Abstract
Positional characteristics can be found in many manufactured products. This type of quality characteristics requires the use of capability indices to measure their performance regarding a circular tolerance region and the center of this region. Different indices have been proposed in the literature. However, the main assumption of these indices is that the positional characteristic follows a bivariate normal distribution. In this sense, two process capability indices are proposed in this paper. These indices are based on the Gaussian copula in the aims of allowing the marginal components to follow any type of probability density function. Thus, these copula-based indices can be applied when the normality assumption cannot be followed. The construction of the indices is based on the Cpm and Cpmk indices, given the interest that the positional characteristics to be close to the center of the tolerance region. A simulation study is carried out to analyze the performance of the proposed indices, and a case study about the position of a pin is also analyzed. It is shown in the results of both studies that the proposed indices can be used to measure the performance of positional characteristics with circular tolerance regions.
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References
Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19(6):716–723. https://doi.org/10.1109/tac.1974.1100705
Anis MZ, Tahir M (2016) On some subtle misconceptions about process capability indices. Int J Adv Manuf Technol 87(9-12):3019–3029. https://doi.org/10.1007/s00170-016-8622-4
Bothe DR (2006) Assessing capability for hole location. Qual Eng 18(3):325–331. https://doi.org/10.1080/08982110600719407
Chan LK, Cheng SW, Spiring FA (1988) A new measure of process capability: Cpm. J Qual Technol 20(3):162–175. https://doi.org/10.1080/00224065.1988.11979102
Chatterjee M, Chakraborty A K (2014) A superstructure of process capability indices for circular specification region. Communications in Statistics - Theory and Methods 44(6):1158–1181. https://doi.org/10.1080/03610926.2012.763095
Davis RD, Kaminsky FC, Saboo S (1992) Process capability analysis for processes with either a circular or a spherical tolerance zone. Qual Eng 5(1):41–54. https://doi.org/10.1080/08982119208918949
Dharmasena L, Zeephongsekul P (2015) A new process capability index for multiple quality characteristics based on principal components. Int J Prod Res 54(15):4617–4633. https://doi.org/10.1080/00207543.2015.1091520
Dianda DF, Quaglino MB, Pagura JA (2016) Performance of multivariate process capability indices under normal and non-normal distributions. Qual Reliab Eng Int 32(7):2345–2366. https://doi.org/10.1002/qre.1939
Dianda DF, Quaglino MB, Pagura JA (2017) Distributional properties of multivariate process capability indices under normal and non-normal distributions. Qual Reliab Eng Int 33(2):275–295. https://doi.org/10.1002/qre.2003
Dianda DF, Quaglino MB, Pagura JA (2018) Impact of measurement errors on the performance and distributional properties of the multivariate capability index n m c pm. AStA Advances in Statistical Analysis 102(1):117–143
de Felipe D, Benedito E (2017) A review of univariate and multivariate process capability indices. Int J Adv Manuf Technol 92(5-8):1687–1705. https://doi.org/10.1007/s00170-017-0273-6
Ganji ZA, Gildeh BS (2015) A modified multivariate process capability vector. Int J Adv Manuf Technol 83(5-8):1221–1229. https://doi.org/10.1007/s00170-015-7654-5
Genest C, Favre AC (2007) Everything you always wanted to know about copula modeling but were afraid to ask. J Hydrol Eng 12(4):347–368. https://doi.org/10.1061/(asce)1084-0699(2007)12:4(347)
Karl DP, Morisette J, Taam W (1994) Some applications of a multivariate capability index in geometric dimensioning and tolerancing. Qual Eng 6(4):649–665. https://doi.org/10.1080/08982119408918756
Kotz S, Johnson NL (2002) Process capability indices—a review, 1992–2000. J Qual Technol 34(1):2–19. https://doi.org/10.1080/00224065.2002.11980119
Krishnamoorhi KS (1990) Capability indices for processes subject to unilateral and positional tolerances. Qual Eng 2(4):461–471. https://doi.org/10.1080/08982119008962740
Lourme A, Maurer F (2017) Testing the Gaussian and Student’s t copulas in a risk management framework. Econ Model 67:203–214. https://doi.org/10.1016/j.econmod.2016.12.014
Nelsen RB (2007) An introduction to copulas. Springer Science & Business Media
Pan JN, Lee CY (2010) New capability indices for evaluating the performance of multivariate manufacturing processes. Qual Reliab Eng Int 26(1):3–15. https://doi.org/10.1002/qre.1024
Pan JN, Li CI (2014) New capability indices for measuring the performance of a multidimensional machining process. Expert Syst Appl 41(5):2409–2414. https://doi.org/10.1016/j.eswa.2013.09.039
Peruchi RS, Junior PR, Brito TG, Largo JJJ, Balestrassi PP (2017) Multivariate process capability analysis applied to AISI 52100 hardened steel turning. Int J Adv Manuf Technol 95(9-12):3513–3522. https://doi.org/10.1007/s00170-017-1458-8
Salinas-Gutiérrez R, Hernández-Aguirre A, Rivera-Meraz MJJ, Villa-Diharce ER (2010) Using Gaussian copulas in supervised probabilistic classification. In: Studies in computational intelligence. Springer, Berlin, pp 355–372. https://doi.org/10.1007/978-3-642-15534-5_22
Shahriari H, Abdollahzadeh M (2009) A new multivariate process capability vector. Qual Eng 21(3):290–299. https://doi.org/10.1080/08982110902873605
Shi L, He Q, Liu J, He Z (2014) A modified region approach for multivariate measurement system capability analysis. Qual Reliab Eng Int 32(1):37–50. https://doi.org/10.1002/qre.1724
Taam W, Subbaiah P, Liddy JW (1993) A note on multivariate capability indices. J Appl Stat 20(3):339–351. https://doi.org/10.1080/02664769300000035
Vännman K (1995) A unified approach to capability indices. Statistica Sinica pp 805–820
Vännman K, Kotz S (1995) A superstructure of capability indices: distributional properties and implications. Scandinavian Journal of Statistics pp 477–491
Wang FK, Hubele NF, Lawrence FP, Miskulin JD, Shahriari H (2000) Comparison of three multivariate process capability indices. J Qual Technol 32(3):263–275. https://doi.org/10.1080/00224065.2000.11980002
Yan J (2007) Enjoy the joy of copulas: with a package copula. Journal of Statistical Software 21(4). https://doi.org/10.18637/jss.v021.i04
Žežula I (2009) On multivariate Gaussian copulas. Journal of Statistical Planning and Inference 139 (11):3942–3946. https://doi.org/10.1016/j.jspi.2009.05.039
Zhang M, Wang GA, He S, He Z (2014) Modified multivariate process capability index using principal component analysis. Chinese Journal of Mechanical Engineering 27(2):249–259. https://doi.org/10.3901/cjme.2014.02.249
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Rodríguez-Picón, L.A., Méndez-González, L.C., Flores-Ochoa, V.H. et al. Capability indices for circular tolerance regions based on a Gaussian copula. Int J Adv Manuf Technol 104, 4143–4153 (2019). https://doi.org/10.1007/s00170-019-04197-w
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DOI: https://doi.org/10.1007/s00170-019-04197-w