.22}, exp {-t0.452.7}], t<1,

then

S(t) [0,1,δ]={[exp {-t0.42.2}, exp {-t0.252.5}], t≥1,[exp {-t0.252.2}, exp {-t0.42.5}], t<1.

The GIFR bands for α₁ = 0 and α₂ = 1 in Figure 2 show decreasing behavior with increasing t.

Considering t = 0.2, the cut sets of GIFR are given by

Sμ (0.2) [α1,δ]=[exp {-0.20.25+0.05α1δ2.2+0.1α1δ},exp{-0.20.4+0.05α1δ2.5-0.1α1δ}],Sν (0.2) [α2,δ]=[exp {-0.20.3-0.1α2δ2.2-0.3α2δ},exp{-0.20.35+0.1α2δ2.4+0.3α2δ}].

The different cut sets of GIFR for δ = 2 and different values of (α₁, α₂) are represented in Table 2, indicating reduced bandwidth for large and small values of α₁ and α₂, respectively.

Figure 3 depicts the membership and non-membership functions of GIFR for δ = 2.

The reliability bands S(t) [0.3, 0.1, δ] for δ = 0.5, 1, and 2 are computed, respectively, as

S(t) [0.3,0.1,0.5]={[exp{-t0.3732.225},exp{-t0.2772.445}],t≥1,[exp{-t0.2772.255},exp{-t0.3732.445}],t<1,,S(t) [0.3,0.1,1]={[exp{-t0.362.27},exp{-t0.292.43}],t≥1,[exp{-t0.292.27},exp{-t0.362.43}],t<1,,S(t) [0.3,0.1,2]={[exp{-t0.3512.297},exp{-t0.2992.403}],t≥1,[exp{-t0.2992.297},exp{-t0.3512.403}],t<1.,

Accordingly, the reliability bands for δ = 0.5, 1, and 2 in Figure 4 reveal that by increasing δ, the width of reliability bands is decreased.

The reliability bands S(t) [α₁, α₂, 1] for (α₁, α₂) = (0, 1), (1, 0), (0.8, 0.2) respectively provided as

S(t) [0,1,1]={[exp{-t0.42.2},exp{-t0.252.5}],t≥1,[exp{-t0.252.2},exp{-t0.42.5}],t<1,S(t) [1,0,1]={[exp{-t0.352.3},exp{-t0.32.4}],t≥1,[exp{-t0.32.3},exp{-t0.352.4}],t<1,,S(t) [0.8,0.2,1]={[exp{-t0.362.28},exp{-t0.292.42}],t≥1,[exp{-t0.292.28},exp{-t0.362.42}],t<1.,

The reliability bands for (α₁, α₂) = (0, 1), (1, 0), (0.8, 0.2) are depicted in Figure 5. The width of the reliability band reduces as α₁ and α₂ approaches one and zero, respectively.

The cut sets of GIFCR are given as follows:

Sμ(t∣τ) [α1,δ]=[exp(-(t+τ)0.4-0.05α1δ-τ0.4-0.05α1δ2.2+0.1α1δ),exp(-(t+τ)0.25+0.05α1δ-τ0.25+0.05α1δ2.5-0.1α1δ)],Sν(t∣τ) [α2,δ]=[exp(-(t+τ)0.35+0.1α2δ-τ0.35+0.1α2δ2.3-0.3α2δ),exp(-(t+τ)0.3-0.1α2δ-τ0.3-0.1α2δ2.4+0.3α2δ)].

The GIFCR bands with τ = 1 for α₁ = 0 and α₂ = 1 are as follows:

Sμ(t∣1) [0,δ]=[exp(-(t+1)0.4-12.2),exp(-(t+1)0.25-12.5)],Sν(t∣1) [1,δ]=[exp(-(t+1)0.45-12),exp(-(t+1)0.2-12.7)].

GIFCR bands for α₁ = 0 and α₂ = 1 in Figure 6 demonstrate their decreasing behavior.

Let t₀ = 2, τ = 1, the membership and non-membership functions of S̃ (2| 1) are as follows:

Sμ(2∣1) [α1,δ]=[exp(-30.4-0.5α1δ-12.2+0.1α1δ),exp(-30.25+0.05α1δ-12.5-0.1α1δ)],Sν(2∣1) [α2,δ]=[exp(-30.35+0.1α2δ-12.3-0.3α2δ),exp(-30.3-0.1α2δ-12.4+0.3α2δ)].

Both membership and non-membership functions of GIFCR are presented in Figure 7 for δ = 2.

The upper and lower bounds of α_i-cut of the GIFH function are given by

hjL(t) [αi]=inf{βθtβ-1∣θ∈θj[αi,δ],β∈βj[αi,δ]},hjU(t) [αi]=sup{βθtβ-1∣θ∈θj[αi,δ],β∈βj[αi,δ]}.

Corollary 5.1

For a₂₂ ≥ 1, h_j(t)[α_i, δ] are increasing bands, which means W(θ̃, β̃) belongs to the IFR class, and for d₂₂ ≤ 1, h_j(t)[α_i, δ] are decreasing bands; in other words, W(θ̃, β̃) belongs to a DFR class. In addition, it belongs to the NBU class.

Let

h1=(0.25+0.05α1δ)t-0.75+0.05α1δ,h2=(0.4-0.05α1δ)t-0.6-0.05α1δ,h*=β*tβ*-1,β*=(-ln t)-1,h3=(0.3-0.1α2δ)t-0.7-0.1α2δ,h4=(0.35+0.1α2δ)t-0.65+0.1α2δ.

The GIFH bands, for t ≥ 1, are given as

hμ(t) [α1,δ]=[(0.25+0.05α1δ2.5-0.1α1δ)t-0.75+0.05α1δ,(0.4-0.05α1δ2.2+0.1α1δ)t-0.6-0.05α1δ],hν(t) [α2,δ]=[(0.3-0.1α2δ2.4+0.3α2δ)t-0.7-0.1α2δ,(0.35+0.1α2δ2.3-0.3α2δ)t-0.65+0.1α2δ].

For t < 1, the GIFH bands are represented by

hμ(t) [α1,δ]={[h22.5-0.1α1δ,h12.2+0.1α1δ],0<t<e-10.25+0.05α1δ,={[min(h1,h2)2.5-0.1α1δ,h*2.2+0.1α1δ],e-10.25+0.05α1δ<t<e-10.4-0.05α1δ,[h12.5-0.1α1δ,h22.2+0.1α1δ],e-10.4-0.05α1δ<t<1,hν(t)[α1,δ]={[h42.4+0.3α2δ,h32.3-0.3α2δ],0<t<e-10.3-0.1α2δ,[min(h3,h4)2.4+0.3α2δ,h*2.3-0.3α2δ],e-10.3-0.1α2δ<t<e-10.35+0.1α2δ,[h32.4+0.3α2δ,h42.3-0.3α2δ],e-10.35+0.1α2δ<t<1.

In 8, we provide the membership and non-membership functions of the GIFH for t = δ = 2.

GIFH bands for α₁ = 0 and α₂ = 1 at t > 1 and t ≤ 1 are computed, respectively, as

hμ(t) [0,δ]=[t-0.7510,0.4t-0.62.2]hν(t)[1,δ]=[0.2t-0.82.7,0.45t-0.552].

and

hμ(t) [0,δ]={[0.4t-0.62.5,0.25t-0.752.2],0<t<e-10.25,[min(0.4t-0.6,0.25t-0.75)2.5,β*tβ*-12.2],e-10.25<t<e-10.4,[0.25t-0.752.2,0.4t-0.62.5],e-10.4<t<1,hν(t) [1,δ]={[0.45t-0.552.7,0.2t-0.82],0<t<e-10.2,[min(0.2t-0.8,0.45t-0.55)2.7,β*tβ*-12],e-10.2<t<e-10.45,[0.2t-0.82.7,0.45t-0.552],e-10.45<t<1.

Figure 9 confirm the decreasing behavior of GIFH bands for α₁ = 0 and α₂ = 1.

The cut sets of GIFUF are given by

Fj(x) [αi,δ]={1-exp (-xβθ)∣θ∈θj[αi,δ],β∈βj[αi,δ]},Fμ(x) [α1,δ]={[1-exp {-x0.25+0.05α1δ2.5-0.1α1δ},1-exp {-x0.4-0.05α1δ2.2+0.1α1δ}],x≥1,[1-exp {-x0.4-0.05α1δ2.5-0.1α1δ},1-exp {-x0.25+0.05α1δ2.2+0.1α1δ}],x<1,Fν(x) [α2,δ]={[1-exp {-x0.3-0.1α2δ2.4+0.3α2δ},1-exp {-x0.35+0.1α2δ2.3-0.3α2δ}],x≥1,[1-exp {-x0.35+0.1α2δ2.4+0.3α2δ},1-exp {-x0.3-0.1α2δ2.3-0.3α2δ}],x<1.

GIFUF bands for α₁ = 0 and α₂ = 1 are computed as follows:

Fμ(x) [0,δ]={[1-exp {-x0.252.5},1-exp {-x0.42.2}],x≥1,[1-exp {-x0.42.5},1-exp {-x0.252.2}],x<1,Fv(x) [1,δ]={[1-exp {-x0.22.7},1-exp {-x0.452}],x≥1,[1-exp {-x0.452.7},1-exp {-x0.22}],x<1,

then

F(x) [0,1,δ]={[1-exp {-x0.252.5},1-exp {-x0.42.2}],x≥1,[1-exp {-x0.42.5},1-exp {-x0.252.2}],x<1.

The GIFUF bands for α₁ = 0 and α₂ = 1 in Figure 10 reveal that the bands increase with increasing x.

Based on our findings, we infer that increasing α₁ and decreasing α₂ reduces the ambiguity of GIFR, GIFCR, and GIFH bands.

6. Conclusion

Go to

Abstract
1. Introduction
2. Preliminaries
3. Generalized Intuitionistic Fuzzy Distribution
4. Reliability Characteristics
5. Numerical Example
6. Conclusion
Conflict of Interest
Figures
Tables
References

This paper is an extended analysis of the reliability characteristics of the MB family of lifetime distributions, where lifetime parameters are considered GIFNs. In our approach, we obtain GIF reliability measures, including reliability, conditional reliability, hazard, and unreliability functions, through GIFN_B. The obtained measures are consistent with the probability measures proposed by Buckley [45] for FSs. In addition, we derive the cut sets of reliability characteristics as two-variate functions in terms of α_i and t. For t₀, they are generalized intuitionistic fuzzy numbers, and for every special α₁₀ and α₂₀, the curves are the same as bands with upper and lower bounds. Our method is more comprehensive than previous methods. For δ = 1, the results are compatible with intuitionistic fuzzy reliability evaluation, and for δ = 1, α1δ=1-α2δ, a = a₁ and d = d₁, a fuzzy reliability evaluation is obtained due to [2] and [3]. The reliability characteristics based on the Weibull lifetime distribution with GIF parameters are depicted as numerical examples, along with their cut sets with several combinations of α₁ and α₂ values. This paper is limited to the GIF reliability functions without discussing the estimation approaches. Hence, the estimation technique of the GIF reliability functions based on classical and Bayesian approaches can be an interesting topic for subsequent studies. For future studies, the fuzzy MB family can be extended by the PFS, FFS, and aggregation operators. Moreover, the GIF analysis in LP, DEA, MCDM, and SPP contexts is a prominent research area in fuzzy environments.

Conflict of Interest

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Abstract
1. Introduction
2. Preliminaries
3. Generalized Intuitionistic Fuzzy Distribution
4. Reliability Characteristics
5. Numerical Example
6. Conclusion
Conflict of Interest
Figures
Tables
References

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

Figures

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Abstract
1. Introduction
2. Preliminaries
3. Generalized Intuitionistic Fuzzy Distribution
4. Reliability Characteristics
5. Numerical Example
6. Conclusion
Conflict of Interest
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Tables
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Fig. 1.

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

Fig. 2.

GIFR bands for α₁ = 0 and α₂ = 1.

Fig. 3.

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

Fig. 4.

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

Fig. 5.

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

Fig. 6.

GIFCR bands for α₁ = 0 and α₂ = 1.

Fig. 7.

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

Fig. 8.

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

Fig. 9.

The GIFH bands for α₁ = 0 and α₂ = 1.

Fig. 10.

GIFUF bands for α₁ = 0 and α₂ = 1.

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Abstract
1. Introduction
2. Preliminaries
3. Generalized Intuitionistic Fuzzy Distribution
4. Reliability Characteristics
5. Numerical Example
6. Conclusion
Conflict of Interest
Figures
Tables
References

Table. 1.

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

(α₁, α₂)	δ = 1			δ = 2

	P_μ[α₁]	P_ν[α₂]	P[α₁, α₂]	P_μ[α₁]	P_ν[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.

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

(α₁, α₂)	S_μ (0.2) [α₁, 2]	S_ν (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]

References

Go to

Abstract
1. Introduction
2. Preliminaries
3. Generalized Intuitionistic Fuzzy Distribution
4. Reliability Characteristics
5. Numerical Example
6. Conclusion
Conflict of Interest
Figures
Tables
References

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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

(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

Abstract

Keywords: (&alpha,_1, ,&alpha,₂)-cut set, Generalized intuitionistic fuzzy distribution, Generalized intuitionistic fuzzy number, Generalized intuitionistic fuzzy reliability, Moore and Bilikam.

1. Introduction

2. Preliminaries

This section briefly reviews the MB family and definitions and terminologies related to GIFN_B used throughout the paper.

Definition 2.1

The probability density function (PDF) and cumulative distribution function (CDF) of the MB family are demonstrated as

\begin{array}{l} f (x, θ, β) = \frac{β}{θ} g^{'} (x) g^{β - 1} (x) exp (- \frac{g^{β} (x)}{θ}), x, θ, β \geq 0, \\ F (x, θ, β) = 1 - exp (- \frac{g^{β} (x)}{θ}), x, θ, β \geq 0, \end{array}

where g(x) is a real-valued, strictly increasing function of x with g(0⁺) = 0, g(∞) = ∞. g′(x) denotes the derivative of g(x) with respect to x.

Definition 2.2 ([43])

Consider the non-empty set X, the GIFN_BA in X is defined as

A = {⟨ x, μ_{A} (x), ν_{A} (x) ⟩ : x \in X},

where μ_A : X → [0, 1] and ν_A : X → [0, 1], denote the degree of membership and non-membership functions of x in A, respectively, and $0 \leq μ_{A}^{δ} (x) + ν_{A}^{δ} (x) \leq 1$ , for each x ∈ X and δ = n or $\frac{1}{n}$ , n = 1, 2, …, ℕ.

Definition 2.3 ([8])

An especial class of the GIFN_BA is defined as

\begin{array}{l} μ_{A} (x) = {\begin{array}{l} {(\frac{x - a}{b - a})}^{\frac{1}{δ}}, & a \leq x \leq b, \\ 1, & a \leq x \leq c, \\ {(\frac{d - x}{d - c})}^{\frac{1}{δ}}, & c \leq x \leq d, \\ 0, & o.w, \end{array} \\ ν_{A} (x) = {\begin{array}{l} {(\frac{b - x}{b - a_{1}})}^{\frac{1}{δ}}, & a_{1} \leq x \leq b, \\ 0, & b \leq x \leq c, \\ {(\frac{x - c}{d_{1} - c})}^{\frac{1}{δ}}, & c \leq x \leq d_{1}, \\ 1, & o.w, \end{array} \end{array}

where a₁ ≤ a ≤ b ≤ c ≤ d ≤ d₁ and $0 \leq μ_{A}^{δ} (x) + ν_{A}^{δ} (x) \leq 1$ , for each x ∈ X. The GIFN_BA is denoted as A = (a₁, a, b, c, d, d₁, δ).

Notation 2.1

Definition 2.4 ([44])

Let α₁, α₂ ∈ [0, 1] be fixed numbers such that $0 \leq α_{1}^{δ} + α_{2}^{δ} \leq 1$ . A set of (α₁, α₂)-cut generated by a GIFN_BA is defined by

A [α_{1}, α_{2}, δ] = {⟨ x, μ_{A} (x) \geq α_{1}, ν_{A} (x) \leq α_{2} ⟩ : x \in X} .

The α₁-cut set GIFN_BA is a crisp subset of ℝ, which is defined as

\begin{array}{l} \begin{array}{l} A_{μ} [α_{1}, δ] & = {⟨ x, μ_{A} (x) \geq α_{1} ⟩ : x \in X} \\ = [A_{μ}^{L} [α_{1}], A_{μ}^{U} [α_{1}]], 0 \leq α_{1} \leq 1, \end{array} \\ A_{μ}^{L} [α_{1}] = a + (b - a) α_{1}^{δ}, \\ A_{μ}^{U} [α_{1}] = d - (d - c) α_{1}^{δ} . \end{array}

Similarly, a α₂-cut set GIFN_BA is a crisp subset of ℝ, and we have

\begin{array}{l} \begin{array}{l} A_{ν} [α_{2}, δ] & = {⟨ x, ν_{A} (x) \geq α_{2} ⟩ : x \in X} \\ = [A_{ν}^{L} [α_{2}], A_{ν}^{U} [α_{2}]], 0 \leq α_{2} \leq 1, \end{array} \\ A_{ν}^{L} [α_{2}] = b (1 - α_{2}^{δ}) + a_{1} α_{2}^{δ}, \\ A_{ν}^{U} [α_{2}] = c (1 - α_{2}^{δ}) + d_{1} α_{2}^{δ} . \end{array}

Therefore the (α₁, α₂)-cut set of the GIFN_B is represented as follows:

A [α_{1}, α_{2}, δ] = {x, x \in [A_{μ}^{L} [α_{1}], A_{μ}^{U} [α_{1}]] \cap [A_{ν}^{L} [α_{2}], A_{ν}^{U} [α_{2}]]} .

The GIFN_B based on the α₁-cut and α₂-cut sets is given by

A (α_{1}, α_{2}, δ) = (A_{μ} [α_{1}, δ], A_{ν} [α_{2}, δ]) .

Definition 2.5

Let [a, b] and [c, d] be two α-cut sets. We define the following relations and operations on α-cut sets as follows:

1. [a, b] ≼ [c, d] ⇔ a ≤ c and b ≤ d,
2. k ⊗ [a, b] = [ka, kb], k > 0; and k ⊗ [a, b] = [kb, ka], k < 0,
3. k⊕[a, b] = [k+a, k+b]; and k⊖[a, b] = [k−b, k−a],
4. [a, b] ⊕ [c, d] = [a + c, b + d].

Definition 2.6

We define relations and operations on the GIFN_B as follows:

1. A(α₁, α₂, δ) ⊕ B(α₁, α₂, δ) = (A_μ[α₁, δ] ⊕ B_μ[α₁, δ], A_ν[α₂, δ] ⊕ B_ν[α₂, δ]),
2. k ⊗ A(α₁, α₂, δ) ⊕ b = (k ⊗ A_μ[α₁, δ] ⊕ b, k ⊗ A_ν[α₂, δ] ⊕ b),
3. b ⊝ A(α₁, α₂, δ) = (b ⊝ A_μ[α₁, δ], b ⊝ A_ν[α₂, δ]),
4. A(α₁, α₂, δ) ≼ B(α₁, α₂, δ), if and only if A_μ[α₁, δ] ≼ B_μ[α₁, δ] and A_ν[α₂, δ] ≼ B_ν[α₂, δ],
5. A(α₁, α₂, δ) = B(α₁, α₂, δ), if and only if A_μ[α₁, δ] = B_μ[α₁, δ] and A_ν[α₂, δ] = B_ν[α₂, δ],

where A(α₁, α₂, δ) and B(α₁, α₂, δ) are two GIFN_Bs.

Throughout the paper, we consider the following relations 0 ≤ α₁, α₂ ≤ 1, $0 \leq α_{1}^{δ} + α_{2}^{δ} \leq 1$ and (i, j) = (1, μ), (2, ν).

3. Generalized Intuitionistic Fuzzy Distribution

\begin{array}{l} F_{j} (x) [α_{i}, δ] = {F (x, θ, β) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} \\ = F [F_{j}^{L} (x) [α_{i}], F_{j}^{U} (x) [α_{i}]], \end{array}

where F (x, θ, β) is the crisp CDF,

\begin{array}{l} F_{j}^{L} [α_{i}] = inf_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} F (x, θ, β), \\ F_{j}^{U} [α_{i}] = sup_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} F (x, θ, β) . \end{array}

Finally, it can be concluded that

\begin{array}{l} \tilde{F} (x) = F (x) (α_{1}, α_{2}, δ) \\ = (F_{μ} (x) [α_{1}, δ], F_{ν} (x) [α_{2}, δ]), \end{array}

and a set of (α₁, α₂)-cut GIF distribution function is given by

F (x) [α_{1}, α_{2}, δ] = {w, w \in F_{μ} (x) [α_{1}, δ] \cap F_{ν} (x) [α_{2}, δ]},

The distribution bands satisfy the following properties

1. F_j (0) [α_i₀, δ] = [0, 0],
2. F_j (∞) [α_i₀, δ] = [1, 1],
3. F_j (x₁) [α_i₀, δ] ≼ F_j (x₂) [α_i₀, δ] ⇔ x₁ ≤ x₂, i.e., bands of F_j(x) [α_i₀, δ] are non-decreasing.

Based on the definition of GIFUF, we define a set of α₁-cut of membership and α₂-cut set of non-membership functions of the GIFP of C as follows:

\begin{array}{l} P_{j} (C) [α_{i}, δ] = {P (C, θ, β) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} \\ = [P_{j}^{L} (C) [α_{i}], P_{j}^{U} (C) [α_{i}]], \end{array}

where P (C, θ, β) = ∫_Cf (x, θ, β) dx, $P_{j}^{L} [α_{i}] = {inf}_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}}$ P (C, θ, β), and $P_{j}^{U} [α_{i}] = {sup}_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} P (C, θ, β)$ . Finally, we have

\begin{array}{l} \tilde{P} (C) = P (C) (α_{1}, α_{2}, δ) \\ = (P_{μ} (C) [α_{1}, δ], P_{ν} (C) [α_{2}, δ]) . \end{array}

The set of (α₁, α₂)-cut GIFP of C is represented as

P (C) [α_{1}, α_{2}, δ] = {w, w \in P_{μ} (C) [α_{1}, δ] \cap P_{ν} (C) [α_{2}, δ]} .

[0, 0] ≼ P_μ (C) [α₁, δ], P_ν (C) ≼ [1, 1], and

\begin{array}{l} [0, 0] = P (\emptyset) [α_{1}, α_{2}, δ] ≼ P (C) [α_{1}, α_{2}, δ] \\ ≼ P (S) [α_{1}, α_{2}, δ] = [1, 1], \end{array}

where ∅︀ and S, are the empty and universal sets, respectively.

Consider the lifetime random variable X from the MB family with GIF lifetime parameters θ̃ = (a₁₁, a₁, b₁, c₁, d₁, d₁₁, δ) and β̃ = (a₂₂, a₂, b₂, c₂, d₂, d₂₂, δ), say MB(θ̃, β̃). Since $F (x, θ, β) = 1 - exp (- \frac{g {(x)}^{β}}{θ})$ is a monotonically decreasing function of θ, for g(x) ≥ 1, the function F (x, θ, β) is increasing and otherwise decreasing with respect to β, and

\begin{array}{l} F_{μ} (x) [α_{1}, δ] = & {[1 - exp {- \frac{g {(x)}^{a_{2} + (b_{2} - a_{2}) α_{1}^{δ}}}{d_{1} - (d_{1} - c_{1}) α_{1}^{δ}}}, 1 - exp {- \frac{g {(x)}^{d_{2} - (d_{2} - c_{2}) α_{1}^{δ}}}{a_{1} + (b_{1} - a_{1}) α_{1}^{δ}}}], g (x \geq 1) \\ {[1 - exp {- \frac{g {(x)}^{d_{2} - (d_{2} - c_{2}) α_{1}^{δ}}}{d_{1} - (d_{1} - c_{1}) α_{1}^{δ}}}, 1 - exp {- \frac{g {(x)}^{a_{2} + (b_{2} - a_{2}) α_{1}^{δ}}}{a_{1} + (b_{1} - a_{1}) α_{1}^{δ}}}], g (x) < 1, \\ F_{ν} (x) [α_{2}, δ] = & {\begin{array}{l} [1 - exp {- \frac{g {(x)}^{b_{2} + (1 - α_{2}^{δ}) + a_{22} α_{2}^{δ}}}{c_{1} (1 - α_{2}^{δ}) + d_{11} α_{2}^{δ}}}, 1 - exp {- \frac{g {(x)}^{c_{2} (1 - α_{2}^{δ}) + d_{22} α_{2}^{δ}}}{b_{1} (1 - α_{2}^{δ}) + a_{11} α_{2}^{δ}}}], g (x) \geq 1 \\ [1 - exp {- \frac{g {(x)}^{c_{2} (1 - α_{2}^{δ}) + d_{22} α_{2}^{δ}}}{c_{1} (1 - α_{2}^{δ}) + d_{11} α_{2}^{δ}}}, 1 - exp {- \frac{g {(x)}^{b_{2} (1 - α_{2}^{δ}) + a_{22} α_{2}^{δ}}}{b_{1} (1 - α_{2}^{δ}) + a_{11} α_{2}^{δ}}}], g (x) < 1. \end{array} \end{array}

Corollary 3.1

P (C^c) (α₁, α₂, δ) = 1 ⊝ P(C) (α₁, α₂, δ), where C^c means complement set of C.

