## 5 Application of ZPLTS-ELECTRE II Method in Ontology Ranking
### Experimental Results and Discussions
#### 5.2.7 Discordance Relations  
In order to determine the discordance relations between all criteria pairs, the differences in their performance for all 10 criteria need to be calculated. When considering quantitative criteria, that is, criteria 1 to 5, the difference between the performance of alternative \(x\) and alternative \(y\) at criterion \(j\) is the difference between \(x_{j}\) and \(y_{j}\), that is, \(x_{j}-y_{j}\). The calculated differences between all alternatives at each criterion are presented in the heatmaps in Figure 2.  
When considering qualitative criteria, that is, criteria 6 to 10, the difference between the performance of alternative \(x\) and alternative \(y\) at criterion \(j\) is the distance between \(x_{j}\) and \(y_{j}\), as given by Equation (6). The calculated distances between all alternatives at each criterion is presented in the heatmaps in Figure 3.  
\begin{table} \
\begin{tabular}{c 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 & 0.70 & 0.90 & 1.00 & 0.60 & 0.80 & 0.70 & 0.90 & 0.30 \\ \(\mathbf{O_{2}}\) & 0.30 & 0 & 0.60 & 0.80 & 0.30 & 0.60 & 0.30 & 0.80 & 0.20 \\ \(\mathbf{O_{3}}\) & 0.10 & 0.40 & 0 & 0.90 & 0.00 & 0.20 & 0.40 & 0.70 & 0.10 \\ \(\mathbf{O_{4}}\) & 0.00 & 0.20 & 0.10 & 0 & 0.10 & 0.20 & 0.10 & 0.60 & 0.10 \\ \(\mathbf{O_{5}}\) & 0.40 & 0.80 & 1.00 & 0.90 & 0 & 1.00 & 0.70 & 0.80 & 0.20 \\ \(\mathbf{O_{6}}\) & 0.20 & 0.40 & 0.80 & 0.80 & 0.00 & 0 & 0.40 & 0.70 & 0.10 \\ \(\mathbf{O_{7}}\) & 0.30 & 0.70 & 0.70 & 0.90 & 0.30 & 0.60 & 0 & 0.90 & 0.20 \\ \(\mathbf{O_{8}}\) & 0.10 & 0.20 & 0.30 & 0.40 & 0.20 & 0.30 & 0.10 & 0 & 0.20 \\ \(\mathbf{O_{9}}\) & 0.70 & 0.80 & 0.90 & 0.90 & 0.80 & 0.90 & 0.80 & 0.8\end{tabular}\end{table}
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After analyzing the differences and distances between the criteria for all alternatives, the discordance relations are determined. For comparing each alternative pair, \(x\) and \(y\), the comparison between their criteria performances are partitioned into one of three discordance sets: high discordance \(\mathcal{Q}^{+}(x,y)\), medium discordance \(\mathcal{Q}^{0}(x,y)\), or low discordance  
Figure 2: Differences in quantitative criterion performance for all alternative pairs. (**a**) Difference in criterion performance for ARC. (**b**) Difference in criterion performance for ALC. (**c**) Difference in criterion performance for AD. (**d**) Difference in criterion performance for AB. (**e**) Difference in criterion performance for ANP  
\(\mathcal{Q}^{-}(x,y)\). To illustrate this, when considering the comparison between alternative 1 and 3, the comparative sets are formed as follows.  
\[\mathcal{Q}^{+}(1,3)=\{\}\]  
The set \(\mathcal{Q}^{+}(1,3)\) is an empty set, which implies that there is no criteria for which the comparison of alternatives 1 and 3 yields high discordance.  
\[\mathcal{Q}^{0}(1,3)=\{3\}\]  
Since the set \(\mathcal{Q}^{0}(1,3)\) has one element, that is, criterion 3, this means that there is medium discordance when comparing alternative 1 and 3 against criterion 3.  
\[\mathcal{Q}^{-}(1,3)=\{1,2,4,5,6,7,8,9,10\}\]  
By observing the set \(\mathcal{Q}^{-}(1,3)\), it can be seen that comparing alternative 1 with 3 yields the set comprising nine out of 10 criteria that have low discordance levels.