# Categorization of the Condition of Railway Embarments Using a Multi-Attribute Utility Theory
## 3 Case Study Example
### Results and Discussion  
The terminology used in this section includes the following terms: 'line' as the railway line,'section' as part of the line under investigation, and'subsection' is a 100 m length of the investigated line. Therefore, for the total of 181.1 km of 18 investigated sections on 18 different railway lines, we had 1811 subsections, which were analyzed and represented the alternatives for the implemented multi-attribute utility theory model. Taking into consideration that sections selected for investigation were part of different lines, the analysis for 18 sections were done independently, bearing in mind that maintenance on those sections would also be performed independently. Sections differed from 1.0 km of length, which had 10 alternatives (10 subsections of 100 m length) to the largest one, which was 31.3 km long and consisted of 313 alternatives. This division was done in agreement with the Supervision Centers, since it was rational to separately evaluate these groups, rather than implementing categorization procedure for all the lines together. The division will enable infrastructure managers' better insight into the condition of each section within the relevant line, where a subsequent resource optimization can be conducted. As a first step in implementing the developed methodology in the case study, attributes were evaluated for all alternatives within each section. The quantification values were determined as given by Equations (1)-(5). An example of attribute evaluation is given for a 1000 m long section, divided into 100 m subsections, in Table 5.  
Next, a multi-attribute decision-making problem was expressed by an \(n\times m\) decisional matrix, \(n\) being the number of attributes (5 in our case) and \(m\) being the number of alternatives within one section. Considering the 18 investigated lines, the alternatives were divided into 18 groups (\(k=1\), 2,..., 18), leading to the formation of 18 \(n\times m\) decisional matrices, the smallest one being \(5\times 10\) and the largest one being \(5\times 313\). As a first step in calculating the overall utility function of each alternative \(\mathrm{U_{i}}\big{(}\mathrm{S_{j}^{k}}\big{)}\), the normalized utility functions were determined using Equation (6). For example, the normalized utility function value for the GPR fouling Attribute (\(\mathrm{QV_{C4}}\)) and Alternative (\(\mathrm{S_{5}^{1}}\)) which stands for 'fifth alternative within the first section' was calculated as:  
\[\overline{\mathrm{U_{4}}}\big{(}\mathrm{S_{5}^{1}}\big{)}=\frac{\mathrm{QV_{ C4}}\big{(}\mathrm{S_{5}^{1}}\big{)}}{\sum_{j=1}^{\mathrm{m}}\mathrm{QV_{C4}} \big{(}\mathrm{S_{j}^{1}}\big{)}} \tag{11}\]  
Figure 6: Conduction of GPR investigations. with  
\[\sum_{i=1}^{5}\overline{\mathrm{U}_{i}}\big{(}\mathrm{S}_{5}^{1}\big{)}=1. \tag{12}\]  
Further, to define the importance of each of the five attributes in the overall embankment categorization procedure, a developed questionnaire was used to obtain information from experienced experts working in the railway maintenance sector. To overcome the mentioned subjectivity aspect, a questionnaire was delivered to 12 experts, and the results are given in Table 6.  
Taking into consideration the mean and standard deviation (SD) data from Table 6, the weight of importance for each attribute could be calculated. For example, the importance of ballast fouling Attribute (C\({}_{4}\)) was calculated using Equation (8):  
\[\mathrm{w}_{4}=\frac{\mathrm{QV}_{\mathrm{C}4}}{\sum_{i=1}^{5}\mathrm{QV}_{ \mathrm{C},i}}=\frac{8.70}{9.30+5.60+8.40+8.70+6.30}=0.227. \tag{13}\]