# Plum Tree Algorithm and Weighted Aggregated Ensembles for Energy Efficiency Estimation
## 4 Results  
The PTA was evaluated and compared to the CSO algorithm, the PSO algorithm, the GWO, the CS algorithm, the CSA, and the HOA using 24 objective functions. Out of the 24 objective functions, 23 were from the EvoloPy library [92; 93; 94], whereas one was from [95]. The implementations of the PSO algorithm, the GWO, and the CS algorithm were the ones from the EvoloPy library, whereas the implementations of the CSO algorithm, the CSA, and the HOA were developed in-house. The CSO algorithm version used in experiments was an adapted version in which the equations for the updating of the positions of the hens were modified such that the values of the \(S_{1}\) and \(S_{2}\) parameters were set to 1 in cases in which the fitness of the hen was better than the fitness of the associated rooster or hen, respectively. The mother hens were considered to be the top MP (mothers percent) hens. The experiments were written in Python using the EvoloPy library and a machine with the following properties: Intel(R) Core(TM) i7-7500U CPU processor, 8.00 GB installed RAM, 64-bit operating system, and Windows 10 Pro N operating system. Tables 2-4 present the unimodal objective functions, the high-dimensional multi-objective functions, and the fixed-dimensional multimodal objective functions, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Objective Function** & **Mathematical Formula** & **Range** & **N** & **Optimal Value** \\ \hline \(h_{1}\) & \(\left(\frac{1}{500}+\sum_{j=1}^{25}\frac{1}{j+\sum_{j=1}^{25}\left(x-a_{j} \right)^{2}}\right)^{-1}\) & \([-65356,\,65.356]\) & 2 & 0.998 \\ \hline \(h_{2}\) & \(\sum_{i=1}^{11}\left[b_{i}-\frac{n_{i}\left(x_{i}^{2}+c_{i}\right)}{c_{i}^{2}+c _{i}\left(x_{i}+3x_{i}\right)^{2}}\right]^{2}\) & \([-5,\,5]\) & 4 & 0.00030 \\ \hline \(h_{3}\) & \(4x_{1}^{2}-2.1x_{1}^{2}+\frac{1}{34}+x_{1}x_{2}-4x_{2}^{2}+4x_{2}^{4}\) & \([-5,\,5]\) & 2 & \(-1.0316\) \\ \hline \(h_{4}\) & \(\left(x_{2}-\frac{51}{4\pi^{2}}x_{1}^{2}+\frac{5}{8}x_{1}-6\right)^{2}+10 \left(1-\frac{1}{8\pi}\right)\cos x_{1}+10\) & \([-5,\,5]\) & 2 & 0.3978 \\ \hline \(h_{5}\) & \(\left[1+(x_{1}+x_{2}+1)^{2}\left(19-14x_{1}+3x_{1}^{2}-14x_{2}+6x_{1}x_{2}+3x_ {2}^{2}\right)\right]\) & \([-2,\,2]\) & 2 & 2.999 \\ \(\times\left[30+(2x_{1}-3x_{2})^{2}\times\left(18-32x_{1}+12x_{1}^{2}+48x_{2}- 36x_{1}x_{2}+27x_{2}^{2}\right)\right]\) & \([0,\,1]\) & 3 & \(-3.86\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fixed-Dimensional Multimodal Objective Functions (Objective Functions \(h_{1}-h_{10}\)). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Objective Function** & **Mathematical Formula** & **Range** & **N** & **Optimal Value** \\ \hline \(g_{1}\) & \(\sum_{i=1}^{N}x_{i}^{2}\sin\left(\sqrt{\left|x\right|}\right)\) & \([-500,500]\) & 30 & \(-12,569.487\) \\ \hline \(g_{2}\) & \(\sum_{i=1}^{N}\left[x_{i}^{2}-10\times\cos(2\pi x_{i})+10\right]\) & \([-51,25,12]\) & 30 & 0 \\ \hline \(g_{3}\) & \(-20\times\exp\left(-20\sqrt{\frac{1}{2}\sum_{i=1}^{N}x_{i}^{2}}\right)-exp\left( \frac{1}{8}\sum_{i=1}^{N}\cos(2\pi x_{i})\right)+20+\epsilon\) & \([-32,32]\) & 30 & 0 \\ \hline \(g_{4}\) & \(\frac{1}{500}\times\sum_{i=1}^{N}x_{i}^{2}-\Pi_{i=1}^{N}\cos\left(\frac{x_{i} }{\sqrt{\frac{1}{4}}}\right)+1\) & \([-600,600]\) & 30 & 0 \\ \hline \(g_{5}\) & \(\frac{1}{500}\times\sum_{i=1}^{N}x_{i}^{2}-\Pi_{i=1}^{N}\cos^{\left(\frac{x_{i} }{4}\right)}+1\) & \([-300,300]\) & 30 & 0 \\ \hline \(g_{6}\) & \(\frac{\pi}{500}\times\left\{10\times\left|x^{2}(\pi y_{1})+\sum_{i=1}^{N}(y_{1} -1)^{2}\times\left[1+10\times\sin^{2i}(2\pi y_{1})\right]+(y_{N}-1)^{2}\right\}\) & \([-50,50]\) & 30 & 0 \\ \(+\sum_{i=1}^{N}\left\{U(x_{i},10,100,4),\text{where}_{i}-1+\frac{51}{2},U(x_{i},\mu k,n)-\right\right\}\) & \(\frac{\left(k(x_{i}-a)^{n},\quad-a\leq k\leq a;\quad-a\leq k\leq a;\quad-a\leq k \leq a;\quad-a\leq k For each algorithm and objective function, 30 runs were performed. The number of iterations was set to 1000, and the population size was set to 50 for all runs.