A goal programming-TOPSIS approach to multiple response optimization using the concepts of non-dominated solutions and prediction intervals Majid Ramezani a , Mahdi Bashiri a,⇑ , Anthony C. Atkinson b a Department of Industrial Engineering, Shahed University, P.O. Box 18155/159, Tehran, Iran b Department of Statistics, London School of Economics and Political Sciences, UK article info Keywords: Multi-response optimization Confidence and prediction intervals Non-dominated solution GP TOPSIS abstract 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 (alterna- tives) 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 oth- ers 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. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction The control of production processes in an industrial environ- ment requires the selection and correct setting of the input vari- ables, so that on-specification product is produced at minimum cost. But first, the relationship between input and output variables must be determined. The series of techniques used in the empirical study of the association between response variables and several in- put variables is called Response Surface Methodology (RSM). See, for example, (Box & Draper, 1987 or Myers & Montgomery, 2002). Much of the emphasis in RSM has been on building models for one response, whereas industrial processes often have many re- sponses, the values of which ideally require simultaneous optimi- zation. Usually optimizing all responses simultaneously would cause conflicts of interest. Our method of resolution of this conflict is based on realizing that the problem has three stages: data gathering, model building and optimization. At first the data are collected, using an experimental design. Then the techniques of RSM are applied for estimating the relation between response (output) and explanatory (input) variables and the model is con- structed. Finally, the model is optimized. At this stage the purpose is to obtain optimum conditions on the input variables so that all responses concurrently will be as near as possible to their optima. Khuri (1996) discusses such multiple response problems. Most research in multi-response optimization ignores the sta- tistical uncertainty in the results of the modeling stage; the model is taken as given and subjected to multi-objective optimization. When conflicts exist between optimization of the various re- sponses, subjective methods are typically used for trade off be- tween responses to obtain a variety of alternative solutions. On the contrary, in our paper we use the Prediction Intervals (PIs) from the model building stage to trade off between responses in an objective manner. We then find the non-dominated solutions of the problem by modified goal programming. An advantage of our method is that the continuous region of solutions is transformed into a discrete region. Using the Technique for Order Preference by Similarity to the Ideal Solution (TOPSIS) we rank the non-dom- inated solutions from this discrete region. This automatic proce- dure sometimes suggests impractical settings of the explanatory variables. But the ranking is a powerful tool for Decision Makers (DMs) who may need to modify the solutions for practical pur- poses. Our approach is distinguished from others in this field by 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.01.139 ⇑ Corresponding author. Tel.: +98 9123150355; fax: +98 2151212021. E-mail addresses: majid.ramezani@ymail.com (M. Ramezani), Bashiri@shahed. ac.ir (M. Bashiri), A.C.Atkinson@lse.ac.uk (A.C. Atkinson). Expert Systems with Applications 38 (2011) 9557–9563 Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa