## 4 Optimization Problem
### Results and Discussion  
The results of optimization were obtained by the use of random tool, subproblem approximation method and first-order method with various settings. The variants accepted for the comparison are summarized in Table 5. The best results of the minimized objective function, which was the rail wheel mass, are presented in Figure 7. The number of iteration loops to obtain the feasible solution is shown in Figure 8. Concerning the use of ER8 steel, the best optimization result was obtained applying first-order method with the resulted rail wheel mass of 38.1 kg. In turn, for the use of subproblem approximation method the obtained mass was 0.5%-5.5% higher. However, the number of iteration loops was significantly lower. Regarding to the use of 42 CrMo4 steel, the best optimization result was achieved applying subproblem approximation method preceded by the use of random tool with 15 iterations. The resulting rail wheel mass of 33.9 kg was obtained. \begin{table}
\begin{tabular}{c c} \hline R5 + Sub & subproblem approximation method preceded by the use of random tool with 5 iterations \\ \hline R15 + Sub & subproblem approximation method preceded by the use of random tool with 15 iterations \\ \hline R30 + Sub & subproblem approximation method preceded by the use of random tool with 30 iterations \\ \hline First & first-order method with an initial point at the upper limit of the design variables \\ \hline \end{tabular}
\end{table} 
Table 5: Optimization methods variants. Concerning the optimization using the subproblem approximation method, it was expected that increasing the initial number of random results should lead to a better solution. However, as it could be noticed for optimization of wheel made of 42 CrMo4 steel, such a consequence did not occur. Instead, the mass obtained for the optimization case R30 + Sub was higher than for the case R15 + Sub, which requires an explanation. This could be caused by insufficient number of feasible initial data or by an objective function tolerance that affected the termination of calculations. During the optimization process, the first approximation of objective function was made on the basis of initial random set generated and the next approximations also included results of optimization iterations. However, due to strict constrains on the independent state variables, all initial random results were infeasible (the restrictions on state variables were not met) mainly due to constraints on \(\Delta\sigma\). For all reported cases, the same software default solution convergence criteria were accepted. The convergence of the solution was obtained when two consecutive results were feasible and the difference in the value of the objective function was less than accepted tolerance equal to 0.01 of the current set. Further investigations--including the impact of the feasible initial results number and the range of objective function tolerance that terminates calculations--should be developed to indicate the relevant cause of the lack of result improvement for the case of R30 + Sub. The optimal geometry of rail wheel profile is shown in Figure 9, whereas design and state variables were summarized in Tables 6 and 7. Comparing the best optimization results for both considered materials, the mass of rail wheel made of 42 CrMo4 steel was over 11% lower. Figure 8: Number of iteration loops for various variants of optimization method. Figure 7: Optimized rail wheel mass for various variants of optimization method. Figure 10 and 11 show the von Mises stress distribution and maximum principal stress distribution for the optimal rail wheel geometry made of 42 CrMo4 steel, which was the best result of the entire analysis. The greatest stress concentration of von Mises reduced stress occurs at the edge of the bottom bolt hole (Figure 10b). However, the safety factor related to the yield strength was \(k=3.2\). Therefore, no plastic deformation occurred. In turn, the highest value of the maximum principal stress occurred above the upper bolt hole (Figure 11b).