Fuzzy testing model for the lifetime performance of products under consideration with exponential distribution

Abstract

With intense competition in industry today, product quality has become a crucial factor influencing whether a firm can achieve sustainable operations and maintain competitiveness. Process capability indices are effective tools often used in manufacturing to determine whether products meets requirements, and most assume that the quality characteristics of products follow normal distributions. However, not all quality characteristics necessarily follow normal distributions; for example, product lifetime, a time-oriented quality characteristic, generally follows an exponential distribution or other associated non-normal distributions. The lifetime performance index $C_{L}$ was thus developed to gauge the lifetime performance of products, and most related studies use the precise values of time data to evaluate product lifetime. However, in practice, measurement errors may hinder the accuracy of the observed values of quality characteristics, and the time at which the lifetime of a product ends becomes imprecise, which may result in uncertainty in the evaluation method and lead to errors in judgment. For this reason, this study thus proposes a triangular shaped fuzzy number for $C_{L}^{*}$ to deal with imprecise data, and further develops a fuzzy testing model for lifetime performance index $C_{L}$, to assist manufacturers in evaluating product lifetime performance more cautiously and precisely. Finally, we provide an illustration of how the proposed approach can be implemented through a numerical example.

Appendix: Triangular shaped fuzzy number of \( C_{L}^{*} \)

From Eq. (13), we can know that $W = {{V\left( Z \right)} \mathord{\left/ {\vphantom {{V\left( Z \right)} \lambda }} \right. \kern-0pt} \lambda }$ is distributed as $Gamma\left( {n,1} \right)$. In view of this, we can further derive the following:

$$ \begin{aligned} 1 - \alpha & = p\left\{ {Gamma_{{{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}}} \left( {n,1} \right) \le W \le Gamma_{{1 - \left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} \left( {n,1} \right)} \right\} \\ & = p\left\{ {1 - \frac{L}{V\left( Z \right)}Gamma_{{1 - \left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} \left( {n,1} \right) \le 1 - \frac{L}{\lambda } \le 1 - \frac{L}{V\left( Z \right)}Gamma_{{{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}}} \left( {n,1} \right)} \right\} \\ & = p\left\{ {1 - \left( {1 - C_{L}^{*} } \right)\frac{{Gamma_{{1 - \left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} \left( {n,1} \right)}}{n - 1} \le C_{L} \le 1 - \left( {1 - C_{L}^{*} } \right)\frac{{Gamma_{{{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}}} \left( {n,1} \right)}}{n - 1}} \right\} \\ \end{aligned} $$

(29)

Thus, the $100\left( {1 - \alpha } \right)\%$ confidence interval of $C_{L}$ is $\left[ {LC_{L} ,UC_{L} } \right]$, where

$$ LC_{L} = 1 - \left( {1 - C_{L}^{*} } \right)\frac{{Gamma_{{1 - \left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} \left( {n,1} \right)}}{n - 1} $$

(30)

$$ UC_{L} = 1 - \left( {1 - C_{L}^{*} } \right)\frac{{Gamma_{{{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}}} \left( {n,1} \right)}}{n - 1} $$

(31)

In adopting Buckley’s approach (Buckley 2005), the α-cuts of the triangular shaped fuzzy number $\tilde{C}_{L}^{*}$ are $\tilde{C}_{L}^{*} \left[ \alpha \right] = \left[ {C_{L1}^{*} \left( \alpha \right),C_{L2}^{*} \left( \alpha \right)} \right]$, as shown below:

$$ \tilde{C}_{L}^{*} \left[ \alpha \right] = \left\{ {\begin{array}{*{20}lll} {\left[ {C_{L1}^{*} \left( \alpha \right),C_{L2}^{*} \left( \alpha \right)} \right],\quad { 0} . 0 1\le \alpha \le 1} \hfill \\ {\left[ {C_{L1}^{*} 0.01),C_{L2}^{*} \left( {0.01} \right)} \right],\quad { 0} \le \alpha \le 0.01} \hfill \\ \end{array} } \right. $$

