## 4 An Approach to Decision Making with Complex Pythagorean Fuzzy Information
### Decision-Making Approach  
Consider a MADM problem containing a discrete set of \(m\) alternatives \(A=\{A_{1},A_{2},\ldots,A_{m}\}\). Let \(Z=\{r_{1},r_{2},\ldots,r_{n}\}\) be a set of attributes characterized by a CPFS \(\{r,(u_{\mathcal{C}}(r)e^{i\alpha_{\mathcal{C}}(r)},v_{\mathcal{C}}(r)e^{i \beta_{\mathcal{C}}(r)})\mid r\in Y\}\). Also each attribute classified with a vertex and links between attributes with relations (edges) in CPFG. In a CPFG \(\mathcal{G}=(\mathcal{C},\mathcal{D})\), for an alternative, assume that if an attribute \(r_{i}\in Z\) is most important, we choose \(r_{i}=1\), otherwise \(r_{i}=0\). Then the importance of each attribute \(r_{i}\) can be calculated by using  
\[I_{i} = (\mu_{\mathcal{C}}(r_{i})e^{i\alpha_{\mathcal{C}}(r_{i})},v_{ \mathcal{C}}(r_{i})e^{i\beta_{\mathcal{C}}(r_{i})})x_{i}+\tilde{x}_{N_{i}}\ \ \ i=1,2,\ldots,n. \tag{1}\]  
where \(\mathcal{N}_{i}\) is the set of the attributes \(r_{i}\)'s neighbors and  
\[\tilde{x}_{N_{i}}=\sum_{j\in N_{i}}\left(\mu_{\mathcal{G}}(r_{i}r_{j})e^{i \alpha_{\mathcal{G}}(r_{i}r_{j})},v_{\mathcal{G}}(r_{i}r_{j}e^{i\beta_{ \mathcal{G}}(r_{i}r_{j})})\right)\xi(r_{i}r_{j})r_{j},\]  
\(\xi(r_{i}r_{j})\in[0,1]\) is the influence coefficient between relevant attributes. In the CPFG based MADM problems, if there are prioritization relations among the attributes, we will solve this problem by using the prioritized aggregation operators [29] together with the necessary complex Pythagorean fuzzy graphic structure. For a CPFG, suppose that we have a collection of attributes (vertices) partitioned into \(t\) distinct categories \(C_{1},C_{2},\ldots,C_{t}\) such that \(C_{i}=\{r_{i1},r_{i2},\ldots,r_{in_{i}}\}\), where \(r_{ij}\)\((j=1,2,...,n_{i})\) are the attributes in the category \(C_{i}\) and suppose \(C_{1}>C_{2}>...>C_{t}\) is a prioritization relationship among these categories. In the category \(C_{i}\), the attributes have a higher priority than those in \(C_{j}\) if \(i<j\). Then the universal set of attributes is \(Z=\bigcup_{i=1}^{t}C_{i}\) and the total number of attributes is \(n=\sum_{i=1}^{t}n_{i}\). The prioritized hierarchy structure of \(Z\) is shown in Figure 11. We put forward an approach to handle the CPFG based MADM problems according to the prioritized complex Pythagorean fuzzy aggregation operators well-organized with the degrees of attributes. Compute the degrees of all attributes \(d(r_{i})\)\((i=1,2,\ldots,n)\) which can be normalized by  
\[\bar{d}(r_{i})=\left(\frac{d(\mu_{\mathcal{C}}(r_{i}))}{\sum_{j=1}^{m}d(\mu_ {\mathcal{C}}(r_{j}))},\frac{d(\
u_{\mathcal{C}}(r_{i}))}{\sum_{j=1}^{m}d(\
u _{\mathcal{C}}(r_{j}))}\right),i=1,2,\ldots,n\]  
The weights are associated with attributes dependent upon the satisfaction of the higher priority attribute by designing the prioritization between attributes. Then for each category \(C_{i}\), firstly we define  
\[l_{i}=\left\{ \begin{array}{ll}(1,0),(1,0),&i=0\\ \varphi(d(r_{i1}),d(r_{i2}),\ldots,d(r_{in_{i}})),&i=1,2,\ldots,t\end{array} \right. \tag{3}\]  
where \(\varphi\) is an alternative function such as the maximum or minimum function, for calculating \(l_{i}\) based on which we determine the weight of each category:  
\[w_{i}=\prod_{k=1}^{i}l_{l-1},\;\;i=1,2,\ldots,t \tag{4}\]  
Using the complex Pythagorean fuzzy weighted combination (CPFWC) operator, the overall importance of the alternative can be calculated as:  
\[I^{(r_{i})}=\text{CPFWC}(I_{1},I_{2},\ldots,I_{m})=\vee_{i=1}^{t}(\vee_{j=1}^{ n_{i}}(w_{i}\wedge I_{ij}^{(r)})) \tag{5}\]  
And finally to select an optimal alternative, determine the score functions \(s(I^{(r_{i})})=(\mu_{\mathcal{E}}^{2}-\
u_{\mathcal{E}}^{2})+\frac{1}{4\pi^{2 }}(\alpha_{\mathcal{E}}^{2}-\theta_{\mathcal{E}}^{2})(i=1,2,\ldots,m)\), and rank all the alternatives \(A_{i}(i=1,2,\ldots,m)\) in accordance with \(s(I^{(r_{i})})(i=1,2,\ldots m)\). The approach involves the following steps:  
1.