# Using Markov Random Field and Analytic Hierarchy Process to Account for Interdependent Criteria
## 2 Literature Review
### Methods for Interdependent Criteria  
The growing complexity of decision-making problems has necessitated the advancement of MCDM methods that take into account interdependencies among criteria. Ref. [9] offered an extensive analysis of MCDM methods that model the interactions among criteria, stressing the importance of considering preference, criterion, and utility interactions. A prominent MCDM method that tackles interdependencies is the ANP, which is introduced by [7]. This method expands the AHP by addressing interdependencies and feedback loops among criteria and alternatives, providing a structured approach to managing intricate decision-making problems. Nonetheless, the ANP has certain drawbacks, including significant computational complexity and potential inconsistencies in pairwise comparisons [15]. In addition, the massive use of subjective judgments in the ANP can lead to bias, impacting the overall trustworthiness of the results. Researchers have proposed hybrid models that combine various MCDM methods to overcome these challenges. For example, ref. [10] suggested a model that merges the DEMATEL method with ANP, while ref. [16] presented an integrated hybrid model that combines a modified TOPSIS with a PGP model for supplier selection with interdependent criteria. These hybrid models addressed the limitations of traditional MCDM methods concerning interdependencies and showcased their effectiveness through real-world applications [16; 10]. Despite the benefits of these methods in handling interdependencies, some challenges remain. For instance, the integration of DEMATEL within the ANP framework can lead to inconsistencies and ambiguity for decision-makers due to the absence of a cohesive approach [11]. Moreover, hybrid models, such as the FCM-AHP approach proposed by [17], introduced extra complexity and dependence on expert opinions, which can result in subjectivity and reduced practical usability. In their recent publication, ref. [18] presented a novel FCM-based MCDM method for addressing complex decision-making problems. The FCM approach, introduced by [19], builds upon the cognitive maps concept [20] by incorporating fuzzy degrees of interrelationships between concepts. FCMs illustrate the influence of vertices, referred to as concepts, on one another through cause-effect relationships, which are quantified and typically normalized to the \([-1,1]\) interval. Let \(w_{ij}\in[-1,1]\) be the degree of influence from the \(i\)th concept, \(C_{i}\), to the \(j\)th concept, \(C_{j}\), where the sign signifies a positive or negative influence. A value of \(-1\) denotes a full negative impact, while \(1\) signifies a full positive impact. The influence of concept \(x\) can be calculated by the following equation:  
\[x_{i}(t+1)=f(\sum_{j=1,j\
eq i}^{n}x_{j}(t)w_{j,i}) \tag{1}\]  
where \(n\) represents the number of concepts and \(f(\cdot)\) is the transfer function, which compresses the multiplication result into a specific range, e.g., \([0,1]\) or \([-1,1]\). Commonly, bivalent, trivalent, and sigmoid functions are used in the FCM. Ref. [21] proposed a modified version of [PERSON]'s FCM that accounts for the previous value of each concept, thereby examining the self-loop effect. Consequently, Equation (1) can be altered and expanded as follows:  
\[x_{i}(t+1)=f(\sum_{j=1,j\
eq i}^{n}x_{j}(t)w_{j,i}+x_{i}(t)) \tag{2}\]  
Another FCM variant used for rescale inference is presented below:  
\[x_{i}(t+1)=f(\sum_{j=1,j\
eq i}^{n}(2x_{j}(t)-1)w_{j,i}+(2x_{i}(t)-1)) \tag{3}\]  
In an FCM, the presence of positive and negative influences suggests that the centrality of a vertex within a graph should consider two opposing forces to determine the final centrality. Ref. [12] highlighted the need for ongoing research in MCDM methods that consider the dependency between criteria and evaluate their advantages and disadvantages in various decision-making contexts.