Point, Multi-criteria decision making problem.

Fuzzy numbers are real numbers with inaccurate precision that may deal with numerical amount impreciseness. Generalizing the idea of fuzzy numbers, Intuitionistic Fuzzy Numbers (IFNs) are highly helpful in modeling situations with imprecise and insufficient data. The literature proposes a variety of IFNs to model real-life problems using different concepts. These include Real-Valued Intuitionistic Fuzzy Numbers (RVIFNs), Interval-Valued Intuitionistic Fuzzy Numbers (IVIFNs), Triangular Intuitionistic Fuzzy Numbers (TIFNs), and Conventional Trapezoidal Intuitionistic Fuzzy Numbers (CTrIFNs). Each variety of IFNs has a distinct benefit for modeling real-world scenarios, and they are all necessary. For applying suitable ranking procedures for real-time problems, it is necessary to order them.

A new parametric method for ranking IFNs was proposed, and its superiority over other methods was demonstrated using appropriate examples in [1]. Total ordering on the entire class of IFN using upper lower dense sequence was proposed and compared with existing techniques using illustrative examples in [2]. A new ranking based on considering a fuzzy origin and measures distance of each IFN from fuzzy origin in [3]. The method based on the incircle of the membership function and non-membership function of TIFN uses lexicographical order to rank IFNs was investigated by Atalik and Senturk in [4]. A new type-2 intuitionistic exponential triangular fuzzy number was introduced in [5] and some of properties and theorems of this type of fuzzy number with graphical representations have been studied and some examples are given to show the effectiveness of the proposed method. Also, the ranking function based on the centroid concept and Euclidean distance of the generalized exponential TIFN is computed. The defuzzification measure based on area of region for intuitionistic triangular fuzzy numbers was introduced in [6]. Christi [7] applied Ranking technique called Accuracy Function and Russell’s Method to find the optimal solution of the transportation problem in 2015. Salahshour [8] suggested a new defuzzification for obtained TFNs using their values and ambiguities. The mean value of intuitionistic fuzzy variables is obtained with the help of credibility distribution. Then, a more comprehensive ranking method based on mean value of intuitionistic fuzzy variable was developed [9]. A new ranking method is developed on the basis of the concept of a ratio of the value index to the ambiguity index and applied to multiattribute decision making problems in which the ratings of alternatives on attributes are expressed with TIFNs [10]. Ummusalma and Selvakumari [11] presents a solution methodology for Multi Criteria Decision Making (MCDM) problems in intuitionistic fuzzy environment in which the ratings of choice on attributes are represented by TIFNs. In 2014, Liang et al. [12] developed the method for ranking TIFNs based on score and accuracy functions. Later it was applied to solve MCDM problems. Stanujkic et al. [13] used Hamming distance to rank TIFNs, which was applied to a case study of hotels’ websites evaluation. An extended model of DEA is proposed under the triangular intuitionistic fuzzy environment where the inputs and outputs of DMUs are TIFNs in [14]. A new ranking based on preference degree of TFNs is defined combining the probability with possibility of TFN, which considers all the possible values in TFN. Based on this ranking, a new method is proposed to solve triangular fuzzy MAGDM problem [15]. A ranking algorithm for TIF-MCDM issue for the accessible options by processing the various distances between the perfect option and all the accessible options has been proposed [16]. An extension of fuzzy TOPSIS for MCDM problem in intuitionistic fuzzy environment based on magnitude ranking was studied by Gautam and Singh [17]. Selvaraj and Majumdar [18] proposed a non-membership score function which has been utilized in IVIF-TOPSIS using numerical examples. For many decision problems with uncertainty, TIFN is a useful tool in expressing ill-known quantities. Xu et al. [19] develops a novel decision method based on zero-sum game for multiple attribute decision making problems where the attribute values take the form of TIFNs and the attribute weights are unknown.

Novelty and motivation: The motivation behind research work in the ranking of IFNs lies in addressing the challenges posed by uncertainty and vagueness in decision-making processes. The ranking of IFNs becomes crucial in various applications, particularly in decision-making and optimization problems, where the decision-makers need to compare and prioritize alternatives under uncertainty. Here are some key motivations for research in this area:

I. Decision support systems: In real-world decision-making scenarios, decision-makers often deal with incomplete or imprecise information. The ranking of IFNs is essential for developing decision support systems that can handle uncertainty effectively, enabling better decision outcomes.

