Fuzzy Stress and Strength Reliability Based on the Generalized Mixture Exponential Distribution

Abstract

This paper discusses the reliability of stress and strength,R, and fuzzy stress and strength reliability, $R_F$, based on generalized mixtures of exponential distributions. We propose several estimation methods, such as the maximize likelihood estimation, the weighted least-squares estimation, and the percentile estimation, to estimate the corresponding measures. Simulation studies have been conducted to compare the proposed estimators’ performance using different settings. These comparisons are based on biases (Bias) and mean squared errors (MSEs), and we find that $MSE\left(PE\right)>MSE\left(MLE\right)>MSE\left(WLE\right)$ and $\left|Bias\right(PE\left)\right|>\left|Bias\right(WLE\left)\right|>\left|Bias\right(MLE\left)\right|$ in most cases. Moreover, the values of $R_F$ have the same pattern as R, and the values of MSEs and biases for $R_F$ are smaller than R. As the sample size increases, the values of biases for both reliabilities decrease and approach 0. Ultimately, we apply the proposed methods to a data set to illustrate its significance. We find that the estimated values of R are greater than those of $R_F$ for all the estimation methods. Moreover, the fuzzy estimators of $R_F$ are approximately equal to the estimators R.

1. Introduction

A stress and strength reliability analysis frequently employs probability theory and statistics to quantify the likelihood of failure under various scenarios. For two independent random variables, X (representing stress) and Y (representing strength), the reliability is defined as $R=P(X>Y)$, where $P(X>Y)$ signifies the probability that stress is less than strength. If the stress X exceeds the strength Y, it can lead to component failure or system malfunction. Accurately predicting stress and strength reliability is crucial for safeguarding these systems’ safety, performance, and longevity. For instance, by studying stress and strength reliability, engineers can design engines that minimize the risk of failure, thereby ensuring the safety of passengers and crew. On the other hand, manufacturers can reduce the likelihood of component failures, improve vehicle safety, and ultimately, enhance customer satisfaction. Researchers in this field utilize diverse analytical and experimental techniques to investigate material and component stress and strength behavior, aiming to optimize designs and improve overall system reliability.

In the context of stress–strength models, Kelley et al. [1] and Tong [2,3] considered the estimation of stress and strength reliability for two independent exponential random variables. Awad et al. [4] studied the stress and strength reliability of systems involving two bivariate exponential distributions. Kundu and Gupta [5] addressed the estimation of $P(X>Y)$ when X and Y are two independent generalized exponential distributions, each having a different shape parameter but sharing the same scale parameter. Saraçoğlu et al. [6] investigated the estimation of $R=P(X>Y)$ for an exponential distribution under progressive type-II censoring conditions. Jafari and Bafekri [7] made inferences using the stress–strength model for the two-parameter exponential distribution under the assumption of order statistics. Al-Babtain et al. [8] provided a stress–strength model for the power-modified Lindley distribution under classical and Bayesian principles. Liu et al. [9] discussed the stress–strength reliability of the two-parameter power function distribution. Kumari et al. [10] studied inverse exponentiated distribution stress–strength reliability. Ma et al. [11] studied the stress–strength model for an inverted exponential Rayleigh distribution when the latent failure times are progressive Type-II censored. Sultana et al. [12] obtained the stress–strength reliability estimation under a balanced joint Type-II progressive censoring scheme for independent samples from two different populations.

However, uncertainty is an inherent characteristic of most engineering processes, which necessitates effective handling and representation. Sometimes, the data cannot be reported precisely due to unexpected situations. This uncertainty stems from diverse sources, including measurement errors, variations in material properties, environmental factors, and the intrinsic complexity of the engineered systems. To address these challenges, several researchers directed their attention to applying the fuzzy set theory. Fuzzy stress–strength reliability, initially proposed by Zadeh [13], is a concept that seamlessly integrates fuzzy set theory with stress–strength interference analysis within the realm of reliability engineering. This concept tackles the inherent uncertainty in engineering processes by offering a robust framework capable of managing imprecise or vague data about stress and strength variables. By modeling stress and strength as fuzzy sets and defining suitable fuzzy membership functions, we can effectively account for the variability and uncertainty inherent in these parameters, yielding a more realistic and nuanced assessment of the system’s reliability. For example, Huang [14] introduced a novel methodology for reliability modeling that accommodates fuzziness, extending traditional reliability concepts into fuzzy reliability using probability measures associated with fuzzy events. Cai [15] offered a comprehensive review of the application of fuzzy methodology, showcasing the representation of failure through fuzzy sets. Huang et al. [16] investigated the estimation of parameters and the reliability function of multi-parameter lifetime distributions, employing fuzzy measures to represent lifetime data. Li and Kapur [17] proposed fuzzy reliability measures that consider both success and failure as fuzzy states. Eryilmaz and Tütüncü [18] introduced the fuzzy stress–strength reliability for a single unit and defined some properties of fuzzy reliability by using a fuzzy membership function. Yazgan et al. [19] considered estimating stress–strength reliability in the presence of fuzziness when the two random variables follow the WE distribution. Hassan and Muse [20] discussed the Bayesian and non-Bayesian estimation of the stress–strength model when there is fuzziness for stress and strength random variables having Lindley’s distribution with different parameters. Milošević and Stanojević [21] introduced a novel membership function for the definition of fuzzy stress–strength reliability in a two-component system.

The generalized mixture exponential (GME) distribution, proposed by Yang et al. [22], offers more flexible distributions with applications in lifetime modeling. A random variable X is said to have a GME distribution if its probability density function (pdf) is of the following form:

$$f(x;\lambda ,\alpha ,\beta )={\displaystyle \frac{(\alpha +1)\lambda }{\alpha +1+\alpha \beta }}{e}^{-\lambda x}[1+\beta (1-e^{-\alpha \lambda x})],x>0,$$

(1)

where $\alpha >0$ is the scale parameter, $\lambda >0$ and $\beta \ge -1$ are the shape parameters, and we denote it as $X\sim GME(\lambda ,\alpha ,\beta )$. Moreover, the cumulative distribution function (cdf) of $X\sim GME(\lambda ,\alpha ,\beta )$ is given by

$$\begin{array}{ccc}\hfill F(x;\lambda ,\alpha ,\beta )& =& 1+{\displaystyle \frac{\beta e^{-(\alpha +1)\lambda x}}{\alpha +1+\alpha \beta }}-{\displaystyle \frac{(\alpha +1)(\beta +1)e^{-\lambda x}}{\alpha +1+\alpha \beta }}.\hfill \end{array}$$

