# Fuzzy Relationship between Kansei Images: A Grey Decision-Making Method for Product Form
## 4 Case Study
### Result Analysis  
By comparing the ranking results of the three models, it can be seen that the relatively optimal design sample is sample 2, and it is always among the top four samples in the ranking order along with sample 9, sample 3, and sample 5. Sample 1, sample 4, sample 8, and sample 10 are in the last four places in the order. Samples 6 and 7 are always in the middle positions. Therefore, the decision-making trends of the three decision-making models are consistent, so the grey decision-making model for product form driven by a fuzzy relationship between the Kansei images proposed in this paper is reasonable and has certain theoretical and practical significance.  
Because the dimension of the decision-making model based on TOPSIS is different from the dimensions of the other two decision-making models, and the difference coefficient is a relative index used to analyse the degree of data dispersion through the standard deviation and average. It is often used to compare data with different measurement units. To establish the superiority of the proposed method, this paper introduces the difference coefficient to conduct a comparative analysis of the three decision-making methods. A higher difference coefficient signifies greater discrimination in sample evaluations, indicating that the decision-making method can help designers more conveniently select multiple Kansei image products that meet a user's Kansei image needs. The expression [32] is as follows:  
\[Cv=\frac{S}{\overline{E}} \tag{19}\]  
\[\overline{E}=\frac{1}{m}\sum\limits_{i=1}^{m}E_{i} \tag{20}\]  
\[S=\sqrt{\frac{\sum\limits_{i=1}^{m}\left(E_{i}-\overline{E}\right)^{2}}{m}} \tag{21}\]  
where \(E_{i}\) is the decision coefficient or relative closeness of the \(i\)-th sample, \(\overline{E}\) is the average value of the decision coefficient or relative closeness, and \(S\) represents the variance.  
The decision coefficient \(\tau\) of the grey decision-making model proposed in this paper, the decision coefficient \(G\) of the CGTDM for multiple Kansei images, and the relative closeness \(H\) of the decision-making model based on TOPSIS were substituted into Equations (19)-(21). The results are shown in Table 8.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample** & \(\mathbf{d}_{i}^{*}\) & \(\mathbf{d}_{i}^{*}\) & \(\mathbf{H}_{i}\) \\ \hline
1 & 0.5068 & 0.1396 & 0.2160 \\ \hline
2 & 0.2582 & 0.4977 & 0.6584 \\ \hline
3 & 0.4032 & 0.3024 & 0.4286 \\ \hline
4 & 0.5225 & 0.1343 & 0.2045 \\ \hline
5 & 0.3580 & 0.2675 & 0.4277 \\ \hline
6 & 0.4552 & 0.2041 & 0.3096 \\ \hline
7 & 0.4052 & 0.2098 & 0.3411 \\ \hline
8 & 0.4738 & 0.1888 & 0.2849 \\ \hline
9 & 0.3514 & 0.3516 & 0.5002 \\ \hline
10 & 0.4841 & 0.2018 & 0.2942 \\ \hline \hline \end{tabular}
\end{table}
Table 7: The relative closeness between each sample and the ideal solution.  
According to Table 8, the difference coefficients of the CGTDM for multiple Kansei images and the decision-making model based on TOPSIS are small. Therefore, the results of these two decision-making models have great similarity and low discrimination. Compared with the other two decision-making models, the grey decision-making model for product form driven by a fuzzy relationship between the Kansei images has the highest difference coefficient, which indicates that the model can enlarge the difference between the decision-making results. By improving the discrimination between the decision-making results, the model can help designers more conveniently select multiple Kansei image products that meet a user's Kansei image needs, so the decision-making results are closer to actual decision-making outcomes. Therefore, the model proposed in this paper has more advantages.