## Further Tests Using Nonlinear Constrained Engineering Applications  
After testing the methods with fifteen unconstrained high-dimensional benchmark functions, four non-smooth constrained engineering design GO problems were selected to examine the effectiveness of these methods in practice and identify their advantages and drawbacks. These comprised four structural design applications, involving the Welded Beam Design (WBD), the Tension/Compression Spring Design (TSD), the Pressure Vessel Design (PVD) and the Speed Reducer Design (SRD) [1]. Several authors in [15; 95; 96; 97; 98] have solved these problems using different methods such as the hybrid PSO-GA algorithm, the artificial bee colony, etc. All four of these selected constrained engineering problems have been considered to be computationally-expensive black-box problems. The problems chosen have design variables ranging from 3 to 7 with 4, 7, 4 and 11 constraints, respectively. An average of 25 independent runs were performed per problem, with a total of 20,000 NEF in each run. All algorithms were tested with the same stopping conditions to ensure that their convergence rate, efficiency and computation time would be examined using the same parameters. Figure 11 shows the welded beam design problem where ABC started the search with lower function values; however, it required additional time to converge, along with the algorithm, to the global minimum.  
In relation to the tension/compression design problem, Figure 12 shows that GWO outperformed other optimization algorithms, converging to global solution after only a few NFE. In addition, when compared to the other algorithms, GWO converged to global optimal with the least computation time and yielded better minimum values. In the pressure vessel problem, FPA had the best performance, with the lowest standard deviation value. FPA yielded the best global optimum function value, taking more computation time to converge in comparison to BA which had the lowest CPU time.  
Figure 13 compares the required CPU time in dealing with the four selected constrained optimization problems. The obtained statistical results were analysed and compared to find out the performance of the algorithms when dealing with constrained problems. The results in Table 5 have shown that:  
* ABC consistently requires lower NFE than the other algorithms.
* GWO performs well in TSD, WBD and PVD cases, but it seems that the GWO needs more iterations to reach the exact global solution in the SRD case.  
Figure 11: The welded beam problem results.  
Figure 12: The tension/compression spring problem results.  
* Results demonstrate that the other tested algorithms are sensitive when the number of variables increases, but they can provide competitive search efficiency in some problems.  
We conclude that there is no specific superior optimization algorithm that can perform well for all engineering design problems. Clearly, an algorithm may work well for a specific engineering design problem and fail on another problem. The statistical results obtained are shown in Table 5.  
Figure 13: Required CPU time by each algorithm for all constrained problems.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multirow{2}{*}{**Prob.**} & \multicolumn{2}{c}{**Analytical**} & \multicolumn{2}{c}{**\(\mathbf{x_{1}}\)**} & \multicolumn{2}{c}{**\(\mathbf{x_{2}}\)**} & \multicolumn{2}{c}{**\(\mathbf{x_{3}}\)**} & \multicolumn{2}{c}{**\(\mathbf{x_{4}}\)**} & \multicolumn{2}{c}{**\(\mathbf{x_{5}}\)**} & \multicolumn{2}{c}{**\(\mathbf{x_{6}}\)**} & \multicolumn{2}{c}{**\(\mathbf{x_{7}}\)**} & **Obtained F** & **NFE** & **CPU Time** \\ \hline \multirow{4}{*}{_BA_} & TSD & 0.012665 & 0.05123 & 0.34582 & 13.5571 & - & - & - & 0.