# GBUO: \"The Good, the Bad, and the Ugly\" Optimizer
## 2 \"The Good, the Bad, and the Ugly\" Optimizer (GBUO)
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & **GA** & **PSO** & **GSA** & **TLBO** & **GWO** & **COA** & **SHO** & **MPA** & **GRUO** \\ \hline \hline \multirow{2}{*}{F} & Ave & 13.240 & 1.740 \(\times 10^{-3}\) & 2.0265 \(\times 10^{-3}\) & 8.0370 \(\times 10^{-3}\) & 1.0365 \(\times 10^{-3}\) & 2.1440 \(\times 10^{-3}\) & 0 & 3.2738 \(\times 10^{-3}\) & 0 \\  & std & 4.7664 \(\times 10^{-3}\) & 1.5636 \(\times 10^{-3}\) & 1.5898 \(\times 10^{-3}\) & 1.5433 \(\times 10^{-3}\) & 2.3985 \(\times 10^{-3}\) & 0 & 4.6183 \(\times 10^{-3}\) & 0 \\ \hline \multirow{2}{*}{F} & Ave & 2.4211 & 4.5761 & 1.625 \(\times 10^{-3}\) & 1.2458 \(\times 10^{-3}\) & 5.6446 \(\times 10^{-3}\) & 4.2011 \(\times 10^{-3}\) & 0 & 0 & 0 \\  & std & 2.5421 \(\times 10^{-3}\) & 4.6335 \(\times 10^{-3}\) & 3.1776 \(\times 10^{-3}\) & 5.500 \(\times 10^{-3}\) & 5.6446 \(\times 10^{-3}\) & 4.4627 \(\times 10^{-3}\) & 0 & 0 & 0 \\ \hline \multirow{2}{*}{F} & Ave & 3.2218 & 2.4250 & 3.3573 \(\times 10^{-3}\) & 0.2232 & 1.7608 \(\times 10^{-3}\) & 0.2539 & 8.8888 \(\times 10^{-3}\) & 9.4987 \(\times 10^{-3}\) & 8.8882 \(\times 10^{-3}\) \\ \cline{2-11}  & std & 5.1636 \(\times 10^{-3}\) & 2.5441 \(\times 10^{-3}\) & **3.4002** \(\times 10^{-3}\) & 2.5641 \(\times 10^{-3}\) & 2.2541 \(\times 10^{-3}\) & 1.8840 \(\times 10^{-3}\) & 5.4248 \(\times 10^{-3}\) & 6.1235 \(\times 10^{-3}\) & 2.0082 \(\times 10^{-3}\) \\ \hline \multirow{2}{*}{F} & Ave & 7.202 & 0.0265 & 3.7573 & 0.0262 & 0.0007 & 0.1189 & 0 & 0 & 0 \\ \cline{2-11}  & std & 8.4406 \(\times 10^{-3}\) & 3.1031 \(\times 10^{-3}\) & 2.7804 \(\times 10^{-3}\) & 1.9800 \(\times 10^{-3}\) & 1.2606 \(\times 10^{-3}\) & 5.9991 \(\times 10^{-3}\) & 0 & 0 & 0 \\ \hline \multirow{2}{*}{F} & Ave & 6.0070 \(\times 10^{-3}\) & 0.0880 & 0.0362 & 0.0013 & 0.0302 & 1.7241 \(\times 10^{-3}\) & 0 & 0.0881 \(\times 10^{-3}\) & 0.081 \(\times 10^{-3}\) \\ \cline{2-11}  & std & 4.6547 \(\times 10^{-3}\) & 1.8619 \(\times 10^{-3}\) & 6.2038 \(\times 10^{-3}\) & 7.2799 \(\times 10^{-3}\) & 4.3444 \(\times 10^{-3}\) & 1.8347 \(\times 10^{-3}\) & 1.5464 \(\times 10^{-3}\) & 0.0862 & 7.1455 \(\times 10^{-3}\)

### Statistical Testing

The optimization of standard test functions was presented as the average and standard deviation of the best solutions. However, these results alone are not enough to guarantee the superiority of the proposed algorithm. Even after twenty independent performances, this superiority may occur by chance despite its low probability. Therefore, the Friedman rank test is used to statistically evaluate the algorithms and further analyze the optimization results. The Friedman rank test is a non-parametric statistical test developed by [PERSON]. Nonparametric means the test does not assume data comes from a particular distribution. The procedure involves ranking each row (or block) together, then considering the values of ranks by columns [90]. The steps for implementing the Friedman rank test are as follows:

Start. Step1: Determine the results of different groups. Step2: Rank each row of results based on the best result (here from 1 to 9). Step3: Calculate the sum of the ranks of each column for different algorithms. Step4: Determine the strongest algorithm to the weakest algorithm based on the sum of the ranks of each column. End. The [PERSON] rank test results for all three different objective functions: unimodal, multimodal, and fixed-dimension multimodal objective functions are presented in Table 6. Based on the results presented, for all three types of objective functions, the proposed GBUO has the first rank compared to other optimization algorithms. The overall results on all the objective functions (\(F_{1}\)-\(F_{23}\)) show that GBUO is significantly superior to other algorithms. ## 4 Discussion

Optimization algorithms based on random scanning of the search space have been widely used by researchers for solving optimization problems. Exploitation and exploration capabilities are two important indicators in the analysis of optimization algorithms.