The reliability analysis based on the generalized intuitionistic fuzzy two-parameter Pareto distribution

The fuzzy sets (FSs) theory as a generalization of the classical theory of sets provides the uncertainty associated with classification or imprecision. In the FSs, elements are defined by their membership function, which represents the possibility of the occurrence of an event to accommodate the uncertainty. In the last decades, several developments of the FSs are recommended, containing the L-fuzzy, interval-valued fuzzy, rough and intuitionistic fuzzy sets (IFSs). The application of IFSs instead of FSs means providing another degree of freedom into a set description. In other words, the IFSs are equipped by the degree of hesitation, which handles the ambiguity and vagueness along with the membership, non-membership and hesitancy functions.

The IFSs conception has been applied in a wide range of branches, such as reliability (Shu et al. 2006; Aikhuele 2020), transportation problem (Mahmoodirad et al. 2019; Mishra and Kumar 2020), data envelopment analysis (Puri and Yadav 2015; Arya and Yadav 2019) and decision making (Yang et al. 2021; Pękala et al. 2021).

Atanassov (2017) provided a comparison study among the type-1 fuzzy sets and IFSs and transformed some concepts from the IFSs theory to the type-1 fuzzy sets theory. The theory included new operations, relations, and operators that extend the operators defined over the type-1 fuzzy sets.

The triangular intuitionistic fuzzy number (TIFN) is introduced by Mahapatra and Roy (2009) for reliability analysis purposes. Afterward, Mahapatra and Mahapatra (2010) reported the intuitionistic fuzzy fault tree using the arithmetic operation of trapezoidal intuitionistic fuzzy number (TrIFN), which are evaluated based on the \((\alpha ,\beta )\)-cuts method.

Varghese and Rosario (2021) introduced the Pendant, Hexant and Octant fuzzy numbers along with the \(\alpha \)-cuts are defined and mathematical operations. The reliability analysis based on different fuzzy numbers was compared via the numerical examples, and defuzzification was performed using various approaches, including the signed distance, graded mean integration and centroid methods, with special attention to the reliability of the weaving machine.

Feng et al. (2020) concentrated on the generalizations of the expectation score function called Minkowski score functions of intuitionistic fuzzy values (IFV) and ranking IFV from a geometric perspective in decision-making issues. They provided a new algorithm for solving decision-making problems based on the Minkowski weighted score function and the maximizing deviation method under the IFSs.

Due to uncertainty in medical diagnosis, incomplete evidence and imprecise information, Kozae et al. (2020) introduced a new definition of IFS and evaluated its implementation in the Covid-19 pandemic.

Citakoglu et al. (2014) estimated the monthly mean reference evapotranspiration through the adaptive network-based fuzzy inference system and artificial neural network models, and Cobaner et al. (2014) estimated the means of maximum, minimum and average monthly temperatures as a function of geographical coordinates and month number for any location in Turkey by the artificial neural networks, adaptive neuro-fuzzy inference system and multiple linear regression models (see also Citakoglu 2017, 2015).

The classical reliability analysis is based on the crisp information on lifetime data and cannot cover the uncertainty environments regarding the randomness, vagueness, ambiguity, and imprecision with different and specific characteristics. The uncertainties in the reliability fields are concerned with the components, parameters, phenomena and underlying assumptions. The estimation methods for reliability characteristics must be modified based on the fuzzy lifetimes to attain a more realistic analysis and exploit the uncertainty or imprecision in the data. The concept of the FS has also received considerable attention from system reliability analysis researchers such as Mahapatra and Roy (2012), Pan et al. (2015), Pramanik et al. (2019) and El-Damcese et al. (2014).

The fuzzy reliability analysis is illustrated based on various lifetime distributions, for instance, exponential (Baloui Jamkhaneh 2011), Weibull (Baloui Jamkhaneh 2014), Rayleigh (Pak et al. 2014) and three-parameter Weibull, Pareto and Gamma (Shafiq et al. 2017).

Liu et al. (2007) illustrated the fuzzy reliability analysis and mean time to failure of series, parallel, series-parallel, parallel-series and cold standby systems. Kumar et al. (2013) extended the fuzzy set semantics to IFS and analyzed IFS reliability based on the profust reliability theory, where the failure rate is represented by a time-dependent TIFN. Sharma et al. (2012) provided the fuzzy reliability of systems by IFS and implemented the TIFN and its arithmetic operations. Akbari and Hesamian (2020) considered the intuitionistic fuzzy random variable with crisp parameters and reported a procedure for constructing time-dependent reliability systems. The uncertainty of the number of failures is modulated with the aid of a fuzzy framework by Husniah and Supriatna (2021), where Weibull failure distribution is considered with the fuzzy shape parameter.

The new generalized intuitionistic fuzzy sets (GIFS\(_B\)) along with some operators over GIFS\(_B\) and the new generalized intuitionistic fuzzy number (GIFN\(_B\)) based on the GIFS have been, respectively, introduced by Baloui Jamkhaneh and Nadarajah (2015) and Shabani and Baloui Jamkhaneh (2014). Baloui Jamkhaneh (2016) represented the values and indeterminacy of the degree of membership and non-membership functions of GIFS\(_B\) and Baloui Jamkhaneh (2017) considered the generalized intuitionistic fuzzy exponential lifetime distribution and the reliability analysis based on GIFS\(_B\). Ebrahimnejad and Baloui Jamkhaneh (2018) and Roohanizadeh et al. (2021), respectively, considered system reliability of Rayleigh and Pareto distributions with GIFN\(_B\).

