The Fuzzy DEMATEL-ANP (standing for Decision-Making Trial and Evaluation Laboratory – Analytic Network Process) is considered as a hybrid MCDM ( representing multi-criteria decision-making ) approach that i incorporate two powerful techniques, namely DEMATEL and ANP, increased by using fuzzy set theory to manage both the uncertainty and vagueness which are inherent in human judgment and decision Processes. The hybrid methods can be employed to model the complicated systems and decision problems involving interconnected and mutually dependent criteria, which are common in the realistic and relevant situations. The robustness is added to the decision-making process using a fuzzy element in order to capture inaccuracy in human evaluations, making it especially appropriate for the situations where subjective judgments and decisions dominate.

In this detailed description, we will address the basic concepts behind DEMATEL, ANP, and fuzzy set theory, followed by the incorporation of these techniques into the Fuzzy DEMATEL-ANP method. We will also discuss its mathematical framework, applications, and advantages over other MCDM methods. .

**DEMATEL Technique **

The Fundamentals of DEMATEL Technique

DEMATEL is considered as a technique designed for both the analysis and visualization of the structure of complicated problems by determining the causal relations among factors. It was first developed to deal with the very complicated and interdependent nature of real-world systems, especially in areas, such as engineering, management, and social sciences. DEMATEL technique can assist decision-makers to understand how various factors affects one another and which factors can play a significant l role in the SD (standing for system dynamics).

The crucial stages involved in the DEMATEL process are as follows:

– The determination of factors: The first stage is to identify all factors or criteria that can affect the decision- making process or system.

– Pairwise comparisons: Decision-makers are asked to identify similarities and differences among these factors contributing to the pairs using a scale to define the strength of effect one factor on another. For instance, a scale ranging from 0 (no effect) to 4 (very strong effect) can be employed.

– Matrix of direct influence (MDI): These pairwise comparisons can be applied to create a MDI, where the elements denote the effect one factor has on another.

– Matrix of total influence: The MDI can be employed to compute a matrix of the total influence, integrating both direct and indirect influences.

– The determination of threshold rate: A threshold value is set to filter out less significant effects , allowing the decision-makers to concentrate on the most important relations. .

– The creation of causal diagram: Finally, a causal diagram or network is created, indicating which factors are influential (cause) and which influenced (effect) are. The diagram presents a visual understanding of a system structure.

**ANP** **Technique**

**An Overview of ANP Technique**

ANP, as an extension of the AHP (representing Analytic Hierarchy Process), was developed by Thomas L. Saaty, to explore more complicated decision-making problems involving interdependencies among criteria. While AHP technique considers a hierarchy where criteria are independent, it relaxes this assumption and takes the interrelated criteria into consideration. This can be obtained by creating a network, where elements can affect each other in a feedback loop.

** ANP method includes the following crucial stages**

– Modeling of the decision problem: The first stage is to identify the goal, criteria, sub-criteria, and alternatives. In ANP technique, these elements are organized in a network rather than a hierarchal structure.

– Pairwise comparisons: like AHP method, decision-makers are asked to carry out pairwise comparisons between elements for determining their relative influence or significance. However, in ANP method, feedback and interdependencies among elements are examined by these comparisons.

– The creation of supermatrix: The results obtained from pairwise comparisons are employed to create a supermatrix, capturing the effect of each element on others. The supermatrix is a major part of ANP method because it indicates the network of interdependencies in the decision problem.

– Limiting the supermatrix: To put the elements in order of their relative importance and get global weights, the supermatrix can be normalized and raised to a limiting power until it converges, resulting in the final rankings of elements.

** Fuzzy Set Theory**

**Introduction to Fuzzy Sets**

Traditional decision-making approaches usually depend on booth crisp values and definitive judgments. However, real-world problems are filled with uncertainty, ambiguity, and vagueness. Fuzzy set theory, proposed by Lotfi Zadeh in 1965, presents a mathematical framework for managing the uncertainty by enabling partial membership in a set, as contrasted with the binary membership utilized in classical set theory.

**In fuzzy set theory**

– A fuzzy set is characterized by a membership function that allocates a value between 0 and 1 to each element, indicating the degree of membership in the set.

– Linguistic variables (low, medium, and high) can be modeled by fuzzy sets, making it easier to show subjective judgments.

– Fuzzy numbers, especially both TFNs ( standing for the triangular fuzzy numbers) and trapezoidal fuzzy numbers, can be employed to model the numerical uncertainty.

In decision-making, fuzzy set theory provide for for the integration of inexact and vague information, making it a useful tool for modeling human judgments and decisions, which are often subjective and uncertain.

