Fuzzy linear regression models for QFD using optimized h values
Introduction
The quality function deployment (QFD), originated from Japan in 1960s Akao (1990), was firstly proposed to assist companies to proceed the product design planning and competitiveness analysis. So far, besides the design of new products, the methodology of QFD has been successfully applied in many other industries, including software development process (Barnett and Raja, 1995, Herzwurm and Schockert, 2003), supply chain management (Onut and Tosun, 2014, Prasad et al., 2014), project selection and assignment (Chen and Zhang, 2014, Jiang et al., 2014, Mohanty et al., 2005), investment (Celik et al., 2009, McLaughlin and Stratman, 1997), etc. The core idea of QFD suggests that the customer satisfaction is the main source of competitiveness of products (Chien and Su, 2003, Griffin and Hauser, 1993, Herrmann et al., 2000, Karsak et al., 2003, Liu et al., 2014, Zhong et al., 2014), and the producers should design or improve the performances of the products based on the detailed and quantitative requirements for the engineering characteristics of the products which are transferred from the ambiguous and qualitative voice of customers. This transformation is the key and foundation of QFD, and is usually achieved based on the identification of the functional relationships between the customer requirements (CRs) and the engineering characteristics (ECs) (Chen et al., 2005, Jiang et al., 2014, Ko and Chen, 2014, Kwong et al., 2011).
Until now, there are mainly two branches of methods to identify the functional relationships. The first one is determining the functional relationships with crisp or fuzzy numbers by product experienced specialists (Chen and Weng, 2006, Vanegas and Labib, 2001). Actually, in order to handle the uncertain information that the decision makers face, the fuzzy set theory has got a tremendous development in recent years, and has been applied in many areas, including product design (Liu et al., 2014, Zhong et al., 2014), product pricing (Zhao et al., 2012), game theory (Wang et al., 2008), etc. However, no matter the crisp or fuzzy numbers are used to express the relationship coefficients, although maybe the existence of the relationships between the CRs and the ECs is relatively simple to identify, the determination for the strength of the relationships is tough for the specialists, especially when the number of ECs of the product increases, e.g., there are usually hundreds of ECs in the design of a car. Meanwhile, the identification process is inevitably subjective. Apart from the method of determining by specialists, some regression approaches have also been introduced into QFD to identify the functional relationships, including the traditional statistical regression approach and the fuzzy regression approach. Nowadays, the fuzzy regression approach which was firstly proposed by Tanaka et al. (1982) using the fuzzy functions defined by the extension principle in Zadeh (1965) becomes to be a more and more common and adaptable method to handle the uncertain problems including QFD.
In the QFD programming, due to the characteristic of ambiguous and imprecise in the relationships, fuzzy numbers are frequently utilized to express the relationship coefficients. Besides, in practice, the relationships are usually assumed to be linear for simplicity. Kim et al. (2000) first introduced the FLR model into QFD to estimate functional relationships, and they proposed a fuzzy multi-criteria modeling approach for product planning. Since then, the fuzzy linear regression (FLR) model has been widely applied in QFD (see, e.g., Chen et al., 2004, Karsak, 2008, Sener and Karsak, 2011, Wu, 2011). In the FLR model, there is a vital parameter, i.e., the h value, which is used to constrain that the observed crisp output values should be included in the h intervals of the fuzzy outputs obtained through the regression model. Moskowitz and Kim (1993) and Kim et al. (1996) studied the relations among the h value, the shapes of the membership functions, and the spreads of fuzzy parameters in the FLR model with symmetric triangular fuzzy numbers, which were extended to general fuzzy numbers with a decreasingly concave or increasingly convex membership function. The studies showed that whether an h value is appropriate or not would affect the effectiveness of the obtained fuzzy relationship expression directly. However, in practical situations, the h value in the FLR model is usually pre-selected by a design team subjectively (Chen and Ngai, 2008). The concepts of credibility and system credibility were initially proposed by Liu and Chen (2013), and subsequently a systematic method was presented to determine the optimized h value with a maximum system credibility of the FLR model with symmetric triangular fuzzy coefficients.
In this paper, we introduce the FLR model with an optimized h value in Liu and Chen (2013) into QFD, and propose a systematic approach to identifying the functional relationships between the CRs and the ECs. The fuzzy coefficients of the functional relationships are assumed as symmetric triangular fuzzy numbers, which are the most commonly used fuzzy numbers. An example of packing machine is also provided to show that the fuzzy coefficients obtained through the FLR models with optimized h values are more effective than those through the FLR models with arbitrary h values selected by the design team.
The rest of the paper is organized as follows. Section 2 briefly introduces the FLR model and the solution procedure of the optimized h value. Then Section 3 describes the systematic approach to identifying the functional relationships between the CRs and the ECs in a QFD programming through FLR models with optimized h values. Section 4 gives an illustrative example about the packing machine, and some analysis on the corresponding results of the obtained fuzzy functional relationships is also presented.
Section snippets
Preliminaries
In this section, we give a brief introduction about the FLR model together with a systematic method proposed by Liu and Chen (2013) for determining an optimized h value with a maximum system credibility.
Identification of the functional relationships in QFD
There is no doubt that the identification of the concrete and tangible functional relationships between the CRs and the ECs is quite important. In this section, the method in Section 2.2 is introduced into QFD to identify the functional relationships through some well-defined FLR models with optimized h values.
An illustrative example
In this section, a case study about the packing machine is given to illustrate the performance and the effectiveness of the approach to identify the functional relationship matrix between the CRs and the ECs in QFD. The related information and data come from Chen and Chen (2006) and are shown in Fig. 2, where denote ‘improve the quality of packing dynamite’, ‘increase efficiency of packing dynamite’, ‘reduce the packing noise’, and ‘increase the rigidity of the machine’, respectively,
Conclusion
In this paper, we introduced an effective approach for the determination of the optimized h value in the FLR firstly, and then proposed a systematic approach to identifying the functional relationships between the CRs and the ECs in QFD by using the FLR model with an optimized parameter h. Compared with the traditional QFD programming, our approach could obtain more objective and reasonable relationships, based on which, product competitors could understand their real advantages and
Acknowledgments
This work was supported by Grants from the National Natural Science Foundation of China Grant (no. 71272177), the National Social Science Foundation of China (no. 13CGL057), and the Innovation Program of Shanghai Municipal Education Commission (no. 13ZS065).
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