The heavy-tailed univariate Pareto distribution has been used often to model reliability and continuous lifetime data, which was first proposed as a model for rare events as the survival function slowly decreases in comparison to other life distributions. The Pareto distribution has been applied in modeling various phenomena in the description of hydrology, insurance, scientific, finance, and actuarial science, which can be found in the works of Amin (2008), Fu et al. (2012), Prakash (2017), Lee and Kim (2018) and Ghitany et al. (2018). According to practical purposes and research, several kinds of Pareto distribution are introduced. In this paper, we concentrate on two-parameter Pareto distribution with fuzzy scale and fuzzy shape parameters.

The purpose of the paper has twofold. First, we provide a new generalized L-R type intuitionistic fuzzy number with corresponding \((\alpha _1,\alpha _2)\) cut sets. The second principal aim is extending the reliability characteristics in the GIFS environment, which was introduced by Baloui Jamkhaneh and Nadarajah (2015), with special attention to the two-parameter Pareto distribution. We consider Pareto distribution, which has the uncertainty in its lifetime scale and shape parameters by the GIFN. The vagueness in the reliability characteristics is represented perfectly by parameter fuzzification into the GIFN\(_B\), and the generalized intuitionistic fuzzy reliability (GIFR) modeling is introduced via the generalized intuitionistic fuzzy probabilities (GIFP). Several characteristics such as conditional reliability, hazard and mean time to failure functions are obtained via generalized intuitionistic fuzzy parameters. Also, the fuzzy reliability of the series and parallel system has been represented separately.

The structure of the present paper is organized as follows. In Sect. 2, we report some basic concepts of GIFN\(_B\). The GIFP is introduced in Sect. 3, where parameters are set as the GIFN\(_B\). In Sect. 4, we obtain the GIFR characteristics, which include the reliability, conditional reliability, hazard and mean time to failure functions, and as a special case, we consider the two-parameter Pareto distribution with GIFN\(_B\) scale and shape parameters. Section 5 concentrates on GIFR for both series and parallel systems. Finally, in Sect. 6, the graphical illustration and numerical example confirm the theoretical outcomes.

In this section, we concentrate on the GIFS\(_B\) with basic GIFN\(_B\) elements that are summarized in the next Definitions. Also, a new generalized L-R type intuitionistic fuzzy number is provided, which is used throughout the paper.

Definition 1

(Baloui Jamkhaneh and Nadarajah 2015) The generalized intuitionistic fuzzy set (GIFS\(_B(X)\)) A in X, is defined as follows

where X is a non-empty set and \(\mu _{A}:X \rightarrow \left[ 0,1\right] \), \(\nu _{A}:X \rightarrow \left[ 0,1\right] \) denote the degree of membership and non-membership functions of x in A, respectively. Also, \(0\le \mu _A^{\delta }(x) +\nu _A^{\delta }(x)\le 1,\; \forall \, x\in X\) and \(\delta =n\) or \(\frac{1}{n},\ n=1,2,\dots ,N\).

Afterward, Shabani and Baloui Jamkhaneh (2014) introduced the GIFN\(_B\) based on the GIFS\(_B(X)\) defined in Definition 1. We review the GIFN\(_B\) in the next Definition.

Definition 2

(Shabani and Baloui Jamkhaneh 2014) Consider GIFS\(_B\left( X\right) \) from the real number domain, a generalized L-R type intuitionistic fuzzy number A is defined with the following membership \(\mu _{A}\left( x\right) \) and non-membership \(\nu _A\left( x\right) \) functions

such that bound values must be satisfied in \(a_1\le a\le b\le c \le d\le d_1\) constraint and

The basis left \(\big (f^L\left( x\right) , g^L\left( x\right) \big )\) and right \(\big (f^R\left( x\right) , g^R\left( x\right) \big )\) are continuous monotone membership and non-membership functions, where \(f^L\left( x\right) , g^R\left( x\right) \) are increasing and \(f^R\left( x\right) , g^L\left( x\right) \) are decreasing functions.

A class of generalized L-R type intuitionistic fuzzy number (GIFN\(_B\)) A is defined as

with the condition \( \mu +\nu \le 1 \).

The GIFN\(_B\) A is denoted as \(A=(a_1,a,b,c,d,d_1,\mu ,\nu ,\delta )\), where \( \mu _A\left( x\right) \) and \(1-\nu _A\left( x\right) \) are fuzzy numbers. Two parameters \(\mu ^{\frac{1}{\delta }}\) and \(\nu ^{\frac{1}{\delta }}\) reflect the confidence level and non-confidence level of the A, respectively.

The \(\alpha \)-cut of a fuzzy set is the classical set that includes all the elements of the set with greater than or equal to the specified value of \(\alpha \) membership degree. Baloui Jamkhaneh (2016) introduced the \((\alpha _1,\alpha _2)\)-cut of GIFN\(_B\), which is briefly explained in Definition 3.

Hereafter in the paper, we select the fixed numbers \(\alpha _1,\alpha _2\in [0,1]\) such that both hold in the constraint \( 0\le \alpha _1\le \mu ^{\frac{1}{\delta }}, \nu ^{\frac{1}{\delta }}\le \alpha _2\le 1\) and \(0\le \alpha _1^{\delta }+\alpha _2^{\delta }\le 1\), to avoid repetition.