(2)

Drawing from the research of Yang et al. [22], it is evident that the GME distribution offers a versatile framework, encompassing both the exponential and weighted exponential distributions as exceptional cases. Chesneau et al. [23] expanded this concept by introducing the modified weighted exponential distribution and its various applications. Additionally, Bean et al. [24] utilized the GME distribution to analyze stochastic fluid–fluid models in transient and stationary states. However, to our knowledge, a gap exists in the literature regarding exploring stress and strength reliability within a fuzziness-based framework, explicitly leveraging the GME distribution. This article aims to bridge this gap by introducing such an approach. The rest of the article is organized as follows. Section 2 presents the stress and strength reliability concept, utilizing the GME distribution as the underlying model. Section 3 delves into various estimation methods suitable for this GME-based framework for stress and strength reliability and fuzzy stress and strength reliability. Section 4 presents simulation studies conducted to investigate and comprehensively compare the performance of the proposed estimation methods. To illustrate the practical applicability and usefulness of our proposed method, a real-world data application is analyzed in Section 5. Some conclusions and potential avenues for future research are presented in Section 6.

2. Stress and Strength Reliability

In this section, we discuss the measure of the stress and strength reliability, R, and fuzzy stress and strength reliability, $R_F$.

Theorem 1.

Let $X\sim GME({\lambda }_1,{\alpha }_1,{\beta }_1)$ and $Y\sim GME({\lambda }_2,{\alpha }_2,{\beta }_2)$ be independent and define $R=P(X>Y)$. Then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill R& ={\displaystyle \frac{({\alpha }_1+1){\lambda }_1({\alpha }_2+1){\lambda }_2}{({\alpha }_1+1+{\alpha }_1{\beta }_1)({\alpha }_2+1+{\alpha }_2{\beta }_2)}}\left[{\displaystyle \frac{(1+{\beta }_1)(1+{\beta }_2)}{{\lambda }_1({\lambda }_1+{\lambda }_2)}}-{\displaystyle \frac{{\beta }_2(1+{\beta }_1)}{{\lambda }_1({\lambda }_1+{\lambda }_2+{\alpha }_2{\lambda }_2)}}\right.\hfill \\ & -\left.{\displaystyle \frac{{\beta }_1(1+{\beta }_2)}{{\lambda }_1(1+{\alpha }_1)({\lambda }_1+{\lambda }_2+{\alpha }_1{\lambda }_1)}}+{\displaystyle \frac{{\beta }_1{\beta }_2}{{\lambda }_1(1+{\alpha }_1)({\lambda }_1+{\lambda }_2+{\alpha }_1{\lambda }_1+{\alpha }_2{\lambda }_2)}}\right]\hfill \end{array}\end{array}$$

(3)

Proof. 

Based on the definition of the stress and strength reliability, we obtain

$$\begin{array}{ccc}\hfill R& =& P(X>Y)=\underset{x>y}{\int \int }d{F}_X\left(x\right)d{F}_Y\left(y\right)\hfill \\ & =& {\int }_0^{\infty }{\int }_y^{\infty }f(x;{\lambda }_1,{\alpha }_1,{\beta }_1)f(y;{\lambda }_2,{\alpha }_2,{\beta }_2)dxdy\hfill \\ & =& {\int }_0^{\infty }{\int }_y^{\infty }{\displaystyle \frac{({\alpha }_1+1){\lambda }_1}{{\alpha }_1+1+{\alpha }_1{\beta }_1}}{e}^{-{\lambda }_{1}x}[1+{\beta }_1(1-e^{-{\alpha }_1{\lambda }_{1}x})]{\displaystyle \frac{({\alpha }_2+1){\lambda }_2}{{\alpha }_2+1+{\alpha }_2{\beta }_2}}{e}^{-{\lambda }_{2}y}[1+{\beta }_2(1-e^{-{\alpha }_2{\lambda }_{2}y})]dxdy\hfill \\ & =& {\displaystyle \frac{({\alpha }_1+1){\lambda }_1({\alpha }_2+1){\lambda }_2}{({\alpha }_1+1+{\alpha }_1{\beta }_1)({\alpha }_2+1+{\alpha }_2{\beta }_2)}}{\int }_0^{\infty }{\int }_y^{\infty }{e}^{-{\lambda }_{1}x}[1+{\beta }_1(1-e^{-{\alpha }_1{\lambda }_{1}x})]e^{-{\lambda }_{2}y}[1+{\beta }_2(1-e^{-{\alpha }_2{\lambda }_{2}y})]dxdy\hfill \\ & =& {\displaystyle \frac{({\alpha }_1+1){\lambda }_1({\alpha }_2+1){\lambda }_2}{({\alpha }_1+1+{\alpha }_1{\beta }_1)({\alpha }_2+1+{\alpha }_2{\beta }_2)}}\left[{\displaystyle \frac{(1+{\beta }_1)(1+{\beta }_2)}{{\lambda }_1({\lambda }_1+{\lambda }_2)}}-{\displaystyle \frac{{\beta }_2(1+{\beta }_1)}{{\lambda }_1({\lambda }_1+{\lambda }_2+{\alpha }_2{\lambda }_2)}}\right.\hfill \\ & -& \left.{\displaystyle \frac{{\beta }_1(1+{\beta }_2)}{{\lambda }_1(1+{\alpha }_1)({\lambda }_1+{\lambda }_2+{\alpha }_1{\lambda }_1)}}+{\displaystyle \frac{{\beta }_1{\beta }_2}{{\lambda }_1(1+{\alpha }_1)({\lambda }_1+{\lambda }_2+{\alpha }_1{\lambda }_1+{\alpha }_2{\lambda }_2)}}\right]\hfill \end{array}$$

□

Corollary 1.