** Fuzzy DEMATEL-ANP-Based Technique**

The strengths of DEMATEL, ANP, and fuzzy set theory are combined by the Fuzzy DEMATEL-ANP-based technique to explore the complicated decision-making problems involving interdependencies among criteria and uncertainty in judgments. The following is a step-by-step outline of how this hybrid method works:

**Determining Criteria and Building a Decision Model**

The first stage involved in the Fuzzy DEMATEL-ANP technique is to define the decision problem by determining the criteria, sub-criteria, and alternatives. The decision model is built in the form of a network, indicating the interdependencies among criteria. These interdependencies are considered as a major feature of the ANP technique and provide for a more realistic representation of complicated systems.

** Fuzzy Pairwise Comparisons**

Fuzzy pairwise comparisons are performed When the criteria are determined.. Instead of employing the crisp values to compare the significance or influence of one criterion over another, linguistic terms(low, medium, and high) are used by decision-makers in order to express their judgments. These linguistic terms are then converted into fuzzy numbers, usually triangular fuzzy numbers, demonstrating the uncertainty in human judgment.

For instance, a linguistic judgment of “medium” might be denoted by a triangular fuzzy number (2, 3, 4), where 2 represents the lower bound, 3 denotes the most likely value, and 4 indicates the upper bound. The the pairwise comparisons matrices with fuzzy elements are is created for both the DEMATEL and ANP processes.

** DEMATEL Phase: Fuzzy Matrix of Direct Influence**

In the DEMATEL phase, the fuzzy pairwise comparisons of criteria are used to constructa fuzzy matrix of direct influence... The matrix elements are fuzzy numbers showing the degree of effect of one criterion over another. The matrix is employed to compute the fuzzy matrix of total influence, incorporating both direct and indirect effects. .

For simplifying an analysis, the fuzzy matrix of total influence is defuzzified (i.e., converted back to crisp values) via techniques used for defuzzification, including the centroid method or the mean of maximum. This crucial step is used to determine the most significant criteria and relations in the system.

** Threshold and Causal Diagram**

A threshold value can be determined to filter out less important relations in the influence matrix. This enables decision-makers to concentrate on the most significant causal relations. A causal diagram is then created based on the defuzzified matrix of total influence. The diagram visually displays the causal structure of the decision problem, indicating which criteria are the most influential.

** ANP Phase: Formation and Normalization of A Supermatrix**

In the ANP phase, the fuzzy pairwise comparisons are employed to create a fuzzy supermatrix, capturing the effect of criteria on one another in the network. To achieve a crisp supermatrix, the fuzzy supermatrix is then normalized and defuzzified.

The supermatrix is raised to a limiting power until it converges, resulting in the final ranking or weight of each criterion . These weights show the relative significance of criteria, taking the interdependencies among them into consideration.

** Synthesis and Decision-Making**

The results obtained from the DEMATEL and ANP phases are synthesized to get a a deep, thorough understanding of the decision problem. The causal relations determined in the DEMATEL phase can assist decision-makers to understand the system structure, while the priorities of the ANP phase presents a quantitative ranking of each criterion.

The final decision is made based on these insights, taking into account both the causal relations and the relative significance of each criterion.

**Applications of Fuzzy DEMATEL-ANP Technique**

The Fuzzy DEMATEL-ANP technique can be applied in many fields, including:

– Supply chain management: (SCM) To determine the crucial factors influencing supply chain performance and arrange the improvement strategies.

– Project management: To examine the interdependencies among project risks and arrange strategies of risk reduction.

– Sustainability evaluation: To assess the sustainability of products or services by taking environmental, social, and economic factors into consideration.

– Healthcare decision-making process: To evaluate healthcare policies or interventions by inveatigating the complex relations among healthcare criteria.

** Benefits of Fuzzy DEMATEL-ANP Technique **

– Managing Interdependencies: dissimilar to traditional methods, the Fuzzy DEMATEL-ANP technique clearly takes the interdependencies among criteria into consideartion, presenting a more realistic accuracy model model of decision problems.

– Integration of Uncertainty: Fuzzy set theory enables decision-makers to present their judgments using linguistic terms, which are more intuitive and better adapted for managing both uncertainty and inaccuracy .

– Causal Analysis: The DEMATEL phase of the method presents the useful insights into the causal structure of the problem, which can assist decision-makers to understand how different criteria affect one another.

**Conclusion**

The Fuzzy DEMATEL-ANP-based technique is a useful tool used for decision-making process in complicated , interdependent systems with inherent uncertainty. By incorporating the strengths of DEMATEL, ANP, and fuzzy set theory, this hybrid approach provides a thorough and robust framework to analyze and solve real-world decision problems. It has the ability to model interdependencies, manage uncertainty, and present a visual representation of causal relations makes it an ideal option for decision-makers involved in many fields.