II. Optimization problems: Optimization problems under uncertainty require methodologies that can handle fuzzy or uncertain data. Ranking IFNs is a key step in developing optimization algorithms that can work with intuitionistic fuzzy information, leading to more robust and adaptable solutions.

III. Risk management: In various fields such as finance, engineering, and environmental science, decisions are often made in the presence of risk. IFNs can model uncertainty and risk more accurately than traditional methods. Research in ranking IFNs aids in developing tools for risk assessment and management.

The definitions and ideas pertaining to Intuitionistic Fuzzy Sets (IFSs) and intuitionistc fuzzy numbers are briefly summarized in this section.

Definition 1 ([20]). An IFS A in X is given by , where the functions define respectively, the degree of membership and degree of non-membership of the element to the set A, which is a subset of X, and for every Obviously, every fuzzy set has the form

For each IFS A in X, we will call the intuitionistic fuzzy index of x in A. It is obvious that .

Definition 2 ([21]). An IFS A = {(x, μA(x), γA(x)|x ∈ X)} is called IF-normal, if there exist at least two points x0, x1 ∈ X such that μA(x0) = 1, γA(x1) = 1, it is easily seen that given IFS A is IF-normal if there is at least one point that surely belongs to A and atleast one point which does not belong to A.

Definition 3 ([21]). An IFS A = {(x, μA(x), γA(x)|x ∈ X)} of the real line is called IF-convex, if ∀ 1, 2 ∈

ℝ, ∀ ∈ [0,1].

μA(λx1 + (1 − λ)x2 ) ≥ μA(x1)⋀μA(x2),

γA(λx1 + (1 − λ)x2 ) ≥ γA(x1)⋀γA(x2).

Thus A is IF –convex if its membership function is fuzzy convex and its non-membership function is fuzzy concave.

Definition 4 ([21]). An IFS A = {(x, μA(x), γA(x)|x ∈ X)} of the real line is called an IFN if I. A is IF-normal.

II. A is IF-convex.

III. μA is upper semicontinuous and γA is lower semicontinuous.

IV. A = {(x ∈ X|γA(x) < 1} is bounded.

Definition 5 ([22]). A TIFN is an IFS in R with the following membership function μ_A (x) and non-membership function ϑ_A (x)

x − a1 , a1 ≤ x ≤ a2, a2 − a1 μ (x) = x − a3 , a2 ≤ x ≤ a3, a2 − a3 {0, otherwise.

a2 − x , a1 ≤ x ≤ a2, a2 − a1 ϑ (x) = x − a2 , a2 ≤ x ≤ a3, a3 − a2 { 1, otherwise,

where a1 ≤ a1 ≤ a2 ≤ a3 ≤ a3 and μA(x) + ϑA(x) ≤ 1 or μA(x) = ϑA(x) for all x ∈ R. This TIFN is denoted by A = (a1, a2, a3; a1, a2, a3).

Definition 6 ([22]). Let and , then II.

III.

This section details the method of finding coordinates of Nagel point and algorithm to order TIFNs using Nagel points.

Let be a TIFN. The Cartesian coordinates that corresponds to the vertices of

two triangles ABC and DBE representing the membership function and non-membership function

are

Then

The trilinear coordinates of Nagel point of triangle ABC can be found as follows.

The trilinear coordinates of Nagel point for triangle DBE is calculated as follows.

Finally the Cartesian coordinates of Nagel Point related to non-membership function and membership function are produced below.

The Nagel points and corresponding to the non-membership and membership has to be combined which enables to propose ranking of TIFNs.

Hence For simpler ranking,

The new ordering of TIFNs using Nagel point is given by The proposed ranking procedure is well explained by the forthcoming algorithm.

Prakash et al. [23] provide a set of axioms as appropriate characteristics for assessing an order technique's rationality. In this part, it will be demonstrated that the suggested method satisfies Prakash et al.'s [23] reasonable qualities.

Theorem 1. Let be the set of all TIFN’s and E is a subset of . Then for any A ̃ ∈ E, A ̃ ≼ A ̃.

Theorem 2. For any arbitrary finite subset E of and (A ̃, B ̃) ∈ E × E, if A ̃ ≼ B ̃, B ̃ ≽ A ̃, we have A ̃ ≅ B ̃.