Let $X\sim GME({\lambda }_1,{\alpha }_1,{\beta }_1)$ and $Y\sim GME({\lambda }_2,{\alpha }_2,{\beta }_2)$ be independent, we have,

(i)

For ${\beta }_1={\beta }_2=-1$,

$$\begin{array}{c}\hfill R={\displaystyle \frac{{\lambda }_2(1+{\alpha }_2)}{{\lambda }_1+{\lambda }_2+{\alpha }_1{\lambda }_1+{\alpha }_2{\lambda }_2}}.\end{array}$$

(ii)

For ${\beta }_1\to \infty$ and ${\beta }_2\to \infty$,

$$\begin{array}{ccc}\hfill R& =& {\displaystyle \frac{({\alpha }_1+1)({\alpha }_2+1){\lambda }_2^2}{{\alpha }_1}}\left[{\displaystyle \frac{1}{({\lambda }_1+{\lambda }_2)({\lambda }_1+{\lambda }_2+{\alpha }_2{\lambda }_2)}}\right.\hfill \\ & -& \left.{\displaystyle \frac{1}{(1+{\alpha }_1)({\lambda }_1+{\lambda }_2+{\alpha }_1{\lambda }_1)({\lambda }_1+{\lambda }_2+{\alpha }_1{\lambda }_1+{\alpha }_2{\lambda }_2)}}\right].\hfill \end{array}$$

(iii)

For ${\lambda }_1={\lambda }_2$,

$$\begin{array}{ccc}\hfill R& =& {\displaystyle \frac{({\alpha }_1+1)({\alpha }_2+1)}{({\alpha }_1+1+{\alpha }_1{\beta }_1)({\alpha }_2+1+{\alpha }_2{\beta }_2)}}\left[{\displaystyle \frac{(1+{\beta }_1)(1+{\beta }_2)}{2}}\right.\hfill \\ & -& \left.{\displaystyle \frac{{\beta }_2(1+{\beta }_1)}{2+{\alpha }_2}}-{\displaystyle \frac{{\beta }_1(1+{\beta }_2)}{2+{\alpha }_2}}+{\displaystyle \frac{{\beta }_1{\beta }_2}{(1+{\alpha }_1)(2+{\alpha }_1{\alpha }_2)}}\right].\hfill \end{array}$$

(iv)

For ${\lambda }_1={\lambda }_2$, ${\alpha }_1={\alpha }_2$, and ${\beta }_1={\beta }_2$, we have $R={\displaystyle \frac{1}{2}}.$

Moreover, if we set ${\alpha }_1={\alpha }_2=\alpha$, we can obtain

$$\begin{array}{ccc}\hfill R& =& {\displaystyle \frac{{\lambda }_2^2\left[(3+3\alpha +{\alpha }^2){\lambda }_1+(1+\alpha ){\lambda }_2\right]}{({\lambda }_1+{\lambda }_2)({\lambda }_1+{\lambda }_2+\alpha {\lambda }_1)({\lambda }_1+{\lambda }_2+\alpha {\lambda }_2)}},\hfill \end{array}$$

(4)

which is the same as Equation (7) in Yazgan et al. [19].

For different values of ${\lambda }_1$, ${\beta }_1$, ${\lambda }_2$, and ${\beta }_2$, the values of R are plotted in Figure 1, which indicate that R can generate curves with various shapes. According to Figure 1, we see that the values of R increase as the values of $\beta$ increase; however, the trend is opposite for $\alpha$.

In the following, we discuss the $R_F$ of two independent GME distribution, which is defined as follows:

$$\begin{array}{c}\hfill R_F=P(X\succ Y)={\int \int }_{x>y}{\mu }_{A\left(y\right)}\left(x\right)d{F}_X\left(x\right)d_Y\left(y\right),\end{array}$$

(5)

where $A\left(y\right)=\{x:x>y\}$ is a fuzzy set, and ${\mu }_{A\left(y\right)}\left(x\right)$ is an appropriate membership function on $A\left(y\right)$. Eryilmaz and TütTüncTü [18] discussed the detailed information for ${\mu }_{A\left(y\right)}\left(x\right)$, and they chose ${\mu }_{A\left(y\right)}\left(x\right)=1-e^{k(x-y)},k\in R$. Hussam et al. [25] considered ${\mu }_{A\left(y\right)}\left(x\right)=1-e^{-k({\displaystyle \frac{1}{x}}-{\displaystyle \frac{1}{y}})}$ for $k>0$ to study the fuzzy stress and strength model for the Lindley distribution. In this paper, we would use the same ${\mu }_{A\left(y\right)}\left(x\right)$ as in Eryilmaz and TütTüncTü [18].

Theorem 2.

Let $X\sim GME({\lambda }_1,{\alpha }_1,{\beta }_1)$ and $Y\sim GME({\lambda }_2,{\alpha }_2,{\beta }_2)$ be independent. Then, the fuzzy stress and strength reliability is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill R_{F,k}& ={\displaystyle \frac{k({\alpha }_1+1){\lambda }_1({\alpha }_2+1){\lambda }_2}{({\alpha }_1+1+{\alpha }_1{\beta }_1)({\alpha }_2+1+{\alpha }_2{\beta }_2)}}\left[{\displaystyle \frac{(1+{\beta }_1)(1+{\beta }_2)}{{\lambda }_1({\lambda }_1+k)({\lambda }_1+{\lambda }_2)}}-{\displaystyle \frac{{\beta }_2(1+{\beta }_1)}{{\lambda }_1({\lambda }_1+k)({\lambda }_1+{\lambda }_2+{\alpha }_2{\lambda }_2)}}\right.\hfill \\ & -{\displaystyle \frac{{\beta }_1(1+{\beta }_2)}{{\lambda }_1(1+{\alpha }_1)({\lambda }_1+{\alpha }_1{\lambda }_1+k)({\lambda }_1+{\lambda }_2+{\alpha }_1{\lambda }_1)}}\hfill \\ & +\left.{\displaystyle \frac{{\beta }_1{\beta }_2}{{\lambda }_1(1+{\alpha }_1)({\lambda }_1+{\alpha }_1{\lambda }_1+k)({\lambda }_1+{\lambda }_2+{\alpha }_1{\lambda }_1+{\alpha }_2{\lambda }_2)}}\right].\hfill \end{array}\end{array}$$

(6)

Proof. 

