Intuitionistic fuzzy exponential divergence: Application in multi-attribute decision making

Abstract

The aim of this paper is to introduce an exponential methodology for measuring the degree of divergence between two intuitionistic fuzzy sets. For this, an intuitionistic fuzzy exponential divergence measure is proposed and its important properties are discussed axiomatically. In addition, the applicability and efficiency of the proposed intuitionistic fuzzy exponential divergence measure have been demonstrated by comparing with existing intuitionistic fuzzy divergence measures using a numerical example in the framework of pattern recognition. Further, a method to solve multi-attribute decision making problem using the proposed divergence measure in intuitionistic fuzzy environment is presented with an illustrative example. Lastly, a comparative study of the proposed method with the existing TOPSIS and MOORA methods of multi-attribute decision making in an intuitionistic fuzzy environment is presented.

Keywords

Intuitionistic fuzzy set discrimination information measure intuitionistic fuzzy divergence measure multi-attribute decision making (MADM)

1 Introduction

The fuzzy set theory introduced by Zadeh [19] has received a vital attention from researchers for its application in various fields such as pattern recognition, image processing, speech recognition, bioinformatics, fuzzy aircraft control, feature selection, decision making, etc. Entropy, as a measure of fuzziness or uncertain information was presented by Zadeh [20]. The concept of divergence measure was first developed by kullback and Leibler [22]. The corresponding divergence measure based on logarithm operation and an axiomatic structure for the directed divergence measure of fuzzy sets (FSs) was introduced by Bhandari and Pal [23]. In 1999, an exponential divergence measure of FSs is introduced by Fan and Xie [7]. Later on, many researchers have made efforts to define entropy and divergence measures of information in the fuzzy environment and find their applications in a variety of fields. For example, Tomar and Ohlan [32, 33] studied a sequence of fuzzy mean difference divergence measures and a parametrically generalized exponential fuzzy divergence measure respectively, and provided their applications in the context of pattern recognition and decision making.

The notion of Atanassov’s intuitionistic fuzzy sets (IFSs) was first originated by Atanassov [1 –4] which found to be well suited to deal with both fuzziness and lack of knowledge or non-specificity. It is noted that the concept of an IFS is the best alternative approach to define a FS in cases where existing information is not enough for the definition of imprecise concepts by means of a conventional FS. Thus, the concept of Atanassov IFSs is the generalization of the concept of FSs. In 1993, Gau and Buehrer [24] introduced the notion of vague sets. But, Bustince and Burillo [25] presented that the notion of vague sets was equivalent to that of Atanassov IFSs. As a very significant content in fuzzy mathematics, the study on the divergence measure between IFSs has received more attention in recent years. Divergence measures of IFSs have been widely applied to many fields such as pattern recognition [9, 10, 12, 14, 42], linguistic variables [8], medical diagnosis [5, 26], logical reasoning [35] and decision making [11 , 21]. Since the divergence measures of IFSs have been applied to many real-world situations, it is expected to have an efficient divergence measure which deals with the aspect of uncertainty i.e., fuzziness and non specificity or lack of knowledge.

In recent years, some definitions of divergence measures for IFSs have been proposed by the researchers. For example, Szmidt and Kacprzyk [13] provided the methods of distance between IFSs. Li [6] introduced the intuitionistic fuzzy dissimilarity measure between IFSs. Hung and Yang [8] defined an axiomatic structure of the divergence measure between IFSs using the hausdorff distance. In 2005, an axiomatic definition of distance measure in IFS is introduced by Wang and Xin [36] and applied to pattern recognition. Vlachos and Sergiadis [14] provided an intuitionistic fuzzy divergence measure in analogy with Shang and Jiang [27]. Zhang and Jiang [26] presented an entropy and divergence measure between vague sets with the application to pattern recognition and medical diagnosis. In 2008, Hung and Yang [9] constructed J-divergence between IFSs and applied it to pattern recognition. Further, Papakostas et al. [12] provided a comparative analysis of distance and similarity measures between IFSs from a pattern recognition point of view.

From the significant studies, it is noted that IFS theory is well suited to deal with vagueness and inadequate knowledge. Recently, many researchers have used IFSs to build the decision making models which can hold imprecise information. However, there is little investigation on multi-attribute decision making (MADM) using the intuitionistic fuzzy divergence measures. Briefly motivated by the above mentioned works, in this paper, we apply the exponential approach on IFSs and propose a new information-theoretic divergence measure, called intuitionistic fuzzy exponential divergence, to measure the difference between two IFSs.

