Abstract
While the Taguchi capability index developed by Chan (J Qual Technol 20(3):162–175, 1988) takes the process targeting issue into consideration, it fails to account for processes with asymmetric tolerances, which are common in practice. Thus, Chen (Int J Reliab Qual Saf Eng 6(4):383–398) modified this index to include processes with asymmetric tolerances. This index is an important tool for the assessment of quality characteristics with asymmetric tolerances, which are common in practice. As the probability density function of the index is complex, statistical inference can be fairly difficult for quality or process engineers. Furthermore, sample sizes are often small in practice to increase decision-making efficiency, but this can decrease assessment accuracy. To address this issue, we employed a mathematical programming approach to make it more convenient for quality or process engineers to derive the upper confidence limit of the index. We also adopted the suggestion put forward by previous studies to incorporate historical data or expert experience in confidence-interval-based fuzzy testing. The proposed approach therefore has increased assessment accuracy, is convenient to apply in practice, and meets the need for swift responses.
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Abbreviations
- \(C_{pm}\) :
-
Taguchi capability index
- X :
-
Quality characteristic
- N :
-
Normal distribution
- \(\mu\) :
-
Process mean
- \(\sigma\) :
-
Standard deviation
- \(L\left( X \right)\) :
-
Taguchi loss function
- k :
-
Multiplier of the loss
- T :
-
Target value
- \(E\left( {X - T} \right)^{2}\) :
-
Expected value of Taguchi loss function
- \(USL\) :
-
Upper specification limit
- \(LSL\) :
-
Lower specification limit
- d :
-
Half-length of the specification interval
- \(Yield\%\) :
-
Process yield
- \(\Phi \left( \cdot \right)\) :
-
Cumulative distribution function of normal distribution
- \(C_{PMA}\) :
-
Taguchi capability index for symmetric or asymmetric tolerances
- A :
-
\(Max\left\{ {d_{1} \left( {T - \mu } \right),d_{2} \left( {\mu - T} \right)} \right\}\)
- \(d_{m}\) :
-
\(Min\left\{ {D_{1} ,D_{2} } \right\}\)
- \(d_{1}\) :
-
\({{d_{m} } \mathord{\left/ {\vphantom {{d_{m} } {D_{1} }}} \right. \kern-0pt} {D_{1} }}\)
- \(d_{2}\) :
-
\({{d_{m} } \mathord{\left/ {\vphantom {{d_{m} } {D_{2} }}} \right. \kern-0pt} {D_{2} }}\)
- \(X_{1} , \ldots ,X_{j} , \ldots ,X_{n}\) :
-
A random sample
- MLE:
-
Maximum likelihood estimate
- \(\sigma^{2}\) :
-
Process variance
- \(\mu^{*}\) :
-
MLEs of \(\mu\)
- \(\sigma^{*2}\) :
-
MLEs of \(\sigma^{2}\)
- \(Z\) :
-
Standardized normal distribution
- \(\phi_{Z} \left( t \right) = \exp \left\{ {itz - {{t^{2} } \mathord{\left/ {\vphantom {{t^{2} } 2}} \right. \kern-0pt} 2}} \right\}\) :
-
Characteristic function of Z
- \(K\) :
-
Chi-square distribution with \(n - 1\) degrees of freedom
- \(\phi_{K} \left( t \right) = \left( {1 - 2it} \right)^{{{{ - \left( {n - 1} \right)} \mathord{\left/ {\vphantom {{ - \left( {n - 1} \right)} 2}} \right. \kern-0pt} 2}}}\) :
-
Characteristic function of K
- P :
-
Probability
- \(\alpha\) :
-
Significance level
- \(Z_{{{{\alpha^{\prime}} \mathord{\left/ {\vphantom {{\alpha^{\prime}} 2}} \right. \kern-0pt} 2}}}\) :