Theorem 3. For any arbitrary finite subset E of and (A ̃, B ̃) ∈ E × E, if A ̃ ≼ B ̃, B ̃ ≼ C ̃, we have A ̃ ≼ C ̃.

Theorem 4. Let A ̃ and B ̃ beany two TIFN’s. Then N = N + N .

A ̃+B ̃ A ̃ B ̃

In this section, an example illustrating the proposed algorithm in the previous section was initiated first. Then a comparative study with existing ordering procedures in the literature was made and results were tabulated.

Example 1. Consider any two TIFN’s .

Now

Now

Then The final ranking is given by

Similarly calculations, shows that Since

The comparison between the suggested ordering and the current approaches is displayed in the following table. Our suggested approach is consistent with the current approaches, demonstrating the reliability of the new ranking.

Consider the TIFN’s

This section presents a MCDM problem utilizing TIFNs as the decision-making quantifiers, based on the suggested ranking approach.

The multi criteria decision procedure is framed with the criterions to assess the malternatives . The best option that maximizes the benefit criterion and minimizes the cost criterion will be chosen by the decision maker. The weight vector of the criterion is expressed by . The linguistic expression of the expert to represent criteria in alternative is given by the TIFNs

Therefore MCDM problem can be constructed in matrix form as

Now the detail step by step procedure to solve the constructed MCDM problem is as follows.

Step 1. Compute the normalized TIFNs decision matrix.

The primary step to avoid certain mismatches in measures of criterions like scaling, comparing etc., can be neutralized by the process of normalization. The normalized decision matrix has been constructed, using the following formula, the normalization of values in decision matrix can be obtained.

Step 2. Compute the weighted normalized TIFNs decision matrix.

The elements of weighted normalized TIFNs decision matrix are computed by using the formula

The transformed decision matrix is given by Step 3. Compute the aggregated values of the attributes

Step 4. Now rank the alternatives.

Apply the proposed Nagel point ranking for for each alternatives and conclude the problem by ranking procedure.

Step 5. Depending up on the alternatives and criterions nature conclude the decision by selection of maximum or minimum values.

This section discusses a real-world example to illustrate the suggested ranking process. Selecting the ideal site for an aerospace research organization center installation is crucial. Five areas were discovered by the space scientists group after a preliminary survey: A1, A2, A3, A4, and A5. Five criteria were used by the space scientists group to evaluate the locations: Geographic position (C1), climatic condition (C2), safety factor (C3), functional area (C ), and pollution factor (C ). Let w = (0.2,0.15, 0.25,0.1,0.3)T represent the criterion weight factor. The following table shows how the alternatives were rated using the specified criteria.

Applying Nagel point concept, the final decision can be obtained as, Comparative ranking with existing methods.

From Table 5, it is evidently shows that the proposed ranking method align with the existing ranking methods.

This article suggests a new ordering of TIFNs using nagel points as a basis. Some numerical examples to illustrate the applicability of the strategy are provided. Moreover, a comparison analysis is carried out to verify the proposed methodology. A comparison with a few other methods in use is used to demonstrate the ranking method's superiority. This research has some theoretical and managerial ramifications.

In conclusion, the process of ranking TIFNs is a complex and crucial task, given the inherent uncertainty and imprecision associated with these numbers. The study aimed to develop a systematic approach for ranking TIFNs, taking into account both the membership and non-membership degrees, to enhance decision-making processes in various real-world applications.

Through an extensive review of existing methods and a critical evaluation of their limitations, this research proposed a novel ranking method based on a comprehensive consideration of the centroid deviation and area deviation between TIFNs. The proposed method successfully addresses the challenges posed by existing approaches, offering a more robust and accurate means of ranking TIFNs.

The significance of this research extends to diverse fields where decision-making involves uncertainty and imprecision. Whether in finance, engineering, or other domains, the ability to rank TIFNs accurately contributes to more informed and confident decision-making processes. The proposed method not only fills a gap in existing methodologies but also opens avenues for further research and refinement in the realm of ranking under uncertainty.

In conclusion, the developed ranking method for TIFNs offers a valuable contribution to the field, enhancing our ability to navigate decision-making in situations characterized by vagueness and ambiguity. As decision-makers continue to grapple with complex and uncertain information, the proposed method provides a reliable tool for ranking TIFNs, promoting more effective and robust decision outcomes.

K. A. P. verified the manuscript, and M. S. wrote the manuscript.

we assure that there in no funding support we have received.

Not Applicable.

The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.