Optimization of weighted correlated multiple responses using a probabilistic index

Dealing with more than one response in the process optimization has been a great issue in recent years; therefore, multiple-response optimization studies have grown in the published works. In the common problems, there are some input variables which can affect output responses, but optimization can be more complex and more real when the responses have correlation with each other. In such problems, the analyst should consider the correlation structure in addition to the effects of input variables. In some cases, response variables may emerge by different distributions from the normal ones, which can be analyzed by the proposed method. Moreover, in some problems, response variables may have different levels of importance for the decision maker. In this study, we try to propose an effcient method to find the best treatment in an experimental design, which has different weights for correlated responses ,either cardinal or ordinal. Also, a heuristic method is proposed to deal with problems that have a considerable number of correlated responses, or treatments. The results of some numerical examples confrm the validity of the proposed method. Moreover, a real case of air pollution in Tehran is studied to show the applicability of the proposed method in the real problems.