Semi-Parametric Techniques for Multi-Response Optimization

The multi-response optimization (MRO) problem in response surface methodology (RSM) is quite common in industry and in many other areas of science. During the optimization stage in MRO, the desirability function method, one of the most flexible and popular MRO approaches and which has been utilized in this research, is a highly nonlinear function. Therefore, we have proposed use of a genetic algorithm (GA), a global optimization tool, to help solve the MRO problem. Although a GA is a very powerful optimization tool, it has a computational efficiency problem. To deal with this problem, we have developed an improved GA by incorporating a local directional search into a GA process.

In real life, practitioners usually prefer to identify all of the near-optimal solutions, or all feasible regions, for the desirability function, not just a single or several optimal solutions, because some feasible regions may be more desirable than others based on practical considerations. We have presented a procedure using our improved GA to approximately construct all feasible regions for the desirability function. This method is not limited by the number of factors in the design space.

Before the optimization stage in MRO, appropriate fitted models for each response are required. The parametric approach, a traditional RSM regression technique, which is inflexible and heavily relies on the assumption of well-estimated models for the response of interests, can lead to highly biased estimates and result in miscalculating optimal solutions when the user's model is incorrectly specified. Nonparametric methods have been suggested as an alternative, yet they often result in highly variable estimates, especially for sparse data with a small sample size which are the typical properties of traditional RSM experiments.

Therefore, in this research, we have proposed use of model robust regression 2 (MRR2), a semi-parametric method, which combines parametric and nonparametric methods. This combination does combine the advantages from each of the parametric and nonparametric methods and, at the same time, reduces some of the disadvantages inherent in each.