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.
Similar content being viewed by others
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
References
Buckley, J. J. (2005). Fuzzy statistics: Hypothesis testing. Soft Computing, 9(7), 512–518.
Chan, L. K., Cheng, S. W., & Spiring, F. A. (1988). A new measure of process capability Cpm. Journal of Quality Technology, 20(3), 162–175.
Chang, Y. C. (2009). Interval estimation of capability index Cpmk for manufacturing processes with asymmetric tolerances. Computers & Industrial Engineering, 56(1), 312–322.
Chang, Y. C., & Wu, C. W. (2008). Assessing process capability based on the lower confidence bound of Cpk for asymmetric tolerances. European Journal of Operational Research, 190(1), 205–227.
Chen, K. S. (1998). Incapability index with asymmetric tolerances. Statistica Sinica, 8(1), 253–262.
Chang, T. C., & Chen, K. S. (2022). Statistical test of two Taguchi Six-Sigma quality indices to select the supplier with optimal processing quality. Journal of Testing and Evaluation, 50(1), 674–688.
Chen, K. S. (2022). Fuzzy testing of operating performance index based on confidence intervals. Annals of Operations Research, 311(1), 19–33.
Chen, K. S., & Chang, T. C. (2020). Construction and fuzzy hypothesis testing of Taguchi Six Sigma quality index. International Journal of Production Research, 58(10), 3110–3125.
Chen, K. S., Huang, C. F., & Chang, T. C. (2017). A mathematical programming model for constructing the confidence interval of process capability index Cpm in evaluating process performance: An example of five-way pipe. Journal of the Chinese Institute of Engineers, 40(2), 126–133.
Chen, K. S., & Pearn, W. L. (2001). Capability indices for processes with asymmetric tolerances. Journal of the Chinese Institute of Engineers, 24(5), 559–568.
Chen, K. S., Pearn, W. L., & Lin, P. C. (1999). A new generalization of Cpm for processes with asymmetric tolerances. International Journal of Reliability, Quality and Safety Engineering, 6(4), 383–398.
Chen, K. S., Wang, C. H., Tan, K. H., & Chiu, S. F. (2019). Developing one-sided specification Six-Sigma fuzzy quality index and testing model to measure the process performance of fuzzy information. International Journal of Production Economics, 208, 560–565.
Chen, K. S., & Yang, C. M. (2018). Developing a performance index with a Poisson process and an exponential distribution for operations management and continuous improvement. Journal of Computational and Applied Mathematics, 343, 737–747.
Chen, K. S., & Yu, C. M. (2020). Fuzzy test model for performance evaluation matrix of service operating systems. Computers & Industrial Engineering, 140, 106240.
Chen, K. S., Yu, C. M., & Huang, M. L. (2022). Fuzzy selection model for quality-based IC packaging process outsourcers. IEEE Transactions on Semiconductor Manufacturing, 35(1), 102–109.
Cheng, S. W. (1994). Practical implementation of the process capability indices. Quality Engineering, 7(2), 239–259.
Kaya, İ, & Kahraman, C. (2011). Fuzzy process capability indices with asymmetric tolerances. Expert Systems with Applications, 38(12), 14882–14890.
Li, W., & Liu, G. (2022). Dynamic failure mode analysis approach based on an improved Taguchi process capability index. Reliability Engineering & System Safety, 218, 108152.
Lin, G. H., Pearn, W. L., & Yang, Y. S. (2005). A Bayesian approach to obtain a lower bound for the C pm capability index. Quality and Reliability Engineering International, 21(6), 655–668.
Pearn, W. L., Lin, P. C., & Chen, K. S. (2004). The index for asymmetric tolerances: Implications and inference. Metrika, 60(2), 119–136.
Ruczinski, I. (1996). The Relation Between Cpm and the Degree of Includence. Ph.D. dissertation, University of Würzburg, Würzburg, Germany.
Shu, M. H., Wang, T. C., & Hsu, B. M. (2022). Generalized quick-switch sampling systems indexed by Taguchi capability with record traceability. Computers & Industrial Engineering, 172, 108577.
Wang, C. H., & Chen, K. S. (2020). New process yield index of asymmetric tolerances for bootstrap method and six sigma approach. International Journal of Production Economics, 219, 216–223.
Yu, C. M., & Chen, K. S. (2022). Fuzzy evaluation model for attribute service performance index. Journal of Intelligent & Fuzzy Systems, 43(4), 4849–4857.
Yu, C. M., Chen, K. S., & Guo, Y. Y. (2021). Production data evaluation analysis model: A case study of broaching machine. Journal of the Chinese Institute of Engineers, 44(7), 673–682.
Yu, C. M., Lai, K. K., Chen, K. S., & Chang, T. C. (2020). Process-quality evaluation for wire bonding with multiple gold wires. IEEE Access, 8(1), 106075–106082.
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
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interest
The authors have no competing interests to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Chen, KS., Yu, CM. Developing a novel fuzzy testing model for capability index with asymmetric tolerances. Ann Oper Res 340, 149–162 (2024). https://doi.org/10.1007/s10479-024-05948-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-024-05948-z