## 3 Method
### Construction of Judgment Matrices and Consistency Tests
#### 3.3.1 Construction of Judgment Matrices
For example, in Formula (1), \(w_{i}\) represents the normalized weight, \(a_{ij}\) denotes the element in the judgment matrix that indicates the relative importance between the \(i\)-th and \(j\)-th elements, and \(n\) is the order of the matrix. To calculate the weight of each indicator, the judgment matrix must first be normalized. Then, the rows of the normalized matrix are summed to obtain a column vector. The weight vector is obtained by dividing each element in the column vector by \(n\):  
\[w_{i}=\frac{\sqrt[n]{\prod_{j=1}^{n}a_{ij}}}{\sum\limits_{i=1}^{n}\sqrt[n]{ \prod_{j=1}^{n}a_{ij}}} \tag{1}\]  
Step 2: Calculating the maximum eigenvalue \(\lambda_{\text{max}}\) is a crucial step in evaluating the consistency of the judgment matrix. The calculation of the maximum eigenvalue of the judgment matrix is detailed in Formula (2), where \(n\) is the order of the matrix, \(A\) represents  
Figure 6: Questionnaire results: (**a**) the questionnaire results involved pairwise comparisons between the criterion of the target layer and the criteria layer, (**b**) pairwise comparison of indicators C1 to C3, (**c**) pairwise comparison of indicators C4–C6, (**d**) pairwise comparison of indicators C7–C11, (**e**) pairwise comparison of indicators C12–C14, and (**f**) pairwise comparison of indicators C15–C17. the pairwise comparison matrix, and w is the corresponding eigenvector. The term \(\left(Aw\right)_{i}\) denotes the \(i\)-th element of the product of matrix \(A\) and vector \(w\), while \(w_{i}\) refers to the \(i\)-th component of the eigenvector \(w\):  
\[\lambda_{max}=\frac{1}{n}\sum_{i=1}^{n}\frac{\left(Aw\right)_{i}}{w_{i}} \tag{2}\]  
Step 3: The consistency test verifies the conclusions' rationality and calculates the indices' importance weights relative to the higher criterion based on the judgment matrix. Formula (2). represents the consistency index, defined as the ratio of the maximum eigenvalue minus the matrix order to the matrix order. A consistency index of 0 indicates that the judgment matrix is perfectly consistent. A higher \(CI\) value indicates a greater degree of inconsistency in the judgment matrix:  
\[CI=\frac{\lambda_{max}-n}{n-1} \tag{3}\]  
Formula (4) calculates the consistency ratio (CR), which is the ratio of the consistency index (CI) to the random consistency index (RI). The value of RI is obtained from the random consistency index (RI) table, which is derived from simulations conducted by [PERSON] involving 1000 iterations, as shown in Table 3[39]. The judgment matrix passes the consistency test if the CR value is less than 0.10; otherwise, the matrix must be readjusted. This involves recalculating and revising the pairwise comparisons until consistency is achieved. The judgment matrix and consistency test are constructed according to the abovementioned steps. The results of these tests are presented in Tables 4-9:  
\[CR=\frac{CI}{RI} \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Matrix 1**} & \multirow{2}{*}{\begin{tabular}{c} **Interactive** \\ **Behavior (B1)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Color** \\ **Transmission** \\ **(B2)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Sensory** \\ **Experience (B3)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Scene** \\ **Experience (B4)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Packaging** \\ **Function (B5)** \\ \end{tabular} } & \multirow{2}{*}{**Weight**} \\ \hline Interactive behavior (B1) & 1 & 3 & 1/3 & 1/3 & 2 & 0.164 \\ Color transmission (B2) & 1/3 & 1 & 1/3 & 1/2 & 1/3 & 0.080 \\ Sensory experience (B3) & 3 & 3 & 1 & 1/2 & 2 & 0.275 \\ Scene experience (B4) & 3 & 2 & 2 & 1 & 2 & 0.335 \\ Packaging function (B5) & 1/2 & 3 & 1/2 & 1/2 & 1 & 0.146 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Judgment matrix for pairwise comparison of criteria B1 to B5.