What can fuzziness do for capability analysis based on fuzzy data

Document Type : Research Note


Department of Mathematics, Fu-Jen Catholic University, No. 510, Zhongzheng Rd., Xinzhuang Dist., New Taipei City 24205, Taiwan, R.O.C.


An advantage of process incapability index over the classical process capability
index is that it provides uncontaminated separation between process accuracy and process precision. However, the value of the index is hard to be obtained accurately when process parameters cannot be determined precisely. In such case fuzzy set theory can be applied to get more flexible and more sensitive information. In this article a fuzzy process incapability index is proposed when the specification limits are assumed to be type-2 fuzzy. When the process mean and variance are unknown and when the data collected is fuzzy, three fuzzy estimators are considered. A simulation study is conducted based on a TFT-LCD fuzzy process. By employing the total integral value method to the membership function of a fuzzy index, the comparisons of quality among different processes become easier.


1. Zadeh, L.A. \Fuzzy sets", Information and Control, 8,
pp. 338{353 (1965).
2. Black, M. \Vagueness: An exercise in logical analysis",
Philosophy of Science, 4, pp. 427{455 (1937).
3. Rabieha, M., Modarresb, M., and Azar, A. \Robustfuzzy
model for supplier selection under uncertainty:
An application to the automobile industry", Scientia
Iranica, E, 25(4), pp. 2297{2311 (2018).
4. Kane, V.E. \Process capability indices", J. Quality
Technology, 18, pp. 41{52 (1986).
5. Chan, L.K., Cheng, S.W., and Spiring, F.A. \A new
measure of process capability", Journal of Quality
Technology, 20(3), pp. 162{175 (1988).
6. Pearn, W.L., Kotz, S., and Johnson, N.L. \Distribution
and inferential properties of capability indices",
Journal of Quality Technology, 24, pp. 216{231 (1992).
7. Vanman, K. \A uni ed approach to the capability
indices", Statistica Sinica, 5, pp. 805{820 (1995).
8. Greenwich, M. and Jahr-Scha rath, B.L. \A process
incapability index", International Journal of Quality
Reliability Management, 12, pp. 58{71 (1995).
9. Chen, K.S. \Incapability index with asymmetric tolerances",
Statistica Sinica, 8, pp. 253{262 (1998).
10. Zadeh, L.A. \The concept of a linguistic variable and
its application to approximate reasoning", Information
Sciences, Part I, 8, pp. 199{249; Part II, 8, pp. 301{
357 (1975).
11. Parchami, A., Onar, S.C.,  Oztaysi, B., and Kahraman,
C. \Process capability analysis using interval type-2
fuzzy sets", International Journal of Computational
Intelligence Systems, 10, pp. 721{733 (2017).
12. Parchami, A., Mashinchi, M., Yavari, A.R., et al. \Process
capability indices as fuzzy numbers", Austrian
Journal of Statistics, 34(4), pp. 391{402 (2005a).
13. Parchami, A., Mashinchi, M., and Maleki, H.R. \Con-
dence interval for fuzzy process capability index",
Transaction on System, 4(5), pp. 546{551 (2005b).
14. Moeti, M.T., Parchami, A., and Mashinchi, M. \A note
on fuzzy process capability indices", Scientia Iranica,
13(4), pp. 379{385 (2006).
15. Parchami, A., Mashinchi, M., and Nia, P. \A consistent
con dence interval for fuzzy capability index",
Appl. Comput. Math, 7(1), pp. 143{161 (2008).
16. Parchami, A. and Mashinchi, M. \Testing the capability
of fuzzy processes", Quality Technology &
Quantitative Management, 6(2), pp. 125{136 (2009).
17. Wu, C.W. \Decision-making in testing process performance
with fuzzy data", European Journal of Operational
Research, 193, pp. 499{509 (2009).
18. Hsu, B.M., Chiang, C.Y., and Shu, M.H. \Supplier selection
using fuzzy quality data and their applications
to touch screen", Expert Systems with Application, 37,
pp. 6192{6200 (2010).
19. Chen, C.C., Lai, C.M., and Nien, H.Y. \Measuring
process capability index with fuzzy data", Qua.l
Quant, 44, pp. 529{535 (2010).
20. Wu, C.W. and Liao, M.Y. \Measuring process yield
