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

Significance statement

Many engineering applications involve more than one objectives demanding optimization, which are influenced by many factors.  Multi-response optimization based on weighted principal component analysis (WPCA) procedure with unique solution is presented. Moreover, the reason for obtaining different optimal solutions from the application of PCA by others is disclosed. A methodology to obtain a unique solution to this problem is presented.

The components with eigenvalues larger than 1 are usually chosen to replace the original response variables. When more than one components with larger than one eigenvalues are selected, the trade-offs are needed, but there is no standard method for selecting a feasible trade-off solution. Each component is multiplied by a weight, which is the proportion of its corresponding variance over the total variance. The weights are used to emphasize the contribution of components based on their corresponding variation. All the weighted components are combined into one multi-response performance index (MPI) through summation, and the choice of optimal factor-level combination is based on the value of MPI. The main procedures for applying multi-response optimization based on WPCA with a detailed analysis and improvement to obtain a unique solution is given.

As the need of seeking the unique solution is revealed, in this paper, an indexing procedure is proposed in order to get the optimal solution with minimum information distortions caused by directions of eigenvectors. The procedure is based on comparison of assigned indices and determination of a set of eigenvectors which represents the new coordinate axes, such that the relative magnitudes of the projections of data in the new coordinate system have minimal total difference between each pair of indices. A method is presented to obtain a unique combination of eigenvectors leading to a factor level combination with the optimal solution in a multi-response optimization.

A unique Solution for Principal Component Analysis-Based Multiple-Response Optimization Problems. Advances In Engineering












Journal Reference

The International Journal of Advanced Manufacturing Technology, pp 1-13. (June 2015).

Nasser Fard , Huyang Xu, Yuanchen Fang.
Northeastern University, Boston, MA, USA


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.

Go To The International Journal of Advanced Manufacturing Technology


About the author

Dr. Fard is an associate professor of Industrial Engineering at the Northeastern University.    He served as an associate editor of the IEEE Transactions on Reliability and he has been on the editorial board of the International Journal of Reliability, Quality and Safety Engineering. His experience and most recent research and professional activities are in Survival Data Analysis, Reliability Predication and Estimation, and Big Data Analytics. He has been principal investigator for public and private research projects.

Huyang Xu and Yuanchen Fang are PhD candidates in the Industrial Engineering Program at Northeastern University.  Their research interests are in the areas of Robust Design of Experiments, Big Data Analytics, and Reliability.





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