Based on Equation (5), we can obtain

$$\begin{array}{ccc}\hfill R_{F,k}& =& {\displaystyle \frac{({\alpha }_1+1){\lambda }_1({\alpha }_2+1){\lambda }_2}{({\alpha }_1+1+{\alpha }_1{\beta }_1)({\alpha }_2+1+{\alpha }_2{\beta }_2)}}\hfill \\ & & {\int }_0^{\infty }{\int }_y^{\infty }(1-e^{-k(x-y)})e^{-{\lambda }_{1}x}[1+{\beta }_1(1-e^{-{\alpha }_1{\lambda }_{1}x})]e^{-{\lambda }_{2}y}[1+{\beta }_2(1-e^{-{\alpha }_2{\lambda }_{2}y})]dxdy\hfill \\ & =& {\displaystyle \frac{k({\alpha }_1+1){\lambda }_1({\alpha }_2+1){\lambda }_2}{({\alpha }_1+1+{\alpha }_1{\beta }_1)({\alpha }_2+1+{\alpha }_2{\beta }_2)}}\left[{\displaystyle \frac{(1+{\beta }_1)(1+{\beta }_2)}{{\lambda }_1({\lambda }_1+k)({\lambda }_1+{\lambda }_2)}}-{\displaystyle \frac{{\beta }_2(1+{\beta }_1)}{{\lambda }_1({\lambda }_1+k)({\lambda }_1+{\lambda }_2+{\alpha }_2{\lambda }_2)}}\right.\hfill \\ & -& {\displaystyle \frac{{\beta }_1(1+{\beta }_2)}{{\lambda }_1(1+{\alpha }_1)({\lambda }_1+{\alpha }_1{\lambda }_1+k)({\lambda }_1+{\lambda }_2+{\alpha }_1{\lambda }_1)}}\hfill \\ & +& \left.{\displaystyle \frac{{\beta }_1{\beta }_2}{{\lambda }_1(1+{\alpha }_1)({\lambda }_1+{\alpha }_1{\lambda }_1+k)({\lambda }_1+{\lambda }_2+{\alpha }_1{\lambda }_1+{\alpha }_2{\lambda }_2)}}\right].\hfill \end{array}$$

□

For different values of k, the values of $R_{F,k}$ versus $\lambda$, $\alpha$, and $\beta$ are presented in Figure 2, which indicate that the $R_{F,k}$ can generate curves with various shapes.

Corollary 2.

Let $X\sim GME({\lambda }_1,{\alpha }_1,{\beta }_1)$ and $Y\sim GME({\lambda }_2,{\alpha }_2,{\beta }_2)$ be independent, then we can have

(i)

For ${\beta }_1={\beta }_2=-1$,

$$\begin{array}{c}\hfill R_{F,k}={\displaystyle \frac{k{\lambda }_2(1+{\alpha }_2)}{({\lambda }_1+{\alpha }_1{\lambda }_1+k)({\lambda }_1+{\lambda }_2+{\alpha }_1{\lambda }_1+{\alpha }_2{\lambda }_2)}}.\end{array}$$

(7)

(ii)

For ${\beta }_1\to \infty$,${\beta }_2\to \infty$

$$\begin{array}{ccc}\hfill R_{F,k}& =& {\displaystyle \frac{k({\alpha }_1+1)({\alpha }_2+1){\lambda }_2^2}{{\alpha }_1}}\left[{\displaystyle \frac{1}{({\lambda }_1+k)({\lambda }_1+{\lambda }_2)({\lambda }_1+{\lambda }_2+{\alpha }_2{\lambda }_2)}}\right.\hfill \\ & -& \left.{\displaystyle \frac{1}{(1+{\alpha }_1)({\lambda }_1+{\alpha }_1{\lambda }_1+k)({\lambda }_1+{\lambda }_2+{\alpha }_1{\lambda }_1)({\lambda }_1+{\lambda }_2+{\alpha }_1{\lambda }_1+{\alpha }_2{\lambda }_2)}}\right].\hfill \end{array}$$

According to Yang et al. [22], the $GME(\lambda ,\alpha ,\beta )$ distribution is reduced to $E\left(\lambda \right(\alpha +1\left)\right)$ distribution as $\beta =-1$, then Equation (7) is the same as Example 1 in Eryilmaz and Tütüncü [18]. Additionally, for different k, we denote $R_{F,k}$ as its corresponding values of $R_F$ for a straightforward representation.

3. Methods of Estimation

In this section, we consider the methods maximum likelihood estimation (MLE), percentile estimation (PE), and weighted least-squares estimation (WLE) to estimate the unknown parameters, $\theta =(\lambda ,\alpha ,\beta )$, of the GME distribution. Suppose $x_1,x_2,\cdots ,x_n$ is a random sample from $GME(\lambda ,\alpha ,\beta )$.

3.1. Maximum Likelihood Estimator

The method of maximum likelihood is the most frequently used method for parameter estimation. According to Equation (1), the likelihood function is calculated as

$$L(\lambda ,\alpha ,\beta |x_1,\cdots ,x_n)={\left({\displaystyle \frac{(\alpha +1)\lambda }{\alpha +1+\alpha \beta }}\right)}^n{e}^{-\lambda {\displaystyle \sum _{i=1}^n}{x}_i}\prod _{i=1}^n[1+\beta (1-e^{-\alpha \lambda x_i})].$$

(8)

The log-likelihood function is given by

$$\begin{array}{cc}\hfill \ell (\lambda ,\alpha ,\beta |x_1,\cdots ,x_n)& =n[log(\alpha +1)+log\left(\lambda \right)-log(\alpha +1+\alpha \beta )]-\lambda \sum _{i=1}^n{x}_i\hfill \\ \hfill & +\sum _{i=1}^{n}log[1+\beta (1-e^{-\alpha \lambda x_i})].\hfill \end{array}$$

(9)

We denote the first partial derivatives of (9) by ${\ell }_{\lambda }$, ${\ell }_{\alpha }$ and ${\ell }_{\beta }$. Setting ${\ell }_{\lambda }=0$, ${\ell }_{\alpha }=0$, and ${\ell }_{\beta }=0$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\ell }_{\lambda }& ={\displaystyle \frac{n}{\lambda }}-\sum _{i=1}^n{x}_i+\sum _{i=1}^n{\displaystyle \frac{\beta \alpha x_i{e}^{-\alpha \lambda x_i}}{1+\beta (1-e^{-\alpha \lambda x_i})}}=0,\hfill \\ \hfill {\ell }_{\alpha }& ={\displaystyle \frac{n}{\alpha +1}}-{\displaystyle \frac{n(1+\beta )}{\alpha +1+\alpha \beta }}+\sum _{i=1}^n{\displaystyle \frac{\beta \lambda x_i{e}^{-\alpha \lambda x_i}}{1+\beta (1-e^{-\alpha \lambda x_i})}}=0,\hfill \\ \hfill {\ell }_{\beta }& =-{\displaystyle \frac{n\alpha }{\alpha +1+\alpha \beta }}+\sum _{i=1}^n{\displaystyle \frac{1-e^{-\alpha \lambda x_i}}{1+\beta (1-e^{-\alpha \lambda x_i})}}=0.\hfill \end{array}\end{array}$$

(10)

The MLE of $\widehat{\theta }$ for the unknown parameters $\theta$ can be obtained by optimizing the log-likelihood function concerning the involved parameters. Due to the non-linearity of these equations, the MLEs of the parameters can be obtained numerically. These estimators can be easily obtained using the functions from the R statistical software 4.0.

