## 6 Discussion
### The Accuracy with Limited Number of Iterations  
In this experiment, the impact of a limit of the number of iterations as well as the number of function evaluations (NFE) on the accuracy of the results of the algorithms are analyzed. Figures 2-4 show the optimal solution found by BA, CSA, FFA, FPA, ABC and GWO methods versus the iteration number for three computationally-expensive problems. The selected functions are Sphere (\(f_{1}\)), Griewank (\(f_{7}\)) and Cigar (\(f_{14}\)), which represent unimodal, multimodal and tough shape functions. The primary goal is to measure the performance of these algorithms based on different surface topologies/structures. The accuracy is measured by comparing the obtained optimum with the known actual optimum of the function. The evaluated efficiency is based on the convergence rate of these algorithms under the given conditions. The algorithm that yields accurate global solutions in a shorter computation time is the efficient and robust method.  
As illustrated in Figures 2-4 (i.e., Sphere, Griewank and Cigar), GWO, followed by ABC, have shown the best performance among the examined algorithms regardless of the function form used. GWO starts to quickly converge to the global solution within approximately 100 iterations, while the other algorithms require a higher number of iterations to find the region where the global solution may exist. The ABC algorithm also demonstrates its superior efficiency in terms of the accuracy of the solution. This is, perhaps, because of the GWO's and ABC's search mechanism that prevents them from easily getting trapped in local optima. GWO reaches the global solution (\(6.14\times 10^{-56}\), 0 and \(3.51\times 10^{-101}\) respectively) with a very smooth convergence rate, and ABC is the second best (\(5.60\times 10^{-5}\), \(1.96\times 10^{-10}\) and \(2.14\times 10^{-12}\)) as shown in Table 4. On the other hand, BA and FPA show the lowest convergence rates with comparably low efficiencies. It appears that BA and FPA require a higher number of iterations to converge to the global solution. This may be because of the poor exploration ability of BA. In addition, the multidimensional objective problem can affect the convergence rate and the accuracy of the solutions obtained by PFA [17]. This result implies that BA, FPA and also CSA are potentially more efficient in solving low-dimensional computationally-expensive optimization problems.  
Figure 2: Convergence speed for Sphere (\(f_{1}\)) function (50D).  
The performance of any algorithm typically depends upon how well the algorithm can balance the exploitation and exploration mechanisms. With intensive exploitation and poor exploration ability, algorithms such as BA easily get trapped into the local minima, whereas GWO and ABC show a much better performance, especially on the functions with a large number of local minima. CSA and FFA do not perform as well as GWO and ABC in terms of convergence rate and search efficiency and need more iterations to reach the global optima. The poor effectiveness of FFA may be due to the fact that its parameters are fixed during the search process [44] making it more likely to get trapped in local solutions. ABC shows the best performance in all cases except in \(f_{2},f_{4}\) and \(f_{12}\) functions, where GWO is the best performing approach, except in \(f_{13},f_{15}\). Using the Zakharov problem as an example, GWO manages to reach a minimum value of \(3.85\times 10^{-38}\), which can be considered to be zero, whereas ABC finds a local solution of \(105.886\). From the results shown, it is clear that GWO shows a superior performance and robustness in handling high dimensional optimization problems when the number of function evaluations is limited.