## 2 Methodology
### Motivation for Employing MCDM-AHP in Enhancing Cybersecurity Resilience for Smart Grids
In AHP, the problem is constructed as a hierarchy and broken down into elements. The overall goal is located at the top level, the criteria and sub-criteria are at the middle level, and the alternatives are at the bottom of the hierarchy [123, 124]. The use of AHP typically involves five fundamental steps to facilitate a structured decision-making process. These steps are as follows [125, 126]:  
1. **Define the decision problem and establish the hierarchy**: The first step in using AHP is to decompose the decision-making problem into a hierarchy of elements, with the goal to be achieved at the highest level, criteria and sub-criteria at intermediate levels, and alternatives to be considered at the lowest level. This hierarchical structure (Figure 1) helps organize and decompose the problem into manageable components. 2. **Perform Pairwise Comparisons**: Pairwise comparison matrices are performed between elements at the same level of the hierarchy. These matrices capture the relative importance or preference of one element compared to another at a specific level. Specifically, pairwise comparison matrices are performed: * **Between Criteria**: decision-makers compare the criteria to determine their relative importance in achieving the overall goal. Each criterion is compared to every other criterion to assess their relative weights or priorities. The resulting pairwise comparison matrix represents the importance of criteria relative to one another. * **Between Alternatives**: decision-makers compare the alternatives against each other based on specific criteria. Each alternative is compared to every other alternative to evaluate its relative performance or desirability. The pairwise comparison matrix represents the preference of alternatives relative to one another for each criterion. * **Between Criteria and Sub-Criteria**: In some cases, the AHP hierarchy may include sub-criteria within each criterion. Pairwise comparison matrices can also be performed between the criteria and their respective sub-criteria. This allows decision-makers to assess the relative importance or preference of each sub-criterion within its parent criterion. These matrices serve as the basis for deriving priority weights and aggregating preferences within the AHP process. Figure 1: Hierarchical evaluation framework for cybersecurity measures in smart grids, which includes distinct levels: a top-level overarching goal, intermediate evaluation criteria, and a bottom level containing various cybersecurity measures. Source: The author’s own elaboration. Let \(C\) be the pairwise comparison matrix with elements \(c_{ij}\) representing the importance of criterion \(i\) compared to criterion \(j\) (1). \[C=[C_{ij}]_{nxxn}= \begin{bmatrix}c_{11}&c_{12}&c_{13}&\dots&c_{1n}\\ c_{21}&c_{22}&c_{23}&\dots&c_{2n}\\ \vdots&\dots&\dots&\dots&\vdots\\ c_{n1}&c_{n2}&c_{n3}&\dots&c_{nn}\end{bmatrix}  \tag{1}\]  
\(C_{ij}\) is the pairwise comparison rating for the \(i\)th and the \(j\)th criteria. \(n\) is the number of criteria. The matrix \(C\) is reciprocal, meaning that the elements satisfy the property of reciprocity: \(C_{ij}=1/C_{ij}\) for all \(i\) and \(j\). Reciprocity ensures that the judgments made in the pairwise comparisons are consistent. It implies that if element A is considered, for example, twice as important as element B, then B is considered half as important as A.  
Additionally, the diagonal elements of the pairwise comparison matrix are in unity (\(C_{ij}=1\) for \(i=j\)). This indicates that an element is perfectly equal to itself, which is a logical requirement. The entries in the pairwise comparison matrix C are typically taken from a ratio-scale based on the values 1/9 to 9 [121]. In fact, in the AHP, decision-makers assign numerical values to represent the relative importance or preference between elements in the pairwise comparisons. The scale is as follows: 1 (equally important), 3 (moderately more important), 5 (strongly more important), 7 (very strongly more important), 9 (absolutely more important).