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

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 CL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_{L} $$\end{document} 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 CL∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_{L}^{*} $$\end{document} to deal with imprecise data, and further develops a fuzzy testing model for lifetime performance index CL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_{L} $$\end{document}, 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.