# Using Markov Random Field and Analytic Hierarchy Process to Account for Interdependent Criteria
## 4 Numerical Example  
Consider a decision problem that involves choosing the most suitable location for a new store from three alternatives: A1 (City Center), A2 (Suburbs), and A3 (Industrial Zone). The decision is based on four criteria: C1 (Foot Traffic), C2 (Rent Cost), C3 (Proximity to Competitors), and C4 (Accessibility). We begin by constructing pairwise comparison matrices for each criterion in relation to the goal and for each alternative with respect to each criterion. Let us assume the decision-makers provided the following matrices (using the standard AHP 1-9 scale): \begin{tabular}{c c c c c} \hline \hline Criteria & C1 & C2 & C3 & C4 \\ \hline C1 & 1 & 1/3 & 1 & 1/5 \\ C2 & 3 & 1 & 3 & 1/3 \\ C3 & 1 & 1/3 & 1 & 1/5 \\ C4 & 5 & 3 & 5 & 1 \\ \hline \hline \end{tabular}   
Alternatives matrices for each criterion: \begin{tabular}{c c c c c c c} \hline \hline C1 & A1 & A2 & A3 & C2 & A1 & A2 & A3 \\ \hline A1 & 1 & 3 & 1/3 & A1 & 1 & 1/5 & 5 \\ A2 & 1/3 & 1 & 1/5 & A2 & 5 & 1 & 1/5 \\ A3 & 3 & 5 & 1 & A3 & 1/5 & 5 & 1 \\ \hline C3 & A1 & A2 & A3 & C4 & A1 & A2 & A3 \\ \hline A1 & 1 & 5 & 1/5 & A1 & 1 & 6 & 1/6 \\ A2 & 1/5 & 1 & 1/2 & A2 & 1/6 & 1 & 1/3 \\ A3 & 5 & 2 & 1 & A3 & 6 & 3 & 1 \\ \hline \hline \end{tabular}   
After calculating the eigenvectors and normalizing them, we obtain local priorities for criteria and alternatives as shown in Table 3:  
With the given interdependencies, we define the structure of the MRF as shown in Figure 1. We can represent the interdependencies with an undirected graph, where nodes represent the criteria and edges represent the direct connections between criteria. First, let us assume the dependency and pairwise potentials between connected criteria are provided by expert opinion as follows:  
The pairwise potentials between criteria can be represented as (0.1, 0.3, 0.5, 0.7, 0.9) to indicate the degree of dependency between criteria as (very slightly dependent, slightly dependent, moderately dependent, highly dependent, very highly dependent). Take the MRF (1) for example; we can formulate the pairwise potential matrix as:  
\[\mathbf{W}= \begin{bmatrix}0&0&0.9&0\\ 0&0&0&0.5\\ 0.9&0&0&0.7\\ 0&0.5&0.7&0\\ \end{bmatrix} \]  
We combine the local probability distributions (obtained from the pairwise comparisons) with the pairwise potentials between connected criteria to compute the joint \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Decision Table** & **A1** & **A2** & **A3** & **Priorities** \\ \hline C1 & 0.16 & 0.68 & 0.16 & 0.15 \\ C2 & 0.16 & 0.68 & 0.16 & 0.40 \\ C3 & 0.49 & 0.16 & 0.35 & 0.15 \\ C4 & 0.65 & 0.16 & 0.19 & 0.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Decision table for the numerical example. probability distribution over all criteria using MRFs. This provides the criteria's overall priorities, considering their interdependencies, as shown in Table 4. The joint probability distribution results in the following adjusted criteria priorities, which can be derived from Gibbs sampling. The iteration of the Gibbs sampling can refer to Figure 2 and ensure the convergence of the final priorities. Using the overall priorities of the criteria obtained from the MRF model, we can calculate the global priorities of the alternatives. This is accomplished by aggregating the local priorities of the alternatives concerning each criterion, considering the overall priorities of the criteria. With the new adjusted criteria priorities, the ranking of the alternatives has changed. The MRF-adjusted AHP now ranks the alternatives as B, A, and C, while the original AHP ranked them as A, B, and C, as demonstrated in Table 5.  
This indicates that considering the interdependencies between the criteria with the adjusted criteria priorities affects the overall ranking of the alternatives. In this numerical