# Using Markov Random Field and Analytic Hierarchy Process to Account for Interdependent Criteria
###### Abstract  
The Analytic Hierarchy Process (AHP) has been a widely used multi-criteria decision-making (MCDM) method since the 1980s because of its simplicity and rationality. However, the conventional AHP assumes criteria independence, which is not always accurate in realistic scenarios where interdependencies between criteria exist. Several methods have been proposed to relax the postulation of the independent criteria in the AHP, e.g., the Analytic Network Process (ANP). However, these methods usually need a number of pairwise comparison matrices (PCMs) and make it hard to apply to a complicated and large-scale problem. This paper presents a groundbreaking approach to address this issue by incorporating discrete Markov Random Fields (MRFs) into the AHP framework. Our method enhances decision making by effectively and sensibly capturing interdependencies among criteria, reflecting actual weights. Moreover, we showcase a numerical example to illustrate the proposed method and compare the results with the conventional AHP and Fuzzy Cognitive Map (FCM). The findings highlight our method's ability to influence global priority values and the ranking of alternatives when considering interdependencies between criteria. These results suggest that the introduced method provides a flexible and adaptable framework for modeling interdependencies between criteria, ultimately leading to more accurate and reliable decision-making outcomes. A +
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In response to this challenge, the research community has explored and developed numerous methodologies to account for criteria interdependence in MCDM. Prominent among these are the Analytic Network Process (ANP), a natural extension of AHP proposed by [PERSON] himself, which explicitly considers interdependencies and feedback loops among decision elements ([7]). Similarly, the Decision-Making Trial and Evaluation Laboratory (DEMATEL) method has been instrumental in visualizing and analyzing complex causal relationships among criteria [8]. Further enriching the landscape are Fuzzy Cognitive Maps (FCMs), which combine elements of fuzzy logic and neural networks to model systems where human judgment and imprecision play a significant role [9]. These developments, coupled with integrated fuzzy MCDM approaches such as Fuzzy Analytic Hierarchy Process (FAHP) and Fuzzy Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), have significantly advanced the field [10]. Moreover, hybrid MCDM methodologies have emerged, such as the integration of ANP with Fuzzy ELECTRE methods, offering nuanced and sophisticated decision-making frameworks [11]. These innovative approaches have been applied to various domains, ranging from resource allocation to strategic planning, highlighting their versatility and effectiveness. [PERSON]. [12] provided a comprehensive review of methods considering criteria dependencies, including ANP, DEMATEL, and Fuzzy Gray Relational Analysis (Fuzzy GRA). [PERSON]. [13] proposed a method for selecting interdependent information system projects, employing ANP to evaluate interdependencies and Goal Programming (GP) for project selection. These methods, while robust, often rely on elaborate pairwise comparison matrices (PCMs), which can become cumbersome and computationally intensive, particularly in scenarios involving a large number of criteria and alternatives. This research gap has led us to explore the potential of discrete Markov Random Fields (MRFs) as an innovative means to model criteria interdependencies within the AHP framework. MRFs, a cornerstone in the field of probabilistic graphical models, offer a robust and versatile tool for capturing dependencies in a structured manner. Widely employed in areas such as computer vision, natural language processing, and social network analysis, MRFs provide a mathematical framework for representing complex relational structures in a probabilistic context [14].