## 4 Application of Methodology for Satellite Images
### Verification of Methodology  
Alternative methods were used to validate experiment results to confirm the dependability of the ranking results of PROMETHEE-IFS. PROMETHEE rankings of each of the six test images are compared to the expert rankings. The experts evaluated the visual quality of each segmentation and ranked segmentation methods from 1 (best) to 5 (worst) in Table 7. The original image was provided for reference. For example, expert 1 rankings for square_1 are 4, 3, 1, 2, and 5; expert 2 rankings for square_1 are 1, 4, 3, 2, and 5, etc. Since the ranks are restricted to a small range of values and rank scores do not contain outliers, the arithmetic mean was used for the average score/rating of all expert evaluations. The average rank value by the PROMETHEE-IFS method is then compared to the average expert rank value in Table 8.  
The ranks created by PROMETHEE-IFS and experts can be categorized as ordinal data, so the rank difference sign in Table 8 does not necessarily have a meaningful interpretation in this context. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\
\begin{tabular}{c} **Image** \\ **Segmentation** \\ **Algorithms** \\ **(Alternatives)** \\ \end{tabular} } & \multicolumn{4}{c}{**Satellite Images**} \\ \cline{2-7}  & **square\_1,** & **desert\_23,** & **beach\_19,** & **port\_19,** & **farmland\_159,** & **river\_237,** \\ \hline \multirow{2}{*}{
\begin{tabular}{c} FCM \\ GMM \\ \end{tabular} } & **PROMETHEE-** & **PROMETHEE-** & **PROMETHEE-** & **PROMETHEE-** & **PROMETHEE-** & **PROMETHEE-** \\  & **IFSScore \(\mid\) Rank** & **IFSScore \(\mid\) Rank** & **IFSScore \(\mid\) Rank** & **IFSScore \(\mid\) Rank** & **IFSScore \(\mid\) Rank** & **IFSScore \(\mid\) Rank** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Intuitionistic fuzzy PROMETHEE method ranking. The best values (first places are in bold). Figure 14: Satellite image “river_237” after segmentation: (**a**) RSSFCA; (**b**) SFFCM; (**c**) FRFCM; (**d**) FCM; (**e**) GMM; (**f**) original color image from AID. The average expert rank can be calculated in several ways. First, calculate the average value for each image and then find the final average between these averages. In addition to the final average, this approach allows the assessment of individual averages for each image individually. The final average (FA) score can be expressed as follows:  
\[FA=\frac{1}{n}\cdot\sum_{i=1}^{n}\left(\frac{1}{m_{i}}\cdot\sum_{j=1}^{m_{i}}x_{ ij}\right) \tag{31}\]  
where \(m_{i}\) is the total number of experts, \(x_{ij}\) represents the evaluation of the \(j\)-th image by \(i\)-th expert, and \(n\) is the number of images in the set. The percentage difference in Table 8 represents the relative difference between the two values and can help assess the magnitude of the difference meaningfully. When calculating the percentage difference, the average of the two scores was used in the denominator for relative comparison:  
\[\text{Percentage difference}\ =\frac{|A-B|}{\frac{(A+B)}{2}}\cdot 100 \tag{32}\]  
Here, \(A\) is the expert rank and \(B\) is the PROMETHEE-IFS rank from Table 8. We aim for the ratings obtained using the PROMETHEE-IFS method to be as close as possible to experts, as expert ratings are selected for validation. In Table 8, the experts' ranking differs from PROMETHEE-IFS because it is challenging to decide which ones to prioritize when a person evaluates a segmented result against a set of features. The MCDM method helps to overcome this problem and evaluate the result according to individual criteria by combining them to calculate the ranking. Comparing MCDM methods can be challenging because their performance often depends on the alternatives used (in our case, algorithms), the set of criteria, the application area, and the problem being solved. An additional experiment was conducted to assess the sensitivity of image content evaluation and validate the proposed methodology. Sensitivity analysis [56] is an important step in the assessment of the reliability of MCDM methods.