A Fuzzy Multiple Regression Approach for Optimizing Multiple Responses in the Taguchi Method

The fuzzy regression has been found effective in modeling the relationship between the dependent variable and independent variables when a high degree of fuzziness is involved and only a few data sets are available for model building. This research, therefore, proposes an approach for optimizing multiple responses in the Taguchi method using fuzzy regression and desirability function. The statistical regression is formulated for the signal to noise (S/N) ratios of each response replicate. Then, the optimal factor levels for each replicate are utilized in building fuzzy regression model. The desirability function, pay-off matrix, and the deviation function are finally used for formulating the optimization models for the lower, mean, and upper limits. Two case studies investigated in previous literature are employed for illustration; where in both case studies the proposed approach efficiently optimized processes performance. DOI: 10.4018/ijfsa.2012070102 14 International Journal of Fuzzy System Applications, 2(3), 13-34, July-September 2012 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. multiple responses in the Taguchi method has received significant research attention. Several techniques are adopted for this purpose, including a trade-off between quality loss and productivity (Phadke, 1989), multiple response S/N ratio (Antony, 2001; Lin, 2002; Al-Refaie et al., 2010), data envelopment analysis (AlRefaie et al., 2009), grey relational analysis and desirability function approach (Lin, 2004; Palanikumar et al., 2006; Pan et al., 2007; AlRefaie et al., 2008; Lin & Lee, 2009; Chen et al., 2010), principal components analysis (Tong et al., 2005; Fung & Kang, 2005; Huang & Lin, 2008; Gauri & Chakraborty, 2009), fuzzy logic approach (Tarng et al., 2000; Lin & Lin, 2005; Wang & Jean, 2006; Chang & Lu, 2007), and multiple regression-based integration approach (Gopalsamy et al., 2009; Pal & Gauri, 2010). Although statistical regression has many applications (Shu et al., 2006; Gopalsamy et al., 2009), problems can still occur in the following situations, including number of observations is inadequate (small data set), difficulties verifying distribution assumptions, vagueness in the relationship between input and output variables, ambiguity of events or degree to which they occur, and inaccuracy and distortion introduced by linearization. The fuzzy regression analysis uses fuzzy numbers, which can be expressed as intervals with membership values as the regression coefficients, which was first introduced by Tanaka et al. (1982). The fuzzy regression (Toly, 2011a, 2011b; Al-Refaie & Li, 2011) has been found to be more appealing than statistical regression in estimating the relationship between the dependent variable and independent variables when a high degree of fuzziness is involved and only a few data sets are available for model building. In a fuzzy regression model, the deviations between the observed values and the estimated values are expressed as the possibilities that the system has. The relationship between dependent and independent variables is defined by using fuzzy concept rather than statistical concept. In other words, a fuzzy regression model aims to build a model which contains all observed data in the estimated fuzzy numbers. In most manufacturing applications on the Taguchi method, it is difficult to find probability distributions for dependent variables. If the behavior of processes is vague and the observed data is irregular, the statistical regression models have an unnaturally wide possibility range. In fact, many manufacturing processes tend to be very complex in behavior and have inherent system fuzziness; such as, fluctuations of process pressure and temperature due to environmental effects. Sometimes, the observed values from the processes may not be regular. Few published research consider such fuzzy situations when optimizing multiple responses in manufacturing applications on the Taguchi method. Further, the desirability function approach (Derringer & Suich, 1980) is one of the most widely used methods in industry for the optimization of multiple response processes. This research, therefore, proposes and implements an optimization approach of multiple responses in the Taguchi method using fuzzy regression and desirability function approach. The remaining of this paper is organized as follows. First, two outlines the proposed approach. Then, a section provides illustrative two case studies and followed by a section that provides research results and limitations. Finally, the conclusion is summarized. PROPOSED FUZZY REGRESSION APPROACH The proposed approach for optimizing multiple responses in the manufacturing application on the Taguchi method utilizing fuzzy regression approach is outlined in the following steps: Step 1: Suppose q responses are of main concern measured in each experiment. To investigate f process factors concurrently by conducting n experiments of Taguchi’s OA. Typically, the quality response, y, is divided into three main types involving: (1) the smaller-the-better (STB) type response, in which the response is continuous, nonnegative, and its most desired value is zero; 20 more pages are available in the full version of this document, which may be purchased using the "Add to Cart" button on the product's webpage: www.igi-global.com/article/fuzzy-multiple-regressionapproach-optimizing/68990?camid=4v1 This title is available in InfoSci-Journals, InfoSci-Journal Disciplines Computer Science, Security, and Information Technology. 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