Proof

Based on GIFP definition, it can be concluded that

\begin{array}{l} P_{j} (C^{c}) [α_{i}, δ] = {1 - P (C, θ, β) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} \\ = [P_{j}^{L} (C^{c}) [α_{i}], P_{j}^{U} (C^{c}) [α_{i}]] \\ = [inf_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} (1 - P (C, θ, β)), sup_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} (1 - P (C, θ, β))] \\ = [1 - sup_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} P (C, θ, β), 1 - inf_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} P (C, θ, β)] \\ = 1 ⊝ [P_{j}^{L} (C) [α_{i}], P_{j}^{U} (C) [α_{i}]], \end{array}

by Definition 2.6; thus, the proof is completed.

Corollary 3.2

If C₁ ⊂ C₂, then P(C₁)(α₁, α₂, δ) ≼ P(C₂)(α₁, α₂, δ).

Proof

Since P (C₁, θ, β) ≤ P(C₂, θ, β), we have

\begin{array}{l} P_{j} (C_{1}) [α_{i}, δ] = [inf_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} P (C_{1}, θ, β), sup_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} P (C_{1}, θ, β)] \\ ≼ [inf_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} P (C_{2}, θ, β), sup_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} P (C_{2}, θ, β)] \\ = P_{j} (C_{2}) [α_{i}, δ], \end{array}

regarding Definition 2.6; thus, the proof is accomplished.

4. Reliability Characteristics

\begin{array}{l} ψ_{j} (t) [α_{i}, δ] = {ψ (t, θ, β) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} \\ = [ψ_{j}^{L} (t) [α_{i}], ψ_{j}^{U} (t) [α_{i}]], \end{array}

where

\begin{array}{l} ψ_{j}^{L} (t) [α_{i}] = inf_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} ψ (t, θ, β), ψ_{j}^{U} (t) [α_{i}] \\ = sup_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} ψ (t, θ, β) . \end{array}

The function ψ(t, θ, β) can be considered as the reliability, conditional reliability, hazard rate, cumulative risk, and reverse hazard functions.

Finally, as ψ(t) (α₁, α₂, δ) = (ψ_μ(t) [α₁, δ], ψ_ν(t) [α₂, δ]) and (α₁, α₂)-cut set of GIFRC is given by

ψ (t) [α_{1}, α_{2}, δ] = {w, w \in ψ_{μ} (t) [α_{1}, δ] \cap ψ_{ν} (t) [α_{2}, δ]},

4.1 Generalized Intuitionistic Fuzzy Reliability

The GIF reliability (GIFR) function, denoted by S̃(t), is the GIF probability of a unit surviving for more than time t. The cut sets of GIFR function are indicated as

\begin{array}{l} S_{j} (t) [α_{i}, δ] = {S (t, θ, β) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} \\ = [S_{j}^{L} (t) [α_{i}], S_{j}^{U} (t) [α_{i}]], \end{array}

where $S (t, θ, β) = \int_{t}^{\infty} f (x, θ, β) d x, S_{j}^{L} [α_{i}] = {inf}_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} S (t, θ, β)$ and $S_{j}^{U} (t) [α_{i}] = {sup}_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} S (t, θ, β)$ .

Finally, the GIFR is shown as S(t)(α₁, α₂, δ) = (S_μ(t)[α₁, δ], S_ν(t)[α₂, δ]) and the set of (α₁, α₂)-cut GIFR is represented by

S (t) [α_{1}, α_{2}, δ] = {w, w \in S_{μ} (t) [α_{1}, δ] \cap S_{ν} (t) [α_{2}, δ]} .

The reliability bands have the following properties

1. S_j (0) [α_i₀, δ] = [1, 1]; i.e,. no one starts off dead,
2. S_j (∞) [α_i₀, δ] = [0, 0],; i.e., everyone dies ultimately,
3. S_j (t₁) [α_i₀, δ] ≽ S_j (t₂) [α_i₀, δ] ⇔ t₁ ≤ t₂ ; i.e. bands of S_j(t) [α_i₀, δ] decline monotonically.

Let the lifetime random variable X has MB(θ̃, β̃); the cut sets of GIFR function are as follows:

S_{j} (t) [α_{i}, δ] = {exp (- \frac{g {(t)}^{β}}{θ}) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} .

Subsequently, the α₁-cut set and α₂-cut set of GIFR are

\begin{array}{l} S_{μ} (t) [α_{1}, δ] = & {\begin{array}{l} [exp {- \frac{g {(t)}^{d_{2} - (d_{2} - c_{2}) α_{1}^{δ}}}{a_{1} + (b_{1} - a_{1}) α_{1}^{δ}}}, exp {- \frac{g {(t)}^{a_{2} + (b_{2} - a_{2}) α_{1}^{δ}}}{d_{1} - (d_{1} - c_{1}) α_{1}^{δ}}}], g (t) \geq 1, \\ [exp {- \frac{g {(t)}^{a_{2} + (b_{2} - a_{2}) α_{1}^{δ}}}{a_{1} + (b_{1} - a_{1}) α_{1}^{δ}}}, exp {- \frac{g {(t)}^{d_{2} + (d_{2} - c_{2}) α_{1}^{δ}}}{d_{1} - (d_{1} - c_{1}) α_{1}^{δ}}}], g (t) < 1, \end{array} \\ S_{ν} (t) [α_{2}, δ] = & {[exp {- \frac{g {(t)}^{c_{2} (1 - α_{2}^{δ}) + d_{22} α_{2}^{δ}}}{b_{1} (1 - α_{2}^{δ}) + a_{11} α_{2}^{δ}}}, exp {- \frac{g {(t)}^{b_{2} (1 - α_{2}^{δ}) + a_{22} α_{2}^{δ}}}{c_{1} (1 - α_{2}^{δ}) + d_{11} α_{2}^{δ}}}], g (t) \geq 1, \\ {[exp {- \frac{g {(t)}^{b_{2} (1 - α_{2}^{δ}) + a_{22} α_{2}^{δ}}}{b_{1} (1 - α_{2}^{δ}) + a_{11} α_{2}^{δ}}}, exp {- \frac{g {(t)}^{c_{2} (1 - α_{2}^{δ}) + d_{22} α_{2}^{δ}}}{c_{1} (1 - α_{2}^{δ}) + d_{11} α_{2}^{δ}}}], g (t) < 1. \end{array}

Corollary 4.1

For every δ, we have

\begin{array}{l} S (t) [1, 0, δ] = & {\begin{array}{l} [exp {- \frac{g {(t)}^{c_{2}}}{b_{1}}}, exp {- \frac{g {(t)}^{b_{2}}}{c_{1}}}], g (t) \geq 1, \\ [exp {- \frac{g {(t)}^{b_{2}}}{b_{1}}}, exp {- \frac{g {(t)}^{c_{2}}}{c_{1}}}], g (t) < 1, \end{array} \\ S (t) [0, 1, δ] = & {\begin{array}{l} [exp {- \frac{g {(t)}^{d_{2}}}{a_{1}}}, exp {- \frac{g {(t)}^{a_{2}}}{d_{1}}}], g (t) \geq 1, \\ [exp {- \frac{g {(t)}^{a_{2}}}{b_{1}}}, exp {- \frac{g {(t)}^{d_{2}}}{d_{1}}}], g (t) < 1. \end{array} \end{array}

4.2 Generalized Intuitionistic Fuzzy Conditional Reliability

\begin{array}{l} S_{j} (t ∣ τ) [α_{i}, δ] = {S (t, θ, β ∣ τ) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} \\ = [S_{j}^{L} (t ∣ τ) [α_{i}], S_{j}^{U} (t ∣ τ) [α_{i}]], \end{array}

where $S (t, θ, β ∣ τ) = \frac{S (t + τ, θ, β)}{S (τ, θ, β)}, S_{j}^{L} [t ∣ τ] [α_{i}] = {inf}_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} S (t, θ, β ∣ τ)$ and $S_{j}^{U} (t ∣ τ) [α_{i}] = {sup}_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} S (t, θ, β ∣ τ)$ . Hence, GIFCR can be given as S (t| τ )(α₁, α₂, δ) = (S_μ( t| τ ) [α₁, δ], S_ν( t| τ )[α₂, δ]) and a set of (α₁, α₂)-cut GIFCR function is described as

S (t ∣ τ) [α_{1}, α_{2}, δ] = {w, w \in S_{μ} (t ∣ τ) [α_{1}, δ] \cap S_{ν} (t ∣ τ) [α_{2}, δ]} .