3.2. Percentile Estimator

The percentile estimators are mainly obtained by minimizing the distance between the sample percentile and population percentile points. We apply this approach to the GME distribution in the following to obtain the parameter estimators. Suppose $F\left(x_{\left(j\right)}\right)$ denotes the distribution function of the ordered random variables $x_{\left(1\right)}<\cdots <x_{\left(n\right)}$. Let $P_i={\displaystyle \frac{i}{n+1}}$ be the unbiased estimator of $F(x_{\left(i\right)};\lambda ,\alpha ,\beta )$. Define the following function:

$$h(\lambda ,\alpha ,\beta )=\sum _{i=1}^n{\left[ln{P}_i-ln\left(F(x_{\left(i\right)};\lambda ,\alpha ,\beta )\right)\right]}^2.$$

(11)

The percentile estimators of $\theta$ can be obtained by minimizing $h(\lambda ,\alpha ,\beta )$. Therefore, $\widehat{\theta }$ can be obtained by solving the following equations:

$$\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle \frac{\partial h(\lambda ,\alpha ,\beta )}{\partial \lambda }}=\sum _{i=1}^n\left\{2Q_i\left\{{\displaystyle \frac{\beta }{\alpha +1+\alpha \beta }}\left[-(1+\alpha )C_i+B_i\right]+B_i\right\}\right\}=0,\hfill \\ & {\displaystyle \frac{\partial h(\lambda ,\alpha ,\beta )}{\partial \alpha }}=\sum _{i=1}^n\left\{2Q_i\left\{{\displaystyle \frac{-(1+\beta )}{{(\alpha +1+\alpha \beta )}^2}}[e^{-\lambda (1+\alpha )X_{\left(i\right)}}-e^{-\lambda X_{\left(i\right)}}]-{\displaystyle \frac{\beta \lambda }{\alpha +1+\alpha \beta }}{C}_i\right\}\right\}=0,\hfill \\ & {\displaystyle \frac{\partial h(\lambda ,\alpha ,\beta )}{\partial \beta }}=\sum _{i=1}^n\left\{2Q_i\left\{{\displaystyle \frac{1-\alpha }{{(\alpha +1+\alpha \beta )}^2}}\left[e^{-\lambda (1+\alpha }\right)X_{\left(i\right)}-e^{-\lambda X_{\left(i\right)}}]\right\}\right\}=0,\hfill \end{array}\end{array}$$

(12)

where ${\displaystyle Q_i=\frac{\beta }{\alpha +1+\alpha \beta }[e^{-\lambda (1+\alpha )x_{\left(i\right)}}-e^{-\lambda x_{\left(i\right)}}]-e^{-\lambda x_{\left(i\right)}}+1-\frac{i}{n+1}}$, $B_i=x_{\left(i\right)}{e}^{-\lambda x_{\left(i\right)}}$ and $C_i=x_{\left(i\right)}{e}^{-\lambda (1+\alpha )x_{\left(i\right)}}$.

3.3. Weighted Least-Square Estimator

The weighted least-square estimator is an extension of the least-square estimator and proposed by Swain et al. [26]; it is obtained by minimizing the following function:

$$W(\lambda ,\alpha ,\beta )=\sum _{i=1}^n{\displaystyle \frac{{(n+1)}^2(n+2)}{i(n-i+1)}}{\left[F(x_{\left(i\right)};\lambda ,\alpha ,\beta )-{\displaystyle \frac{i}{n+1}}\right]}^2,$$

(13)

where the $F(·)$ function is given in Equation (2). Therefore, the WLE of $\theta$ can be obtained by

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W(\lambda ,\alpha ,\beta )}{\partial \lambda }}& =\sum _{i=1}^n\left\{{\displaystyle \frac{2{(n-1)}^2(n+2)}{i(n-i+1)}}{Q}_i\left\{{\displaystyle \frac{\beta }{\alpha +1+\alpha \beta }}\left[-(1+\alpha )C_i+B_i\right]+B_i\right\}\right\}=0,\hfill \\ \hfill {\displaystyle \frac{\partial W(\lambda ,\alpha ,\beta )}{\partial \alpha }}& =\sum _{i=1}^n\left\{{\displaystyle \frac{2{(n-1)}^2(n+2)}{i(n-i+1)}}{Q}_i\right\{{\displaystyle \frac{-(1+\beta )}{{(\alpha +1+\alpha \beta )}^2}}[e^{-\lambda (1+\alpha )x_{\left(i\right)}}-e^{-\lambda x_{\left(i\right)}}]\hfill \\ \hfill & -{\displaystyle \frac{\beta \lambda }{\alpha +1+\alpha \beta }}{C}_i\left\}\right\}=0,\hfill \\ \hfill {\displaystyle \frac{\partial W(\lambda ,\alpha ,\beta )}{\partial \beta }}& =\sum _{i=1}^n\left\{{\displaystyle \frac{2{(n-1)}^2(n+2)}{i(n-i+1)}}{Q}_i\left\{{\displaystyle \frac{1-\alpha }{{(\alpha +1+\alpha \beta )}^2}}[e^{-\lambda (1+\alpha )x_{\left(i\right)}}-e^{-\lambda x_{\left(i\right)}}]\right\}\right\}=0,\hfill \end{array}$$

(14)

where $Q_i,B_i$ and $C_i$, $i=1,\cdots ,n$, are defined as above.