Definition 3

(Baloui Jamkhaneh (2016)) Consider the set of \((\alpha _1,\alpha _2)\)-cut generated by a GIFN\(_B\) A defined by

\( A\left[ \alpha _1,\alpha _2,\delta \right] \) is defined as the crisp set of elements x which belong to A at least \(\alpha _1 \) degree and does not belong to A at most \( \alpha _2 \) degree. The \(\alpha _1\)-cut set of a GIFN\(_B\) A is a crisp subset of real number domain \({\mathbb {R}}\), which is defined as

where

Analogously, the \(\alpha _2\)-cut set of a GIFN\(_B\) A is a crisp subset of \({\mathbb {R}}\) as

where

Similarly,

If set \( \alpha _1=1-\alpha _2=\alpha \), then

The GIFN\(_B\) based on the \(\alpha _1\)-cut and \(\alpha _2\)-cut sets is shown as

Definition 4

Consider two \(\alpha \)-cut sets \(\left[ a,b\right] \) and \(\left[ c,d\right] \), some relations and operations on \(\alpha \)-cut sets are defined as follows:

1. (i)

   The relation \(\left[ a,b\right] \preccurlyeq [c,d]\) is hold, if and only if \( a\le c\) and \(b\le d\),

2. (ii)

   If \(k>0\), then \(k\otimes [a,b]=[ka,kb]\) and if \(k<0\), then \(k\otimes [a,b]=\left[ kb,ka\right] \),

3. (iii)

   \(k\oplus \left[ a,b\right] =\left[ k+a,k+b\right] \) and \(k\ominus \left[ a,b\right] =[k-b,k-a]\),

4. (iv)

   \(\left[ a,b\right] \oplus \left[ c,d\right] =\left[ a+c,b+d\right] .\)

Definition 5

Several relations and operations on GIFN\(_B\)s are listed as below:

1. (i)

   \(A\left( \alpha _1,\alpha _2,\delta \right) \oplus B\left( \alpha _1,\alpha _2,\delta \right) =\big (A_{\mu }\left[ \alpha _1,\delta \right] \oplus \) \(B_{\mu }\left[ \alpha _1,\delta \right] ,A_{\nu }\left[ \alpha _2,\delta \right] \oplus B_{\nu }\left[ \alpha _2,\delta \right] \big ),\)

2. (ii)

   \(k\otimes A\left( \alpha _1,\alpha _2,\delta \right) \oplus b=\big (k\otimes A_{\mu }\left[ \alpha _1,\delta \right] \oplus b,\) \(k\otimes A_{\nu }\left[ \alpha _2,\delta \right] \oplus b\big ),\)

3. (iii)

   \(b\ominus A\left( \alpha _1,\alpha _2,\delta \right) =(b\ominus A_{\mu }\left[ \alpha _1,\delta \right] ,b\ominus A_{\nu }\left[ \alpha _2,\delta \right] )\),

4. (iv)

   \(A\left( \alpha _1,\alpha _2,\delta \right) \preccurlyeq B\left( \alpha _1,\alpha _2,\delta \right) \), if and only if \(A_{\mu }\left[ \alpha _1,\delta \right] \preccurlyeq B_{\mu }\left[ \alpha _1,\delta \right] \) and \(A_{\nu }\left[ \alpha _2,\delta \right] \preccurlyeq B_{\nu }\left[ \alpha _2,\delta \right] \),

5. (v)

   \(A\left( \alpha _1,\alpha _2,\delta \right) =B(\alpha _1,\alpha _2,\delta )\), if and only if \(A_{\mu }\left[ \alpha _1,\delta \right] =B_{\mu }\left[ \alpha _1,\delta \right] \) and \(A_{\nu }\left[ \alpha _2,\delta \right] =B_{\nu }\left[ \alpha _2,\delta \right] \).

where \(A\left( \alpha _1,\alpha _2,\delta \right) \) and \(B\left( \alpha _1,\alpha _2,\delta \right) \) are two GIFN\(_B\)s.

The uncertainty in lifetime data may be caused by the random variables or parameters of the model. Here, we focus on the imprecise parameters modeled by fuzzy numbers. We introduce the fuzzy probability where parameters of the model are considered as the GIFN\(_B\).

Consider the continuous random variable X from a density function \(f(x,{\tilde{\theta }},{\tilde{\beta }})\) where \({\tilde{\theta }}\) and \({\tilde{\beta }}\) are GIFN\(_B\). Then, \(\alpha _1\)-cut set of membership and \(\alpha _2\)-cut set of non-membership functions of the GIFP of C is defined as

where \(P\left( C\right) \) is the crisp probability defined as \(P\left( C\right) =\int _C f\left( x,\theta \right) dx,\) and

Consequently,

which is the GIFN\(_B\) and \((\alpha _1,\alpha _2)\)-cut set of GIFP of C is defined as

Corollary 1

Consider the GIFP as P(C), then

1. (i)

   \(P\left( C^c\right) \left( \alpha _1,\alpha _2, \delta \right) =1\ominus P(C)\left( \alpha _1,\alpha _2,\delta \right) \),

2. (ii)

   If \(C_1\subset C_2\) then \(P\left( C_1\right) \left( \alpha _1,\alpha _2,\delta \right) \preccurlyeq P\left( C_2\right) \left( \alpha _1,\alpha _2,\delta \right) \).

Proof

(i) Regarding to the definition of GIFP, for \( \left( i,j\right) =\left( 1,\mu \right) ,(2,\nu ) \) we have

which is verified by Definition 5-v. (ii) Since \(P\left( C_1\right) \le P(C_2)\), so

and based on Definition 5-iv, the proof is completed. \(\square \)

The fuzzification of some statistical concepts, including expectation and variance, can be induced by the specification of GIFP.

