A unique solution for principal component analysis-based multi-response optimization problems

Abstract

A procedure to find a unique solution for multi-response optimization problems based on indexing is presented. The procedure utilizes principal component analysis to map the original data to a new vector of component scores, transforming the original response variables into uncorrelated principal components. This process involves loadings that are the elements of the eigenvectors corresponding to the eigenvalues of response variables in the correlation matrix. It is shown that for a given eigenvalue λ, its corresponding eigenvectors are not unique, which could lead to different “optimal” parametric (factor-level) settings and will further mislead the process or product improvement strategy. The proposed indexing method will determine a unique optimal solution in the presence of (2p)(p!) combinations of eigenvectors.