4. Simulation Studies

In this section, we would like to study the reliability, R, and fuzzy reliability, $R_F$, of the GME distribution based on different estimation methods. The R programming conducts all the programs. The study is designed with $N=1000$ repetitions for various values of $(\lambda ,\alpha ,\beta )$. To address the performance of the proposed method, the bias and mean squared error(MSE) are obtained by the following:

$$\begin{array}{c}\hfill Bias\left(\widehat{\theta }\right)={\displaystyle \frac{1}{N}}\sum _{i=1}^N({\widehat{\theta }}_i-\theta ),\\ \hfill MSE\left(\widehat{\theta }\right)={\displaystyle \frac{1}{N}}\sum _{i=1}^N{({\widehat{\theta }}_i-\theta )}^2,\\ \hfill SD\left(\widehat{\theta }\right)=\sqrt{{\displaystyle \frac{1}{N-1}}{\sum }_{i=1}^N{({\widehat{\theta }}_i-\theta )}^2}.\end{array}$$

We take the sample size $n=\{20,50,100,200,300,400,500\}$ for each simulation and consider $k=\{1,10,100\}$ for $R_F$. Furthermore, we consider three scenarios, such as $\left(i\right)$${\theta }_X=(1,5,-0.5)$ and ${\theta }_Y=(2,1,1.5)$; (ii) ${\theta }_X=(1,5,-0.5)$ and ${\theta }_Y=(3,10,5)$; (iii) ${\theta }_X=(2,1,1.5)$ and ${\theta }_Y=(3,10,5)$, where ${\theta }_X$ and ${\theta }_Y$ are the parameters of $\theta$ for data X and Y, respectively. The simulation results are shown in

Table 1. The mean value, SD, bias, and MSE for scenario 1 of estimators under different n.

Sample Size	Method	R				${\mathit{R}}_{\mathit{F}}$
		-				$\mathit{k}=\mathbf{1}$				$\mathit{k}=\mathbf{10}$				$\mathit{k}=\mathbf{100}$
		Mean	SD	Bias	MSE	Mean	SD	Bias	MSE	Mean	SD	Bias	MSE	Mean	SD	Bias	MSE
$n=20$	MLE	0.512	0.094	−0.037	0.100	0.247	0.072	−0.018	0.074	0.454	0.095	−0.037	0.102	0.505	0.095	−0.037	0.101
	PE	0.456	0.151	−0.093	0.177	0.198	0.104	−0.067	0.123	0.394	0.150	−0.098	0.179	0.448	0.151	−0.094	0.178
	WLE	0.516	0.079	−0.033	0.085	0.260	0.061	−0.005	0.061	0.460	0.079	−0.032	0.085	0.509	0.079	−0.033	0.086
$n=50$	MLE	0.540	0.048	−0.009	0.048	0.257	0.037	−0.008	0.038	0.480	0.048	−0.012	0.050	0.533	0.048	−0.010	0.048
	Per	0.504	0.074	−0.045	0.086	0.231	0.058	−0.034	0.067	0.444	0.075	−0.048	0.089	0.497	0.074	−0.045	0.087
	WLE	0.531	0.048	−0.018	0.051	0.260	0.037	−0.005	0.037	0.473	0.048	−0.019	0.051	0.524	0.048	−0.018	0.051
$n=100$	MLE	0.546	0.037	−0.003	0.037	0.266	0.029	0.001	0.029	0.489	0.037	−0.003	0.037	0.540	0.037	−0.003	0.037
	PE	0.521	0.049	−0.028	0.060	0.246	0.041	−0.019	0.045	0.462	0.051	−0.029	0.058	0.514	0.050	−0.029	0.057
	WLE	0.539	0.039	−0.010	0.040	0.263	0.030	−0.002	0.030	0.481	0.039	−0.010	0.040	0.532	0.039	−0.010	0.040
$n=200$	MLE	0.547	0.027	−0.002	0.027	0.263	0.019	−0.002	0.019	0.489	0.027	−0.003	0.027	0.540	0.027	−0.002	0.027
	PE	0.530	0.034	−0.019	0.039	0.254	0.028	−0.011	0.030	0.472	0.035	−0.019	0.040	0.523	0.034	−0.019	0.039
	WLE	0.540	0.026	−0.009	0.028	0.261	0.019	−0.004	0.020	0.482	0.026	−0.010	0.028	0.532	0.026	−0.010	0.028
$n=300$	MLE	0.548	0.022	−0.001	0.022	0.264	0.017	−0.001	0.017	0.450	0.023	−0.002	0.023	0.541	0.022	−0.001	0.022
	PE	0.536	0.029	−0.013	0.031	0.258	0.024	−0.007	0.025	0.478	0.029	−0.013	0.032	0.529	0.027	−0.013	0.031
	WLE	0.543	0.023	−0.006	0.024	0.263	0.017	−0.002	0.017	0.485	0.023	−0.007	0.024	0.536	0.023	−0.006	0.024
$n=400$	MLE	0.547	0.020	−0.002	0.020	0.263	0.014	−0.002	0.014	0.489	0.020	−0.003	0.020	0.540	0.020	−0.002	0.020
	PE	0.538	0.023	−0.011	0.026	0.259	0.021	−0.006	0.022	0.480	0.024	−0.011	0.026	0.531	0.023	−0.011	0.026
	WLE	0.545	0.020	−0.004	0.021	0.263	0.015	−0.002	0.015	0.487	0.020	−0.005	0.021	0.538	0.020	−0.005	0.021
$n=500$	MLE	0.548	0.018	−0.001	0.018	0.264	0.013	−0.001	0.013	0.490	0.018	−0.002	0.018	0.541	0.018	−0.001	0.018
	PE	0.540	0.024	−0.009	0.025	0.261	0.024	−0.004	0.024	0.482	0.024	−0.009	0.025	0.533	0.024	−0.009	0.025
	WLE	0.547	0.018	−0.002	0.018	0.265	0.013	0.000	0.013	0.489	0.017	−0.003	0.018	0.540	0.018	−0.002	0.018

,

Table 2. The mean value, SD, Bias, and MSE for scenario 2 of estimators under different n.

