## 2 Methodology
### Maintenance  
Wheel wear rate was calculated using 150 rail cars as a sample. Wheel measurements were evaluated during the one-year period in which hopper cars ran at approximately 100,000 km. However, it should be noted that the travel distance used for the rate was the actual one for each car in the sample. \[\text{Wear Rate}=\left|\frac{f_{\text{final}}-f_{\text{initial}}}{km_{\text{ final}}-km_{\text{initial}}}\right| \tag{2}\]  
Wear Rate: wheel flange wear rate (mm/km);  
\(f_{\text{final}}\): final flange thickness (mm);  
\(f_{\text{init}}\): initial flange thickness (mm). This approach considers the friction wedge to have a greater correlation with the bogie performance. Therefore, friction wedge height was used as a parameter to evaluate wheel wear behavior. Figure 5 shows the increase in the friction wedge height. VLI's maintenance parameter limit was fixed in accordance with the [16] AAR standard criteria. These criteria focus on railway safety and usually do not establish criteria for performance analysis. The model's purpose was to define additional maintenance parameters for the hopper cars' fleet, considering the maintenance costs for bogies and wheels, in addition to the cost of a new wheel. The mathematical model that was proposed to solve this problem is based on integer programming, as represented by expressions (1)-(7). Set V represents the wagons (index \(i\)), and set N represents the years (index \(j\)). The used parameters were the cost of bogie maintenance \(C_{1}\), the cost of turning the wheels \(C_{2}\), the cost of replacing a new wheel \(C_{3}\), and \(C_{4}\), the cost of doing nothing. In addition, the decision variables are binary, where \(t_{ij}=1\) if bogie maintenance is performed on wagon \(i\) in year \(j\) and 0 otherwise; \(r_{ij}=1\) if wagon wheel turning is performed in year \(j\) and 0 otherwise; \(s_{ij}=1\) if the wheel is replaced on wagon \(i\) in year \(j\); and \(n_{ij}=1\) if nothing is performed on wagon \(i\) in year \(j\), and 0 otherwise. To project the flange thickness (f), the wear rates were parameterized as \"taxam\" and \"taxaa\"; these represent the wear rate of bogie maintenance at the current rate of the car. By design, the AAR-1B narrow flange thickness (fn) was 32 mm, and the limit (fl) was 21 mm, according to the VLI's manual. The costs used in the modeling are defined below:  
Figure 4: Methodology scheme. Figure 5: Friction wedge rise [19]. \(C_{1}\): cost of fixing the bogie (BRL 3175.00);  
\(C_{2}\): turning cost (BRL 304.00);  
\(C_{3}\): wheelset average unit price (BRL 7290.00);  
\(C_{4}\): doing nothing (BRL 0.00). \[\text{Minimize }z=\sum_{i\in V}\sum_{j\in N}C_{1}t_{ij}+\sum_{i\in V}\sum_{j\in N }C_{2}r_{ij}+\sum_{i\in V}\sum_{j\in N}C_{3}s_{ij}+\sum_{i\in V}\sum_{j\in N}C_ {4}n_{ij} \tag{3}\]  
\[t_{ij}+r_{ij}+s_{ij}\leq 1\quad\forall i,j\;inN \tag{4}\]  
\[r_{ij}+s_{ij}\leq 1\quad\forall i,j\in N \tag{5}\]  
\[\sum_{j\in N}r_{ij}\leq 2 \tag{6}\]  
\[(f_{i-1}-\text{taxam})\times t_{ij}+(f_{i-1}\times\text{taxaa})\times n_{ij}+ f_{n}\times r_{ij}+f_{n}\times s_{ij}\geq f_{l}\forall i,j\in N \tag{7}\]  
\[(f_{i-1}-\text{taxam})\times t_{ij}+(f_{j-1}\times\text{taxaa})\times n_{ij}+ f_{n}\times r_{ij}+f_{n}\times s_{ij}\leq f_{n}\forall i,j\in N \tag{8}\]  
Constraint (2) guarantees that the wheelset cannot be turned and replaced at the same time in the same year. Constraint (3) states that the wheel cannot be turned more than twice, since it is a wheel with three lives (including the initial one). Constraints (4) and (5) calculate the flange thickness, a result of the rate and strategies used, and compare this with the lower and upper limits. The first is based on operational safety, and the second is based on the maximum thickness for this wheel design. The variable \"ratea\" is the value of the regression rate. Analogously, the \"taxam\" is calculated and multiplied by the average total km traveled per year. For the post-maintenance condition of the bogie, the lowest regression rate is considered.