Performance of Multivariate Process Capability Indices Under Normal and Non‐Normal Distributions

Abstract

In the context of process capability analysis, the results of most processes are dominated by two or even more quality characteristics, so that the assessment of process capability requires that all of them are considered simultaneously. In recent years, many researchers have developed different alternatives of multivariate capability indices using different approaches of construction.In this paper, four of them are compared through the study of their ability to correctly distinguish capable processes from incapable processes under a diversity of simulated scenarios, defining suitable minimum desirable values that allow to decide whether the process meets or does not meet specifications. In this sense, properties analyzed can be seen as sensitivity and specificity, assuming that a measure is sensitive if it can detect the lack of capability when it actually exists and specific if it correctly identifies capable processes. Two indices based on ratios of regions and two based on the principal component analysis have been selected for the study. The scenarios take into account several joint distributions for the quality variables, normal and non-normal, several numbers of variables, and different levels of correlation between them, covering a wide range of possible situations.The results showed that one of the indices has better properties across most scenarios, leading to right conclusions about the state of capability of processes and making it a recommendable option for its use in real-world practice.

Distributional Properties of Multivariate Process Capability Indices under Normal and Non-normal Distributions

“…Table presents the probabilities of correctly identifying the state of the process, computed considering the value Φ p ; 0.0027 = 0.9973 ∀ p = 2, 3, 5 as the cutoff point. In the study of sensitivity and specificity carried out by Dianda et al , the index Mp 2 was found specific but not always sensitive, unless the anomaly in the process is very noticeable. When the index is computed from sampling information, the true value of the index is underestimated and so the index losses specificity.…”

“…Specificity of this index could be affected because of the bias of its estimator. However, the ability of the estimator to correctly detect processes that meet specifications is not affected as the index is overvalued under this type of heavy‐tailed distributions . The magnitudes of the bias are not great enough to cancel out the overvaluation that the departure from normality causes in the index.…”

“…Table shows the probability that the estimator properly detects the true process capability state, computed from the empirical distributions and the cutoff points defined by Dianda et al detailed also in this table. From these results, it can be seen that in some situations the index losses its population properties of specificity and sensitivity.…”

“…In the same way, the index was found not sensitive for processes more spread than allowed. 13 Although one might expect that the negative bias corrects this problem, this does not happen. Again, the magnitudes of bias are not great enough to compensate the overvaluation.…”

A review of univariate and multivariate process capability indices

“…As we have seen, authors that suggest using the PCA method to transform the original variance-covariance matrices, which describe the original PR, into a diagonal and uncorrelated-matrix are dealing with eliminating the correlation between the measured product characteristics. With this approach, the whole problem is moved to a new system of coordinate axes dened by the eigenvectors [70]. Once the product characteristics are uncorrelated, these authors suggest combining univariate PCIs to calculate the capability of each direction described by each eigenvector.…”