# Using Markov Random Field and Analytic Hierarchy Process to Account for Interdependent Criteria
## 5 Discussion  
This paper introduces an innovative approach to address the issue of interdependency among criteria in the AHP by incorporating discrete MRFs. The traditional AHP assumes that criteria are independent, which may not always be the case in real-world decision-making scenarios. Overlooking these interdependencies can result in misleading or suboptimal decisions. Our proposed method aims to enhance the decision-making process by capturing and representing the interdependencies between criteria more efficiently and intuitively.  
In the numerical example presented, we showcased the application of the MRF-adjusted AHP in an MCDM scenario. The example demonstrated how criteria interdependencies could be considered using pairwise potentials, representing the strength of dependencies between criteria. The findings underlined the discrepancies in global priority values and the potential shifts in alternative rankings when interdependencies are considered. Furthermore, both the MRF-adjusted AHP and FCM methods were employed to address criteria interdependencies. The results revealed that the overall ranking of alternatives differed when compared to the conventional AHP, which assumes criteria independence. This underscores the significance of accounting for criteria interdependencies in real-world decision-making situations. Although both methods effectively capture interdependencies, the MRF-adjusted AHP provides a more intuitive and straightforward representation through pairwise potentials, simplifying the decision-makers' understanding and interpretation of the results. The two methods exhibit distinct applications and  
Figure 2: The convergence of criteria priorities.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Scores** & **A** & **B** & **C** & **Rank** \\ \hline AHP & 0.3270 & 0.3260 & 0.3470 & C, A, B \\ \hline MRF-adjusted weights (1) & 0.2617 & 0.3948 & 0.3435 & B, C, A \\ FCM-adjusted weights (1) & 0.2700 & 0.3840 & 0.3460 & B, C, A \\ \hline MRF-adjusted weights (2) & 0.2644 & 0.3920 & 0.3436 & B, C, A \\ FCM-adjusted weights (2) & 0.2718 & 0.3816 & 0.3466 & B, C, A \\ \hline MRF-adjusted weights (3) & 0.2605 & 0.3978 & 0.3417 & B, C, A \\ FCM-adjusted weights (3) & 0.2680 & 0.3860 & 0.3460 & B, C, A \\ \hline MRF-adjusted weights (4) & 0.2626 & 0.3922 & 0.3452 & B, C, A \\ FCM-adjusted weights (4) & 0.2695 & 0.3815 & 0.3490 & B, C, A \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison between the proposed method and the AHP.  
data requirements. Moreover, the FCM method requires setting multiple parameters to obtain accurate results, which is not necessary for the proposed method.  
Compared to other approaches that address criteria interdependence in the AHP, such as ANP and Bayesian networks, the MRF-adjusted AHP offers several advantages. First, it provides a more intuitive and straightforward representation of criteria interdependencies through the pairwise potentials matrix, which decision-makers can easily understand and interpret. In contrast to ANP, the MRF-adjusted AHP simplifies the process of comparing the criteria's relative weights. Next, the proposed method can account for both positive and negative interdependencies between criteria to apply them to more complicated problems. In addition, decision-makers can effortlessly alter the dependency strength between criteria by adjusting the pairwise potentials matrix, enabling sensitivity analysis and scenario exploration. Furthermore, the MRF-adjusted AHP is built on the well-established foundation of the original AHP, ensuring compatibility with existing AHP-based tools and methodologies. If interdependencies between criteria are disregarded, the proposed method reverts to the conventional AHP.