# Energy of Pythagorean Fuzzy Graphs with Applications
## 5 Applications of the Energy of PFGs in Decision-Making
### Evaluation of the Schemes of Reservoir Operation  
This section focuses on evaluating the schemes of reservoir operation. It is a water resource system led by Jiudianxia reservoir with a complex condition and multipurpose use along with the Tao River basin and cascaded power stations in the Tao River. The reservoir was designed for many purposes, such as power generation, irrigation, total water supply for agriculture, industry, residents and environment. Due to different requirements for the partition of the amount of water, five reservoir operation schemes \(z_{1},z_{2},z_{3},z_{4}\) and \(z_{5}\) are recommended. 1. Maximum plant output, enough supply of water used in the Tao River basin, lower and higher supply for society and the economy;
2. Maximum plant output, enough supply of water used in the Tao River basin, lower and higher supply for society and the economy, lower supply for the ecosystem;\(z_{3}\): Maximum plant output, enough supply of water used in the Tao River basin, lower and higher supply for society and the economy, total supply for the ecosystem and environment, 90% of which is passed down for flushing sands during low water periods; \(z_{4}\): Maximum plant output, enough supply of water used in the Tao River basin, lower and higher supply for society and the economy, total supply for the ecosystem and environment, 50% of which is passed down for flushing sands during low water periods; \(z_{5}\): Maximum plant output, enough supply of water used in the Tao River basin, lower and higher supply for society and the economy, total supply for the ecosystem and environment during level and flood periods. To  
Utilize the aggregation operator to fuse all the individual PEPRs \(R_{k}=(r_{ij}^{(k)})_{5\times 5}\) (\(k=1,2,3,4\)) into the collective PFPR \(R=(r_{ij})_{5\times 5}\). Here, we apply the Pythagorean fuzzy weighted averaging (PFWA) operator [2] to fuse the individual PFPR. Thus, we have:  
\[\text{PFWA}(r_{ij}^{(1)},r_{ij}^{(2)},\ldots,r_{ij}^{(s)})=\left(\sqrt{1- \prod_{k=1}^{s}\left(1-(\mu_{ij}^{2})^{(k)}\right)^{n_{k}}},\prod_{k=1}^{s} \left(v_{ij}^{(k)}\right)^{n_{k}}\right).\]  
\[R=\left[ \begin{array}{cccc}(0.5000,0.5000)&(0.6892,0.5160)&(0.7212,0.3085)&(0. 7118,0.4108)&(0.6892,0.2493)\\ (0.6544,0.6690)&(0.5000,0.5000)&(0.5614,0.4264)&(0.5828,0.4347)&(0.5645,0.4500 )\\ (0.4712,0.6805)&(0.4905,0.5413)&(0.5000,0.5000)&(0.6895,0.4938)&(0.6623,0.4589 )\\ (0.4446,0.6978)&(0.5535,0.3811)&(0.6078,0.5763)&(0.5000,0.5000)&(0.6991,0.34 86)\\ (0.4284,0.6317)&(0.4974,0.5437)&(0.5051,0.6438)&(0.3567,0.6391)&(0.5000,0.5000 )\end{array} \right].\]  
Calculate their scores using the score function \(s_{ij}=\mu_{ij}^{2}-v_{ij}^{2}\):  
\[R=\left[ \begin{array}{cccc}0.0000&0.2088&0.4250&0.3379&0.4129\\ -0.0194&0.0000&0.1334&0.1507&0.1161\\ -0.2411&-0.0524&0.0000&0.2316&0.2280\\ -0.2893&0.1611&0.0372&0.0000&0.3672\\ -0.2155&-0.0481&-0.1593&-0.2812&0.0000\\ \end{array} \right].\]  
Figure 7: Pythagorean fuzzy digraphs. \[R=\left[ \begin{array}{cccccc}(0.5000,0.5000)&(0.6872,0.5138)&(0.7225,0.3127)&(0.70 99,0.4129)&(0.6913,0.2463)\\ (0.6542,0.6674)&(0.5000,0.5000)&(0.5629,0.4289)&(0.5845,0.4375)&(0.5618,0.4491) \\ (0.4723,0.6816)&(0.4930,0.5425)&(0.5000,0.5000)&(0.6834,0.5004)&(0.6641,0.4613 )\\ (0.4466,0.6961)&(0.5572,0.3790)&(0.6108,0.5719)&(0.5000,0.5000)&(0.6961,0.3475 )\\ (0.4218,0.6356)&(0.4964,0.5416)&(0.5068,0.6462)&(0.3556,0.6357)&(0.5000,0.5000 )\end{array} \right].\]  
Calculate their scores using the score function:  
\[R=\left[ \begin{array}{cccccc}0.0000&0.2082&0.4242&0.3335&0.4173\\ -0.0174&0.0000&0.1329&0.1503&0.1139\\ -0.2415&-0.0512&0.0000&0.2167&0.2283\\ -0.2851&0.1668&0.0460&0.0000&0.3638\\ -0.2261&-0.0469&-0.1608&-0.2777&0.0000\end{array} \right].\]  
The net flows of the five alternatives are:  
\[\Phi = (\phi(z_{1}),\phi(z_{2}),\phi(z_{3}),\phi(z_{4}),\phi(z_{5}))^{T}= (2.1533,0.1028,-0.2900,-0.1313,-1.8348)^{T}\]  
which gives the ranking of \(z_{1}\succ z_{2}\succ z_{4}\succ z_{3}\succ z_{5}\). Thus, the best scheme is \(z_{1}\).