## 5 Application of ZPLTS-ELECTRE II Method in Ontology Ranking
### Experimental Results and Discussions
#### 5.2.9. Exploitation of Outranking Relations  
After constructing the outranking graphs, the outranking relations are exploited by applying the forward and reverse procedures, as shown in Section 4.9. Firstly, the forward procedure is performed as follows. In the first iteration, the strong and weak outranking graphs, \(G_{1}^{F}\) and \(G_{1}^{f}\), are observed, and the sets \(N_{1}^{F}\) and \(N_{1}^{f}\) are formed. The elements in these sets represent the non-dominant alternatives in \(G_{1}^{F}\) and \(G_{1}^{f}\). \(N_{1}^{F}\) is formed as:  
\[N_{1}^{F}=\{1,2,5,9\}\]  
and \(N_{1}^{f}\) is formed as:  
\[N_{1}^{f}=\{1,2,4,5,7,9\}\]  
Thereafter, the intersection between the non-dominant sets is formed as \(N_{1}\):  
\[N_{1}=\{1,2,5,9\}\]  
After determining the intersection \(N_{1}\), the alternatives in \(N_{1}\) are assigned a rank, that is, \(\psi_{1}(1)=1\), \(\psi_{1}(2)=1\), \(\psi_{1}(5)=1\), and \(\psi_{1}(9)=1\). All alternatives assigned a rank are then removed from the graphs along with their associated edges. The new graphs are then constructed, as shown in Figure 5.  
The non-dominant sets and their intersection are then formed for the second iteration as follows:  
\[N_{2}^{F}=\{3,6,7\}\]  
and \(N_{2}^{f}\) is formed as:  
\[N_{2}^{f}=\{3,4,7\}\]  
Thereafter, the intersection between the non-dominant sets are formed as \(N_{2}\):  
\[N_{2}=\{3,7\}\]  
The alternatives in the intersection are assigned the rank position of 2, that is, \(\psi_{1}(3)=2\) and \(\psi_{1}(7)=2\). The third iteration then proceeds with the new graphs in Figure 6.  
Figure 5. Strong and weak outranking graphs for the second iteration in the forward procedure. (a) Strong outranking graph \(G_{2}^{F}\). (b) Weak outranking graph \(G_{2}^{f}\). Figure 6. Strong and weak outranking graphs for the third iteration in the forward procedure. (a) Strong outranking graph \(G_{3}^{F}\). (b) Weak outranking graph \(G_{3}^{f}\). The non-dominant sets and their intersection are then formed for the third iteration as follows:  
\[N_{1}^{f}=\{1,2,4,5,7,9\}\]0  
and \(N_{3}^{f}\) is formed as:  
\[N_{1}^{f}=\{1,2,4,5,7,9\}\]1  
Thereafter, the intersection between the non-dominant sets are formed as \(N_{3}\):  
\[N_{1}^{f}=\{1,2,4,5,7,9\}\]2  
The alternatives in the intersection are assigned the rank position of 3, that is, \(\psi_{1}(4)=3\) and \(\psi_{1}(6)=3\). Since there are still nodes in the graph, the fourth iteration then proceeds with the new graphs in \[N_{1}^{f}=\{1,2,4,5,7,9\}\]3  
Since both the graphs in Figure 7 have only one node which is not dominated by any other nodes, their intersection will be equal, and hence, the alternative \(O_{8}\) is assigned the \(4^{th}\) rank position, that is, \(\psi_{1}(8)=4\). All alternatives in the graphs have been ranked, and the forward procedure is completed. After performing the forward procedure, the reverse procedure is performed. The first step in the reverse procedure is to construct the mirror image of the strong and weak outranking graphs by reversing the direction of the edges. Thereafter, the same steps follow, as performed in the forward procedure. The resulting ranking is then transformed by applying Equation (28). Finally, to generate the final ranking of the alternatives, the forward ranking and the transformed reverse ranking are combined by applying Equation (29). The ranking obtained from the reverse procedure, \(\psi_{2}(O_{i})\), the transformed ranking, \(\psi_{3}(O_{i})\), and the final ranking, \(\bar{\psi}(O_{i})\) are shown in \[N_{1}^{f}=\{1,2,4,5,7,9\}\]4  
The final ranking for the nine ontologies is determined as:  
\[N_{1}^{f}=\{1,2,4,5,7,9\}\]5  
Accordingly, ontologies 5 and 9 are the best, and the worst ontology is ontology 8.