Elipses de confiança para soluções pareto-ótimas multivariada

Abstract

This study presents a non-linear bi-objective optimization method for correlated responses of Robust parameter design optimization (RPD) using Normal Boundary Intersection (NBI) method. Even in capable region for multiple and conflicting objectives optimization Pareto frontiers could be formed by indistinguishable points which may require a highly confirmatory sample sizes to verify non-dominance. Taking advantage of uniformly spread Pareto-frontiers, some propositions are established to treat a trade-off between mean and variance. In this approach, Response Surface Methodology (RSM) is applied for modeling the quality characteristics of process, using propagation of error to extract the implicit variance. Moreover, in order to avoid correlated variables on subsequent optimization, Factor Analysis rotated by Equimax is applied replacing original data by factor scores regressions. In contemplation of distinguishing Pareto-solutions are formed (1-α) confidence ellipse region for centrality and dispersion of every solution, which variability is quantified by variance covariance matrix. These ellipses are especially important to understand the stochastic nature of Pareto-optimal solutions obtained when NBI is used coupled with RSM. As a key result, this study conceives the Fuzzy decision-maker, which is a smart Pareto filter based on Fuzzy logic, combining confidence ellipses volume (variability) and Mahalanobis distance (mean shift) as a quality indicator. This approach becomes possible to synchronously minimize accuracy and precision. The adequacy of the proposal is illustrated with two real cases of hardened steel turning process, optimizing cost and tool life. The quality of practical results motivates us to suggest the method may be extended to applications on similar manufacturing processes problems.