Affiliations: Department of International Business, Chang Jung Christian University, Tainan, Taiwan, R.O.C | Department of Applied Mathematics, National Chiayi University, Chiayi, Taiwan, R.O.C
Note: [] Corresponding author. Jong-Wuu Wu, Department of Applied Mathematics, National Chiayi University, 300 Syuefu RD., Chiayi City 60004, Taiwan, R.O.C. E-mail: [email protected]
Abstract: The recent paper by Parchami et al. [2] proposes an open problem: Let $\( \hat{\tilde{C\!}}_p = T\left( {\frac{{a_u - c_l }} {{6s}},\frac{{b_u - b_l }} {{6s}},\frac{{c_u - a_l }} {{6s}}} \right)\)$ be a point estimate of fuzzy process capability index $\(\tilde{C}_p\)$ as definition of Parchami et al. [2], where $\( s = \sqrt {\frac{1}{{n - 1}}\sum\limits_{i = 1}^n {\left( {x_i - \bar x} \right)^2}}\)$. Is it true that: $\( \mathop {\lim }\limits_{n \to \infty } \left[ {\hat{\tilde{C\!}}_{p \otimes } \sqrt {\frac{{\chi _{n - 1,\alpha /2}^2 }} {{n - 1}}} ,\hat{\tilde{C\!}}_{p \otimes } \sqrt {\frac{{\chi _{n - 1,1 - \alpha /2}^2 }} {{n - 1}}}} \right] = \left\{ {\hat{\tilde{C\!}}_p } \right\}?\)$ We modify their open problem and prove that “$\( \mathop {\lim }\limits_{n \to \infty } \left[ {\hat{\tilde{C\!}}_{p \otimes } \sqrt {\frac{{\chi _{n - 1,\alpha /2}^2 }} {{n - 1}}} ,\hat{\tilde{C\!}}_{p \otimes } \sqrt {\frac{{\chi _{n - 1,1 - \alpha /2}^2 }} {{n - 1}}} } \right] \cong \left\{ {\hat{\tilde{C\!}}_p } \right\}\)$ {for large} n” is true.
Keywords: Fuzzy process capability index, fuzzy confidence interval, chi-square distribution
