## 2 Technical Background  
In this section, we will present the background terminologies and some fundamental properties related to the goal of the paper. Considering a reference set \(R=\{r_{1},r_{2},\ldots,r_{n}\}\) with cardinality \(n\), pairwise comparisons in the form of ratios between the weights of the elements of the reference set can be collected into a mathematical structure called the pairwise comparison matrix (PCM). **Definition 1** (Pairwise comparison matrix). : _A real matrix \(\mathbf{A}=[a_{ij}]_{n\times n}\) is said to be a pairwise comparison matrix (PCM) if it is reciprocal and positive, i.e., \(a_{ji}=1/a_{ij}\) and \(a_{ij}>0\) for all \(i,j=1,2,\ldots,n\)._  
Semantically, each entry of \(\mathbf{A}\) is an estimation of the ratio between two positive weights, i.e., \(a_{ij}\approx w_{i}/w_{j}\), where \(w_{i}\) and \(w_{j}\), are the weights associated with the \(i\)th and the \(j\)th elements of the reference set, respectively. The reciprocity of \(\mathbf{A}\) is a minimal coherence condition which is always required. Nevertheless, as it is desirable to ask experts to discriminate with a sufficient level of rationality, it is important to determine whether the pairwise comparisons contained in a PCM represent rational preferences. The consistency condition corresponds to the condition of rationality. For the matrix entries \(a_{ij}\) of \(\mathbf{A}\), [PERSON] [2,3] has suggested adopting a discrete scale, i.e., \(a_{ij}\in\{1/9,1/8,\ldots,1/2,1,2,3,\ldots,8,9\}\) for all \(i,j=1,2,\ldots,n\). **Definition 2** (Consistency). : _A pairwise comparison matrix \(\mathbf{A}=[a_{ij}]_{n\times n}\) is said to be consistent if and only if the transitivity property \(a_{ik}=a_{ij}a_{jk}\) holds for all \(i,j,k=1,2,\ldots,n\). Otherwise, it is called inconsistent._  
However, in light of our cognitive limits, it is hardly ever possible for an expert to express consistent preferences. When one uses [PERSON]'s discrete scale, it is very rare for the pairwise comparison matrix to become consistent. The appearance of the consistency ratio for all \(4\times 4\) PCMs is \(0.001421\%\)[31]. For this reason, as discussed in a recent survey [17], proposals of inconsistency indices are abundant in the literature. In spite of the variety of proposals, the inconsistency index \(CR\) proposed by [PERSON] [2] in their works in the AHP has gained and maintained prominence and to date continue to represent the standard in the field. **Definition 3** (Inconsistency Ratio). : _[PERSON]'s inconsistency ratio (CR) [2,3] is defined by_  
\[CR(\mathbf{A})=\frac{(\lambda_{max}(\mathbf{A})-n)/(n-1)}{RI_{n}}\]_where \(\lambda_{max}(\mathbf{A})\) is the Perron eigenvalue of the complete PCM \(\mathbf{A}\), and \(RI_{n}\) is the random index value associated with the matrix size \(n\) reported in Table 1._  
Indeed, as a quantification of inconsistency, the greater the value of \(CR\), the greater the estimated inconsistency of the judgments contained in the pairwise comparison matrix. Moreover, as it is unavoidable that a certain level of inconsistency must be tolerated, [PERSON] [2] proposed the cut-off rule \(CR<0.1\) to define the set of acceptable PCMs. Note that the inconsistency of a PCM may arise due to the redundancy of the judgments in the PCM itself. **Definition 4** (Incomplete pairwise comparison matrix). : _A pairwise comparison matrix \(\mathbf{A}=[a_{ij}]_{n\times n}\) is called an incomplete pairwise comparison matrix if one or more elements are missing, i.e., \(a_{ji}=1/a_{ij}\) if \(a_{ij}\) is known, otherwise \(a_{ji}=a_{ij}=*\), where \(*\) represents the unknown elements. In short, it can be represented in the form of:_  
\[\mathbf{A}= \begin{pmatrix}a_{11}&*&\cdots&a_{1n}\\ *&a_{22}&\cdots&*\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&*&\cdots&a_{nn}\end{pmatrix} . \tag{2}\]  
It is conventional to visualize the structure of an incomplete pairwise comparison matrix using an associated directed or undirected graph. However, by targeting our study in the paper, we will use the concept of an undirected graph.