A set of \(\alpha _1\)-cut of membership and \(\alpha _2\)-cut set of non-membership functions of generalized intuitionistic fuzzy expectation (GIFE) \({\tilde{E}}(g\left( X\right) )\) is determined as

where \( \left( i,j\right) =\left( 1,\mu \right) ,(2,\nu ) \) and based on the crisp expectation \(E(g\left( X\right) )\), we have

Consequently, it is concluded that

and \((\alpha _1,\alpha _2)\)-cut set of GIFE of \(g\left( X\right) \) is described as

Remark 1

The GIFE of \(X({\tilde{\mu }})\) and generalized intuitionistic fuzzy variance of \(X({\tilde{\sigma }}^2)\) are obtained by the assumptions \(g\left( X\right) =X\) and \(g\left( X\right) =(X-E\left( X\right) )^2\), respectively.

Corollary 2

Consider a, b, c as constant numbers, then

1. (i)

   \({\tilde{E}}\left( c\right) =c\),

2. (ii)

   \({\tilde{E}}\left( ag\left( X\right) +b\right) =a\otimes {\tilde{E}}(g(X))\oplus b\),

3. (iii)

   \({\tilde{\sigma }}^2\left( c\right) =0\),

4. (iv)

   \({\tilde{\sigma }}^2\left( aX+b\right) =a^2\otimes {\tilde{\sigma }}^2\left( X\right) \).

Proof

(i) and (iii) are obvious and the proofs are omitted. The proof of (ii) is obtained as follows:

Also, (iv) is concluded by

and the proof is completed. \(\square \)

The intuitionistic fuzzy approach for reliability parameter analysis leads to more flexible information that can capture subjective, uncertain, and ambiguous information.

Consider X as a lifetime variable of a component with a density function \(f(x,{\tilde{\theta }})\) where the vector of the parameters \({\tilde{\theta }}\) is the GIFN\(_B\) and the GIFR characteristic (GIFRC) denoted by \({\tilde{g}}\left( t\right) \). A set of \(\alpha _1\)-cut of membership and \(\alpha _2\)-cut set of non-membership functions of GIFRC are denoted by \(g_j(t)\left[ \alpha _i,\delta \right] \) as

where

The function g(t) can be considered as the reliability, conditional reliability, hazard rate, cumulative risk and reverse hazard functions. It can be shown that \(g\left( \alpha _1,\alpha _2,\delta \right) =(g_{\mu }(t)\left[ \alpha _1,\delta \right] , g_{\nu }(t)\left[ \alpha _2,\delta \right] )\) and the \((\alpha _1,\alpha _2)\)-cut set of GIFRC is defined as

In the next subsection, we provide different reliability characteristics, comprehensively. Also, the fuzzy reliability characteristics of the two-parameter Pareto lifetime distribution with the scale parameter \(\lambda \) and shape parameter \(\gamma \) is provided as a special case.

4.1 Generalized intuitionistic fuzzy reliability function

The fuzzy reliability accounts for the uncertainty of the membership and non-membership grades of the component’s reliability. In this section, the GIFR as the GIFP of surviving beyond time t, denoted by \({\tilde{S}}\left( t\right) \), is constructed based on the lifetime GIFN\(_B\) parameters.

The \(\alpha _1\)-cut set of membership and \(\alpha _2\)-cut set of non-membership functions of GIFR of component denoted by \(S_j(t)\left[ \alpha _i,\delta \right] \), are obtained as

where S(t) is the crisp reliability function and

and it can be shown as

The \((\alpha _1,\alpha _2)\)-cut set of GIFR is defined as follows:

where \(S_j(t)\left[ \alpha _i,\delta \right] ,\;\left( i,j\right) =\left( 1,\mu \right) ,(2,\nu )\) are two-variate functions in terms of \(\alpha _i,\; i=1,2\) and t. For \(t_0\), \({\tilde{S}}\left( t_0\right) \) is the GIFN\(_B\).

As a special case, we consider the two-parameter Pareto distribution and provide each corresponding reliability characteristic, respectively. Consider the random variable X from the two-parameter Pareto lifetime distribution

which has the uncertainty in both scale and shape parameters and the vagueness are represented by fuzzifying the parameter values into a GIFN\(_B\). Set the generalized intuitionistic fuzzy lifetime scale parameter

and shape parameter

then, the cut sets of GIFR function for \(\left( i,j\right) =\left( 1,\mu \right) ,(2,\nu )\) is obtained as follows

Since \((\frac{\gamma }{t})^{\lambda }\) is a monotonically decreasing with respect to \(\lambda \) and increasing with respect to \(\gamma \), the reliability bands are given by

In this method, for every specially \(\alpha _{10}\) and \(\alpha _{20}\), shapes of \(S_j\left( t\right) \left[ \alpha _{i0},\delta \right] ,\; (i,j)=(1,\mu ),(2,\nu )\) are like bands with upper and lower curves. For \( (i,j)=(1,\mu ),(2,\nu )\), this reliability bands has the following properties

1. (i)

   \(S_j\left( 0\right) \left[ \alpha _{i0},\delta \right] =[1,1]\), i.e., no one starts off dead,

2. (ii)

   \(S_j\left( \infty \right) \left[ \alpha _{i0},\delta \right] =[0,0],\) i.e., everyone dies eventually,

3. (iii)

   \(S_j\left( t_1\right) \left[ \alpha _{i0},\delta \right] \succcurlyeq S_j\left( t_2\right) \left[ \alpha _{i0},\delta \right] \) if and only if \( t_1\le t_2,\) i.e., bands of \(S_j\left( t\right) \left[ \alpha _{i0},\delta \right] \) declines monotonically.

