## 3 Multi-Attribute Evaluation System for Product Form Design
### Decision-Making for the Product Form Design Scheme Based on Expert Knowledge  
Step 1. Construct an evaluation matrix based on expert knowledge. 1 The set of alternatives \(P\) and the set of attributes \(F\) are identified. 2 Experts evaluate each scheme's performance on various attributes using Pythagorean hesitant fuzzy numbers, creating an initial \(m\times n\) decision matrix \(C=\left(c_{ij}\right)_{m\times n}\). \[C=\left[ \begin{array}{cccc}c_{11}&c_{12}&\cdots&c_{1n}\\ c_{21}&c_{22}&\cdots&c_{2n}\\ \vdots&\vdots&\vdots&\\ c_{m1}&c_{m2}&\cdots&c_{mn}\end{array} \right] \tag{5}\]  
3 Normalize matrix \(\overline{C}\) to refine the evaluations. \[\overline{C}=\left[ \begin{array}{cccc}\overline{c_{11}}&\overline{c_{12}}& \cdots&\overline{c_{1n}}\\ \overline{c_{21}}&\overline{c_{21}}&\cdots&\overline{c_{2n}}\\ \vdots&\vdots&\vdots&\\ \overline{c_{m1}}&\overline{c_{m2}}&\cdots&\overline{c_{mn}}\end{array} \right] \tag{6}\]  
4 The overall score for each attribute of the program is derived by applying the PHFWG to integrate the expert evaluation matrix \(C^{\prime}\). Step 2. Establish a gray relational coefficient decision matrix and determine the weights of attributes. 1 For each attribute in the decision matrix, its gray correlation coefficient with the ideal sequence is calculated by the GRA method. All the calculated gray correlation coefficients are combined into a gray correlation coefficient decision matrix \(\xi\). Calculate the correlation between attributes to determine the importance of each attribute to the overall scheme and the degree of influence between attributes. The formula is as follows:  
\[ \begin{cases}\text{PIS}_{x}^{+}=max\left(c_{i}\right)\\ \text{NIS}_{x}^{-}=min\left(c_{i}\right)\end{cases}  \tag{7}\]  
In Equation (7), \(PIS_{x}^{+}\) and \(NIS_{x}^{-}\) denote the positive and negative ideal solutions for the jth attribute of the ith scheme, respectively. \[\xi_{ij}=\frac{min\ min|\text{C}_{0}(j)-\text{c}(i,j)|+\rho max\ max|\text{C}_ {0}(j)-\text{c}(i,j)|}{|\text{C}_{0}(j)-\text{c}(i,j)|+\rho max\ max|\text{C}_ {0}(j)-\text{x}(i,j)|} \tag{8}\]  
In Equation (8), \(\xi_{ij}\) represents the gray relational coefficient between the \(j\)th attribute of the \(i\)th scheme and the ideal solution, \(\text{C}_{0}(j)\) is the value of the \(j\)th attribute in the ideal solution, \(\text{x}(i,j)\) is the value of the \(j\)th attribute for the \(i\)th scheme in the matrix, \(\rho\) is the resolution coefficient, typically set at 0.5 [44], and \(min\ min\) and \(max\ max\), respectively, represent the min min and max max operations, taken over all \(i\) and \(j\). 2 Form the gray relational coefficient decision matrix \(\xi\) and calculate attribute correlations \(r_{j}\). \[\xi=\left[ \begin{array}{cccc}\xi_{11}&\xi_{12}&\cdots&\xi_{1n}\\ \xi_{21}&\xi_{22}&\cdots&\xi_{2n}\\ \vdots&\vdots&&\vdots\\ \xi_{m1}&\xi_{m2}&\cdots&\xi_{mn}\end{array} \right] \tag{9}\]  
In Equation (9), \(\xi\) is the gray correlation coefficient decision matrix. \[r_{j}=\frac{1}{m(n-1)}{\sum_{i=1}^{m}\sum_{k=1,\ k\
eq i}^{n}}\left(\xi_{ij} \right) \tag{10}\]In Equation (10), \(r_{j}\) is the gray correlation of the attribute. By normalization, the attribute weights \(w_{j}\) are as follows:  
\[w_{j}=r_{j}/\sum\limits_{j=1}^{m}r_{j} \tag{11}\]  
Step 3. Obtain the expert evaluation values. The VIKOR method was first used to determine the group utility value \(S_{i}\), individual regret \(R_{i}\) value, and benefit ratio \(Q_{i}\) for each scheme. Then, the Improved Closeness Method was applied to enhance the variability between the schemes, in order to more accurately assess the degrees of program strengths and weaknesses [45]. A more comprehensive and balanced decision can be made by combining the two approaches. 1 Calculate the group utility value \(S_{i}\), individual regret value \(R_{i}\), and benefit ratio \(Q_{i}\) for each scheme to evaluate proximity to the ideal scheme. \[\left\{