# Using Markov Random Field and Analytic Hierarchy Process to Account for Interdependent Criteria
## 2 Literature Review
### Ahp  
AHP was introduced by [1] as a decision-making methodology for complex, multi-criteria problems. AHP is based on pairwise comparisons in a hierarchical structure, which simplifies decision making by breaking it down into smaller, more manageable parts [4]. The mathematical model of AHP involves constructing the pairwise comparison matrix (PCM), calculating the weights of criteria or alternatives, and computing the consistency ratio [1; 22]. Since its introduction, AHP has seen significant development in both theory and application. Researchers have focused on improving the mathematical foundations, extending their applicability, and integrating it with other methods to address various decision-making problems. To quantify the PCMs, ref. [1] introduced a scale ranging from 1 to 9, where 1 represents equal importance, and 9 represents extreme importance. The decision-maker assigns values from this scale to express the relative importance of one element over another. This results in a reciprocal PCM, which is used to calculate the eigenvector corresponding to the largest eigenvalue. This eigenvector represents the normalized weights of the criteria or alternatives [1], which can be represented as solving the following eigenvalue problem:  
\[\mathbf{A}\mathbf{w}=\lambda_{\text{max}}\mathbf{w} \tag{4}\]  
where \(\mathbf{A}\) is the PCM, \(\mathbf{w}\) denotes the weight vector of criteria, and \(\lambda_{\text{max}}\) denotes the maximum eigenvalue. AHP's methodology also accounts for the consistency of the decision-maker's judgments. Ref. [1] proposed the consistency ratio (\(CR\)), which compares the consistency index (\(CI\)) to the random index (\(RI\))--an average index derived from randomly generated matrices. A \(CR\) less than or equal to 0.10 is generally considered acceptable, indicating that the decision-maker's judgments exhibit a reasonable level of consistency. \(CR\) and \(CI\) can be formulated as follows:  
\[CR=\frac{CI}{RI}=\frac{\lambda_{\text{max}}-n}{n-1}/RI \tag{5}\]  
where \(n\) denotes the number of criteria; and \(RI\) (random index) is used to estimate the average consistency of a randomly generated pairwise comparison matrix and can refer to Table 1.  
In terms of theoretical foundations, consistency and inconsistency measures have been proposed to evaluate the reliability of pairwise comparisons, such as the consistency ratio (CR) [1] and the geometric consistency index (GCI) by [23]. In recent research, ref. [24] \begin{table}
\begin{tabular}{c c} \hline \hline
**Number of Criteria** & **Random Index** \\ \hline
3 & 0.58 \\
4 & 0.90 \\
5 & 1.12 \\
6 & 1.24 \\
7 & 1.32 \\
8 & 1.41 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: RI table. addressed the critical issue of maintaining consistency, especially ordinal consistency, in AHP-based decision-making models. This study proposes optimization models to achieve transitive preferences and to solve individual consistency and group consensus problems in AHP. It offers a methodological advancement by formulating constraints for ordinal consistency and developing mixed integer linear optimization models to ensure both ordinal and cardinal consistencies in group decision-making settings. Alternative weight derivation methods have been developed, including the row geometric mean method (RGMM) by [25], the logarithmic least squares method (LLSM) by [26], and the least square priorities method (LSM) by [27]. Additionally, group decision making has been incorporated into AHP through aggregation methods like the weighted arithmetic mean (WAM) by [28] and the geometric aggregation operator (GAO) by [29]. Regarding practical applications, AHP has been used in supply chain management for supplier selection and evaluation, incorporating green supplier evaluation by [30] and assessing supply chain resilience by [31]. In the field of environmental decision making, AHP has been applied to prioritize climate change adaptation strategies by [32] and renewable energy source prioritization in Turkey [33].