## 3 Multi-Attribute Evaluation System for Product Form Design
### Decision-Making for the Product Form Design Scheme Based on User Preferences  
Step 1. Construct an evaluation matrix based on user preferences. 1 Each user assigns a preference rating \(r_{ij}\) to the attribute \(F_{j}\) of scheme \(P_{i}\) using PHFNs, reflecting their degree of preference. 2 Compile these evaluations into an individual user evaluation matrix \(R=\left(r_{ij}\right)_{m\times n}\), with the normalized matrix being \(\overline{R}=\left(\overline{r_{ij}}\right)_{m\times n}\). Step 2. Calculate user weights. Due to variations in the needs and priorities of different users, conflicts in evaluation data may occur, complicating the accurate reflection of user satisfaction. Consequently, this paper determines user weights based on the similarity of hesitant Pythagorean fuzzy matrices [46]. The core idea revolves around computing the similarity (SIM) between the evaluation matrices of each user, utilizing the Hamming distance, in order to obtain the support (SUP) for that user's preference towards other users. Higher similarity indicates greater support, suggesting that the evaluation provided by that user is more influential, i.e., user weight \(\gamma\). Accordingly, users with greater influence are assigned higher weights. The formula is as follows:  
\[SIM(q,l)=\frac{1}{m}\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n}w_{j}s\Big{(} \alpha_{ij}^{q},\alpha_{ij}^{l}\Big{)} \tag{17}\]  
\[SIM\Big{(}\alpha_{ij}^{q},\alpha_{ij}^{l}\Big{)}= \begin{cases}0.5&,\alpha_{1} =\alpha_{2}=\alpha_{3}\\ \frac{d\big{(}\alpha_{ij}^{q}\big{(}\alpha_{ij}^{l}\big{)}^{c}\big{)}}{d\big{(} \alpha_{ij}^{q}\alpha_{ij}^{l}\big{)}+d\big{(}\alpha_{ij}^{q}\big{(}\alpha_{ij} ^{l}\big{)}^{c}\big{)}},\text{others}\end{cases}  \tag{18}\]  
\[SIM= \begin{bmatrix}SIM_{11}&SIM_{12}&\cdots&SIM_{1q}\\ SIM_{21}&SIM_{22}&\cdots&SIM_{2q}\\ \vdots&\vdots&&\vdots\\ SIM_{q1}&SIM_{q2}&\cdots&SIM_{qq}\end{bmatrix}  \tag{19}\]  
In Equations (17)-(19), \(SIM(q,l)\) represents the similarity between the evaluation matrices of users \(U_{q}\) and \(U_{l}\), \(SIM\Big{(}\alpha_{ij}^{q},\alpha_{ij}^{l}\Big{)}\) is the similarity between fuzzy numbers \(\alpha_{ij}^{q}\) and \(\alpha_{ij}^{l}\), also known as the Hamming distance, \(w_{j}\) is the attribute weight, and \(SIM\) is the similarity matrix of all users. \[SUP_{q}=\sum\limits_{ \begin{subarray}{c}L\\ l=1\end{subarray} }^{m}SIM_{ql} \tag{20}\]  
\[\gamma_{q}=\frac{SUP_{q}}{\sum\limits_{k=1}^{m}SUP_{k}} \tag{21}\]  
In Equations (20)-(21), \(SUP_{q}\) and \(\gamma_{q}\) are the support and weight of user \(U_{q}\), respectively. Step 3. Calculate the user preference values for the alternative schemes. 1. Aggregate the normalized decision matrix and user weight information to obtain the Pythagorean hesitant fuzzy weighted scores for each attribute of the alternative schemes. 2. Map the distance of each scheme to the positive and negative ideal schemes according to Pythagorean hesitant fuzzy distance mapping. \[ \begin{cases}d^{+}(P_{i})=\sqrt{\sum\limits_{j=1}^{n}w_{j}\bigg{[}\Big{(}s_{ ij}^{+}-s_{ij}\Big{)}^{2}+\Big{(}t_{ij}^{+}-t_{ij}\Big{)}^{2}\bigg{]}}\\ d^{-}(P_{i})=\sqrt{\sum\limits_{j=1}^{n}w_{j}\bigg{[}\Big{(}s_{ij}^{-}-s_{ij} \Big{)}^{2}+\Big{(}t_{ij}^{-}-t_{ij}\Big{)}^{2}\bigg{]}}\end{cases}  \tag{22}\]  
In Equation (22), \(d^{+}(P_{i})\) and \(d^{-}(P_{i})\) are the distances from scheme \(P_{i}\) to the positive and negative ideal schemes, respectively; \(s_{ij}^{+}\) and \(t_{ij}^{+}\) represent the satisfaction and dissatisfaction of the _jth_ attribute in the positive ideal scheme, respectively; \(s_{ij}^{-}\) and \(t_{ij}^{-}\) represent the satisfaction and dissatisfaction of the _jth_ attribute in the negative ideal scheme, respectively; and \(w_{j}\) is the weight of the attribute. 3 Calculate the degree of superiority or inferiority for each scenario based on the ratio of the distances between each scheme and the positive and negative ideal schemes.