# Joint Maintenance Strategy Optimization for Railway
## 2 Problem Statement  
The contact and action between the rolling stock and the rail are carried out through the bogie. The bogie supports the entire train body and enables the train to run fast on the rail. The structure of the bogie is shown in Figure 1. The bogie is mainly composed of the frame, wheelset, axle box, primary suspension, secondary suspension, traction motor and braking unit. The wheelset is an essential part of the bogie and transfers the vehicle weight to the rails directly. Traction and braking force are generated by wheel-rail adhesion, and the vehicle is guided on the rail. The dimensions of important structures of the wheel need to be measured during maintenance. Referring to the standard [36], the wheel profile and main parameters are shown in Figure 2. The wheel diameter is donated as \(D\). The thickness and height of the flange are indicated as \(Sd\) and \(Sh\), respectively. \(Qr\) stands for the flange gradient. The wheelset is one of the main components of the bogie and also one of the most severely worn parts of the vehicle [37]. For rail transit, the wear of the wheel flange mainly occurs in the curved section. The outer flange of the wheel is in contact with the inner side of the outer rail, and the reactive force on the rail leads to flange wear. Wheel diameter wear is mainly caused by contact with the rail during operation. Moreover, the interaction between the brake shoe and the wheel when the train is braking also causes wear, which is more severe for vehicles that frequently brake and stop [38]. Wheel diameter wear rate and flange thickness, as well as flange thickness wear rate and diameter, are independent of each other [14; 35; 38]. The wear of the wheel flange and the diameter can be regarded as uncorrelated. Wheel tread scratches are generally caused by wheels sliding on the rail. Strong friction between the tread and the rail generates a high temperature. It is easy to cause small pieces of metal on the tread to fall off or lift up and to cause the tread to spall after the temperature cools down. Figure 1: The bogie structure. Different vehicle models have strict regulations on wheel parameters. It is required that the vehicle wheelsets of railway operation companies are publicly available, and details are shown in Table 1[39]. Wheel flange and diameter dimensions decrease with wear during operation. Therefore, wheelset parameters are inspected periodically to ensure the safe operation of the train. Reprofiling is performed when the wheel flange thickness is smaller than the specified requirements. Reprofiling leads to increased wheel flange thickness and reduced wheel diameter [34; 37; 40], as shown in Figure 3. Preventive maintenance is adopted to the wheel before reaching the threshold to avoid derailment. To facilitate the description and research of the problem, the following assumptions are given. 1. The wear of the wheel flange and diameter is caused by external shocks, and they are uncorrelated with each other. 2. In this study, it is assumed that the depot is not equipped to automatically measure wheelsets. The wheel flange thickness and wheel diameter are measured manually, and the measurements are recorded by maintenance personnel. The internal failures of the wheel are not considered. 3. A maintenance cycle starts at the end of the previous periodic inspection and ends at the start of the next periodic inspection, and the interval of a maintenance cycle is \(\Delta t\). 4. The effect of reprofiling is imperfect. Reprofiling only recovers the wheel profile but does not eliminate the failure mechanism. After reprofiling, the condition of the wheel is between'repaired as new' and'repaired as old'. Based on the wear mechanisms of the wheel flange and diameter, it is assumed that the external shocks on them are uncorrelated. For many depots, manual inspection is required due to the high cost of automatic measuring equipment and the problem of measurement