Consider the random variable X from MB(θ̃, β̃); cut sets of the GIFCR function are given as

S_{j} (t ∣ τ) [α_{i}, δ] = {exp {- \frac{g {(t + τ)}^{β} - g {(τ)}^{β}}{θ}} ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} .

The α₁-cut set and α₂-cut set of GIFCR are computed as follows:

\begin{array}{l} S_{μ} (t ∣ τ) [α_{1}, δ] = & [exp {- \frac{g {(t + τ)}^{d_{2} - (d_{2} - c_{2}) α_{1}^{δ}} - g {(τ)}^{d_{2} - (d_{2} - c_{2}) α_{1}^{δ}}}{a_{1} + (b_{1} - a_{1}) α_{1}^{δ}}}, exp {- \frac{g {(t + τ)}^{a_{2} + (b_{2} - a_{2}) α_{1}^{δ}} - g {(τ)}^{a_{2} + (b_{2} - a_{2}) α_{1}^{δ}}}{d_{1} - (d_{1} - c_{1}) α_{1}^{δ}}}], \\ S_{ν} (t ∣ τ) [α_{2}, δ] = & [exp {- \frac{g {(t + τ)}^{c_{2} (1 - α_{2}^{δ}) + d_{22} α_{2}^{δ}} - g {(τ)}^{c_{2} (1 - α_{2}^{δ}) + d_{22} α_{2}^{δ}}}{b_{1} (1 - α_{2}^{δ}) + a_{11} α_{2}^{δ}}}, exp {- \frac{g {(t + τ)}^{b_{2} (1 - α_{2}^{δ}) + a_{22} α_{2}^{δ}} - g {(τ)}^{b_{2} (1 - α_{2}^{δ}) + a_{22} α_{2}^{δ}}}{c_{1} (1 - α_{2}^{δ}) + d_{11} α_{2}^{δ}}}] . \end{array}

Corollary 4.2

For every δ, we have

\begin{array}{l} S (t ∣ τ) [1, 0, δ] = & [exp {- \frac{g {(t + τ)}^{c_{2}} - g {(τ)}^{c_{2}}}{b_{1}}}, exp {- \frac{g {(t + τ)}^{b_{2}} - g {(τ)}^{b_{2}}}{c_{1}}}], \\ S (t ∣ τ) [0, 1, δ] = & [exp {- \frac{g {(t + τ)}^{d_{2}} - g {(τ)}^{d_{2}}}{a_{1}}}, exp {- \frac{g {(t + τ)}^{a_{2}} - g {(τ)}^{a_{2}}}{d_{1}}}] . \end{array}

4.3 Generalized Intuitionistic Fuzzy Hazard

\begin{array}{l} h_{j} (t) [α_{i}, δ] = {h (t, θ, β) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} \\ = [h_{j}^{L} (t) [α_{i}], h_{j}^{U} (t) [α_{i}]], \end{array}

where

\begin{array}{l} h_{j}^{L} (t) [α_{i}] = inf_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} h (t, θ, β), \\ h_{j}^{U} (t) [α_{i}] = sup_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} h (t, θ, β), \end{array}

and $h (t, θ, β) = \frac{f (t, θ, β)}{S (t, θ, β)}$ .

The GIFH can be expressed as h(α₁, α₂, δ) = (h_μ(t)[α₁, δ], h_ν(t)[α₂, δ]) and a set of (α₁, α₂)-cut a GIFH function is defined by

h (t) [α_{1}, α_{2}, δ] = {w, w \in h_{μ} (t) [α_{1}, δ] \cap h_{ν} (t) [α_{2}, δ]} .

h_j(t)[α_i, δ], i = 1, 2 is a two-variate function in terms of α_i and t. In addition, cut sets of the GIF cumulative hazard (GIFCH) function are

\begin{array}{l} H_{j} (t) [α_{i}, δ] = {H (t, θ, β) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} \\ = [H_{j}^{L} (t) [α_{i}], H_{j}^{U} (t) [α_{i}]], \end{array}

where

\begin{array}{l} H_{j}^{L} (t) [α_{i}] = inf_{θ \in θ_{j} [α_{i}, δ]} H (t, θ, β), \\ H_{j}^{U} (t) [α_{i}] = sup_{θ \in θ_{j} [α_{i}, δ]} H (t, θ, β), \end{array}

and $H (t, θ, β) = \int_{0}^{t} \frac{f (u, θ, β)}{S (u, θ, β)} d u$ .

Based on the random variable X from MB(θ̃, β̃) family, the cut sets of GIFH function are

\begin{array}{l} h_{j} (t) [α_{i}, δ] = {\frac{β}{θ} g^{'} (x) g^{β - 1} (x) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} \\ = [h_{j} {(t)}^{L} [α_{i}], h_{j} {(t)}^{U} [α_{i}]], \end{array}

where

\begin{array}{l} h_{j}^{L} (t) [α_{i}] = inf {\frac{β}{θ} g^{'} (x) g^{β - 1} (x) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]}, \\ h_{j}^{U} (t) [α_{i}] = sup {\frac{β}{θ} g^{'} (x) g^{β - 1} (x) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} . \end{array}

In addition,

\begin{array}{l} H_{j} (t) [α_{i}, δ] = {\frac{g^{β} (x)}{θ} ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} \\ = [H_{j} {(t)}^{L} [α_{i}], H_{j} {(t)}^{U} [α_{i}]], \end{array}

where

H_{μ} (t) [α_{1}, δ] = {\begin{array}{l} [\frac{g {(t)}^{a_{2} + (b_{2} - a_{2}) α_{1}^{δ}}}{d_{1} - (d_{1} - c_{1}) α_{1}^{δ}}, \frac{g {(t)}^{d_{2} - (d_{2} - c_{2}) α_{1}^{δ}}}{a_{1} + (b_{1} - a_{1}) α_{1}^{δ}}], g (t) \geq 1 \\ [\frac{g {(t)}^{d_{2} + (d_{2} - c_{2}) α_{1}^{δ}}}{d_{1} - (d_{1} - c_{1}) α_{1}^{δ}}, \frac{g {(t)}^{a_{2} - (b_{2} - a_{2}) α_{1}^{δ}}}{a_{1} + (b_{1} - a_{1}) α_{1}^{δ}}], g (t) < 1, \end{array}

and

H_{ν} (t) [α_{2}, δ] = {\begin{array}{l} [\frac{g {(t)}^{b_{2} + (1 - α_{2}^{δ}) + a_{22} α_{2}^{δ}}}{c_{1} (1 - α_{2}^{δ}) + d_{11} α_{2}^{δ}}, \frac{g {(t)}^{c_{2} + (1 - α_{2}^{δ}) + d_{22} α_{2}^{δ}}}{b_{1} (1 - α_{2}^{δ}) + a_{11} α_{2}^{δ}}], g (t) \geq 1, \\ [\frac{g {(t)}^{c_{2} (1 - α_{2}^{δ}) + d_{22} α_{2}^{δ}}}{c_{1} (1 - α_{2}^{δ}) + d_{11} α_{2}^{δ}}, \frac{g {(t)}^{b_{2} (1 - α_{2}^{δ}) + a_{22} α_{2}^{δ}}}{b_{1} (1 - α_{2}^{δ}) + a_{11} α_{2}^{δ}}], g (t) < 1. \end{array}

Corollary 4.3

For every δ, we have

\begin{array}{l} H (t) [1, 0, δ] = & {\begin{array}{l} [\frac{g {(t)}^{b_{2}}}{c_{1}}, \frac{g {(t)}^{c_{2}}}{b_{1}}], g (t) \geq 1, \\ [\frac{g {(t)}^{c_{2}}}{c_{1}}, \frac{g {(t)}^{b_{2}}}{b_{1}}], g (t) < 1, \end{array} \\ H (t) [0, 1, δ] = & {\begin{array}{l} [\frac{g {(t)}^{a_{2}}}{d_{1}}, \frac{g {(t)}^{d_{2}}}{a_{1}}], g (t) \geq 1, \\ [\frac{g {(t)}^{d_{2}}}{d_{1}}, \frac{g {(t)}^{a_{2}}}{a_{1}}], g (t) < 1. \end{array} \end{array}

