## 3 Multi-Objective Design Optimization of Wheel Profile  
A design change of wheel profile not only reduces wear and fatigue that occur on the wheels, but also changes the performance, such as the contact pressure between wheels and the rails, creepage, and the dynamic behaviors of the vehicle. In the present study, derailment, lateral pressure, and vertical force are important in terms of the railway vehicle safety, and have been set as the constraints in relation to the wheel profile optimization to guarantee the safety of the railway vehicle's dynamic performance [23]. Formulation of wheel profile optimization design for city trains has been defined in such a way the  
Figure 1: City train wheel experiencing severe wear. Figure 2: A worn wheel profile after running the distance of approximately 10,000 km. objective functions of wear and fatigue are minimized at the same time, while the design performance constraints of derailment, lateral pressure, and vertical force are satisfied. The objective function of wear is defined as wear number, and is calculated from the creep forces and the creepages in the longitudinal, lateral, and spin directions at the contact point between a wheel and a rail. The wear model [24] and the contact energy wear model [25] of [PERSON] are mostly used as wear estimation methods. The contact energy wear model is a wear theory based on the simple assumption that the frictional energy consumed by the creepage at the contact point between a wheel and a rail removes a certain amount of the metal material on the contact surface. A wear number is the energy diffused when friction occurs and is calculated as the product of creep force and creepage. The unit of wear number is Newton (N), and it allows relative comparison of the frictional energy diffused at the contact point. The objective function of fatigue is defined as a surface fatigue index. The surface fatigue index of the surface on which fatigue starts is based on a shake down map, and the surface fatigue index is dimensionless [26]. Derailment among the design performance constraints is evaluated with a derailment ratio, and is defined using [PERSON]'s formula, which is widely used as the standard for evaluation of running safety [27]. The force acting between a wheel and a rail is divided into the vertical force of wheel and the lateral force of wheel. A derailment ratio is expressed as the ratio of the lateral force (\(Y\)) of the wheel and the vertical force (\(Q\)) of the wheel acting between a wheel and a rail. The tolerable limit of derailment ratio is 0.8 in general. A derailment ratio is defined using the following expression:  
\[\frac{Y}{Q}=\frac{\tan\alpha-\mu}{1+\mu\tan\alpha} \tag{1}\]  
where \(Q\) is vertical force on wheel, \(Y\) is lateral force on wheel, \(\mu\) is friction coefficient, and \(\alpha\) is angle between the wheel flange and horizontal line. Multi-objective function optimization design formulation of the wheel profile for a city train is defined as follows:  
\[\textit{Minimize}\ f_{1}=\text{Wear Number}=T_{x}\gamma_{x}+T_{y}\gamma_{y}+T_{ \phi}\gamma_{\phi} \tag{2}\]  
\[f_{2}=\text{Fatigue Index}=\mu-\frac{2\pi abk}{3F_{z}},\ \mu=\frac{\sqrt{F_{x} {}^{2}+F_{y}{}^{2}}}{F_{z}} \tag{3}\]  
\[\textit{Subject to}\ g_{1}(\text{derailment ratio})=\frac{Y}{Q}\leq 0.8 \tag{4}\]  
\[g_{2}(\text{lateral force})=Y\leq 32.6\ \text{kN} \tag{5}\]  
\[g_{3}(\text{vertical force})=Q\leq 158.6\ \text{kN} \tag{6}\]  
\[-1.5\leq dx_{1}\leq 1.5\ (\text{mm}) \tag{7}\]  
\[-1.5\leq dx_{2}\leq 1.5\ (\text{mm}) \tag{8}\]  
Figure 2:0  
Figure 2:1  
Figure 2:2  
where \(T_{x}\) is longitudinal creep force, \(\gamma_{x}\) is longitudinal creepage, \(T_{y}\) is lateral creep force, \(\gamma_{y}\) is lateral creepage, \(T_{\phi}\) is spin creep force, \(\gamma_{\phi}\) is lateral creepage, \(\mu\) is traction coefficient, \(a,b\) is semiaxes of the Hertzian contact patch, \(k\) is yield stress on pure shear, \(F_{x}\) is longitudinal force, \(F_{y}\) is lateral creep force, and \(F_{z}\) is normal force.