Sample Size	Method	R				${\mathit{R}}_{\mathit{F}}$
		-				$\mathit{k}=\mathbf{1}$				$\mathit{k}=\mathbf{10}$				$\mathit{k}=\mathbf{100}$
		Mean	SD	Bias	MSE	Mean	SD	Bias	MSE	Mean	SD	Bias	MSE	Mean	SD	Bias	MSE
$n=20$	MLE	0.640	0.085	−0.028	0.089	0.297	0.070	−0.023	0.073	0.563	0.090	−0.033	0.094	0.631	0.086	−0.029	0.090
	PE	0.631	0.154	−0.121	0.193	0.214	0.101	−0.106	0.144	0.460	0.154	−0.136	0.203	0.537	0.156	−0.124	0.196
	WLE	0.645	0.066	−0.024	0.069	0.312	0.051	−0.008	0.051	0.568	0.066	−0.029	0.071	0.635	0.066	−0.025	0.070
$n=50$	MLE	0.648	0.047	−0.021	0.051	0.311	0.040	−0.008	0.040	0.576	0.049	−0.020	0.053	0.640	0.048	−0.021	0.052
	PE	0.643	0.057	−0.026	0.062	0.289	0.056	−0.030	0.063	0.562	0.062	−0.034	0.070	0.633	0.058	−0.027	0.063
	WLE	0.643	0.050	−0.026	0.055	0.305	0.040	−0.015	0.042	0.567	0.051	−0.029	0.058	0.634	0.050	−0.026	0.056
$n=100$	MLE	0.662	0.039	−0.007	0.039	0.317	0.031	−0.002	0.031	0.589	0.041	−0.007	0.042	0.654	0.039	−0.007	0.040
	PE	0.638	0.038	−0.031	0.049	0.287	0.040	−0.032	0.051	0.559	0.042	−0.037	0.056	0.629	0.039	−0.032	0.050
	WLE	0.663	0.031	−0.006	0.031	0.321	0.026	0.001	0.026	0.590	0.032	−0.007	0.032	0.654	0.031	−0.006	0.031
$n=200$	MLE	0.666	0.027	−0.003	0.027	0.317	0.024	−0.002	0.024	0.592	0.029	−0.004	0.029	0.657	0.028	−0.004	0.028
	PE	0.660	0.032	−0.009	0.033	0.313	0.027	−0.006	0.028	0.585	0.033	−0.011	0.035	0.651	0.032	−0.009	0.033
	WLE	0.663	0.020	−0.006	0.021	0.318	0.018	−0.002	0.018	0.589	0.021	−0.008	0.022	0.654	0.020	−0.007	0.021
$n=300$	MLE	0.667	0.021	−0.002	0.021	0.318	0.017	−0.001	0.017	0.594	0.022	−0.002	0.022	0.659	0.021	−0.002	0.021
	PE	0.659	0.024	−0.010	0.026	0.312	0.023	−0.008	0.024	0.584	0.0255	−0.012	0.028	0.650	0.024	−0.010	0.026
	WLE	0.664	0.018	−0.005	0.019	0.317	0.017	−0.003	0.017	0.588	0.020	−0.008	0.021	0.655	0.019	−0.006	0.019
$n=400$	MLE	0.667	0.018	−0.002	0.019	0.317	0.015	−0.003	0.016	0.593	0.019	−0.003	0.020	0.658	0.019	−0.002	0.019
	PE	0.663	0.023	−0.006	0.024	0.316	0.020	−0.003	0.020	0.589	0.024	−0.007	0.025	0.654	0.024	−0.006	0.024
	WLE	0.667	0.019	−0.002	0.019	0.318	0.017	−0.002	0.017	0.593	0.020	−0.004	0.020	0.658	0.019	−0.003	0.019
$n=500$	MLE	0.667	0.017	−0.002	0.017	0.317	0.013	−0.002	0.014	0.593	0.018	−0.003	0.018	0.658	0.017	−0.002	0.017
	PE	0.664	0.023	−0.005	0.024	0.317	0.020	−0.003	0.020	0.590	0.024	−0.006	0.025	0.655	0.024	−0.005	0.024
	WLE	0.663	0.016	−0.006	0.017	0.314	0.011	−0.006	0.012	0.588	0.016	−0.008	0.017	0.654	0.016	−0.007	0.017

and

Table 3. The mean value, SD, bias, and MSE for scenario 3 of estimators under different n.

Sample Size	Method	R				${\mathit{R}}_{\mathit{F}}$
		-				$\mathit{k}=\mathbf{1}$				$\mathit{k}=\mathbf{10}$				$\mathit{k}=\mathbf{100}$
		Mean	SD	Bias	MSE	Mean	SD	Bias	MSE	Mean	SD	Bias	MSE	Mean	SD	Bias	MSE
$n=20$	MLE	0.626	0.067	−0.024	0.069	0.220	0.042	−0.019	0.044	0.532	0.065	−0.029	0.069	0.615	0.067	−0.025	0.069
	PE	0.667	0.027	0.018	0.030	0.240	0.032	0.001	0.029	0.571	0.032	0.009	0.030	0.657	0.027	0.017	0.029
	WLE	0.655	0.093	0.005	0.088	0.247	0.066	0.008	0.062	0.566	0.094	0.005	0.089	0.645	0.093	0.005	0.088
$n=50$	MLE	0.642	0.056	−0.008	0.055	0.239	0.045	−0.000	0.044	0.553	0.060	−0.008	0.059	0.632	0.056	−0.008	0.056
	PE	0.635	0.073	−0.015	0.072	0.239	0.045	0.000	0.043	0.550	0.072	−0.012	0.071	0.626	0.073	−0.014	0.072
	WLE	0.657	0.041	0.007	0.041	0.248	0.025	0.008	0.026	0.569	0.040	0.007	0.026	0.569	0.041	0.007	0.041
$n=100$	MLE	0.646	0.033	−0.003	0.032	0.240	0.026	0.001	0.026	0.558	0.035	−0.003	0.034	0.637	0.033	−0.003	0.033
	PE	0.645	0.051	−0.004	0.051	0.240	0.037	0.001	0.036	0.556	0.053	−0.005	0.052	0.635	0.052	−0.005	0.051
	WLE	0.658	0.041	0.009	0.042	0.250	0.030	0.011	0.032	0.571	0.043	0.009	0.044	0.649	0.042	0.009	0.042
$n=200$	MLE	0.645	0.027	−0.004	0.027	0.238	0.018	−0.001	0.018	0.557	0.027	−0.004	0.027	0.636	0.027	−0.004	0.027
	PE	0.651	0.034	0.001	0.034	0.246	0.024	0.007	0.024	0.564	0.035	0.003	0.034	0.641	0.034	0.001	0.034
	WLE	0.651	0.022	0.002	0.022	0.242	0.016	0.003	0.017	0.563	0.023	0.002	0.023	0.642	0.022	0.002	0.022
$n=300$	MLE	0.648	0.022	−0.002	0.022	0.238	0.016	−0.001	0.015	0.559	0.023	−0.002	0.023	0.659	0.022	−0.002	0.022
	PE	0.659	0.065	0.009	0.065	0.251	0.038	0.012	0.040	0.572	0.065	0.011	0.066	0.650	0.065	0.010	0.066
	WLE	0.650	0.021	0.001	0.021	0.241	0.015	0.002	0.015	0.562	0.022	0.000	0.022	0.641	0.021	0.001	0.021
$n=400$	MLE	0.649	0.018	−0.001	0.018	0.239	0.012	0.000	0.012	0.560	0.018	−0.001	0.018	0.649	0.018	−0.001	0.018
	PE	0.653	0.020	0.004	0.020	0.244	0.014	0.006	0.015	0.565	0.020	0.004	0.020	0.644	0.020	0.004	0.020
	WLE	0.650	0.019	0.001	0.019	0.241	0.013	0.002	0.013	0.562	0.019	0.001	0.019	0.641	0.019	0.001	0.019
$n=500$	MLE	0.650	0.017	0.001	0.017	0.240	0.011	0.001	0.011	0.562	0.017	0.001	0.017	0.641	0.017	0.001	0.017
	PE	0.654	0.020	0.004	0.020	0.247	0.014	0.008	0.016	0.567	0.020	0.005	0.021	0.644	0.020	0.004	0.020
	WLE	0.652	0.017	0.002	0.017	0.241	0.011	0.002	0.011	0.563	0.017	0.002	0.017	0.642	0.017	0.002	0.017