Corollary 4.4

Consider ψ(t, θ, β) as the unreliability, reliability, conditional reliability, hazard rate, and cumulative hazard functions,

(i) If δ₁ ≤ δ₂, then ψ_μ(t) [α₁, δ₁] ⊂ ψ_μ(t) [α₁, δ₂] and ψ_ν(t) [α₂, δ₂] ⊂ ψ_ν(t) [α₂, δ₁]; therefore, increasing δ leads to wider bandwidth of ψ_μ(t) [α₁, δ] and narrower bandwidth of ψ_ν(t) [α₂, δ].
(ii) If $(α_{1}^{(1)}, 1 - α_{2}^{(1)}) ≼ (α_{1}^{(2)}, 1 - α_{2}^{(2)})$ , then $ψ_{μ} (t) [α_{1}^{(2)}, δ] \subset ψ_{μ} (t) [α_{1}^{(1)}, δ]$ and $ψ_{ν} (t) [α_{2}^{(2)}, δ] \subset ψ_{ν} (t) [α_{2}^{(1)}, δ]$ . Hence, as (α₁, 1−α₂) increases, bandwidth of ψ_μ(t)[α₁, δ] and ψ_ν(t) [α₂, δ] reduces.
(iii) Let $1 - α_{1}^{δ} = α_{2}^{δ}$ , a_i₁ = a_i and d_i₁ = d_i, i = 1, 2. Then, we have

$ψ_{μ} (t) [α_{1}, δ] = ψ_{ν} (t) [α_{2}, δ] = ψ (t) [α_{1}, α_{2}, δ] .$

Definition 4.1

Let F̃(x) be a GIFUF, then

(i) It belongs to the class of distributions with an increasing failure rate (IFR) (decreasing failure rate [DFR]) if S_j(t)[α_i, δ] increases (decreases) with respect to t.
(ii) It belongs to the class of distributions with an increasing failure rate average (IFRA) if $\frac{1}{t} H_{j} (t) [α_{i}, δ]$ increases with respect to t.
(iii) It belongs to the new better than used (NBU) class, if

$F_{j} (t_{1} + t_{2}) [α_{i}, δ] \underline{≺} F_{j} (t_{1}) [α_{i}, δ] \otimes F_{j} (t_{2}) [α_{i}, δ] .$

4.4 GIFR Function of k out of n Systems

If n-components are connected in k out of n systems with independent and identically distributed lifetime variables, then the α_i-cut (i = 1, 2) of GIFR with GIF distribution is

\begin{array}{l} S_{j} (t) [α_{i}, δ] = {S (t, θ, β) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} \\ = [S_{j}^{L} (t) [α_{i}], S_{j}^{U} (t) [α_{i}]], \end{array}

where

\begin{array}{l} S_{j}^{L} (t) [α_{i}] = inf_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} S (t, θ, β) \\ S_{j}^{U} (t) [α_{i}] = sup_{\begin{array}{l} θ \in θ_{j} [α_{i}, δ] \\ β \in β_{j} [α_{i}, δ] \end{array}} S (t, θ, β), \end{array}

and

S (t, θ, β) = \sum_{r = k}^{n} (\begin{matrix} n \\ r \end{matrix}) S {(t, θ, β)}^{r} F {(t, θ, β)}^{n - r} .

If n-components are connected serially, cut sets in the GIFMB family are represented by

S_{j} (t) [α_{i}, δ] = {exp (- \frac{n g {(t)}^{β}}{θ}) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} .

If n-components are connected parallelly, cut sets with the GIFMB family are given as

S_{j} (t) [α_{i}, δ] = {1 - {(1 - exp (- \frac{g {(t)}^{β}}{θ}))}^{n} ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} .

5. Numerical Example

P_{j} (X \leq 0.1) [α_{i}] = {1 - exp (- \frac{{0.1}^{β}}{θ}) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} .

Hence, for δ = 1, 2,

\begin{array}{l} P_{μ} (X \leq 0.1) [α_{1}, δ] = [1 - exp (- \frac{{0.1}^{0.4 - 0.05 α_{1}^{δ}}}{2.5 - 0.1 α_{1}^{δ}}), 1 - exp {\frac{{0.1}^{0.25 + 0.05 α_{1}^{δ}}}{2.2 + 0.1 α_{1}^{δ}}}] \\ P_{ν} (X \leq 0.1) [α_{2}, δ] = [1 - exp (- \frac{{0.1}^{0.35 + 0.1 α_{2}^{δ}}}{2.4 + 0.3 α_{2}^{δ}}), 1 - exp {\frac{{0.1}^{0.3 - 0.1 α_{2}^{δ}}}{2.3 - 0.3 α_{2}^{δ}}}] . \end{array}

The cut sets of GIFR are given by

\begin{array}{l} S_{j} (t) [α_{i}, δ] = {exp (- \frac{t^{β}}{θ}) ∣ θ \in θ_{j} [α_{j}, δ], β \in β_{j} [α_{i}, δ]}, \\ S_{μ} (t) [α_{1}, δ] = {\begin{array}{l} [exp {- \frac{t^{0.4 - 0.05 α_{1}^{δ}}}{2.2 + 0.1 α_{1}^{δ}}}, exp {- \frac{t^{0.25 + 0.05 α_{1}^{δ}}}{2.5 - 0.1 α_{1}^{δ}}}], t \geq 1, \\ [exp {- \frac{t^{0.25 - 0.05 α_{1}^{δ}}}{2.2 + 0.1 α_{1}^{δ}}}, exp {- \frac{t^{0.04 - 0.05 α_{1}^{δ}}}{2.5 - 0.1 α_{1}^{δ}}}] t < 1, \end{array} \\ S_{ν} (t) [α_{2}, δ] = {\begin{array}{l} [exp {- \frac{t^{0.35 + 0.1 α_{2}^{δ}}}{2.3 - 0.3 α_{2}^{δ}}}, exp {- \frac{t^{0.3 - 0.1 α_{2}^{δ}}}{2.4 + 0.3 α_{2}^{δ}}}], t \geq, \\ [exp {- \frac{t^{- 0.03 - 0.1 α_{2}^{δ}}}{2.3 - 0.3 α_{2}^{δ}}}, exp {- \frac{t^{- 0.35 + 0.1 α_{2}^{δ}}}{2.4 + 0.3 α_{2}^{δ}}}], t < 1 . \end{array} \end{array}

The GIFR bands for α₁ = 0 and α₂ = 1 are computed as follows:

\begin{array}{l} S_{μ} (t) [0, δ] = {\begin{array}{l} [exp {- \frac{t^{0.4}}{2.2}}, exp {- \frac{t^{0.25}}{2.5}}], t \geq 1, \\ [exp {- \frac{t^{0.25}}{2.2}}, exp {- \frac{t^{0.4}}{2.5}}], t < 1, \end{array} \\ S_{ν} (t) [1, δ] = {\begin{array}{l} [exp {- \frac{t^{0.45}}{2}}, exp {- \frac{t^{0.2}}{2.7}}], t \geq 1, \\ [exp {- \frac{t^{0.2}}{2}}, exp {- \frac{t^{0.45}}{2.7}}], t < 1, \end{array} \end{array}

then

S (t) [0, 1, δ] = {\begin{array}{l} [exp {- \frac{t^{0.4}}{2.2}}, exp {- \frac{t^{0.25}}{2.5}}], t \geq 1, \\ [exp {- \frac{t^{0.25}}{2.2}}, exp {- \frac{t^{0.4}}{2.5}}], t < 1. \end{array}

The GIFR bands for α₁ = 0 and α₂ = 1 in Figure 2 show decreasing behavior with increasing t.

Considering t = 0.2, the cut sets of GIFR are given by

\begin{array}{l} S_{μ} (0.2) [α_{1}, δ] = [exp {- \frac{{0.2}^{0.25 + 0.05 α_{1}^{δ}}}{2.2 + 0.1 α_{1}^{δ}}}, exp {- \frac{{0.2}^{0.4 + 0.05 α_{1}^{δ}}}{2.5 - 0.1 α_{1}^{δ}}}], \\ S_{ν} (0.2) [α_{2}, δ] = [exp {- \frac{{0.2}^{0.3 - 0.1 α_{2}^{δ}}}{2.2 - 0.3 α_{2}^{δ}}}, exp {- \frac{{0.2}^{0.35 + 0.1 α_{2}^{δ}}}{2.4 + 0.3 α_{2}^{δ}}}] . \end{array}

The different cut sets of GIFR for δ = 2 and different values of (α₁, α₂) are represented in Table 2, indicating reduced bandwidth for large and small values of α₁ and α₂, respectively.

