## 5 Application of ZPLTS-ELECTRE II Method in Ontology Ranking
### Experimental Design
#### 5.1.2 Quantitative Attributes  
The quantitative metrics used in this study were adopted from the OntoMetrics framework [57], which defines various metrics for evaluating ontologies and provides an online environment for their automatic calculation. Several graph metrics have been defined to evaluate ontologies [57; 58]. Five of these metrics were adopted in this study to demonstrate the effectiveness of the proposed ZPLTS-ELECTRE II method. The graph metrics are calculated from the graph and taxonomy tree of the ontology, thereby providing insights pertaining to the characteristics and attributes of the ontology [58]. The five graph metrics adopted measure the design complexity of ontologies and include the Absolute Leaf Cardinality (ALC), Absolute Root Cardinality (ARC), Average Depth (AD), Average Breadth (AB), and Average Number of Paths (ANP). The metrics are elaborated on as follows.  
1. The _Absolute Leaf Cardinality_ metric, abbreviated ALC, is an indicator of the number of leaf nodes within a graph [58; 59]; it expresses the dispersion of leaf nodes within a graph and is calculated by Equation (30). \[m=n_{LEA\subseteq g}\] (30) where \(m\) is the ALC metric, and \(n\) is the cardinality of the set \(LEA\), which is a subset of the directed graph \(g\).
2. The _Absolute Root Cardinality_ metric, abbreviated ARC, expresses the number of root nodes in a graph [59; 60]. The ARC is calculated by Equation (31). \[m=n_{ROO\subseteq g}\] (31) where \(m\) is the ARC metric, and \(n\) is the cardinality of the set \(ROO\), which is a subset of the directed graph \(g\).
3. Depth is a graph property of an ontology relating to the number of paths within the graph, considering the taxonomy is-a arcs only. The _Average Depth_ metric, abbreviated AD, is determined by dividing the sum of the cardinalities of every path within a graph, by the cardinality of the set of all paths in that graph [58]. The AD is an indicator of the degree to which an ontology has a vertical modeling with many taxonomy is-a relations, which represent richer information content in the ontology [60]. The AD metric is calculated by Equation (32). \[m=\frac{1}{n_{P\subseteq g}}\sum_{P}^{j}N_{j\in P}\] (32) where \(m\) represents the AD value, \(N\) is the cardinality of the paths \(j\), \(P\) represents the set of paths within the graph \(g\), and \(n\) represents the cardinality of \(P\).
4. The _Average Breadth_ metric, abbreviated AB, expresses the average cardinality value of a generation in a graph. It is calculated by dividing the sum of the cardinalities of all generations by the number of generations in the graph [58]. The metric is an indicator of the degree to which the ontology has a horizontal modeling of its hierarchies with less taxonomy is-a relations [60]. The AB metric is calculated by Equation (33). \[m=\frac{1}{n_{L\subseteq g}}\sum_{L}^{j}N_{j\in L}\] (33) where \(m\) represents the AB value, \(N\) is the cardinality of a generation, \(j\) a particular generation, \(n_{L\subseteq g}\) is the cardinality of \(L\), and \(L\) is the set of all generations within a graph \(g\).
5. The _Average Number of Paths_, abbreviated ANP, expresses the relationship between a concept and the root concept within the taxonomy hierarchy of the ontology. An ontology that has a high ANP value contains a large number of taxonomy/inheritance relationships along with a large number of interconnections between the classes withinthe ontology. If the ANP value is 1, then it signifies that the inheritance hierarchy of the ontology is a tree [61]. The ANP is calculated by Equation (34).  
\[m=\frac{\sum_{i=1}^{n}p_{i}}{|C|} \tag{34}\]  
where \(m\) is the ANP value, \(p_{i}\) is the number of paths for a given concept, and \(|C|\) is the total number of concepts.