A goal programming-TOPSIS approach to multiple response optimization using the concepts of non-dominated solutions and prediction intervals

Multiple response problems include three stages: data gathering, modeling and optimization. Most approaches to multiple response optimization ignore the effects of the modeling stage; the model is taken as given and subjected to multi-objective optimization. Moreover, these approaches use subjective methods for the trade off between responses to obtain one or more solutions. In contradistinction, in this paper we use the Prediction Intervals (PIs) from the model building stage to trade off between responses in an objective manner. Our new method combines concepts from the goal programming approach with normalization based on negative and positive ideal solutions as well as the use of prediction intervals for obtaining a set of non-dominated and efficient solutions. Then, the non-dominated solutions (alternatives) are ranked by the TOPSIS (Technique for Order Preference by Similarity to the Ideal Solution) approach. Since some suggested settings of the input variables may not be possible in practice or may lead to unstable operating conditions, this ranking can be extremely helpful to Decision Makers (DMs). The consideration of statistical results together with the selection of the preferred solution among the efficient solutions by Multiple Attribute Decision Making (MADM) distinguishes our approach from others in the literature. We also show, through a numerical example, how the solutions of other methods can be obtained by modifying the relevant approach according to the DM's requirements