Figure 3 depicts the membership and non-membership functions of GIFR for δ = 2.

The reliability bands S(t) [0.3, 0.1, δ] for δ = 0.5, 1, and 2 are computed, respectively, as

\begin{array}{l} S (t) [0.3, 0.1, 0.5] = {\begin{matrix} [exp {- \frac{t^{0.373}}{2.225}}, exp {- \frac{t^{0.277}}{2.445}}], & t \geq 1, \\ [exp {- \frac{t^{0.277}}{2.255}}, exp {- \frac{t^{0.373}}{2.445}}], & t < 1, \end{matrix}, \\ S (t) [0.3, 0.1, 1] = {\begin{matrix} [exp {- \frac{t^{0.36}}{2.27}}, exp {- \frac{t^{0.29}}{2.43}}], & t \geq 1, \\ [exp {- \frac{t^{0.29}}{2.27}}, exp {- \frac{t^{0.36}}{2.43}}], & t < 1, \end{matrix}, \\ S (t) [0.3, 0.1, 2] = {\begin{matrix} [exp {- \frac{t^{0.351}}{2.297}}, exp {- \frac{t^{0.299}}{2.403}}], & t \geq 1, \\ [exp {- \frac{t^{0.299}}{2.297}}, exp {- \frac{t^{0.351}}{2.403}}], & t < 1. \end{matrix}, \end{array}

Accordingly, the reliability bands for δ = 0.5, 1, and 2 in Figure 4 reveal that by increasing δ, the width of reliability bands is decreased.

The reliability bands S(t) [α₁, α₂, 1] for (α₁, α₂) = (0, 1), (1, 0), (0.8, 0.2) respectively provided as

\begin{array}{l} S (t) [0, 1, 1] = {\begin{matrix} [exp {- \frac{t^{0.4}}{2.2}}, exp {- \frac{t^{0.25}}{2.5}}], & t \geq 1, \\ [exp {- \frac{t^{0.25}}{2.2}}, exp {- \frac{t^{0.4}}{2.5}}], & t < 1, \end{matrix} \\ S (t) [1, 0, 1] = {\begin{matrix} [exp {- \frac{t^{0.35}}{2.3}}, exp {- \frac{t^{0.3}}{2.4}}], & t \geq 1, \\ [exp {- \frac{t^{0.3}}{2.3}}, exp {- \frac{t^{0.35}}{2.4}}], & t < 1, \end{matrix}, \\ S (t) [0.8, 0.2, 1] = {\begin{matrix} [exp {- \frac{t^{0.36}}{2.28}}, exp {- \frac{t^{0.29}}{2.42}}], & t \geq 1, \\ [exp {- \frac{t^{0.29}}{2.28}}, exp {- \frac{t^{0.36}}{2.42}}], & t < 1. \end{matrix}, \end{array}

The reliability bands for (α₁, α₂) = (0, 1), (1, 0), (0.8, 0.2) are depicted in Figure 5. The width of the reliability band reduces as α₁ and α₂ approaches one and zero, respectively.

The cut sets of GIFCR are given as follows:

\begin{array}{l} S_{μ} (t ∣ τ) [α_{1}, δ] = [exp (- \frac{{(t + τ)}^{0.4 - 0.05 α_{1}^{δ}} - τ^{0.4 - 0.05 α_{1}^{δ}}}{2.2 + 0.1 α_{1}^{δ}}), exp (- \frac{{(t + τ)}^{0.25 + 0.05 α_{1}^{δ}} - τ^{0.25 + 0.05 α_{1}^{δ}}}{2.5 - 0.1 α_{1}^{δ}})], \\ S_{ν} (t ∣ τ) [α_{2}, δ] = [exp (- \frac{{(t + τ)}^{0.35 + 0.1 α_{2}^{δ}} - τ^{0.35 + 0.1 α_{2}^{δ}}}{2.3 - 0.3 α_{2}^{δ}}), exp (- \frac{{(t + τ)}^{0.3 - 0.1 α_{2}^{δ}} - τ^{0.3 - 0.1 α_{2}^{δ}}}{2.4 + 0.3 α_{2}^{δ}})] . \end{array}

The GIFCR bands with τ = 1 for α₁ = 0 and α₂ = 1 are as follows:

\begin{array}{l} S_{μ} (t ∣ 1) [0, δ] = [exp (- \frac{{(t + 1)}^{0.4} - 1}{2.2}), exp (- \frac{{(t + 1)}^{0.25} - 1}{2.5})], \\ S_{ν} (t ∣ 1) [1, δ] = [exp (- \frac{{(t + 1)}^{0.45} - 1}{2}), exp (- \frac{{(t + 1)}^{0.2} - 1}{2.7})] . \end{array}

GIFCR bands for α₁ = 0 and α₂ = 1 in Figure 6 demonstrate their decreasing behavior.

Let t₀ = 2, τ = 1, the membership and non-membership functions of S̃ (2| 1) are as follows:

\begin{array}{l} S_{μ} (2 ∣ 1) [α_{1}, δ] = [exp (- \frac{3^{0.4 - 0.5 α_{1}^{δ}} - 1}{2.2 + 0.1 α_{1}^{δ}}), exp (- \frac{3^{0.25 + 0.05 α_{1}^{δ}} - 1}{2.5 - 0.1 α_{1}^{δ}})], \\ S_{ν} (2 ∣ 1) [α_{2}, δ] = [exp (- \frac{3^{0.35 + 0.1 α_{2}^{δ}} - 1}{2.3 - 0.3 α_{2}^{δ}}), exp (- \frac{3^{0.3 - 0.1 α_{2}^{δ}} - 1}{2.4 + 0.3 α_{2}^{δ}})] . \end{array}

Both membership and non-membership functions of GIFCR are presented in Figure 7 for δ = 2.

The upper and lower bounds of α_i-cut of the GIFH function are given by

\begin{array}{l} h_{j}^{L} (t) [α_{i}] = inf {\frac{β}{θ} t^{β - 1} ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]}, \\ h_{j}^{U} (t) [α_{i}] = sup {\frac{β}{θ} t^{β - 1} ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]} . \end{array}

Corollary 5.1

Let

\begin{array}{l} h_{1} = (0.25 + 0.05 α_{1}^{δ}) t^{- 0.75 + 0.05 α_{1}^{δ}}, \\ h_{2} = (0.4 - 0.05 α_{1}^{δ}) t^{- 0.6 - 0.05 α_{1}^{δ}}, \\ h^{*} = β^{*} t^{β^{*} - 1}, β^{*} = {(- ln t)}^{- 1}, \\ h_{3} = (0.3 - 0.1 α_{2}^{δ}) t^{- 0.7 - 0.1 α_{2}^{δ}}, \\ h_{4} = (0.35 + 0.1 α_{2}^{δ}) t^{- 0.65 + 0.1 α_{2}^{δ}} . \end{array}

The GIFH bands, for t ≥ 1, are given as

\begin{array}{l} h_{μ} (t) [α_{1}, δ] = [(\frac{0.25 + 0.05 α_{1}^{δ}}{2.5 - 0.1 α_{1}^{δ}}) t^{- 0.75 + 0.05 α_{1}^{δ}}, (\frac{0.4 - 0.05 α_{1}^{δ}}{2.2 + 0.1 α_{1}^{δ}}) t^{- 0.6 - 0.05 α_{1}^{δ}}], \\ h_{ν} (t) [α_{2}, δ] = [(\frac{0.3 - 0.1 α_{2}^{δ}}{2.4 + 0.3 α_{2}^{δ}}) t^{- 0.7 - 0.1 α_{2}^{δ}}, (\frac{0.35 + 0.1 α_{2}^{δ}}{2.3 - 0.3 α_{2}^{δ}}) t^{- 0.65 + 0.1 α_{2}^{δ}}] . \end{array}