(32)

where

$$ C_{L1}^{*} \left( \alpha \right) = 1 - \left( {1 - C_{L}^{*} } \right)\frac{{Gamma_{{1 - \left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} \left( {n,1} \right)}}{n - 1},C_{L2}^{*} \left( \alpha \right) = 1 - \left( {1 - C_{L}^{*} } \right)\frac{{Gamma_{{{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}}} \left( {n,1} \right)}}{n - 1}. $$

Furthermore, for $0. 0 1\le \alpha \le 1$, starting at 0.01 is arbitrary. It is worth noting that $C_{L1}^{*} \left( 1 \right) = C_{L2}^{*} \left( 1 \right) \ne C_{L}^{*}$ when $\alpha = 1$. This leads to considerable inconvenience for management in evaluation of lifetime performance. To resolve this, let $C_{L}^{\prime *}$ be defined as follows:

$$ C_{L}^{\prime *} = 1 - \left( {1 - C_{L}^{*} } \right)\frac{n - 1}{{Gamma_{0.5} \left( {n,1} \right)}} $$

(33)

Thus, the $\alpha {\text{ - cuts}}$ of the triangular shaped fuzzy number $\tilde{C}_{L}^{*}$ after transforming variable can be expressed as follows:

$$\begin{aligned} \tilde{C}_{L}^{\prime *} \left[ \alpha \right] = \left\{ \begin{array}{llll} \left[ {C_{L1}^{\prime *} \left( \alpha \right),C_{L2}^{\prime *} \left( \alpha \right)} \right], \quad 0.01 \le \alpha \le 1 \hfill \\ \left[ {C_{L1}^{\prime *} 0.01),C_{L2}^{\prime *} \left( {0.01} \right)} \right], \quad 0 \le \alpha \le 0.01 \hfill \\ \end{array} \right.\end{aligned} $$

(34)

where

$$ C_{L1}^{\prime *} \left( \alpha \right) = 1 - \left( {1 - C_{L}^{*} } \right)\frac{{Gamma_{{1 - \left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} \left( {n,1} \right)}}{{Gamma_{0.5} \left( {n,1} \right)}},C_{L2}^{\prime *} \left( \alpha \right) = 1 - \left( {1 - C_{L}^{*} } \right)\frac{{Gamma_{{{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}}} \left( {n,1} \right)}}{{Gamma_{0.5} \left( {n,1} \right)}} $$

From the above equation, we can see that $C_{L1}^{\prime *} \left( 1 \right) = C_{L2}^{\prime *} \left( 1 \right) = C_{L}^{*}$ when $\alpha = 1$. Obviously, the triangular shaped fuzzy number of $C_{L}^{*}$ after transforming variable can further be expressed as follows:

$$ \tilde{C}_{L}^{\prime *} = \left( {C_{L}^{\prime L} ,C_{L}^{\prime M} ,C_{L}^{\prime R} } \right) $$

(35)

where

$$ \begin{aligned} & C_{L}^{\prime L} = C_{L1}^{\prime *} \left( {0.01} \right) = 1 - \left( {1 - C_{L}^{*} } \right)\frac{{Gamma_{0.995} \left( {n,1} \right)}}{{Gamma_{0.5} \left( {n,1} \right)}} \\ & C_{L}^{\prime M} = C_{L1}^{\prime *} \left( 1 \right) = C_{L2}^{\prime *} \left( 1 \right) = C_{L}^{*} \\ & C_{L}^{\prime R} = C_{L2}^{\prime *} \left( {0.01} \right) = 1 - \left( {1 - C_{L}^{*} } \right)\frac{{Gamma_{0.005} \left( {n,1} \right)}}{{Gamma_{0.5} \left( {n,1} \right)}}. \\ \end{aligned} $$