Nonconvex optimization of desirability functions

ABSTRACT

Desirability functions (DFs) are commonly used in optimization of design parameters with multiple quality characteristic to obtain a good compromise among predicted response models obtained from experimental designs. Besides discussing multi-objective approaches for optimization of DFs, we present a brief review of literature about most commonly used Derringer and Suich type of DFs and others as well as their capabilities and limitations. Optimization of DFs of Derringer and Suich is a challenging problem. Although they have an advantageous shape over other DFs, their nonsmooth nature is a drawback. Commercially available software products used by quality engineers usually do optimization of these functions by derivative free search methods on the design domain (such as Design-Expert), which involves the risk of not finding the global optimum in a reasonable time. Use of gradient-based methods (as in MINITAB) after smoothing nondifferentiable points is also proposed as well as different metaheuristics and interactive multi-objective approaches, which have their own drawbacks. In this study, by utilizing a reformulation on DFs, it is shown that the nonsmooth optimization problem becomes a nonconvex mixed-integer nonlinear problem. Then, a continuous relaxation of this problem can be solved with nonconvex and global optimization approaches supported by widely available software programs. We demonstrate our findings on two well-known examples from the quality engineering literature and their extensions.

KEYWORDS:

• design parameter optimization
• desirability functions
• global optimization
• multi-response optimization
• nonsmooth optimization

Acknowledgments

The authors thank Professor Murat Koksalan for his helpful comments and suggestions about possible extensions of this study.

Funding

This study is partially supported by The Scientific and Technological Research Council of Turkey (TUBITAK) under project number 105M138.

Additional information

Notes on contributors

Basak Akteke-Ozturk

Basak Akteke-Ozturk received her B.Sc. in Mathematics from Middle East Technical University (METU) in Ankara, Turkey in 2003 and M.Sc. and Ph.D. in Scientific Computing from the Institute of Applied Mathematics of METU in 2005 and 2010, respectively. She currently serves as a Research Assistant Dr. in the Department of Industrial Engineering of METU. Her research interests are in the areas of optimization, and data mining.

Gulser Koksal

Gulser Koksal is a Professor of Industrial Engineering Department at Middle East Technical University (METU) of Turkey. She received her B.Sc. (1985) and M.Sc. (1987) in Industrial Engineering from METU, and Ph.D. (1993) in the same area from North Carolina State University. She has published extensively, supervised projects, and worked as a mentor and consultant in a wide range of industries and areas covering quality management and control, product management and development, product and process design optimization, and data mining.

Gerhard Wilhelm Weber

Gerhard-Wilhelm Weber is Professor at IAM, METU, Ankara, with research on OR, finance, optimization, data-mining, and life-sciences. He received Diploma/Doctorate at RWTH Aachen, Habilitation at TU Darmstadt and held short professorships in Cologne/Chemnitz, and is EURO conference-advisor and chair of IFORS OR for Developing-Countries online-resources.