A New Ranking Method for Interval-Valued Intuitionistic Fuzzy Numbers and Its Application in Multi-Criteria Decision-Making

Ranking of interval-valued intuitionistic fuzzy numbers (IVIFNs) is an important task for solving real-life Decision-Making problems. It is a potential area of research that has attracted the researchers working in fuzzy mathematics. Researchers worldwide are looking for a unique ranking principle that can be used to discriminate any two arbitrary IVIFNs. Various ranking functions on the set of IVIFNs have been proposed. However, every method has some drawbacks in ranking arbitrary IVIFNs due to the partial ordering. This paper introduces a new ranking principle for comparing two arbitrary IVIFNs by defining a new score function based on the non-membership value of IVIFNs. In this paper, firstly, the limitations of a few well-known and existing ranking methods for IVIFNs have been discussed. Secondly, a new non-membership score on the class of IVIFNs has been introduced. Thirdly, the superiority of the proposed score function in ranking arbitrary IVIFNs over the existing methods has been demonstrated. Finally, the proposed non-membership score function has been utilized in interval-valued intuitionistic fuzzy TOPSIS (IVIF-TOPSIS) using numerical examples.


Introduction
Decision-Making problems often involve imprecise information that can be modelled better using fuzzy sets than classical sets. The main advantage of using fuzzy sets is that they give freedom to the decision-maker to assign membership values between 0 and 1, whereas in the case of crisp sets, decision-makers are restricted to providing a crisp value (either 0 or 1). However, in most of the real-life problems, incompleteness occurs in the data in addition to imprecision. Intuitionistic fuzzy set (IFS) was introduced in the literature as a generalization of the classical fuzzy set. The main characteristics of an IFS is that the sum of the degrees of membership and non-membership values of any element in the IFS will be less than or equal to 1. Further, it has been generalized to interval-valued intuitionistic fuzzy set (IVIFS) since the IVIFSs model incompleteness in a better way than the real-valued IFSs. The ranking of IVIFSs will play an essential role in solving any decision-making problem involving imprecision and incompleteness. A lot of ranking procedures have been introduced in the literature for differentiating any two arbitrary IVIFNs. Xu [1] introduced the idea of score and accuracy functions on the set of IVIFNs. Using these two functions, he tried to compare arbitrary IVIFNs. His "score function" was nothing but the difference between the mid-points of the membership and non-membership functions, and the "accuracy function" was the sum of the midpoints of the membership and non-membership functions. However, if A = ([a, b][c, d])

1.
discuss the limitations of a few well-known ranking methods for IVIFNs; 2.
introduce a new non-membership score on the class of IVIFNs and study its mathematical properties; 3.
demonstrate the superiority of the proposed score function, over the existing score functions, in ranking arbitrary IVIFNs; 4.
explain the applicability of the proposed ranking method in solving interval-valued intuitionistic fuzzy multi-criteria decision-making (IVIF-MCDM) problems using numerical examples.
The remainder of the paper is organized in the following manner. After the introduction, a few basic definitions are presented in Section 2. Section 3 discusses the various drawbacks of existing methods in ranking arbitrary IVIFNs. Section 3 also compares different available ranking principles of IVIFNs. Section 4 presents the new score function based on the non-membership value of an IVIFN, along with its essential mathematical properties. Section 4 also encompasses the comparison between the proposed score function and the available ranking procedures of IVIFNs. Section 5 presents the proposed score function's applicability in solving the MCDM problem modelled under an intervalvalued intuitionistic fuzzy environment. Numerical illustrations of the proposed MCDM algorithm are also given in Section 5. Finally, conclusions are presented in Section 6.

Preliminaries
Some basic definitions are given in this section.
Definition 1 (Xu 2007, [1]). Let D[0, 1] be the set of all closed subintervals of the interval [0, 1]. An IVIFS on a set X = φ is an expression given by The intervals µ A (x) and ν A (x) denote, respectively, the degree of belongingness and non-belongingness of the element x to the set A. Thus, for each x ∈ X, µ A (x) and ν A (x) are closed intervals whose lower and upper end points are, respectively, denoted by For each element x ∈ X, we can compute the unknown degree (hesitance degree) of belongingness

Comparison between Various Existing Methods for Ranking IVIFNs
This section compares different methods for ranking arbitrary IVIFNs and discusses the drawbacks of one method over other methods based on their limitations using numerical examples.
Here, we discuss the limitations of Generalized Improved Score function (GIS) [6] in different cases.

