## 2 Design and Methodology
New (\(X_{i,new}\)) values are generated randomly by the multiplication difference of design variable limits and the pitch adjusting rate (PAR) while the process is loading. The selection of the proper way to evaluate the new vector is based on the value of HMCR. Correspondingly, the generation of a random value takes place, and if the random value becomes less than the value of HMCR, the first way is selected; otherwise, the second way is applied. \[X_{i,new}=X_{i,old}+rnd(0,1)\cdot PAR\cdot(X_{i,max}-X_{i,min}) \tag{8}\]  
Step 4: Comparison of the new vector with the vectors stored in a solution matrix is done. With regard to the objective function, the existing vector is replaced with a new one if the next vector is better than the existing vector defined in the matrix. If not, the solution matrix is saved in its current form. The better solution is evaluated by the comparison of the value obtained from the objective function, and the minimum one is chosen. The design constraint is also taken into consideration during comparisons. In addition to this, the amounts of violations are controlled, and if the violations of the design constraints exist, the solution with minimum violation is chosen as the best solution. Step 5: Controlling the stopping criterion. Iterations are continued until the satisfaction of stopping criterion. This criterion can be identified in different ways, and it has been calculated as the maximum number of iterations in the present study. Three different design parameters were considered to apply the HS algorithm to the cantilever soldier pile problem. The variables are related to the cross-sectional dimensions (\(X_{1}\)) and the reinforcement design of the wall (\(X_{2}\), \(X_{3}\)), and the details of the definitions of variables are given in Table 1.  
The reinforced concrete requirements are defined according to ACI 318-05 code [36]. The ACI 318 code proposes to use equivalent rectangular compressive stress distribution rather than other stress distributions such as parabolic. With the use of equivalent compressive stress distribution, the moment capacity of the cross-section of the cantilever soldier pile can be determined. The critical section of the pile is only checked for reinforcement design. The flexural moment (\(M_{u}\)), the shear force (\(V_{u}\)), the spacing between two bars (\(S\)), the area of reinforcing bars (\(A_{s}\)), and the diameter of bars (\(d_{b}\)) are defined. National and international pre-design guidelines are used to identify the lower and upper limits of design constraints. The design constraints on strength and dimensions are given in \[w(x)=\frac{P_{0}l_{0}^{3}}{EI}a_{up}(x)\hskip 28.452756 ptw(x)=\frac{Ml_{0}^{3}}{ EI}a_{up}(x) \tag{1}\]0 \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Symbol** & **Description of Parameter** \\ \hline Variables in relation to & & \\ _Cross-section dimension_ & \(X_{1}\) & Diameter of soldier pile (\(D\)) \\ \hline Variables in relation to & \(X_{2}\) & Diameter of reinforcing bars of soldier pile (\(\phi_{p}\)) \\ _reinforced concrete design_ & \(X_{3}\) & Number of reinforcing bars of soldier pile \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definitions of wall design variables. \begin{table}
\begin{tabular}{c c} \hline \hline
**Description** & **Constraints** \\ \hline Flexural strength capacities of critical sections (\(M_{d}\)) & \(g_{1}(X):M_{d}\geq M_{u}\) \\ Shear strength capacities of critical sections (\(V_{d}\)) & \(g_{2}(X):V_{d}\geq V_{u}\) \\ Minimum reinforcement areas of critical sections (\(A_{smin}\)) & \(g_{3}(X):A_{s}\geq A_{smin}\) \\ Maximum reinforcement areas of critical sections (\(A_{smax}\)) & \(g_{4}(X):A_{s}\leq A_{smax}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design constraints on strength and dimensions. The supposed objective function of the design includes the unit cost of concrete \(C_{c}\), volume of concrete \(V_{c}\), unit cost of reinforcing bars \(C_{s}\), the unit weight of reinforcing bars \(W_{s}\) items.