Multivariate Normal Boundary Intersection based on rotated factor scores: A multiobjective optimization method for methyl orange treatment

Abstract This paper presents the multiobjective optimization of methyl orange treatment with ozone using Normal Boundary Intersection and response surface models of rotated principal component factor scores for the expected value E[f(x)] and prediction variance Var[f(x)] of dye removal (Y 1 ) and chemical oxygen demand removal (Y 2 ). The innovation and the main contribution of this paper consists of building a 2-dimensional equispaced and convex Pareto Frontier for rotated factor scores representing the original multivariate set, reducing the number of objective functions without inverting the correlation among the original responses. Furthermore, this proposal provides a practical way to generate the narrowest possible prediction confidence intervals for a desired optima using the fuzzy membership function criterion in order to select the best compromise solution between E[f(x)] and Var[f(x)]. To illustrate the proposal's feasibility, a central composite design for the ozonation process of methyl orange solution with three factors (x 1  = pH, x 2  = air flow and x 3  = ozone dosage) was run. The optimization results showed a maximum dye removal of 94.1% ± 4.3 with a respective chemical oxygen demand removal of 88.4% ± 5.3 obtained at x* = [9.5; 7.1 l.min-1; 18.4 g h −1 ]. However, this point have presented the largest 95% prediction confidence interval. Based on the fuzzy membership of Pareto set it was possible to select the narrowest 95% confidence intervals with maximum removal rates (Y 1  = 90.5 ± 2.2 and Y 2  = 88.3 ± 2.7), obtained at x* = [7.9, 5.6 l min −1 , 18.4 g h −1 ]. Confirmation runs and comparisons among several optimization methods were done and indicated that the results fell within the respective confidence intervals for predictions, which corroborates the good adequacy of the proposal.