For t < 1, the GIFH bands are represented by

\begin{array}{l} h_{μ} (t) [α_{1}, δ] = {[\frac{h_{2}}{2.5 - 0.1 α_{1}^{δ}}, \frac{h_{1}}{2.2 + 0.1 α_{1}^{δ}}], 0 < t < e^{- \frac{1}{0.25 + 0.05 α_{1}^{δ}}}, \\ = {\begin{array}{l} [\frac{min (h_{1}, h_{2})}{2.5 - 0.1 α_{1}^{δ}}, \frac{h^{*}}{2.2 + 0.1 α_{1}^{δ}}], e^{- \frac{1}{0.25 + 0.05 α_{1}^{δ}}} < t < e^{- \frac{1}{0.4 - 0.05 α_{1}^{δ}}}, \\ [\frac{h_{1}}{2.5 - 0.1 α_{1}^{δ}}, \frac{h_{2}}{2.2 + 0.1 α_{1}^{δ}}], e^{- \frac{1}{0.4 - 0.05 α_{1}^{δ}}} < t < 1, \end{array} \\ h_{ν} (t) [α_{1}, δ] \\ = {\begin{array}{l} [\frac{h_{4}}{2.4 + 0.3 α_{2}^{δ}}, \frac{h_{3}}{2.3 - 0.3 α_{2}^{δ}}], 0 < t < e^{- \frac{1}{0.3 - 0.1 α_{2}^{δ}}}, \\ [\frac{min (h_{3}, h_{4})}{2.4 + 0.3 α_{2}^{δ}}, \frac{h^{*}}{2.3 - 0.3 α_{2}^{δ}}], e^{- \frac{1}{0.3 - 0.1 α_{2}^{δ}}} < t < e^{- \frac{1}{0.35 + 0.1 α_{2}^{δ}}}, \\ [\frac{h_{3}}{2.4 + 0.3 α_{2}^{δ}}, \frac{h_{4}}{2.3 - 0.3 α_{2}^{δ}}], e^{- \frac{1}{0.35 + 0.1 α_{2}^{δ}}} < t < 1. \end{array} \end{array}

In 8, we provide the membership and non-membership functions of the GIFH for t = δ = 2.

GIFH bands for α₁ = 0 and α₂ = 1 at t > 1 and t ≤ 1 are computed, respectively, as

\begin{array}{l} h_{μ} (t) [0, δ] = [\frac{t^{- 0.75}}{10}, \frac{0.4 t^{- 0.6}}{2.2}] \\ h_{ν} (t) [1, δ] = [\frac{0.2 t^{- 0.8}}{2.7}, \frac{0.45 t^{- 0.55}}{2}] . \end{array}

and

\begin{array}{l} h_{μ} (t) [0, δ] \\ = {\begin{array}{l} [\frac{0.4 t^{- 0.6}}{2.5}, \frac{0.25 t^{- 0.75}}{2.2}], 0 < t < e^{- \frac{1}{0.25}}, \\ [\frac{min (0.4 t^{- 0.6}, 0.25 t^{- 0.75})}{2.5}, \frac{β^{*} t^{β^{*} - 1}}{2.2}], e^{- \frac{1}{0.25}} < t < e^{- \frac{1}{0.4}}, \\ [\frac{0.25 t^{- 0.75}}{2.2}, \frac{0.4 t^{- 0.6}}{2.5}], e^{- \frac{1}{0.4}} < t < 1, \end{array} \\ h_{ν} (t) [1, δ] \\ = {\begin{array}{l} [\frac{0.45 t^{- 0.55}}{2.7}, \frac{0.2 t^{- 0.8}}{2}], 0 < t < e^{- \frac{1}{0.2}}, \\ [\frac{min (0.2 t^{- 0.8}, 0.45 t^{- 0.55})}{2.7}, \frac{β^{*} t^{β^{*} - 1}}{2}], e^{- \frac{1}{0.2}} < t < e^{- \frac{1}{0.45}}, \\ [\frac{0.2 t^{- 0.8}}{2.7}, \frac{0.45 t^{- 0.55}}{2}], e^{- \frac{1}{0.45}} < t < 1. \end{array} \end{array}

Figure 9 confirm the decreasing behavior of GIFH bands for α₁ = 0 and α₂ = 1.

The cut sets of GIFUF are given by

\begin{array}{l} F_{j} (x) [α_{i}, δ] \\ = {1 - exp (- \frac{x^{β}}{θ}) ∣ θ \in θ_{j} [α_{i}, δ], β \in β_{j} [α_{i}, δ]}, \\ F_{μ} (x) [α_{1}, δ] \\ = {\begin{array}{l} [1 - exp {- \frac{x^{0.25 + 0.05 α_{1}^{δ}}}{2.5 - 0.1 α_{1}^{δ}}}, 1 - exp {- \frac{x^{0.4 - 0.05 α_{1}^{δ}}}{2.2 + 0.1 α_{1}^{δ}}}], & x \geq 1, \\ [1 - exp {- \frac{x^{0.4 - 0.05 α_{1}^{δ}}}{2.5 - 0.1 α_{1}^{δ}}}, 1 - exp {- \frac{x^{0.25 + 0.05 α_{1}^{δ}}}{2.2 + 0.1 α_{1}^{δ}}}], & x < 1, \end{array} \\ F_{ν} (x) [α_{2}, δ] \\ = {\begin{array}{l} [1 - exp {- \frac{x^{0.3 - 0.1 α_{2}^{δ}}}{2.4 + 0.3 α_{2}^{δ}}}, 1 - exp {- \frac{x^{0.35 + 0.1 α_{2}^{δ}}}{2.3 - 0.3 α_{2}^{δ}}}], & x \geq 1, \\ [1 - exp {- \frac{x^{0.35 + 0.1 α_{2}^{δ}}}{2.4 + 0.3 α_{2}^{δ}}}, 1 - exp {- \frac{x^{0.3 - 0.1 α_{2}^{δ}}}{2.3 - 0.3 α_{2}^{δ}}}], & x < 1. \end{array} \end{array}

GIFUF bands for α₁ = 0 and α₂ = 1 are computed as follows:

\begin{array}{l} F_{μ} (x) [0, δ] \\ = {\begin{array}{l} [1 - exp {- \frac{x^{0.25}}{2.5}}, 1 - exp {- \frac{x^{0.4}}{2.2}}], & x \geq 1, \\ [1 - exp {- \frac{x^{0.4}}{2.5}}, 1 - exp {- \frac{x^{0.25}}{2.2}}], & x < 1, \end{array} \\ F_{v} (x) [1, δ] \\ = {\begin{array}{l} [1 - exp {- \frac{x^{0.2}}{2.7}}, 1 - exp {- \frac{x^{0.45}}{2}}], & x \geq 1, \\ [1 - exp {- \frac{x^{0.45}}{2.7}}, 1 - exp {- \frac{x^{0.2}}{2}}], & x < 1, \end{array} \end{array}

then

\begin{array}{l} F (x) [0, 1, δ] \\ = {\begin{array}{l} [1 - exp {- \frac{x^{0.25}}{2.5}}, 1 - exp {- \frac{x^{0.4}}{2.2}}], & x \geq 1, \\ [1 - exp {- \frac{x^{0.4}}{2.5}}, 1 - exp {- \frac{x^{0.25}}{2.2}}], & x < 1. \end{array} \end{array}

The GIFUF bands for α₁ = 0 and α₂ = 1 in Figure 10 reveal that the bands increase with increasing x.

Based on our findings, we infer that increasing α₁ and decreasing α₂ reduces the ambiguity of GIFR, GIFCR, and GIFH bands.

6. Conclusion

This paper is an extended analysis of the reliability characteristics of the MB family of lifetime distributions, where lifetime parameters are considered GIFNs. In our approach, we obtain GIF reliability measures, including reliability, conditional reliability, hazard, and unreliability functions, through GIFN_B. The obtained measures are consistent with the probability measures proposed by Buckley [45] for FSs. In addition, we derive the cut sets of reliability characteristics as two-variate functions in terms of α_i and t. For t₀, they are generalized intuitionistic fuzzy numbers, and for every special α₁₀ and α₂₀, the curves are the same as bands with upper and lower bounds. Our method is more comprehensive than previous methods. For δ = 1, the results are compatible with intuitionistic fuzzy reliability evaluation, and for δ = 1, $α_{1}^{δ} = 1 - α_{2}^{δ}$ , a = a₁ and d = d₁, a fuzzy reliability evaluation is obtained due to [2] and [3]. The reliability characteristics based on the Weibull lifetime distribution with GIF parameters are depicted as numerical examples, along with their cut sets with several combinations of α₁ and α₂ values. This paper is limited to the GIF reliability functions without discussing the estimation approaches. Hence, the estimation technique of the GIF reliability functions based on classical and Bayesian approaches can be an interesting topic for subsequent studies. For future studies, the fuzzy MB family can be extended by the PFS, FFS, and aggregation operators. Moreover, the GIF analysis in LP, DEA, MCDM, and SPP contexts is a prominent research area in fuzzy environments.

Abstract
1. Introduction
2. Preliminaries
3. Generalized Intuitionistic Fuzzy Distribution
4. Reliability Characteristics
5. Numerical Example
6. Conclusion

Fig 1.

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