4.2 Generalized intuitionistic fuzzy conditional reliability function

In reliability analysis, conditional reliability is the probability of an item surviving for the time t, given that it has already survived until time \(\tau \).

Here, we extend the conditional reliability function to the uncertain case by the GIFS concept. The generalized intuitionistic fuzzy conditional reliability (GIFCR) function of the component is denoted by \({\tilde{S}}\left( t\vert \tau \right) \). The \(\alpha _1\)-cut set of membership and \(\alpha _2\)-cut set of non-membership functions of \({\tilde{S}}\left( t\vert \tau \right) \) are represented as

where \( S( t\vert \tau ) \) is the crisp conditional reliability function and

Subsequently, we have

The \((\alpha _1,\alpha _2)\)-cut set of GIFCR function is defined as

where \(S_j( t\vert \tau )\left[ \alpha _i,\delta \right] , \;\left( i,j\right) =\left( 1,\mu \right) ,(2,\nu )\) are two-variate functions in terms of \(\alpha _i,\;i=1,2\) and t.

For \(t_0\), \({\tilde{S}}(t_0\vert \tau )\) is the GIFN\(_B\). In this method, for every specially \(\alpha _{10}\) and \(\alpha _{20}\), shapes of \(S_j( t\vert \tau )\left[ \alpha _{i0},\delta \right] , \;\left( i,j\right) =\left( 1,\mu \right) ,(2,\nu )\) are like bands with upper and lower curves.

Consider the two-parameter Pareto distribution, the cut sets of GIFCR function, for \(\left( i,j\right) =\left( 1,\mu \right) ,(2,\nu )\) are represented by

Since \({\big (\frac{\tau }{t+\tau }\big )}^{\lambda }\) is a monotonically decreasing function with respect to \(\lambda \), the conditional reliability bands are computed as

For every especial \(t_{0}\), membership and non-membership functions of \({\tilde{S}}\left( t_0\mid \tau \right) \) are given as

4.3 Generalized intuitionistic fuzzy hazard function

Another fuzzy character of the lifetime distribution is the fuzzy hazard function (or fuzzy failure rate). We propose the generalized intuitionistic fuzzy hazard (GIFH) function of component as \({\tilde{h}}\left( t\right) \) and it means the probability of an item failing at the time interval \(\Delta t\) if it operated until t. The \(\alpha _1\)-cut set of membership and \(\alpha _2\)-cut set of non-membership functions of GIFH of the component are illustrated as

where \( h\left( t\right) \) is the crisp hazard rate function and

It can be shown that

and the \((\alpha _1,\alpha _2)\)-cut set of GIFH function is defined by

where \(h_{j}(t)\left[ \alpha _i,\delta \right] ,\left( i,j\right) =\left( 1,\mu \right) ,(2,\nu )\) are two-variate functions in terms of \(\alpha _i, i=1,2\) and t.

Remark 2

Same as the GIFR, for every especially \(\alpha _{10}\) and \(\alpha _{20}\), the shapes of \(S_j(t\vert \tau )[\alpha _{i0},\delta ]\) and \(h_j(t)[\alpha _{i0},\delta ], (i, j) = (1, \mu ) , (2, \nu )\) are like bands with upper and lower curves and for every especially \(t_0\), \({\tilde{S}}(t_0\vert \tau )\) and \({\tilde{h}}(t_0)\) are the GIFB\(_B\)s.

Remark 3

If \(\mu =1\) and \(\nu =0\), then our method changes to its special case; in addition, if \( \delta =1 \), then our method is named intuitionistic fuzzy reliability evaluation. If \(\alpha _1=1-\alpha _2\), \(a=a_1\) and \(d=d_1\), then it changes to fuzzy reliability evaluation. Finally, if assumption \(a=b=c=d\) is added, it agrees to classical reliability theory.

For \(\left( i,j\right) =\left( 1,\mu \right) ,(2,\nu ),\) the cut set of GIFH function for two-parameter Pareto lifetime distribution is demonstrated as

where

Therefore,

As can be seen, the GIFH function for generalized intuitionistic fuzzy two-parameter Pareto distribution is decreasing with respect to time t. The membership and non-membership functions of \({\tilde{h}}\left( t_{0}\right) \) are reported as follows:

4.4 Some fuzzy reliability characteristics properties

In this section, we provided some relations and properties of the reliability characteristics, with special attention to fuzzy two-parameter Pareto reliability.

Corollary 3

If \(\mu _1\le \mu _2\) and \(\nu _1\le \nu _2\), then we have

1. (i)

   \(S_{\mu _1}(t)\left[ \alpha _1,\delta \right] \subset S_{\mu _2}(t)\left[ \alpha _1,\delta \right] \) and \(S_{\nu _2}(t)\left[ \alpha _2,\delta \right] \subset S_{\nu _1}(t)\left[ \alpha _2,\delta \right] \),

2. (ii)

   \(S_{\mu _1}( t\mid \tau )\left[ \alpha _1,\delta \right] \subset S_{\mu _2}( t\mid \tau )\left[ \alpha _1,\delta \right] \) and \(S_{\nu _2}( t\mid \tau )\left[ \alpha _2,\delta \right] \subset S_{\nu _1}( t\mid \tau )\left[ \alpha _2,\delta \right] \),

3. (iii)

   \(h_{\mu _1}(t)\left[ \alpha _1,\delta \right] \subset h_{\mu _2}(t)\left[ \alpha _1,\delta \right] \) and \(h_{\nu _2}(t)\left[ \alpha _2,\delta \right] \subset h_{\nu _1}(t)\left[ \alpha _2,\delta \right] \).

