Análise da convexidade de funções de superfície de resposta em processos de fabricação

Abstract

Response Surface Methodology (RSM) has proven to be highly relevant in the optimization of industrial processes, as it provides a comprehensive understanding of the relationship between input and output variables while minimizing the number of experiments required for analysis. However, the complexity associated with obtaining saddle-type functions represents a critical challenge, demanding effective approaches to overcome limitations in the search for global optima in multivariate experiments. The convexity of objective functions generated by RSM designs can be analyzed through spectral decomposition. This work presents an analysis of the convexity of functions derived from RSM in different manufacturing processes, applying the aforementioned theory. The study involves the calculation and classification of eigenvalues to identify concave, convex, and saddle-type functions, as well as curvature tests to verify the presence of significant quadratic components. The main consequence of using saddle-type objective functions is that, for this type of optimization, the optima—usually constrained—tend to fall in regions far from the experimental design center, where the prediction variance is extremely high, compromising the quality of optimization models. In this work, an extended set of 240 objective functions obtained from RSM with Central Composite Design (CCD) arrangements was analyzed. The investigation revealed that 71.25% of the analyzed functions exhibited a “saddle-type” behavior, while 14.17% were “concave” and 14.58% “convex.” It was found that 80.42% of the models showed statistically significant curvature and that, in approximately 80% of the cases, the experiments were conducted with fewer center points than recommended, which contributed to increased variance and predictive instability. Among the saddle-type functions, 92.40% required active experimental-space constraints, reinforcing the importance of controlling the hypersphere radius (ρ) to ensure more stable predictions. The results demonstrate that an appropriate balance between the expected response value and variance, combined with the use of additional center points and suitable constraints, constitutes an effective strategy to ensure statistical stability and reliability in the application of RSM to manufacturing processes.