Penalty Function Optimization in Dual Response Surfaces Based on Decision Maker’s Preference and Its Application to Real Data

Abstract

The dual response surface methodology is a widely used technique in industrial engineering for simultaneously optimizing both the process mean and process standard deviation functions of the response variables. Many optimization techniques have been proposed to optimize the two fitted response surface functions that include the penalty function method (PM). The PM method has been shown to be more efficient than some existing methods. However, the drawback of the PM method is that it does not have a specific rule for determining the penalty constant; thus, in practice, practitioners will find this method difficult since it depends on subjective judgments. Moreover, in most dual response optimization methods, the sample mean and sample standard deviation of the response often use non-outlier-resistant estimators. The ordinary least squares (OLS) method is also usually used to estimate the parameters of the process mean and process standard deviation functions. Nevertheless, not many statistics practitioners are aware that the OLS procedure and the classical sample mean and sample standard deviation are easily influenced by the presence of outliers. Alternatively, instead of using those classical methods, we propose using a high breakdown and highly efficient robust MM-mean, robust MM-standard deviation, and robust MM regression estimators to overcome these shortcomings. We also propose a new optimization technique that incorporates a systematic method to determine the penalty constant. We call this method the penalty function method based on the decision maker’s (DM) preference structure in obtaining the penalty constant, denoted as PMDM. The performance of our proposed method is investigated by a Monte Carlo simulation study and real examples that employ symmetrical factorial design of experiments (DOE). The results signify that our proposed PMDM method is the most efficient method compared to the other commonly used methods in this study.