## 3 Methods
### Model Derivation  
In the setting described, the average asset quality just before and just after the assets received a maintenance measure is determined. The values are thereby determined for every asset age for each class, and reactive and preventative measures are considered collectively to generate a larger database of maintenance measures. The evaluation of the maintenance measures using the method described shows that both the condition of an asset just before a maintenance measure and just after evolve approximately logarithmically with their age. Therefore, for the construction of the model, a logarithmic fit of the form  
\[\mathrm{a}\cdot\log\cdot(\mathrm{b}\cdot\mathrm{x}+1) \tag{5}\]  
with \(\mathrm{a}\), \(\mathrm{c}\in\mathrm{R}\), and \(\mathrm{b}\geq 0\) is performed on the corresponding data. Thus, these factors differ for the conditions before and after a maintenance measure. For the fitting, particular consideration must be given to the assets' service life. Service life denotes the age at which an asset is normally supposed to be replaced. However, due to limited financial resources, a replacement cannot always be conducted in time. This means that, in all classes, there are assets that exceed this predefined maximum age. Yet, in line with the intended replacement, the number of data points significantly decreases after the service life, and those assets are in relatively good condition when proportionally considered. Therefore, the assets that survive the service life are often in a disproportionately good condition, because the limited resources are first spent on the replacement of assets that are most in need of renewal. For this reason, only the data of all the assets of a class up to the age of the service life that is valid for this class are used for the logarithmic fit. The results of these fits are depicted in \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Asset** & \multirow{2}{*}{**Asset Class**} & \multicolumn{2}{c}{**Year of**} & \multirow{2}{*}{**Service Life**} & **Date of** & **Reason for** & \multirow{2}{*}{**Score**} \\
**Number** & & & **Construction** & & **Measurement** & & \\ \hline
10388089 & platforms & 1980 & 70 & 22 April 2008 & regular & 60 \\
10388089 & platforms & 1980 & 70 & 7 April 2010 & Completion of reactive & 20 \\
10345863 & elevators & 2005 & 14 & 4 August 2008 & regular & 22 \\
10345863 & elevators & 2005 & 14 & 7 March 2011 & Completion of preventive & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Example of data record (reduced to relevant information). The score of the platform drops by 40 points; that of the elevator drops by 17 points. Figure 3 with the example of pedestrian underpasses. The average asset condition before maintenance is shown in green, whereas the average condition after maintenance is shown in red. In each case, the dots refer to the data, and the lines display the corresponding fit. With the fitted curves describing the average condition before and after an asset receives a maintenance measure, the effect of a maintenance measure can be approximated as the difference between these two conditions for a specific age in each case. In addition to the condition at which a maintenance measure is performed on average and the corresponding effect on the asset score, it is necessary to know with which frequency maintenance measures are normally performed. To determine these frequencies, the share of assets that have received a maintenance measure at a certain age is first calculated. As a reference for this share, the number of all assets that reached the age considered at some point within the investigation period are included, since not every asset is assessed every year. Finally, these shares are cumulated, and the cumulative values are used to determine the ages at which an asset receives a maintenance measure on average [15]. These intervals are determined using the fits along the cumulative maintenance frequencies.