## 4 Results and Discussion  
In the present study, we used the multi-objective function optimization method to develop a wheel profile that can reduce wear and fatigue of wheel flange at the same time when a train is running in a curved section. As to the multi-objective function optimization algorithm, we employed the NSGA-II (Non-dominated Sorting Algorithm-II) that uses a \begin{table}
\begin{tabular}{c c} \hline \hline
**Specification** & **Value** \\ \hline Mass of car body & 42,100 kg \\ Height of car body center of gravity from rail level & 2300 mm \\ Mass of bogie frame & 3820 kg \\ Height of bogie frame center of gravity from rail level & 450 mm \\ X-stiffness of primary spring & 5400 kN/m \\ Y-stiffness of primary spring & 5400 kN/m \\ Z-stiffness of primary spring & 1350 kN/m \\ Flange back distance & 1354 mm \\ Gauge distance & 1435 mm \\ Wheel profile (initial) & Conical type 1/20 profile \\ Wheel diameter & 860 mm \\ Rail profile & KS60 \\ Rail inclination & 1/40 \\ Wheel load & 68.6 kN \\ Vehicle speed & 70 km/h \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design specification for wheel profile optimization. Figure 5: Procedure of wheel profile design optimization. Pareto-optimal Solution, of which the usefulness has been verified as a result of having been widely used in various fields. NSGA-II is a fast and top class non-dominated sorting method that guarantees independence of the shared parameter designated by the user while maintaining the diversity of the solutions [17]. The performance function was produced by simulating the movement of the vehicle in a curved section using VAMPIRE [18], which is a program dedicated to railway vehicle dynamics, and the wear number and surface fatigue index, which are the objective functions, were calculated. As for the parameters for multi-objective function optimization analysis, the population size was set to 100 individuals over 100 generations, the crossover rate to 0.9, the mutation rate to 0.1, and the relative tournament size was set to 0.5. A Pareto-optimal Solution produced using NSGA-II for the gene algorithm based on multi-objective optimization is shown in Figure 6.  
As shown in Figure 6, the value that corresponds to the minimization of the wear number is the solution of 'wear_opt', and the value that corresponds to the minimization of the surface fatigue index is the solution of 'fatigue_opt'. The design of 'wear and fatigue_opt' is the one selected among the solutions which satisfy both the minimization of wear number and the surface fatigue index. The optimum design variables of the wheel profile are put in order in Table 2, and are compared with the initial design. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Design Specification**} & \multicolumn{5}{c}{**Value Optimized Design Variables**} \\ \cline{2-6}  & \(dx_{1}\) & \(dx_{2}\) & \(dx_{3}\) & \(dx_{4}\) & \(dx_{5}\) \\ \hline Initial design & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ Wear optimum design & \(-0.475\) & \(1.422\) & \(1.066\) & \(0.582\) & \(-0.151\) \\ Fatigue optimum design & 0.179 & \(1.341\) & \(1.065\) & \(0.587\) & \(-0.588\) \\ Wear and fatigue optimum design & 0.811 & \(1.429\) & \(1.157\) & \(0.710\) & \(-0.213\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimized design variables. Figure 6: Pareto optimum results of multi-objective functions. When a simplified wheel shaft consisting of a conical wheel profile, as shown in Figure 4, runs on rails, only one contact point occurs between the wheel and the rail profiles. As the wheel shaft and the track system are symmetrical, if the center positions of the wheel shaft and the track match each other, the rolling radius of the left wheel (\(r_{2}\)) and the rolling radius of the right wheel (\(r_{1}\)) are the same. If the center of the wheel shaft shifts by \(\Delta y\), the rolling radii of the right and left wheels become different from each other due to the conicity of the wheel, at which time a RRD (rolling radius different) occurs.