## 5 Application of ZPLTS-ELECTRE II Method in Ontology Ranking
### Comparative Analysis of ZPLTS-ELECTRE II
#### 5.3.1 Comparison with Traditional ELECTRE II  
Since the traditional ELECTRE II cannot model linguistic criteria, only the quantitative criteria were used, that is, ALC, ARC, AD, AB, and ANP. Firstly, the alternatives and criteria form a decision matrix, which is the same as the matrix in Table 2. The decision matrix (Table 2) was then normalized to form the matrix in Table 19.  
The criteria were assigned equal weights, as with the application of ZPLTS-ELECTRE II. Since there were five criteria, each criterion received a weight of 0.20. The concordance thresholds were set as \(c_{1}=0.85\), \(c_{2}=0.70\), and \(c_{3}=0.55\). However, since the traditional ELECTRE II does not allow separate discordance thresholds to be specified for each criterion as in ZPLTS-ELECTRE II, the discordance thresholds, \(d_{1}\) and \(d_{2}\), were set as \(d_{1}=0.40\) and \(d_{2}=0.25\).  
The next step was to calculate the concordance values between all alternative pairs, as shown in Table 20.  
Thereafter, the discordance values were determined for every alternative pair, as shown in Table 21  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \(\mathbf{O_{1}}\) & \(\mathbf{O_{2}}\) & \(\mathbf{O_{3}}\) & \(\mathbf{O_{4}}\) & \(\mathbf{O_{5}}\) & \(\mathbf{O_{6}}\) & \(\mathbf{O_{7}}\) & \(\mathbf{O_{8}}\) & \(\mathbf{O_{9}}\) \\ \hline \(\mathbf{O_{1}}\) & 0 & 0.6 & 0.8 & 1.0 & 0.6 & 0.8 & 0.6 & 0.8 & 0.6 \\ \(\mathbf{O_{2}}\) & 0.6 & 0 & 0.8 & 0.8 & 0.6 & 0.8 & 0.4 & 0.6 & 0.4 \\ \(\mathbf{O_{3}}\) & 0.2 & 0.2 & 0 & 0.8 & 0.0 & 0.0 & 0.2 & 0.4 & 0.2 \\ \(\mathbf{O_{4}}\) & 0.0 & 0.2 & 0.2 & 0 & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 \\ \(\mathbf{O_{5}}\) & 0.4 & 0.6 & 1.0 & 0.8 & 0 & 1.0 & 0.4 & 0.6 & 0.4 \\ \(\mathbf{O_{6}}\) & 0.2 & 0.2 & 1.0 & 0.8 & 0.2 & 0 & 0.4 & 0.6 & 0.2 \\ \(\mathbf{O_{7}}\) & 0.4 & 0.6 & 0.8 & 0.8 & 0.6 & 0.6 & 0 & 0.8 & 0.4 \\ \(\mathbf{O_{8}}\) & 0.2 & 0.4 & 0.6 & 0.8 & 0.4 & 0.6 & 0.2 & 0 & 0.4 \\ \(\mathbf{O_{9}}\) & 0.4 & 0.6 & 0.8 & 0.8 & 0.6 & 0.8 & 0.6 & 0.6 & 0 \\ \hline \end{tabular}
\end{table}
Table 20: Concordance matrix for traditional ELECTRE II.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \(\mathbf{O_{i}}\) & \(\mathbf{\Delta C}\) & \(\mathbf{ARC}\) & \(\mathbf{AD}\) & \(\mathbf{AB}\) & \(\mathbf{ANP}\) \\ \hline \(O_{1}\) & 1.00 & 0.13 & 0.29 & 0.85 & 1.00 \\ \(O_{2}\) & 0.27 & 0.14 & 0.29 & 1.00 & 0.47 \\ \(O_{3}\) & 0.04 Thereafter, the strong and weak outranking graphs were constructed by considering the concordance and discordance matrices. The strong and weak outranking relations are shown in Table 22.  
This is followed by the ontologies being ranked using the forward and reverse ranking procedures, and finally, they were combined [29]. The final ranking was as follows:  
\[O_{9}\succ O_{1}\succ O_{7}\succ O_{2}\succ O_{5}\succ O_{6},\,O_{8}\succ O_{3}, \,O_{4}\]  
It can be observed that there are some differences in the ranking by ZPLTS-ELECTRE II with that of the traditional ELECTRE II, which is due to the fact that the traditional ELECTRE II does not model the qualitative linguistic criteria and the decision-makers credibility levels, and it only allows the definition of two discordance thresholds. Therefore, the traditional ELECTRE II method offers limited modeling capabilities in decision-making problems compared to the proposed ZPLTS-ELECTRE II method.