.

From Table 1, Table 2 and Table 3, we find that the values of SD for all three estimators decrease as sample size n increases, and the estimated values are close to the true value, which means that when we increase the sample size n, the values of bias and MES for all estimators decrease. In addition, the values of $MSE\left(PE\right)>MSE\left(MLE\right)>MSE\left(WLE\right)$ and $\left|Bias\right(PE\left)\right|>\left|Bias\right(WLE\left)\right|>\left|Bias\right(MLE\left)\right|$ in most cases. For the values of Bias and MSE, we can see that as n increases, they both approach 0, indicating that these estimators are asymptotically unbiased and consistent with the parameters. Moreover, the values of MSE and Bias for $R_F$ are smaller than these for R for all three estimation methods.

Furthermore, to observe the changes more intuitively, we plotted the values of Bias and MSE for three estimators with different sample sizes n, as shown in Figure 3 and Figure 4. By comparing the three estimation methods, we find that the MLE performs best, followed by the PE and WLE. The plots of $R_F$ have the same pattern as R with different values of k and almost the same when $k=100$.

5. Application

In this section, we consider the data set concerning the breaking strength of jute fiber (BSJF) provided by Xia et al. [27] for illustrative purpose. Each fiber group has 30 fibers. Let X be the breaking strength of jute fiber at 10 mm and Y be the strength at 20 mm. The detailed data information is displayed in

Table 4. The BSJF data set.

No.	10 mm (X)	20 mm (Y)	No.	10 mm (X)	20 mm (Y)	No.	10 mm (X)	20 mm (Y)
1	693.73	71.46	11	671.49	578.62	21	262.90	547.44
2	704.66	419.02	12	183.16	756.70	22	353.24	116.99
3	323.83	284.64	13	257.44	594.29	23	422.11	375.81
4	778.17	585.57	14	727.23	166.49	24	43.93	581.60
5	123.06	456.60	15	291.27	99.72	25	590.48	119.86
6	637.66	113.85	16	101.15	707.36	26	212.13	48.01
7	383.43	187.85	17	376.42	765.14	27	303.90	200.16
8	151.48	688.16	18	163.40	187.13	28	506.60	36.75
9	108.94	662.66	19	141.38	145.96	29	530.55	244.53
10	50.16	45.58	20	700.74	350.70	30	177.25	83.55

.

The MLE for the parameters according to the data sets are obtained as $\widehat{\lambda }=0.0036$, $\widehat{\alpha }=0.9826$, and $\widehat{\beta }=2.1294$. We apply the Anderson–Darling goodness of fit test to check the validity of the GME distribution with data for the 10 mm and 20 mm. The values of Anderson–Darling statistic (AD), adjusted AD statistic (AD*), and the corresponding p-value for the GME distribution are presented in

Table 5. The values of AD, AD*, and the corresponding p-value of the BSJF data.

	X	Y
AD value	0.692	0.572
AD* value	0.711	0.587
p-value	0.064	0.126

.

Table 5 shows that the p-values of X and Y are higher than 0.05, which cannot reject the null hypothesis of distribution. The results indicate that the BSJF data set follows the GME distribution. Therefore, we used the MLE, PE, and WLE methods to calculate the R and $R_F$ at different values of k in

Table 6. The R, $R_{F,1}$, $R_{F,10}$ and $R_{F,100}$ of different estimation methods for the BSJF data.

Method	R	${\mathit{R}}_{\mathit{F},1}$	${\mathit{R}}_{\mathit{F},10}$	${\mathit{R}}_{\mathit{F},100}$
MLE	0.514	0.512	0.513	0.514
PE	0.538	0.537	0.538	0.538
WLE	0.525	0.524	0.525	0.525

. We can see from the table that the estimated values obtained by the three methods are not significantly different, indicating that the three methods perform well.

In Table 6, we find that the estimated values of R are greater than those of $R_{F,k}$ for all the estimation methods. Furthermore, as the values of k increase, the fuzzy estimators of $R_{F,k}$ are approximately equal to the classical estimators R.

6. Conclusions

In this paper, we explore the concept of reliability regarding stress and strength, particularly in fuzzy stress and strength reliability, using generalized mixtures of exponential distributions. We define a fuzzy membership function that depends on the relationship between stress and strength values, increasing whenever the stress (X) exceeds the strength (Y). This approach allows for a more nuanced understanding of reliability that acknowledges real-world applications’ inherent uncertainties and fuzziness. We propose several estimators to estimate the relevant measures, including MLE, WLE, and PE. We then conduct simulation studies to compare the performance of these estimators, focusing on biases and mean squared errors (MSEs) as accuracy metrics. The results indicate that the MLE method outperforms the others in terms of these metrics.

Furthermore, we apply the proposed methods’ practical application to a real-world data set. Overall, this paper contributes to the field of reliability engineering by providing a novel framework for analyzing fuzzy stress and strength reliability using generalized mixtures of exponential distributions. In future work, we plan to investigate interval estimations to provide more comprehensive insights into the system’s reliability. Additionally, we aim to consider different choices for the membership functions used in the fuzzy reliability analysis, as this could lead to more natural interpretations and improve the accuracy of the results.