1.
( Hence the generalized improved score function introduced by Garg [6] fails to rank arbitrary IVIFNs of this type; 2. ( . This implies that k 1 and k 2 do not have any importance in ranking arbitrary intuitionistic fuzzy numbers; 3. ( Here, both the initial assumptions contradict each other. Hence, they cannot be considered (together) in the proof. 5. ( However A 1 = A 2 because their nonmembership values are different, and they are non-zero. If Xu's [1] score function is applied, then we get A 1 > A 2 . In these places, Xu's [1] score function works better which is happening because Garg's method measures the membershipness of an arbitrary IVIFNs, and hence the generalized improved score function maps the IVIFNs with zero membership value to zero; 8.
Let This implies that M = N. This example shows that none of these methods [1,2,6] are better to each other; 9.
Comparison of GIS score function with Novel accuracy score function M in Ye [2]: Using example 2.1. in [6], Garg has shown the inconsistency of Ye's [2] method. However, his method also fails to compare arbitrary IVIFNs which we can be see from the following example. , respectively. This implies that A = B. Hence, both the methods are illogical in comparing arbitrary IVIFNs of the above type. 11. Comparison of GIS function with an improved score function I in Bai [5]: Since Garg's [6] method is the generalization of Bai's [5] improved score function, Garg's method also has the same drawbacks of Bai's [5] improved score function. Hence, from the above examples, we may conclude that none of the familiar methods [1,2,[5][6][7] can rank arbitrary IVIFNs, which makes the pathway for defining a new score function on the class of IVIFNs for comparing any two IVIFNs and also GIS did not define a total order on the set of IVIFNs, which is seen from the incorrectness of Theorem 3.1 (in page no. 5 of [6]). In 2016, Nayagam et al. [10] have defined a non-hesitance score for ranking arbitrary IVIFNs. The ranking principle that was introduced by Nayagam et al. [10] has overcome all the drawbacks mentioned above. The efficiency of Nayagam et al.'s [10] approach in overcoming the Limitations I 1 to I 7 and the drawbacks 1-5 are shown in the following table.
Hence, from  [10] non-hesitance score function is also failed to rank arbitrary IVIFNs in some places, which can be seen from the following example.  Hence, there is a need for another parameter for ranking IVIFNs. In the following section, a new score function has been introduced, which measures the non-membershipness of an IVIFN for ranking arbitrary IVIFNs.

A New Non-Membership Score of IVIFNs
This section proposes a non-membership score function for ranking arbitrary IVIFNs and examines some of its properties using illustrative examples. Proof. Let A, B ∈ IV IFN. Assume: Add and subtract a 1 b 2 from (2), we get,

Proof. Let
Using Definition 8, we get, It is very clear that, Therefore, from Theorem 1, N M (A + C) ≥ N M (B + C). Hence the proof. The efficacy of the proposed non-membership based score function in overcoming the Limitations I 1 to I 6 are shown in Table 2. Table 2. Comparison of some existing score function with the proposed non-membership based score function.

Numerical Example Shortcomings of Existing Methods Proposed Ranking Principle
which favors with human intuition.  Table 2, it is very clear that the shortcomings of all the ranking methods introduced in [1,2,5-7,10] can be eliminated by the proposed ranking principle.

