# Research on Optimized Product Image Design Integrated Decision System Based on Kansei Engineering
## 3 Product Image Design Integrated Decision System
### Product Image Design Quantitative Decision Model  
#### 3.2.1 GRA-Fuzzy Logic Model  
Grey Relation Analysis (GRA) is a method to determine the relationship (similarity) between two sets of random sequences in grey system. One is a reference sequence \((x_{0}\in X)\) and the other is a comparison sequence \(X=\{x_{o}|\sigma=0,1,2,\ldots,n\}\), and \(X=\{x_{o}|\sigma=0,1,2,\ldots,n\}\) is a set of grey related elements [12]. At a certain moment, the grey correlation degree of two sets of series can be expressed by grey correlation coefficient \(r(x_{0}(k),x_{i}(k))\), which is defined as below:  
\[r(x_{0}(k),x_{i}(k))=\min_{i}\min_{k}\left|x_{0}(k)-x_{i}(k)\right|+\xi\max_{ i}\max_{k}\left|x_{0}(k)-x_{i}(k)\right|/ \tag{5}\]  
\(k=1,2,\ldots,n\); \(i=1,2,\ldots,m\)  
The grey correlation coefficients of different design sub-elements are different. If \(r(x_{0},x_{i})>r\Big{(}x_{0},x_{i}\Big{)}\), then design sub-element \(x_{i}\) is more relevant to product image than design sub-element \(x_{j}\). Based on the Kansei image database of each design sub-element affecting the product image, the grey correlation coefficient is used to express the relationship between the product image and design sub-elements. By comparing the numerical value of grey correlation coefficient, the sorting of the influence degree of different design sub-elements on product image is obtained. The most influential design sub-elements are selected as input linguistic variables, and the corresponding product image as output linguistic variable of GRA-Fuzzy logic model. Based on the GRA-Fuzzy theory, combining with the Kansei Engineering experiment, the product design sub-elements with the greater influence are selected first, and the product image of the experimental samples is evaluated by seven-point Likert scale [13]. Then, the influential design sub-elements are taken as the input fuzzy set, and the evaluation values of product image are taken as the output fuzzy set. The triangular fuzzy functions of input and output linguistic variables are obtained. Through the application of fuzzy logic model in MATLAB, the fuzzy rules are constructed, and the center of maximum (CoM) method is used for defuzzification [14]. Finally, the GRA-Fuzzy logic model of product image design is established to determine the specific Kansei image value of product design. Figure 5: The qualitative decision model of product image design. Combined with the product image database in Section 2, the fuzzy rules of GRA-Fuzzy logic model are constructed. In order to reflect the relationship between the multiple design sub-elements and product images involved in the product design process, the \"if... then... \" rules with multiple fuzzy conditions are used as follows:  
\[ \begin{array}{l}\textit{If }X_{1}\textit{ is }A_{1}\textit{ and }X_{2}\textit{ is }A_{2}\cdots\textit{ and }X_{n}\textit{ is }A_{n};\\ \textit{Then }Y_{1}\textit{ is }B_{1}\textit{ and }Y_{2}\textit{ is }B_{2}\cdots\textit{ and }Y_{n}\textit{ is }B_{n}\end{array}  \tag{6}\]  
where \(A_{1},A_{2},\ldots,A_{n}\) and \(B_{1},B_{2},\ldots,B_{n}\) are fuzzy linguistic terms, which are, respectively, represented by input linguistic variables \(X_{1},X_{2},\ldots,X_{n}\) and output linguistic variables \(Y_{1},Y_{2},\ldots,Y_{n}\). The input linguistic variables \(X_{1},X_{2},\ldots,X_{n}\) represent sub-elements of product design, and the output linguistic variables \(Y_{1},Y_{2},\ldots,Y_{n}\) represent product image. Each fuzzy rule of GRA-Fuzzy logic model of product image design is to associate the given combination of product design sub-elements with the corresponding numerical state of product image. Defuzzification is the process of transforming the membership of output linguistic variables into numerical value.