## 1 Introduction  
The geometric shape of a wheel profile is very important in the interaction between wheels and rails when the wheels of a railway vehicle contact the rails [1]. Such a geometric factor changes the contact location between a wheel and a rail, as well as the shape of the contact patch, and has an effect on the dynamic performance of the railway vehicle, such as running safety and curve running performance, and on the wear and fatigue between the wheels and rails [2, 3, 4]. Wear and fatigue of the wheel flange not only increase the maintenance costs of railway vehicles, but also may lead to a significant accident, such as derailment. In particular, as the contact pressure generated between a wheel flange and a rail gauge surface in a curved track is higher than that in a straight track, it causes an increase in the wear and fatigue of wheels, and a decrease in the repprofing and replacement cycles, noise increase, and deterioration of riding comfort, etc. [5]. To reduce such problems, although initial wheel profile designs have been performed mostly based on the experience of railway operators, useful results have been obtained recently using the optimal design technique. [PERSON] et al. [6] created a new profile through a dynamic analysis and a contact pressure analysis of an averagely-worn wheel profile and a profile by extension of rail shape. [PERSON] et al. [7, 8] designed an optimized wheel profile by minimizing the difference between the objective rolling radius difference (RRD) function and the actual RRD function using multipoint approximations based on response surface fitting method (MARS). [PERSON] et al. [9] presented a method of creating a new wheel profile using the function of a contact angle between a wheel and a rail. [PERSON] et al. [10] presented an optimized algorithm capable of performing efficient calculations with a small number of design variables when compared to other papers that use an objective RRD for lateral displacement. [PERSON] et al. [11] presented a method of optimizing a railway wheel profile using the vertical direction interval of the wheel and the rail at the contact point of a wheel and a rail. Although most of the existing studies on wheel profile optimization have been conducted using a single objective function, wear and fatigue occur at the same time in most wheel damage cases. To reduce wear and fatigue at the same time, optimization should be carried out while setting up a multi-objective function design problem and using an algorithm suitable for it. In the case of a multi-objective function design problem, if the multi-objective function has characteristics that are in conflict with each other, it is almost impossible to find the unique optimum solution that satisfies all objective functions [12]. As a way of solving this problem, a method of using a Pareto-optimal Solution is widely employed for the optimization of a multi-objective function, and this method has an advantage that the designer may find as many optimum solutions as needed within the design domain. The multi-objective optimization using various meta-models was efficiently applied to several practical design problems, such as fire resistance design, offshore installation vessel design, and automatic ocean salt collector design [13; 14; 15; 16]. For a vehicle to safely and smoothly run on rails, fatigue and wear damage should be minimized and, at the same time, the performance functions, such as running safety, safety against derailment, and the contact stress between rails and wheels, should satisfy appropriate design limits. In the present study, we intended to reduce the wear and fatigue generated on wheels by setting a multi-objective function design problem and using an algorithm to secure the curved section running safety of city trains.