Non-Gaussian Multivariate Process Capability Based on the Copulas Method: An Application to Aircraft Engine Fan Blades

Abstract

Process capability indices (PCIs) are major tools in geometric dimensioning and tolerancing (GD&T) for quantifying the production quality, monitoring production, or prioritizing projects. Initially, PCIs were constructed for studying each characteristic of the process independently. Then, they have been extended to analyze several dependent characteristics simultaneously. Nowadays, with the increasing complexity of the production parts, for example, in aircraft engines, the conformity of one part may rely on the conformity of hundreds of characteristics. Also, because those characteristics are dependent, it may be misleading to make decisions based only on univariate PCIs. However, classical multivariate PCIs in the literature do not allow treating such amount of data efficiently, unless assuming Gaussian distribution, which may be false. Regarding those issues, we advocate for PCI based on some transformation of the conformity rates. This presents the advantage of being free from distributional assumptions, such as the Gaussian distribution. In addition, it has direct interpretation, allowing it to compare different processes. To estimate the PCIs of parts with hundreds of characteristics, we propose to use Vine Copulas. This is a very flexible class of models, which gives precise estimation even in high dimension. From an industrial perspective, the computation of the estimator can be costly. To answer this point, one explains how to compute a lower bound of the proposed PCI, which is faster to calculate. We illustrate our method adaptability with simulations under Gaussian and non-Gaussian distributions. We apply it to compare the production of fan blades of two different factories.