## 3 Selection of Methods for Methodology
### Prommethee-IFS Method  
A new extension of the PROMETHEE method under the intuitionistic fuzzy set environment, namely PROMETHEE-IFS, is designed and presented in this section to solve the selection of the image segmentation algorithm for satellite images. The main steps of this approach can be expressed as follows:  
1. The solution procedure of the PROMETHEE method starts with the constructed initial algorithm evaluation matrix \(X\), which consists of the elements \(x_{ij}\) and is defined as follows: \[X_{ij}= \begin{pmatrix}x_{11}&x_{12}&\cdots&x_{1j}\\ x_{21}&x_{22}&\cdots&x_{2j}\\ \vdots&\vdots&\ddots&\vdots\\ x_{i1}&x_{i2}&\cdots&x_{ij}\end{pmatrix} ,\;i=1,\;2\;,\ldots,\;n;j=1,\;2\;, \ldots,\;m.\] (22)  
The element \(x_{ij}\) represents the assessment of the \(i\)-th alternative (algorithm) by the \(j\)-th criterion (image features). For this test case, additional information is included by evaluating the image content based on the same criteria. These expert evaluations can also be expressed in linguistic terms (Table 3) and are used to create an image vector \(I_{i}=(I_{1},I_{2}\ldots)\) that has the same number of elements as the columns in the algorithm evaluation matrix \(X_{ij}\). 2. Vector normalization is applied to the algorithm evaluation matrix \(X_{ij}\) elements \(x_{ij}\) to obtain a normalized algorithm evaluation matrix \(\overline{X}_{ij}\): \[\overline{x}_{ij}=\frac{x_{ij}}{\sqrt{\sum_{i=1}^{m}x_{ij}*x_{ij}}};\;i=1,\;2 \;,\ldots,\;n;\;\;j=1,\;2\;,\ldots,\;m.\] (23) Additionally, a normalized image assessment vector \(\overline{I}_{i}=\left(\overline{I_{1}},\overline{I_{2}},\ldots\right)\) is calculated. * (3) Fuzzification is a fundamental process in fuzzy logic that involves converting crisp or precise data, which typically consists of numerical values, into fuzzy data. Fuzzification of the normalized decision matrix \(\overline{X}_{ij}\) is performed by mapping crisp values (expert evaluations) to fuzzy values, as displayed in Table 3. In this step, every member \(\overline{x}_{ij}\) of \(\overline{X}_{ij}\) is replaced with the intuitionistic fuzzy numbers \(\left(IFNs\right)_{ij}\) as in _Definition 2_. Intermediate values are obtained via linear interpolation. As a result, the intuitionistic fuzzy decision matrix \(\widehat{X}_{ij}\) is created. Similarly, we perform the fuzzification of the normalized image assessment vector \(\overline{I}_{i}\) to obtain an intuitionistic fuzzy image assessment vector \(\widetilde{I}_{i}\). A special step is proposed and performed to obtain the final decision matrix \(\overline{\widetilde{X}I}\) by fuzzy column-wise addition (Equation (13)) of the fuzzy image assessment vector \(\widetilde{I}_{i}\) and the fuzzy algorithm evaluation matrix \(\widehat{X}_{ij}\):  
\[\widetilde{X}_{ij}=\widehat{X}_{ij}\oplus\ \widetilde{I}_{i} \tag{24}\]  
This integration improves the selection process of the appropriate segmentation algorithm by considering both the visual image characteristics and the algorithm's sensitivity to these characteristics. These evaluations for the algorithm matrix are left constant throughout the experiment, while the image content vector varies for each satellite image. The final decision matrix is then formed from these two elements by fuzzy fusion. In this way, the decision matrices are computed for all test images. * (4) To rank the alternatives using the PROMETHEE-IFS method, a comparison is made between each pair of alternatives (\(A_{j}\) and \(A_{k}\)) by calculating the aggregated preference function \(\pi\big{(}A_{j},\ A_{k}\big{)}\) using the following equation: \[\pi\big{(}A_{j},\ A_{k}\big{)}=\sum_{i=1}^{m}\omega_{i}\cdot P_{t}\big{(}d_{i} \big{(}A_{j},\ A_{k}\big{)}\big{)}\] (25)  
Here, \(\omega_{i}\) are weights (of the \(i\)-th criterion) obtained from the SWARA method (\(\sum_{i=1}^{m}\omega_{i}=1\)).