## 2 Design and Methodology
In the case of an acting single horizontal force or moment at the edge of the beam (Figure 1c), with the use of the constant and variable coefficient of soil reaction through depth (\(K_{S}\)), the change of deflection, rotation, bending moment, and shear force for different depths can be calculated by Equations (1)-(4). \[w(x)=\frac{P_{0}l_{0}^{3}}{EI}a_{up}(x)\hskip 28.452756 ptw(x)=\frac{Ml_{0}^{3}}{ EI}a_{up}(x) \tag{1}\]  
\[Q(x)=\frac{P_{0}l_{0}^{2}}{EI}a_{tp}(x)\hskip 28.452756 ptQ(x)=\frac{Ml_{0}^{2}}{ EI}a_{tp}(x) \tag{2}\]  
\[M(x)=\frac{P_{0}l_{0}}{EI}a_{mp}(x)\hskip 28.452756 ptM(x)=\frac{Ml_{0}}{EI}a_{mm}(x) \tag{3}\]  
\[V(x)=P_{0}a_{up}(x)\hskip 28.452756 ptV(x)=\frac{M_{0}}{l_{0}}a_{pm}(x) \tag{4}\]  
The characteristic length of the pile is described by \(l_{0}\); it can be calculated by the use of Equation (5) for the constant coefficient of reaction, and it can be calculated by using Equation (6) for a linear change of the coefficient of reaction through the depth. The embedment depth of the pile can be determined with the \"\(nl_{0}\)\" equation for soldier piles to behave similar to beam-supported elastic soil [32]. The total length of the pile can be calculated by the sum of excavation depth and embedment depth. \[l_{0}^{4}=4 EI/k_{h}t \tag{5}\]  
\[l_{0}^{5}=4 EI/\left(\frac{kb}{x}\right) \tag{6}\]  
Depending on the above-mentioned assumption to design cantilever soldier pile walls, the first step of the present study is decided and multivariate parametric analyses are conducted to obtain the effects of design parameters. This method is applied to numeric analyses that are carried out by using Matlab software, the harmony search algorithm (HS), and the effects of the change of excavation depth, shear strength angle, and unit weight of soil, external loading condition, and coefficient of soil reaction are investigated. The harmony search algorithm was developed by [PERSON] et al. in 2005 as a meta-heuristic algorithm that is based on the natural musical performance process to search for a perfect state of harmony similar to during jazz improvisation [33]. The Jazz improvisation investigates to obtain musically pleasing harmony as a perfect state that can be determined by an aesthetic standard, and the optimization process researches to acquire a global solution as a perfect state that can be calculated by an objective function [34]. The HS algorithm process can be arranged with five steps. Step 1: A harmony memory is initialized [35]. The definition of design constants, upper and lower limits of design variables, maximum iteration number, and the values of algorithm-specific parameters are done. Three special parameters are used for HS that are named harmony memory size (HMS), harmony memory consideration rate (HMCR), and the t pitch adjustment rate (PAR), respectively. Step 2: A new harmony is improvised [35] from the creation of a harmony vector with the generation of a random value (\(rnd\)(0,1)) within a range. The upper (upper limit (\(X_{i,max}\)) and lower limits (\(X_{i,min}\))) are defined for each design variable (\(X_{i}\)) (Equation (7)). With the usage of the constants and design variables of the problem, the motion equation is solved; following, the solution of the objective function is procured, and the result of this objective function is stored in harmony vector. \[X_{i}=X_{i,min}+rnd(0,1)\cdot(X_{i,max}-X_{i,min}) \tag{7}\]This process is repetitive, as much as the size of the harmony memory and every harmony vector is stored in a matrix named the initial solution matrix. Step 3: The iteration process is started, and the New Harmony vector is created. This creation can be done in two ways based on the algorithm rules. It is possible to generate the design variables randomly within the identified upper and lower limits, or a new vector can be activated using a chosen vector (\(X_{i,old}\)) from the solution matrix (Equation (8)).