Corollary 4

If \(\delta _1\le \delta _2\), then we have

1. (i)

   \(S_{\mu }(t)\left[ \alpha _1,\delta _1\right] \subset S_{\mu }(t)\left[ \alpha _1,\delta _1\right] \) and \(S_{\nu }(t)\left[ \alpha _2,\delta _2\right] \subset S_{\nu }(t)\left[ \alpha _2,\delta _1\right] \),

2. (ii)

   \(S_{\mu }( t\mid \tau )\left[ \alpha _1,\delta _1\right] \subset S_{\mu }( t\mid \tau )\left[ \alpha _1,\delta _2\right] \) and \(S_{\nu }( t\mid \tau )\left[ \alpha _2,\delta _2\right] \subset S_{\nu }( t\mid \tau )\left[ \alpha _2,\delta _1\right] \),

3. (iii)

   \(h_{\mu }(t)\left[ \alpha _1,\delta _1\right] \subset h_{\mu }(t)\left[ \alpha _1,\delta _2 \right] \) and \(h_{\nu }(t)\left[ \alpha _2,\delta _2\right] \subset h_{\nu }(t)\left[ \alpha _2,\delta _1\right] \).

Corollary 5

For every \(\delta \),

Corollary 6

Consider \(g\left( t_0\right) \left[ \alpha _1,\alpha _2\right] \) as \((\alpha _1,\alpha _2)\)-cut set of reliability characteristics (GIFR or GIFCR or GIFH) and set

then

1. (i)

   \({\small g(t_0)[\alpha _1,\alpha _2]=\left\{ \begin{array}{ll} \left[ g^L_{\nu }\left[ \alpha _2\right] ,g^U_{\nu }\left[ \alpha _2\right] \right] ,&{} \eta <\min (z_1,z_2) \\ \left[ g^L_{\mu }\left[ \alpha _1\right] ,g^U_{\mu }\left[ \alpha _1\right] \right] ,&{} \eta \ge \max (z_1,z_2) \end{array}\right. ,}\)

2. (ii)

   if \(\eta =1\) (i.e., \(1-\frac{1-\alpha ^{\delta }_2}{1-\nu }=1-\frac{\alpha ^{\delta }_1}{\mu }\)), then \(g\left( t_0\right) \left[ \alpha _1,\alpha _2\right] =\left[ g^L_{\mu }\left[ \alpha _1\right] ,g^U_{\mu }\left[ \alpha _1\right] \right] \),

3. (iii)

   if \(z_1=z_2=\eta \) then \(g_{\mu }\left( t_0\right) \left[ \alpha _1\right] =g_{\nu }\left( t_0\right) \left[ \alpha _2 \right] = g\left( t_0\right) \left[ \alpha _1,\alpha _2\right] \),

Corollary 7

Consider the two-parameter Pareto lifetime distribution, if \(\mu _{g\left( t_0\right) }\left( x\right) = \nu _{g\left( t_0\right) }(x)\) and \(z_1=z_2=z\), then we have

1. (i)

   \(S\left( t_0\right) \left[ \alpha _1,\alpha _2\right] =S_{\mu }\left( t_0\right) \left[ \alpha _1\right] =S_{\nu }\left( t_0\right) \left[ \alpha _2\right] =[\big (\frac{ \gamma }{t_0})^{\zeta },(\frac{ \gamma }{t_0}\big )^{\xi }]\),

2. (ii)

   \(h\left( t_0\right) \left[ \alpha _1,\alpha _2\right] =h_{\mu }\left( t_0\right) \left[ \alpha _1\right] =h_{\nu }\left( t_0\right) \left[ \alpha _2\right] =\big [\frac{\zeta }{t_0}\), \(\frac{\xi }{t_0}\big ]\),

3. (iii)

   \(S\left( t_0\mid \tau \right) \left[ \alpha _1,\alpha _2\right] =S_{\mu }\left( t_0\mid \tau \right) \left[ \alpha _1\right] =S_{\nu }\left( t_0\mid \tau \right) \left[ \alpha _2\right] =\big [\big (\frac{\tau }{t_0+\tau }\big )^{\zeta },{\big (\frac{\tau }{t_0+\tau }\big )}^{\xi }\big ]\),

4. (iv)

   \(\alpha _1=\alpha _2=\Big (\dfrac{(1-\nu )z+\nu }{1+z(\frac{1-\nu }{\mu })}\Big )^{\frac{1}{\delta }}\),

where \(\zeta =\frac{d_1(\mu -\nu )+c_1(\nu +(1-\nu )z)}{\mu +(1-\nu )z}\) and \(\xi =\frac{a_1(\mu -\nu )+b_1(\nu +(1-\nu )z)}{\mu +(1-\nu )z}\).

