A Comparative Study of Six Process Capability Indices and Their Applications to Electronic and Food Industries

Abstract In this article, we consider six process capability indices (PCIs) whose quality characteristics have a normal distribution, out of which first PCI <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">𝒞</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\mathcal{C}_{p}} was developed in [J. M. Juran, Juran’s Quality Control Handbook, 3rd ed., McGraw-Hill, New York, 1974], next PCI <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">𝒞</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mi>m</m:mi> </m:mrow> </m:msub> </m:math> {\mathcal{C}_{pm}} was developed in [T. C. Hsiang and G. Taguchi, A tutorial on quality control and assurance, Annual Meeting on the American Statistical Association, Las Vegas 1985], third PCI <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">𝒞</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mi>m</m:mi> <m:mo>⁢</m:mo> <m:mi>c</m:mi> </m:mrow> </m:msub> </m:math> {\mathcal{C}_{pmc}} was developed in [M. Saha, S. Dey and L. Wang, Parametric inference of the loss based index <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mi>m</m:mi> </m:mrow> </m:msub> </m:math> C_{pm} for normal distribution, Qual. Reliab. Eng. Int. 38 2022, 10.1002/qre.2987], fourth PCI <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="script">𝒞</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mi>m</m:mi> </m:mrow> <m:mo>′</m:mo> </m:msubsup> </m:math> {\mathcal{C}^{\prime}_{pm}} was developed in [S. Dey, W. Wang and M. Saha, Modified estimation and confidence intervals of an asymmetric loss based process capability index <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="script">𝒞</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mi>m</m:mi> </m:mrow> <m:mo>′</m:mo> </m:msubsup> </m:math> \mathcal{C}^{\prime}_{pm} , Qual. Reliab. Eng. Int. 38 2022, 8, 4033–4048], fifth PCI <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">𝒞</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mi>c</m:mi> </m:mrow> </m:msub> </m:math> {\mathcal{C}_{pc}} and sixth one <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="script">𝒞</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mi>m</m:mi> <m:mo>⁢</m:mo> <m:mi>c</m:mi> </m:mrow> <m:mo>′</m:mo> </m:msubsup> </m:math> {\mathcal{C}^{\prime}_{pmc}} are newly proposed in the paper. Next, we estimate the cited process capability indices using the method of moment estimation (MOM) approach when the underlying process follows normal distribution. A simulation study is conducted to evaluate the performance of these indices with respect to their mean squared errors. Additionally, confidence intervals are constructed using the asymptotic confidence interval (ACI), the generalized confidence interval (GCI) and parametric bootstrap confidence interval (BCI). Using Monte Carlo simulation, the performance of the GCI and BCI is compared in terms of average width and associated coverage probabilities. Finally, four data sets, two related to electronic industries and two related to food industries are re-analyzed to show the applicability’s of the suggested indices, MOM estimation, GCI and BCI.