Parametric and non-parametric methods for monitoring nonlinear fuzzy profiles

Abstract

In many statistical process control (SPC) applications, the quality of a process or product is represented by a relationship or the so-called quality profile, between a quality characteristic and one or more explanatory variables. The process quality characteristics are sometimes measured with a reasonable degree of approximation. Often, especially when qualitative assessments arise, evaluation of quality characteristics is carried out ambiguously or using linguistic qualifiers. The fuzzy sets theory has proved to be a well-established approach for dealing with uncertainty due to the approximate measurement, ambiguity in subjective evaluations, or vagueness in linguistic variables. Our main purpose was to present and compare four methods of monitoring nonlinear fuzzy profiles, for which different nonlinear fuzzy regression modeling approaches are considered. The first two methods are “a data-driven fuzzy rule-based” and “an extended least square support vector machine (LS-SVM),” for which the profile is characterized without considering a predefined mathematical relationship. However, for the other two methods, a specific form of the profile was needed. The third method, namely “a modified fuzzy regression model,” was initially invented for linear models. Besides, the fourth method employs “the fuzzy least square method” based on linearizing transformation. The exponentially weighted moving average (EWMA) control statistic was used to derive the control statistic to be plotted on the univariate as well as multivariate control charts. An extensive simulation study was conducted to compare the performance of the methods, and the average run length (ARL) criterion was considered to assess the detect-ability of control charts against various out-of-control conditions. Our comparison results indicated that the multivariate EWMA (MEWMA) chart based on the LS-SVM method outperforms the rest in detecting process shifts with smaller values of ARL when the process undergoes out-of-control conditions.