Theorem 1

Consider the lifetime variables \(T_1\) and \(T_2\) with the generalized intuitionistic fuzzy density function \({\tilde{f}}_1(x,{\tilde{\theta }},{\tilde{\beta }})\) and \({\tilde{f}}_2(x,{\tilde{\theta }},{\tilde{\beta }})\), respectively. For every \(t>0\), if the condition \({\tilde{h}}_1(t)\succcurlyeq {\tilde{h}}_2(t)\) and \( {\tilde{S}}_1(\tau )={\tilde{S}}_2(\tau )\) hold, it can be concluded that \({\tilde{S}}_1(t\vert \tau )\preccurlyeq {\tilde{S}}_2(t\vert \tau )\).

Proof

By using \({\tilde{h}}_1\left( t\right) \left( \alpha _1,\alpha _2,\delta \right) \,\succcurlyeq \, {\tilde{h}}_2\left( t\right) \left( \alpha _1,\alpha _2,\delta \right) \) it is induced that

which leads to

Therefore, for every \(\gamma =L,U\), we have

consequently,

Hence, regarding the definition of hazard rate function, we have

and

subsequently,

also, \(S_1\left( t+\tau \right) (\alpha _1,\alpha _2,\delta )\, \preccurlyeq \, S_2\left( t+\tau \right) (\alpha _1,\alpha _2,\delta )\) and \({\tilde{S}}_1(t\vert \tau )\preccurlyeq {\tilde{S}}_2(t\vert \tau )\), which completes the proof. \(\square \)

Theorem 2

The increasing condition on the \({\tilde{S}}(x\vert t)\) function is a necessary and sufficient condition for \(f(x,{\tilde{\theta }},{\tilde{\beta }})\) to belong to a class of distribution with a decreasing failure rate (IFR).

Proof

For every \(t_1<t_2\) we have \({\tilde{S}}(x\vert t_1)\,\preccurlyeq \,{\tilde{S}}(x\vert t_2)\) and

we conclude that

then

and

For every \(\gamma =L,U\), it can be concluded that

and

Therefore, \(S^{\gamma }_{\mu }\) and \(S^{\gamma }_{\nu }\) are increasing functions and by using definition of GIFCR function, for \( \left( i,j\right) =\left( 1,\mu \right) ,\left( 2,\nu \right) \), we have

Due to increasing shape of \(S^{\gamma }_j\) function, so it is induced that \(\dfrac{\partial S^{\gamma }_j\left( x\mid t\right) \left[ \alpha _i\right] }{\partial t}\;\ge \; 0\) and hence

so,

and it is concluded that

Finally, we have \(h(t)(\alpha _1,\alpha _2,\delta )\, \succcurlyeq \, h(x+t)(\alpha _1,\alpha _2,\delta )\) and \({\tilde{h}}(t)\,\succcurlyeq \,{\tilde{h}}(x+t),\) which completes the proof. \(\square \)

4.5 Generalized intuitionistic fuzzy mean time to failure for Pareto distribution

The mean time to failure (MTTF) is a reliability character that indicates the expected time span when an unrepairable system is active. The MTTF can be used to evaluate reliability and to improve maintenance and system management strategies. The generalized intuitionistic fuzzy mean time to failure (GIFMTTF) of components is the expected time to failure of the fuzzy system and is denoted by \(\text {M}\widetilde{\text {TT}}\text {F}\). In this section, the GIFMTTF function of any component is provided under the two-parameter Pareto lifetime distribution, which is defined as follows

then

The reliability of a system depends on the manner of relation of each component such as the series or parallel structure. In a series structure, the reliability of the system is the minimum of the reliability of components and the system fails even if an individual component failed. On contrary, for parallel structure, the system works even only one component works and reliability is equal to the maximum of the reliability of components. In this section, we focus on the GIFR of series and parallel systems, such that the failure of any component does not depend on any other component.

5.1 Series system

If n-components are connected in a series manner, then the \(\alpha _i\)-cut (\(i=1,2\)) of GIFR with generalized intuitionistic fuzzy distribution is given by

where

The \(\alpha _i\)-cut (\(i=1,2\)) of GIFR with generalized intuitionistic fuzzy two-parameter Pareto distribution is given by

and

5.2 Parallel system

If n-components are related in a parallel manner, the \(\alpha _i\)-cut (\(i=1,2\)) of GIFR with generalized intuitionistic fuzzy distribution is provided as

where

The \(\alpha _i\)-cut (\(i=1,2\)) of GIFR with generalized intuitionistic fuzzy two-parameter Pareto distribution is represented as

and

Let the lifetime of electronic component is modeled by the two-parameter Pareto distribution with generalized intuitionistic fuzzy scale and shape parameters

Then cut sets of GIFP of \(X\le 2\) is obtained, for \( (i,j)=(1,\mu ),(2,\nu ) \), as follows:

and

The membership and non-membership functions of the GIFP are represented in Fig. 1. Also, for several values of \(\alpha _1\) and \(\alpha _2\), the \(\alpha _1\)-cut set of membership and \(\alpha _2\)-cut set of non-membership bands of GIFP and \((\alpha _1,\alpha _2)\)-cut bands of GIFP are reported in Table 1, respectively.

The membership and non-membership functions of GIFP

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Regarding Table 1, by increasing \(\alpha _1\) and decreasing \(\alpha _2\), the ambiguity in membership and non-membership bands of GIFP is decreased as well as the bands of GIFP.

The cut sets of GIFR are given by

The surfaces of GIFR function

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Figure 2 represents the surfaces of GIFR for different angles.

