## 5 Numerical Results
### Investigation of the Real Performance of IALO and ALO  
The results from the suggested method and ALO method are summarized in Table 8. In all the cases of Table 8, the best cost of the IALO was always smaller than that of the ALO. Moreover, the time to run the program of the IALO method was always lower than that of the ALO method for all cases. The _ic_ of 0.43% was the highest improvement of IALO over ALO, corresponding to the case of 15-units with POZ for a 2650 MW load demand. In contrast, 0.0001% was the smallest improvement of the suggested IALO over the ALO, corresponding to the case of a 20-unit system considering transmission loses for a 2500 MW load demand. However, in this case, ALO used \(N_{\text{/gs}}=9000\) while IALO used only half of that quantity, i.e., 4500. To demonstrate the superiority of the IALO more persuasively, the convergence characteristics of the IALO and ALO are plotted in Figures 2-6 for different cases from the sections above. Through the figures, it can be seen that the IALO converged faster than the ALO after half of the number of iterations, or even fewer than half. For instance, Figure 2 shows the IALO reached a solution close to the best solution at the fifteenth iteration, whereas the solution of the ALO at the fiftieth iteration was far from the obtained solution of the IALO. In Figure 3, the solution of the IALO at the sixtieth iteration was much more effective than the solution of the ALO at the 200 th iteration. Furthermore, Figures 4-6 indicate that the IALO was at least three times faster than the ALO, and the best solution of the ALO was much worse that the solution of the IALO at one third the number of iterations. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{4}{c}{**Best Cost (\$/h)**} \\ \cline{2-5}  & **15** & **30** & **60** & **90** \\ \hline CSA [29] & 32,544.9704 & 65,084.9949 & 130,170.3949 & 195,258.7847 \\ RCSA [29] & 32,542.5593 & 65,084.9161 & 130,169.9367 & 195,255.9077 \\ ALO & 32,684.0058 & 65,209.0115 & 130,316.2964 & 195,540.6541 \\ IALO & 32,544.9704 & 65,084.9977 & 130,169.8555 & 195,254.8220 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The comparison of results for systems with POZ constraints. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Studied Cases**} & **Method** & **Best Cost (\$/h)** & \(N_{\text{/gs}}\) & _ic (\%)_ \\ \hline
10-unit system with multi-fuel options for a 2400 MW load & ALO & 482.2892 & 1000 & 0.117 \\ IALO & 481.7229 & 1000 & & \\ ALO & 526.9130 & 1000 & 0.128 \\ IALO & 526.2390 & 1000 & & \\ ALO & 574.5782 & 1000 & 0.032 \\ IALO & 574.3917 & 1000 & & \\ ALO & 625.8373 & 1000 & 0.324 \\ IALO & 623.8096 & 1000 & & \\ ALO & 623.9079 & 8000 & 0.012 \\ IALO & 623.8347 & 8000 & & \\ ALO & 624.6170 & 3000 & 0.039 \\ IALO & 624.3747 & 3000 & & \\ ALO & 624.666.647 & 9000 & 0.0001 \\ IALO & 624.4566.631 & 4500 & & \\ ALO & 32,684.0057 & 2000 & 0.425 \\ IALO & 32,544.9704 & 2000 & & \\ ALO & 65,209.0114 & 20,000 & 0.190 \\ IALO & 65,084.9976 & 20,000 & & \\ ALO & 130,316.2964 & 80,000 & 0.112 \\ IALO & 130,169.8555 & 80,000 & & \\ ALO & 195,540.6541 & 160,000 & 0.146 \\ IALO & 195,254.8220 & 160,000 & & \\ ALO & 5001.8871 & 80,000 & 0.194 \\ IALO & 4992.1712 & 80,000 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The comparison of the suggested method and the ALO method. In order to investigate the efficiency of finding optimal solutions, the fitness of the best solution of successful runs was also plotted in Figures 7-11 for comparison. Fitness values of the IALO are black points while those of the ALO are red points. All the figures have the same characteristic that red points have very large fluctuations, but black points have very tiny deviations, even though they are on a line with the same fitness value. This manner implies that approximately all runs of the IALO are close to the best solution, but the ALO fails to reach the same achievement.