-
Upper \({{\alpha^{\prime}} \mathord{\left/ {\vphantom {{\alpha^{\prime}} 2}} \right. \kern-0pt} 2}\) quintile of \(N\left( {0,1} \right)\)
- \(\chi_{{{{\alpha^{\prime}} \mathord{\left/ {\vphantom {{\alpha^{\prime}} 2}} \right. \kern-0pt} 2};n - 1}}^{2}\) :
-
Lower \({{\alpha^{\prime}} \mathord{\left/ {\vphantom {{\alpha^{\prime}} 2}} \right. \kern-0pt} 2}\) quintile of \(\chi_{n - 1}^{2}\)
- \(\alpha^{\prime}\) :
-
\(1 - \sqrt {1 - \alpha }\)
- \(CR\) :
-
Confidence region
- n :
-
Sample size
- \(\sigma_{L}\) :
-
\(\sqrt {\frac{n}{{\chi_{{1 - \left( {{{\alpha^{\prime}} \mathord{\left/ {\vphantom {{\alpha^{\prime}} 2}} \right. \kern-0pt} 2}} \right);n - 1}}^{2} }}} \sigma^{*}\)
- \(UC_{PMA}\) :
-
Upper confidence limit of \(C_{PMA}\)
- \(e_{Z}\) :
-
\(\frac{{Z_{{{{\alpha^{\prime}} \mathord{\left/ {\vphantom {{\alpha^{\prime}} 2}} \right. \kern-0pt} 2}}} }}{{\sqrt {\chi_{{1 - {{\alpha^{\prime}} \mathord{\left/ {\vphantom {{\alpha^{\prime}} 2}} \right. \kern-0pt} 2};n - 1}}^{2} } }}\sigma^{*}\)
- \(I\) :
-
Indicator variable
- \(x_{j}\) :
-
Observed value of \(X_{j}\)
- \(\mu_{0}^{*}\) :
-
Observed values of \(\mu^{*}\)
- \(\sigma_{0}^{*}\) :
-
Observed values of \(\sigma^{*}\)
- \(UC_{PMA0}\) :
-
Observed value of the upper confidence limit \(UC_{PMA}\)
- \(H_{0}\) :
-
Null hypothesis
- \(H_{1}\) :
-
Alternative hypothesis
- C :
-
The value of required level
- \(\tilde{C}_{PMA0} \left[ \alpha \right]\) :
-
The \(\alpha {\text{ - cuts}}\) of \(\tilde{C}_{PMA0}\)
- \(\Delta \left( { \, C_{M} , \, C_{R} } \right)\) :
-
The half-triangular shaped fuzzy number of \(C_{PMA}\)
- \(\eta_{{C_{PMA} }} (x)\) :
-
The membership function of \(\tilde{C}_{PMA0}\)
- \(HA_{T}\) :
-
The area in the graph of \(\eta_{{C_{PMA} }} (x)\)
- \(ha_{T}\) :
-
Total area of \(HA_{T}\)
- \(HA_{Tl}\) :
-
The lth segment of \(HA_{T}\)
- \(d_{l}\) :
-
Lower base of trapezoid HATl
- d l +1 :
-
Upper base of trapezoid HATl
- \( \, ha_{Tl}\) :
-
The area of \(HA_{Tl}\)
- \(A_{R}\) :
-
The area under \(\eta_{{C_{PMA} }} (x)\) to the right of vertical line \(x = C\)
- \(a_{R}\) :
-
Total area of \(A_{R}\)
- \(A_{Rl}\) :
-
The lth segment of AR
- \( \, r_{l}\) :
-
Lower base of trapezoid \(A_{Rl}\)
- \(r_{l - 1}\) :
-
Upper base of trapezoid \(A_{Rl}\)
- \(a_{T}\) :
-
\(2 \times ha_{T}\)
- \(\phi_{1} ,\phi_{2}\) :
-
Decision-making values
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Acknowledgments
This is an expanded version of our 2023 RQD conference paper. The author would like to thank the Editor, Prof. Hoang Pham, and anonymous referees for their helpful comments and careful reading, which significantly improved the presentation of this paper. This work was supported by the National Science and Technology Council, Taiwan, Republic of China, Under Grant No. MOST 111-2221-E-167-011-MY2 and NSTC 112-2221-E-167-030
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Chen, KS., Yu, CM. Developing a novel fuzzy testing model for capability index with asymmetric tolerances. Ann Oper Res (2024). https://doi.org/10.1007/s10479-024-05948-z
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DOI: https://doi.org/10.1007/s10479-024-05948-z