# An Improved Novel Global Harmony Search Algorithm Based on Selective Acceptance
## 4 Experimental Results and Analysis
### Two Benchmarks' Engineering Optimization Problems
#### 4.4.2. Welded Beam Design Optimization Problem  
The second problem is the welded beam design engineering problem. In this problem, there are four variables that need to be determined that are the thickness of the weld (\(x_{1}\)), the length of the clamped bar (\(x_{2}\)), the height of the bar (\(x_{3}\)), and the thickness of the bar (\(x_{4}\)). The main purpose of this problem is to minimize the manufacturing cost when seven constraints are satisfied. The constraints include limits on the shear stress, bending stress, buckling load and end deflection, and several size constraints. The welded beam and its design variables are shown in Figure 4. These seven constraints and the objective function of manufacturing cost are defined as follows. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & **d** & **D** & **N** & **Optimal Cost** \\ \hline HHO [26] & 0.05179639 & 0.35930536 & 11.13885900 & 0.0126654 \\ GWO [27] & 0.05169000 & 0.35673700 & 11.28885000 & 0.0126663 \\ RO [28] & 0.05137000 & 0.34909600 & 11.76279000 & 0.0126786 \\ CDE [29] & 0.05160900 & 0.35471400 & 11.41083100 & 0.0126702 \\ CPSO [30] & 0.05172800 & 0.35674400 & 11.24453400 & 0.0126747 \\ GA3 [31] & 0.05198900 & 0.36396500 & 10.89052200 & 0.0126810 \\ ES [32] & 0.05164300 & 0.35536000 & 11.39792600 & 0.0126980 \\ GSA [33] & 0.05027600 & 0.32368000 & 13.52541000 & 0.0127022 \\ MPO [34] & 0.05194946 & 0.36410932 & 10.86482186 & 0.0126669 \\ CWCA [35] & 0.05170910 & 0.35710734 & 11.27082577 & 0.0126716 \\ LAFBA [36] & 0.05166300 & 0.35607400 & 11.33340000 & 0.0126720 \\ HS & 0.05152035 & 0.35266977 & 11.53040649 & 0.0126659 \\ IHS & 0.0512921 & 0.34725895 & 11.86597566 & 0.0126683 \\ SGHS & 0.05149684 & 0.35211045 & 11.56429455 & 0.0126659 \\ NGHS & 0.05148116 & 0.35173679 & 11.58707057 & 0.0126660 \\
**SANGHS** & **0.05162828** & **0.35525732** & **11.37510196** & **0.0126653** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Experimental results in the tension/compression spring design engineering problem. Figure 4: Welded beam design. problems and compared the experimental results with other state-of-the-art metaheuristic algorithms. The experimental results provide two findings that are worth noting. Firstly, it was found that the SANGHS algorithm could result in improved searching performance as opposed to the NGHS algorithm. In the SANGHS algorithm, the selective acceptance mechanism dynamically adjusts the acceptable probability according to the selective acceptance formula. The most important part of the selective acceptance formula is proportion formula, and the main concept of the proportion formula is convergence degree in the HM. Hence, in Equation (6), \(F_{\textit{tworse}}^{k}-F_{\textit{best}}^{k}\) represents the convergence degree of all harmonies in the HM. In the early iterations, all solutions have not yet converged in the HM. Therefore, the value of \(F_{\textit{tworse}}^{k}-F_{\textit{best}}^{k}\) will be a larger value, and then there is a larger acceptable probability to accept a worse harmony. In the next iteration, there is a higher probability to generate a new harmony in a larger truth region. In other words, there is a strong global exploration ability in the early iterations. However, in the late iterations, the value of \(F_{\textit{tworse}}^{k}-F_{\textit{best}}^{k}\) will be a smaller value while all solutions have converged in the HM. Therefore, there is a smaller acceptable probability to accept a worse harmony. In the next iteration, there is a higher probability to generate a new harmony in a smaller truth region. In other words, there is a strong local exploitation ability in the later iterations. According to the experimental results, the proposed proportion mechanism can balance the global exploration and local exploitation ability in the NGHS algorithm.