The bands for \(\alpha _1=\sqrt{0.15}\) and \(\alpha _2=\sqrt{0.25}\) are given by

The GIFR bands for \(\alpha _1=\sqrt{0.15}\) and \(\alpha _2=\sqrt{0.25}\) are plotted in Fig. 3. As can be seen, the ambiguity of GIFR is increased by increasing time t, due to the increase in both bandwidths of membership and non-membership functions.

The GIFR bands for \(\alpha _1=\sqrt{0.15}\) and \(\alpha _2=\sqrt{0.25}\)

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If set \(t=2\), then cut sets of GIFR are computed as

The membership and non-membership functions of GIFR are depicted in Fig. 4.

In Table 2, the \(\alpha _1\)-cut of membership, \(\alpha _2\)-cut of non-membership bands and \((\alpha _1,\alpha _2)\)-cut bands of GIFR are prepared, for different combinations of cuts \(\alpha _1\) and \(\alpha _2\).

The membership and non-membership functions of GIFR

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a The reliability bands of \(S(t)[0.05,0.5,\delta ]\), b The reliability bands of \(S(t)[\alpha _1,\alpha _2,1]\)

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Based on Table 2, by increasing \(\alpha _1\) and decreasing \(\alpha _2\), the vagueness in membership and non-membership bands of GIFR and bands of GIFR is decreased.

The cut sets of GIFR are computed as follows:

The reliability bands for the different values of \(\delta \) and cut sets (\(\alpha _1,\alpha _2\)) are represented in Fig. 5; the large values of the parameter \(\delta \) lead to less reliability bandwidth and more accurate reliability. Also, by increasing \(\alpha _1\) and decreasing \(\alpha _2\), the uncertainty in reliability bands is reduced. Also, by increasing time t, the uncertainty in GIFR function is increased.

The \(\alpha _i\)-cuts of GIFCR for \(i=1,2\) are given by

Figure 6 shows surfaces of the GIFCR function from different angles.

The surfaces of GIFCR function

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The GIFCR bands with \(\tau =3\) for \(\alpha _1=\sqrt{0.1}\) and \(\alpha _2=\sqrt{0.8}\) are expressed as

The GIFCR bands for \(\alpha _1=\sqrt{0.1}\) and \(\alpha _2=\sqrt{0.8}\)

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The membership and non-membership functions of GIFCR

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The GIFCR bands for \(\alpha _1=\sqrt{0.1}\) and \(\alpha _2=\sqrt{0.8}\) are depicted in Fig. 7, which indicates that increasing time t leads to increasing the bandwidth which is equivalent to increasing in uncertainty. Let \(t_0=3,\tau =3\), the membership and non-membership functions of \({\tilde{S}}\left( t_0\mid \tau \right) \) are obtained as follows:

The \((\sqrt{0.75},\sqrt{0.25})\)-cut set of \({\tilde{S}}\left( t\vert \tau \right) \) is given by

The membership and non-membership functions of GIFCR are represented in Fig. 8.

The \(\alpha _1\)-cut of membership, \(\alpha _2\)-cut of non-membership bands and \((\alpha _1,\alpha _2)\)-cut bands of GIFCR, for different combinations of cut sets \((\alpha _1,\alpha _2)\) are assembled in Table 3. Based on Table 3, the more accurate bands of membership and non-membership of GIFCR and bands of GIFCR are attained by the maximum value of \(\alpha _1\) and minimum of \(\alpha _2\).

The \(\alpha _i\)-cuts of GIFH function for \(i=1,2\) are given by

Figure 9 shows the surfaces of GIFH function from different angles.

The surfaces of the GIFH function

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The GIFH bands for \(\alpha _1=\sqrt{0.1}\) and \(\alpha _2=\sqrt{0.8}\) are computed as

The GIFH bands of membership and non-membership functions for \(\alpha _1=\sqrt{0.1}\) and \(\alpha _2=\sqrt{0.8}\) are exhibited in Fig. 10. Analogously, increasing time t causes the more accuracy in GIFH function.

The GIFH bands for \(\alpha _1=\sqrt{0.1}\) and \(\alpha _2=\sqrt{0.8}\)

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The membership and non-membership functions of \({\tilde{h}}\left( 2\right) \) are represented as below:

The membership and non-membership functions of GIFH are displayed in Fig. 11.

The membership and non-membership functions of GIFH

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Figure 4 reports the \(\alpha _1\)-cut of membership, \(\alpha _2\)-cut of non-membership bands and \((\alpha _1,\alpha _2)\)-cut bands of GIFH, for different combinations of \(\alpha _1,\alpha _2\), which has the same results as other counterpart tables.

Generally based on Tables 1, 2, 3 and 4, it is inferred that increasing \(\alpha _1\) and decreasing \(\alpha _2\) lead to ambiguity decreasing of the fuzzy reliability characteristics, including GIFR, GIFCR and GIFH bands. Moreover, regarding Figs. 3, 7 and 10, the GIFR, GIFCR and GIFH are decreasing functions with respect to t.

In the present paper, we extend the GIFN\(_B\) to analyze the system reliability with the special two-parameter Pareto distribution discussion. Both scale and shape parameters of the two-parameter Pareto distribution are considered as GIFN\(_B\), and various generalized intuitionistic fuzzy reliability characteristics are obtained. The reliability characteristics are represented through bands, which attained their most precise bands for large value of the cut set of membership and small value of the cut set of non-membership functions. The theoretical results are evaluated by a comprehensive numerical approach. In this context, our study covers several research kinds of literature in fuzzy subjects.