by fuzzy lower con dence bounds", 2010 International
Symposium on Computer, Communication, Control
and Automation, pp. 103{106 (2010).
21. Abdolshah, M., Yusu , R.M., Hong, T.S., et al. \Measuring
process capability index Cpmk with fuzzy data
and compare it with other fuzzy process capability
indices", Expert Systems with Applications, 38, pp.
6452{6457 (2011).
22. Abdolshah, M. \Fuzzy process capability indices: A
review", World Applied Sciences Journal, 16(12), pp.
1734{1740 (2012).
23. Yen, C.H. \Fuzzy testing for one-sided process capability
indices", Communications in Statistics-Theory and
Methods, 41, pp. 1603{1616 (2012).
24. Parchami, A., Sadeghpoour-Gildeh, B., Nourbakhsh,
M., et al. \A new generation of process capability
indices based on fuzzy measurements", Journal of
Applied Statistics, 41(5), pp. 1122{1136 (2014).
25. Basu, S., Dan, P.K., and Thakur, A. \Experimental
design in soap manufacturing foroptimization of fuzzi-
ed process capability index", Journal of Manufacturing
Systems, 33, pp. 323{334 (2014).
26. Fayyaz, S., Ebrahimi, M., and Devin, A.G. \Fuzzy
multivariate process capability index for measuring
process capability", Shiraz Journal of System Management,
2(4)(8), pp. 41{56 (2014).
27. Abdolshah, M. \Measuring loss-based process capability
index Le and Its Generation L0e0 with fuzzy numbers",
Hindawi Publishing Corporation, Mathematical
Problems in Engineering, Article ID 217406, pp. 1{8
28. Geng, A., Wang, Z., Peng, C., and Han, Y. \A new
fuzzy process capability estimation method based on
kernel function and FAHP\, IEEE Tranactions on
Engineering Management, 63(2), pp. 177{188 (2016).
29. Kaya, I. \An overview of the fuzzy sets on process
capability analysis", 10th Research /Expert Conference
with International Participations, Quality 2017, Neum,
B&H, May, pp. 17{20 (2017).
1064 S.M. Chen and T.M. Hung/Scientia Iranica, Transactions E: Industrial Engineering 28 (2021) 1049{1064
30. Kaya, I. and Baracli, H. \Fuzzy process incapability
index with asymmetric tolerances", J. of Multiple
Valued Logic & Soft Computing, 18, pp. 493{511
31. Kaya, I. \The process incapability index under fuzziness
with an application for decision making", International
Journal of Computational Intelligence Systems,
7(1), pp. 114{128 (2014).
32. Abbasi, Z. and Sadeghpour Gildeh, B. \Assessing
process performance with incapability index based on
fuzzy critical value", Iranian Journal of Fuzzy Systems,
13(5), pp. 21{34 (2016).
33. Viertl, R. \On the description and analysis of measurements
of continuous quantities", Kybernetika, 38(3),
pp. 353{362 (2002).
34. Filzmoser, P. and Viertl, R. \Testing hypotheses with
fuzzy data: the fuzzy p-value", Metrika, 59, pp. 21{29
35. Puri, M.L. and Ralescu, D.A. \The concept of normality
for fuzzy random variables", The Annals of
Probability, 13(4), pp. 1373{1379 (1985).
36. Kaya, I. and Kahraman, C. \Fuzzy process capability
analyses with fuzzy normal distribution", Expert Systems
with Applications, 37, pp. 5390{5403 (2010).
37. Buckley, J.J., Fuzzy Probability and Statistics, Springer
38. Ser
ing, R.J., Approximation Theorems of Mathematical
Statistics, John Wiley & Sons (1980).
39. Hummel, R., Banga, S., and Hettmansperger, T.P.,
Better Con dence Intervals for the Variance in
a Random Sample, courses.wcupa.edu/rbove/eco252/
OneVariance.pdf (2005).
40. Park, S.H., Lee, J.H. and Pahk, H.J. \In-line critical
dimension measurement system development of LCD
pattern proposed by newly developed edge detection
algorithm", Journal of the Optical Society of Korea,
October, 17(5), pp. 392{398 (2013).
41. Chen, K.S. and Chen, T.W. \Multi-process capability
plot and fuzzy inference evaluation", International
Journal of Production Economics, 111(1), pp. 70{79
42. Lion, T.S. and Wang, M.J.J. \Ranking fuzzy numbers
with integral value", Fuzzy Sets and Systems, 50, pp.
247{255 (1992).