# Energy of Pythagorean Fuzzy Graphs with Applications
## 5 Applications of the Energy of PFGs in Decision-Making
### Designing of a Satellite Communication System
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Exerts** & \multicolumn{2}{c}{**The Overall Results of the Experts**} \\ \hline  & \(p_{1}^{(1)}\) & \((0.5595,0.5233)\) \\ \(e_{1}\) & \(p_{2}^{(1)}\) & \((0.6581,0.5477)\) \\  & \(p_{3}^{(1)}\) & \((0.5360,0.5384)\) \\  & \(p_{4}^{(1)}\) & \((0.6130,0.5785)\) \\ \hline  & \(p_{1}^{(2)}\) & \((0.5145,0.5623)\) \\  & \(p_{2}^{(2)}\) & \((0.5542,0.5595)\) \\  & \(p_{3}^{(2)}\) & \((0.5709,0.5335)\) \\  & \(p_{4}^{(2)}\) & \((0.7189,0.4356)\) \\ \hline  & \(p_{1}^{(3)}\) & \((0.6187,0.4325)\) \\ \(e_{3}\) & \(p_{2}^{(3)}\) & \((0.6031,0.5144)\) \\  & \(p_{3}^{(3)}\) & \((0.4034,0.6117)\) \\  & \(p_{4}^{(3)}\) & \((0.5647,0.4356)\) \\ \hline  & \(p_{1}^{(4)}\) & \((0.5595,0.5144)\) \\ \(e_{4}\) & \(p_{2}^{(4)}\) & \((0.4235,0.5097)\) \\  & \(p_{3}^{(4)}\) & \((0.6610,0.4865)\) \\  & \(p_{4}^{(4)}\) & \((0.6581,0.4681)\) \\ \hline  & \(p_{1}^{(5)}\) & \((0.6884,0.4356)\) \\ \(e_{5}\) & \(p_{2}^{(5)}\) & \((0.5877,0.5384)\) \\  & \(p_{3}^{(5)}\) & \((0.4419,0.5958)\) \\  & \(p_{4}^{(5)}\) & \((0.5715,0.4606)\) \\ \hline  & \(p_{1}^{(6)}\) & \((0.6884,0.3936)\) \\  & \(p_{2}^{(6)}\) & \((0.5982,0.4527)\) \\  & \(p_{3}^{(6)}\) & \((0.5595,0.5791)\) \\  & \(p_{4}^{(6)}\) & \((0.4034,0.5180)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The aggregation results of the experts \(e_{k}\)\((k=1,2,\ldots,6)\). Based on this, we compute a collective PFE \(p_{i}\)\((i=1,2,3,4)\) of the testing venue \(z_{i}\) over all the other testing venues using the PFWA operator:  
\[p_{i}=\text{PFWA}(p_{i}^{(1)},p_{i}^{(2)},\ldots,p_{i}^{(s)})\]  
That is, \(p_{1}=(0.6113,0.4749)\), \(p_{2}=(0.5802,0.5202),p_{3}=(0.5412,0.5553)\), \(p_{4}=(0.6057,0.4812)\). Compute the score functions \(s(p_{i})=\mu_{i}^{2}-\
u_{i}^{2}\) of \(p_{i}(i=1,2,3,4)\), and rank all the testing venues \(z_{i}(i=1,2,3,4)\) according to the values of \(s(p_{i})(i=1,2,3,4)\). \[s(p_{1})=0.1482,s(p_{1})=0.0660,s(p_{3})=-0.0155,s(p_{4})=0.1353.\]  
Then, \(z_{1}\succ z_{4}\succ z_{2}\succ z_{3}\). Thus, the testing venue \(z_{1}\) is the best among the four given testing venues. We present our scheme for this application in the following Algorithm 1. **INPUT:** A discrete set of testing venues (alternatives) \(Z=\{z_{1},z_{2},\ldots,z_{n}\}\), a set of experts \(e=\{e_{1},e_{2},\ldots,e_{s}\}\) and construction of PFPR \(R_{k}=(p_{ij}^{(k)})_{n\times n}\) for each expert. **OUTPUT:** The selection of the optimal testing venue. 1. **begin**
2. Calculate the energy and Laplacian energy of each PFDG \(\mathcal{D}_{k}(k=1,2,\ldots,s)\). 3. Determine the wight vector for experts based on the energy of PFDGs by utilizing \(w_{k}=\left(\frac{\frac{E((\mathcal{D}_{k})_{k})}{s}\frac{E((\mathcal{D}_{k})_ {k})}{E((\mathcal{D}_{k})_{l})}}{\sum\limits_{l=1}^{s}E((\mathcal{D}_{k})_{l })}\right),\) and then, based on the Laplacian energy of PFDGs by utilizing \(w_{k}=\left(\frac{\frac{LE((\mathcal{D}_{k})_{k})}{s}}{\sum\limits_{l=1}^{s} LE((\mathcal{D}_{k})_{l})},\frac{LE((\mathcal{D}_{k})_{k})}{\sum\limits_{l=1}^{s} LE(((\mathcal{D}_{k})_{l}))}\right),k=1,2,\ldots,s\). 4. Aggregate all \(p_{ij}^{(k)}(j=1,2,\ldots,n)\) corresponding to the testing venue \(z_{i}\), and get the PFE \(p_{i}^{(k)}\) of the testing venue \(z_{i}\) over all the other testing venues for the expert \(e_{k}\) by using the PFA operator. 5. Aggregate all \(p_{i}^{(k)}(k=1,2,\ldots,s)\) into a collective PFE \(p_{i}\) for the testing venue \(z_{i}\) using the PFWA operator. 6. Compute the score functions \(s(p_{i})\) of \(p_{i}(i=1,2,\ldots,n)\). 7. Rank all the testing venues \(z_{i}(i=1,2,\ldots,n)\) according to \(s(p_{i})(i=1,2,\ldots,n)\). 8. Output the optimal testing venue. 9. **end**  
**Algorithm 1** The algorithm for the selection of the most important testing venue.