## 3 Materials and Methods  
### The Utilization of Fuzzy Logic in Multi-Criteria Decision-Making  
Over the past few decades, the utilization of fuzzy logic in Multi-Criteria Decision-Making (MCDM) has significantly increased, attributable to its capacity to address issues of uncertainty and inconsistency in expressing and modeling preferences in contexts involving estimated values or uncertain information. Furthermore, fuzzy logic is appropriate for uncertain or approximate reasoning, particularly in systems with a mathematical model that is difficult to derive [43]. The most common approach to fuzzification in MCDM problems is based on type-1 fuzzy sets, allowing membership within an interval defined by two real values. This facilitates adaptation to the ambiguities and uncertainties inherent in expressing preferences. Alternative approaches are also being utilized, which expand the membership function's scope compared to the conventional form, enabling enhanced flexibility. Examples include Pythagorean fuzzy sets [43,44], intuitionistic hesitant fuzzy sets [45], interval-valued spherical fuzzy sets [46,47], trapezoidal bipolar fuzzy sets [48], and q-rung orthopair fuzzy sets [49,50]. The Concept of Interval Type-2 Fuzzy Logic  
Type-2 fuzzy sets are extensions of type-1 fuzzy sets and are represented by an interval-based trapezoidal membership function. This provides a significantly greater degree of flexibility, making them more competent for modeling ambiguities, which, as mentioned by [PERSON] and [PERSON] [51], primarily stem from the following:  
* The meaning of linguistic terms can be ambiguous (people perceive the meaning of these expressions in various ways);
* The measures that activate type-1 fuzzy logic can be ambiguous;
* The data used to set parameters in type-1 fuzzy logic systems can also be ambiguous. [PERSON] et al. [52] attribute this to their normal, convex, and continuous membership function. [PERSON] [53] introduces IT2F sets as an alternative to conventional fuzzy sets, allowing for improved handling of uncertainties related to an element's membership owing to their inherently fuzzy membership function [54]. In addition, the simplicity of its mathematical model renders it suitable for practical applications. Considering these advantages, various MCDM methods have been integrated under IT2F sets to bring up much more convenient models of decision-making problems where uncertainty is dominant; as [PERSON] and [PERSON] [55] conclude, the trend in this research is expected to remain stable in the future. For instance, [PERSON] et al. [56] employed the integral AHP/PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluation) model in the IT2F environment for open innovation partner evaluation. In a similar context, [PERSON] et al. [57] employed the AHP method, incorporating IT2F logic to prioritize criteria. [PERSON] et al. [58] developed the IT2F decision methodology based on MULTIMORA (MULTIplicative form of Multi-Objective Optimization by Ratio Analysis), MOOSRA (Multi-Objective Optimization by Ratio Analysis), and TPOP (Technique of Precise Order Preference) for selecting the project-critical path. Meanwhile, [PERSON] et al. [59] presented an IT2 FPROMETHEE-based framework for investment decisions for a compressed air energy storage project. [PERSON] et al. [60] proposed a consensus framework for evaluating dispute resolution alternatives in international law based on an IVT2 FTOPSIS (Technique for Order Performance by Similarity to Ideal Solution) approach. Below, some concepts and the arithmetical operations of this type of fuzzy set are defined. **Definition 1**.