# Minimization of Material Waste Through Maintenance Interval Optimization in Transport Systems
## 3 Materials and Methods
### Example Requirements According to the Tsi/dsu
This approach allows for the inclusion of real operating conditions, such as route changes, weather conditions, or vehicle load, ensuring more precise mileage estimation. 2. Probability Distribution Application  
Average daily mileage \(\overline{M}_{daily}\) can also be modeled as a random variable that follows a specific probability distribution, such as a normal or log-normal distribution. In this approach, the average daily mileage \(\overline{M}_{daily}\) is variable and can be described as:  
\[\overline{M}_{daily}\sim\mathcal{N}\left(\mu,\sigma^{2}\right) \tag{5}\]  
where \(\mu\) is the average daily mileage, and \(\sigma^{2}\) is the standard deviation, which describes variability. Using Monte Carlo simulation, daily values of \(\overline{M}_{daily}\) can be generated and then summed to determine \(\hat{M}(t)\). This approach takes into account uncertainty and variability in vehicle usage, resulting in more realistic forecasts. 3. Using Artificial Intelligence (AI)  
Artificial intelligence models, such as neural networks (e.g., LSTM--Long Short-Term Memory), can also be used to predict daily mileage \(\hat{M}(t)\). The AI model is trained on historical mileage data, which include information about routes, weather conditions, peak hours, etc. For prediction, a daily mileage simple feedforward ANN network was prepared. We use for this the Machine Learning extension in Visual Studio. The tested structure consists between \(2\div 5\) hidden layers. In each layer were from 2 to 15 neurons. The best results were achieved for a tangensoidonal function of activation in first three hidden layers and linear for the output one. The numbers of neurons in the layers were 4, 10, and 6 in the last one. The training parameters in Matlab were: maximum 1000 epochs, \(1\times 10^{-5}\)minimum gradient, 6 max fail, 0.001 mu. The structure of the network used is presented in Figure 2.  
After training, the model predicts daily mileage \(\Delta M\) for each day:  
\[M_{i}=f_{AI}(x_{i}) \tag{6}\]  
where \(x_{i}\) is a set of features describing day \(i\), such as weather, type of route, number of stops, and \(f_{AI}\) is the trained prediction model. Subsequently, as in previous approaches, the predicted mileage \(M(t)\) is calculated. This approach allows for the consideration of many nonlinear relationships between various factors affecting the daily mileage of a vehicle, leading to more accurate forecasts. 4. ARIMA Time Series Regression (AutoRegressive Integrated Moving Average)  
The ARIMA statistical model can also be used to predict daily mileage based on historical data. The ARIMA model, described by autoregressive and moving average equations, calculates the value of daily mileage as a function of previous values:  
\[M_{i}=c+\sum_{j=1}^{p}\phi_{j}\Delta M_{i-j}+\epsilon_{i}+\sum_{k=1}^{q}\theta _{k}\epsilon_{i-k} \tag{7}\]  
where \(\phi_{j}\) and \(\theta_{k}\) are the model coefficients, \(c\) is a constant, and \(\epsilon_{i}\) is the random error. Each of these methods has its advantages, and their application depends on data availability, required accuracy, and implementation capabilities. The choice of method should be tailored to the specific needs of the transport operator and the type of vehicles subject to maintenance. Figures 2 and 3 present the general principle of the algorithm for predicting the technical review date. In this example, the hierarchy of technical reviews/repairs, starting from the most extensive, is as follows: P5, P4, P3, P2, P1. Figure 1:0 Technical review/repair dates (P4) by algorithm part 1. Figure 1:1 The structure of the best ANN examined. Initially, the technical review with the largest scope for the given rail vehicle, i.e., the P5 technical review, is considered. This is determined based on the DSU data. Subsequently, the preliminary dates for P4 technical reviews are calculated according to the DSU. The last date of calculating the date is before the P5 technical review (Figure 3).