## 3 Materials and Methods
### Methods
#### 3.2.5 Application Belief Structure in Multi-Criteria Decision-Making Analysis  
The Belief Structure model has been integrated into a multi-criteria framework [46; 62], and later extended to a Fuzzy Belief Structure in relation to fuzzy conditions [61; 63; 64; 65], with evaluation grades and their utilities represented by triangular or trapezoidal fuzzy numbers. Both models highlight their versatility and applicability in decision-making processes involving complex criteria.  
[PERSON] et al. [46] proposed the TOPSIS method to address group multi-criteria decision-making problems under uncertainty using the Belief Structure model. They introduce a belief decision matrix, aggregate judgments through evidential reasoning, and define ideal belief solutions. A belief distance measure is then employed to rank alternatives based on their proximity to the ideal solutions, with a numerical example illustrating the methodology.  
In a subsequent paper, [PERSON] et al. [61] present a method that combines the TOPSIS technique with a Fuzzy Belief Structure (FBS) model to tackle Group Belief MCDM challenges. In this approach, the judgments of decision-makers are modeled using FBS frameworks, and a numerical example is provided to demonstrate the effectiveness of the proposed method.  
[PERSON] and [PERSON] [62] analyzed decision-making processes under conditions of risk and uncertainty through the perspective of Sociological Game Theory (SGT). Their work emphasizes key components of SGT, including the belief structure, value complex, action repertoire, and judgment complex. The study identifies correlations between the universal architecture of SGT and the steps of the TOPSIS method based on Belief Structure.  
[PERSON] et al. [63] introduce a novel Failure Mode and Effect Analysis (FMEA) model that integrates the TOPSIS method with a belief structure, thereby overcoming the limitations of traditional FMEA. The fuzzy belief TOPSIS approach allows for the expression of expert knowledge in fuzzy linguistic terms and incorporates variable component weights within the FMEA index. By integrating uncertainty and expert judgment, this enhanced model significantly improves risk evaluation.  
The study by [PERSON] et al. [64] highlights the critical role of public transport in urban settings, focusing on a case study that examines the causes of delays in the Tehran metro system. By employing the VIKOR method under uncertainty using FBS, the study identifies human-related issues and transportation system problems as the primary contributors to delays. The results are compared with those obtained using the TOPSIS with FBS.  
[PERSON] et al. [65] propose a hybrid method that combines a compromise solution approach with Data Envelopment Analysis (DEA) to assess the efficiency of Decision-Making Units (DMUs). This proposed method integrates the advantages of TOPSIS, VIKOR, DEA, and Fuzzy Belief Structures to facilitate group decision-making based on input from multiple experts. It utilizes fuzzy evaluation grades and belief degrees within the multi-objective fuzzy belief structure model, and an illustrative example is included to demonstrate the effectiveness of the method.