## 3 Multi-Attribute Evaluation System for Product Form Design
### Multi-Attribute Evaluation Method Based on PHFS
: _Let \(X\) be a domain, and let \(M\) be a PHFS defined based on \(X\)._  
\[\mathrm{M}=\{<x,\Gamma_{\mathrm{M}}(x),\psi_{\mathrm{M}}(x)>\mid x\in X\} \tag{1}\]  
In Equation (1), \(X\) is the set of all product form design alternatives. \(\mathrm{M}\) is used to denote the ambiguity and uncertainty in the decision-making process for a specific attribute \(x\), where \(x\) is one of the attributes of a product form design alternative scheme. Both \(\Gamma_{M}(x)\) and \(\psi_{M}(x)\) are non-empty finite subsets of \([0,\ 1]\). \(\Gamma_{M}(x)\) denotes the degree of affiliation of \(x\) to \(\mathrm{M}\), and it reflects the extent to which the decision-maker believes that the product form design attribute \(x\) satisfies or exceeds the evaluation criteria. \(\psi_{M}(x)\) denotes the degree of non-affiliation of \(x\) to \(\mathrm{M}\), and it reflects the extent to which the decision-maker believes that the product form design attribute \(x\) is unsatisfactory or inferior to the evaluation criteria. This set satisfies the condition: \(\forall x\in X\), \(\mu_{M}\in\Gamma_{M}(x)\), \(\upsilon_{M}\in\Psi_{M}(x)\), \(\mu_{M}\in[0,1]\), \(\upsilon_{M}\in[0,1]\), \(0\leq\mu_{M}^{2}(x)+\upsilon_{M}^{2}(x)\leq\)1, \(\mu_{M}^{+}=max\{\mu_{M}\}\), \(\mu_{A}^{+}=max\{\mu_{A}\}\). **Definition 2**. : _Let the PHFN be denoted as \(\alpha=\langle\Gamma_{\alpha},\ \Psi_{\alpha}\rangle\), which represents the Pythagorean hesitant fuzzy evaluation linguistic value. The scoring function for each attribute of the alternative schemes is defined as follows:_  
\[\mathrm{S}_{\alpha}=\frac{1}{\mid\Gamma_{\alpha}\mid}{\sum_{\mu\in\Gamma_{ \alpha}}}\mu^{2}-\frac{1}{\mid\psi_{\alpha}\mid}{\sum_{\upsilon\in\psi_{ \alpha}}}\upsilon^{2} \tag{2}\]  
In Equation (2), \(S_{\alpha}\) is a score function of \(\alpha\) that represents the composite score of an attribute in an alternative. \(\mid\Gamma_{\alpha}\mid\) and \(\mid\psi_{\alpha}\), respectively, represent the number of elements in \(\Gamma_{\alpha}\) and \(\psi_{\alpha}\), derived from the designers' and users' ratings of the attribute. **Definition 3**. : _Let \(\alpha_{1}\) and \(\alpha_{2}\) be the evaluative information from two different decision-makers. The Euclidean distance between \(\alpha_{1}\) and \(\alpha_{2}\) is defined as follows:_  
\[d(\alpha_{1},\alpha_{2})=\sqrt{\frac{1}{2}\left[\frac{1}{l(\Gamma)}{\sum_{h=1 }^{l(\Gamma)}}\left(\mu_{\alpha_{1h}}^{2}-\mu_{\alpha_{2h}}^{2}\right)^{2}+ \frac{1}{l(\psi)}{\sum_{h=1}^{l(\psi)}}\left(\upsilon_{\alpha_{1h}}^{2}-\upsilon _{\alpha_{2h}}^{2}\right)^{2}+\frac{1}{l(\prod)}{\sum_{h=1}^\(F=\{F_{1},\ F_{2},\ F_{3},\ldots,\ F_{n}\}\), \(u\) sets of expert groups denoted as \(D=[D_{1},\ D_{2},\ D_{3},\ldots,D_{u}]\), and \(q\) user groups represented as \(U=[U_{1},\ U_{2},\ U_{3},\ldots,\ U_{q}]\). Experts \(D_{u}\) and users \(U_{q}\) evaluate and assign values to the satisfaction and dissatisfaction of attribute \(P_{i}\) corresponding to scheme \(F_{j}\) in the form of Pythagorean hesitant fuzzy numbers denoted as \(\mathbf{\mathsf{a}}_{ij}=\langle\Gamma_{ij},\ \psi_{ij}\rangle\). They construct the Pythagorean hesitant fuzzy expert individual evaluation matrix \(C=\left(c_{ij}^{u}\right)_{m\times n}\) and user individual evaluation matrix \(R=\left(r_{ij}^{q}\right)_{m\times n}\), where \(c_{ij}^{u}\) and \(r_{ij}^{q}\), respectively, represent the assignment to the \(j\)th attribute of scheme \(P_{i}\) by the \(u\)th expert and the \(q\)th user. Upon standardizing the evaluation matrices based on the Pythagorean Hesitant Fuzzy Sets, the normalized individual decision matrices \(\overline{C}\) and \(\overline{R}\) are obtained. By integrating the Pythagorean hesitant fuzzy aggregation operator, the attributes of each scheme are integrated to obtain the comprehensive attribute value of the scheme. Finally, the scoring function value for each attribute is calculated, which determines the final score.