## 3 Application of the ORW Optimization Model--Case Study
### Input Data for the Case Study  
The physical and mechanical properties of soils can vary considerably due to natural processes. To account for these variations, statistical analyses and Markov Chain Monte Carlo (MCMC) simulations were performed according to the theory of [PERSON] [28; 29]. This approach enables the development of a representative soil model based on data from site samples, which are then integrated into the optimization model. To accurately define the random variables, 15 soil samples were taken from five different boreholes on the site, with samples taken at different depths. Laboratory tests, including a triaxial shear strength test, were performed to determine the effective shear angle (\(\varphi^{\prime}_{k}\)) and cohesion (\(c^{\prime}_{k}\)). In addition, the unit weight of soil (\(\gamma_{k}\)) was measured indirectly while performing the mentioned laboratory test. A total of 15 measurements were performed for each soil property, which are listed in Table 2. This comprehensive sampling and testing procedure ensures that the soil model accurately reflects the variability of site conditions. The large number of soil samples and the high-quality laboratory tests that have been performed provide reliable preliminary information on the soil properties of the site. These data are used to create a preliminary distribution for the soil properties. Given the adequacy of the laboratory data, a normal distribution, characterized by its mean and standard deviation, was chosen for the prior construction. The summarized prior information about the soil is presented in Table 3. The coefficients of variation (COV) values were obtained according to the theory provided by several authors [30; 31; 32; 33]. \begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \hline
**Random Variable** & \multicolumn{3}{l}{**Value Estimated with Laboratory Tests and Field Tests**} & & & & & & & \\ \hline \(\varphi^{\prime}_{k}\) (\({}^{\circ}\)) & 34.35 & 34.46 & 34.67 & 34.75 & 35.21 & 35.71 & 35.81 & 35.97 & 36.06 & 36.11 \\ \(c^{\prime}_{k}\) (kPa) & 36.33 & 36.39 & 36.81 & 37.26 & 37.76 & & & & & \\ \(c^{\prime}_{k}\) (kPa) & 5.71 & 5.79 & 5.85 & 5.90 & 5.92 & 5.96 & 5.97 & 5.98 & 6.07 & 6.13 \\ \(c^{\prime}_{k}\) (kPa) & 6.22 & 6.36 & 6.37 & 6.44 & 6.48 & & & & & \\ \(\gamma_{k}\) (kN/m\({}^{3}\)) & 19.61 & 19.79 & 19.91 & 19.92 & 19.98 & 20.54 & 20.68 & 20.84 & 20.99 & 21.02 \\ \(\gamma_{k}\) (kN/m\({}^{3}\)) & 21.34 & 21.49 & 21.55 & 21.78 & 21.82 & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of the laboratory test that were carried out on the soil samples. \begin{table}
\begin{tabular}{c l l l l l} \hline \hline
**Random Variables** & **Min. Value** & **Max. Value** & **COV [\%]** & \
\begin{tabular}{c} **Min. Standard** \\ **Deviation** \\ \end{tabular} &
\begin{tabular}{c} **Max. Standard** \\ **Deviation** \\ \end{tabular} \\ \hline \(\varphi^{\prime}_{k}\) (\({}^{\circ}\)) & 34.35 & 37.76 & 5–10 & 1.72 & 3.78 \\ \(c^{\prime}_{k}\) (kPa) & 5.71 & 6.48 & 20–30 & 1.14 & 1.95 \\ \(\gamma_{k}\) (kN/m\({}^{3}\)) & 19.61 & 21.82 & 4–6 & 0.78 & 1.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Informative prior information of the random variables. To optimize the design with respect to a certain probability of failure, the data were processed using an Excel plugin that uses the Bayesian sampling algorithm known as \"BEST\" (Bayesian Equivalent Sample Toolkit) [27]. This method utilizes Bayesian theory, which provides a framework for updating the probability distribution of a parameter as more evidence or data becomes available. The optimization also used Monte Carlo simulations in combination with Markov Chain Monte Carlo (MCMC) methods to generate samples. Monte Carlo simulations are used to perform numerical experiments to estimate the behavior of complex systems through random sampling, while MCMC methods facilitate the sampling of random variables from probability distributions using a Markov chain approach.