## 5 The Extended Eco-Apostle Model Applied to National Identity  
### Fuzzy TOPSIS and Fuzzy Clustering  
The problem of expressing some form of vagueness arose in various disciplines in the mid-1900s, such as logic, linguistics, physics, and mathematics. Importantly, for the subsequent development of fuzzy set theory, the first attempts to propose logic with three truth values (rather than the classic true/false dichotomy) were made in the 1930s [41]. The study of fuzzy sets obtained an excellent impetus for the mathematical development of the so-called multi-value logic. Similarly, multi-value logic does not only consider true or false as truth values but introduces others. From the simplest case, the \"I don't know\" is added up to infinite values between 0 and 1 [41]. Fuzzy set theory (FST) has been successfully applied to various analyses and disciplines, such as education [42], supplier selection [43], or green energy [44]. The essence of the application of FST in multi-criterion decision-making techniques (MCDM) provides a non-unique multivariate perspective for measuring common hidden concepts in the social sciences [45]. The study uses FST to adequately handle the vagueness of the information provided by the ISSP national identity module. First, we convert semantic ordinal scales to triangular fuzzy numbers (TFNs). The transformation of the semantic ordinal scales provided by the interviewees, from 1 to 4, into TFN is characterized by the interval [0, 100] without loss of generalization. It can be summarized as follows: (1) not at all important (0,0,50); (2) not very important (30,50,70); (3) quite important (50,70,90); (4) very important (70,100,100). Given the vagueness of the information, we represent the semantic scale with 3-tuples \((a_{1},a_{2},a_{3})\), for which the intersection of all the consecutive ordinal semantic points is an interval. The fuzzy hybrid analysis (FHA) is an extension of TOPSIS that can process information at an individual level using the fundaments of FST. The first step is to calculate the ideal solution. For that, the positive and negative ideal solutions are obtained as follows:  
\[A^{+}=\big{\{}\big{(}\max V_{ij}|j\in J\big{)},\big{(}\min V_{ij}|j\in J^{ \prime}\big{)},i=1,2,\ldots,m\big{\}}, \tag{1}\]  
\[A^{-}=\big{\{}\big{(}\min V_{ij}|j\in J\big{)},\big{(}\max V_{ij}|j\in J^{ \prime}\big{)},i=1,2,\ldots,m\big{\}}, \tag{2}\]  
where \(V_{ij}\) stands for the crisp information matrix given by \((a_{1}+2a_{2}+a_{3})/4\), where \(i=1\) to 27,873 (total sample of respondents) and \(j\) stands for the items included in each of the national identity dimensions (5 and 3 for ethnic and civic national identity, respectively). \(J^{\prime}\) is an empty set as all the items included in the national identity module can be considered as a benefit [46]. The positive ideal solution is represented by the largest number observed in the data set, and the opposite logic prevails for the negative ideal solution [46]. Afterwards, each individual observation can be compared with these ideal solutions by using the Euclidean distance and comparing the relative distances between them as follows [47]:  
\[ \begin{split} D_{i}^{+}=dist(V_{i},A^{+})&=\sqrt{ \sum\limits_{j=1}^{n}\big{(}V_{ij}-A_{;j}^{+}\big{)}^{2}}\,i=1,2,\ldots,m\\ D_{i}^{-}=dist(V_{i},A^{+})&=\sqrt{\sum\limits_{j=1 }^{n}\big{(}V_{ij}-A_{;j}^{-}\big{)}^{2}}\,i=1,2,\ldots,m\end{split}  \tag{3}\]  
Therefore, the TOPSIS indicator is calculated as follows:  
\[TOPSIS_{i}=\frac{D_{i}^{-}}{D_{i}^{+}+D_{i}^{-}}\,\,i=1,2,\ldots,m, \tag{4}\]The analysis of the national identity at the individual level will be based on fuzzy clustering [47]. Thus, the membership function can determine the degree of similarity that each citizen has for a representative group [48]. The fuzzy cluster analysis segmentation extends the bagged cluster algorithm introduced by [49]. Hence, the C-means fuzzy algorithm for fuzzy data can be expressed as follows:  
\[\left\{