## 3 Multi-Objective Design Optimization of Wheel Profile
It is virtually impossible to find a solution that minimizes all of the objective functions in the process of simultaneously minimizing all of the objective functions. Therefore, NSGA-II-based multi-objective optimization generally selects a solution that is not dominated by other solutions among the recommended candidate solutions, and this solution is called a non-dominated solution. The NSGA-II algorithm is an algorithm for solving a representative multi-objective optimization problem that finds a Pareto-optimal solution classified as a non-dominant solution. Using this algorithm, we found the Pareto-optimal solution classified as the non-dominated solution, which is not dominated by other solutions, among candidate solutions that minimize both the wear and fatigue. The design shape of the track profile contacting wheels is a conical 1/20 profile, and is shown in Figure 2:3  
In a curved section, contact occurs between a wheel flange root or flange and a rail gauge corner. The area where the wheel is seriously damaged is the location of the inflection point of the flange in contact with the rail on the curved track. Therefore, the major inflection points (dx 1 to dx 5) were considered as design variables to search for the optimal profile that minimizes the wear and fatigue of the flange. To reduce the wear and fatigue which occur at a wheel flange root or a flange, five design variables were set so that the flange root and flange shapes of the wheel profile could be modified, as shown in Figure 2:4  
As shown in Figure 4, in order to reduce the fatigue and wear in curved sections while maintaining the dynamic performance, as well as the fatigue and wear characteristics in straight sections, the shape of the contacting wheel tread during driving in straight sections is maintained as it is. The wheel design variables create a profile with PCHIP (piecewise cubic hermite interpolating polynomial) [28], moving in the direction of the profile normal line. To minimize the time it takes for the calculation, some nodes that do not have any effect on the contact between the tread and flange of the profile are fixed. The optimum design procedure for a wheel profile applied to reduce the wheel gauge wear which occurs on sharp curves is shown in Figure 2:5  
Figure 2:6 Rail profile of KS60. As shown in Figure 5, the wheel profile, existing rail profile, track parameters, dynamics parameters of the railway vehicle, and so on, are entered at the first step, and the design domain required for a wheel profile design is set up. At the next step, a new wheel profile is created, and the dynamic behavior of the vehicle is calculated using a dynamics program. At the optimization step, the procedure is closed if the constraints and the objective function are satisfied and, if not satisfied, the procedure is repeated moving to the step of creating a new profile again until the constraints and objective function are satisfied. To analyze the contact behavior of the wheel profile, a dynamic performance analysis is conducted using VAMPIRE [18], which is a program dedicated to railway vehicle dynamics. VAMPIRE allows users to organize mathematical models of nearly all railway vehicle forms based on the multi-body modeling method, and can efficiently evaluate actual performance of railway vehicle dynamics with high accuracy. The NSGA-II algorithm applied in this study first selects the initial population and determines whether the termination condition is satisfied. If the termination condition is not satisfied, an offspring group is created through tournament selection and reproduction, and a new population is created by combining the population and the offspring group. The non-dominated solution is sorted using the fast non-dominating sorting algorithm, and the sorting process is performed through the sorted non-dominating front and the crowding distance calculations. For the selected population, the population which is to evolve into the next generation is determined through crossover and mutation, and fitness is evaluated.