Interval-Valued Intuitionistic Fuzzy TOPSIS
This section presents a new algorithm for solving MCDM problems using the IVIF-TOPSIS method. TOPSIS is one of the most popular MCDM methods introduced by Hwang and Yoon [15]. Any IVIF-MCDM problem is mathematically defined as follows.
Let M = {M 1 , M 2 , . . . , M r } be the set of r alternatives and let C = {C 1 , C 2 , . . . , C s } be the set of s criteria based on which the alternatives to be evaluated. Let N = (n ij ) r×s be the decision matrix, where n ij represents the performance of M i (i-th alternative ) with respect to the C j (j-th criterion). In this paper, every n ij is represented by an IVIFN, that is, , where n ij represents the membership and nonmembership degree of alternative M i with respect to criterion C j .
Algorithm 1 for solving the IVIF-MCDM problem is given as follows: be the decision matrix. 1.

Score Matrix (N ):
A new r × s score matrix N is formed by applying non-membership score function to the each entry of the decisionmatrix N. i.e., N = (s ij ) r×s , where s ij = N M (n ij ); 2.

Interval-valued intuitionistic fuzzy positive ideal solution (IVIFPIS):
An interval-valued intuitionistic fuzzy positive ideal solution (IVIFPIS) denoted by P IS , is the set of IVIFNs, where the number of IVIFNs in P IS is equal to s (the number of columns in the decision matrix); that is, P IS = {P 1 , P 2 , . . . , P s }. Each P i is an IVIFN and is defined as: . . .

Interval-valued intuitionistic fuzzy negative ideal solution (IVIFNIS):
An interval-valued intuitionistic fuzzy negative ideal solution (IVIFNIS) denoted by N IS , is the set of IVIFNs where the number of IVIFNs in N IS is equal to s (the number of criteria available in the problem), that is, . . , N s }. Each N i is an IVIFN and is defined as: . . .

Distance between aggregated performance of alternatives and IVIFPIS, IVIFNIS:
The aggregated performance of an alternatives, non-membership score functions of IVIFPIS and IVIFNIS are obatined as real numbers which can be seen from Step 1 and Step 4. Hence, the distance between them can be calculated using the following formula: ) represent the distance between alternative M i and IVIFPIS, IVIFNIS, respectively, and w j is the weight of the j-th criterion; 6.
Closeness Coefficient of Alternative (C i (M i )): Relative closeness of an alternative M i is defined as, . The values of the closeness coefficient decides the Ranking of alternatives (M i ). The alternative with the highest closeness coefficient will be ranked first.

Numerical Illustrations
This subsection demonstrates the applicability of the proposed algorithm in solving the IVIF-MCDM problem using numerical examples.

Example 5.
An animation expert wants to select the best computer animation work for an award from four possible alternatives (A 1 , A 2 , A 3 , A 4 ). The expert has to make a decision according to the following three criteria, (a) C 1 is the artistic appeal, (b) C 2 is the visual effect and (c) C 3 is the creative script. The criteria weight is given by W = {0.35, 0.25, 0.40}. Six candidates are evaluated based on their interview performance. Performance of six candidates under six criteria are represented by IVIFNs and which are given in decision matrix M = (m ij ) 6×6 , where m ij represents the performance of candidate A i with respect to criteria Cr j , i, j = 1, 2, . . . , 6.
Algorithm 1 is used to solve the selection problem.

Conclusions and Future Scope
A new non-membership score function has been introduced for the effective comparison of any two arbitrary IVIFNs. Besides, the effectiveness of the proposed nonmembership score function has been demonstrated by comparing it with different existing methods. Finally, the application of the proposed non-membership score function in solving IVIF-MCDM problems using the IVIF-TOPSIS method has been demonstrated through numerical examples. The proposed score function defines a total ordering on the particular subset of IVIFNs and does not define a total order on the entire class of IVIFNs. Therefore, a total ordering principle on the class of IVIFNs would be aimed in further research. Similar to IVIFNs, interval-valued Pythagorean fuzzy numbers (IVPFNs) [16], interval-valued Fermatean fuzzy numbers (IVFFNs) [17], and so forth, are also used in the literature for modelling real-life problems with incomplete information. Hence, the proposed non-membership score function can be extended to the class of IVPFNs and IVFFNs. Using this, a new MCDM algorithm can be developed for solving problems modelled under the IVPFN and IVFFN environment. Researchers can also introduce a new similarity on the set of IVPFNs and IVFFNs by using the idea of the non-membership score function.