The rest of the paper is organized as follows: Section 2 is devoted to briefly review some well-known concepts related to fuzzy set theory and intuitionistic fuzzy set theory. In Section 3, we introduce the intuitionistic fuzzy exponential divergence measure between IFSs with the proof of its validity and establish some more elegant properties of the proposed divergence measure in a number of theorems. In order to show the authenticity and efficiency of the proposed intuitionistic fuzzy exponential divergence in pattern recognition a numerical example is presented in Section 4. In this way, a method to solve MADM problem using the proposed divergence measure in the intuitionistic fuzzy environment is presented in Section 5. Section 6 presents the application of the proposed measure of intuitionistic fuzzy exponential divergence in the existing TOPSIS and MOORA methods of MADM. In the same section a comparative analysis between the proposed method of MADM and the existing methods are provided in order to show the consistency of the proposed method. Finally, Section 7 concludes the paper.

2 Preliminaries

We begin by reviewing some well-known concepts related to fuzzy set theory and intuitionistic fuzzy set theory.

Definition 1. Fuzzy Set [19]: A fuzzy set A′ defined on a finite universe of discourse X = (x₁, x₂, …, x_n) is given as: $A^{'} = {〈 x, μ_{A^{'}} (x) 〉 / x \in X}$ (1) where μ_A′ : X → [0, 1] is the membership function of A′. The membership value μ_A′ (x) describes the degree of the belongingness of x ∈ X in A′. When μ_A′ (x) is valued in {0, 1}, it is the characteristic function of a crisp i.e., non-fuzzy set.

Atanassov [1 –4] introduced the concept of intuitionistic fuzzy set (IFS) as the generalization of the concept of fuzzy set.

Definition 2. Intuitionistic Fuzzy Set (IFS) [1 –4]: An Atanassov intuitionistic fuzzy set (IFS) A on a finite universe of discourse X = (x₁, x₂, …, x_n) is defined as $A = {〈 x, μ_{A} (x), ν_{A} (x) / x \in X 〉}$ (2)

where μ_A : X → [0, 1], ν_A : X → [0, 1] with the condition 0 ≤ μ_A + ν_A ≤ 1 ∀ x ∈ X. The numbers μ (x) , ν_A (x) ∈ [0, 1] denote the degree of membership and non-membership of x to A, respectively.

For each IFS A in X, we will call π_A (x) =1 - μ_A (x) - ν_A (x), the intuitionistic index or degree of hesitation of x in A. It is obvious that 0 ≤ π_A (x) ≤1 for each x ∈ X. For a fuzzy set A′ in X, π_A (x) =0 when ν_A (x) =1 - μ_A (x). Thus, FSs are the special cases of IFSs.

Atanassov [3] further introduced the set operations on IFSs as follows:

Let A, B ∈ IFS (X) be the family of all IFSs in the universe X, given by $\begin{matrix} A & = & {〈 x, μ_{A} (x), ν_{A} (x) / x \in X 〉}, \\ B & = & {〈 x, μ_{B} (x), ν_{B} (x) / x \in X 〉} . \end{matrix}$

A ⊆ B iff μ_A (x) ≤ μ_B (x) and ν_A (x) ≥ ν_B (x) ∀ x ∈ X;

A = B iff A ⊆ B and B ⊆ A;

A^c = {〈 x, ν_A (x) , μ_A (x) 〉/ x ∈ X};

$A \cup B = {〈 \begin{matrix} x, max (μ_{A} (x), μ_{B} (x)), \\ min (ν_{A} (x), ν_{B} (x)) \end{matrix} 〉 / x \in X}$ ;

$A \cap B = {〈 \begin{matrix} x, min (μ_{A} (x), μ_{B} (x)), \\ max (ν_{A} (x), ν_{B} (x)) \end{matrix} 〉 / x \in X}$ .

3 Intuitionistic fuzzy exponential divergence measure

In this Section the intuitionistic fuzzy exponential divergence between IFSs is proposed and some of its properties are proved. Li et al. [28] introduce a method for transforming Atanassov IFSs into FSs as briefly mentioned in Verma and Sharma [17], is used to proposed the intuitionistic fuzzy exponential divergence measure between IFSs.

3.1 Definition of intuitionistic fuzzy exponential divergence measure

Let us assume A and B be two IFSs defined in a finite universe of discourse X = (x₁, x₂, …, x_n). Corresponding to intuitionistic fuzzy entropy _eE (A) [17], the expected amount of information for discrimination of A against B is given by: $I_{1} (A, B) = \sum_{i = 1}^{n} [\begin{matrix} \frac{1}{2} - (1 - \frac{μ_{A} (x_{i}) + 1 - ν_{A} (x_{i})}{2}) \\ e^{(\frac{μ_{A} (x_{i}) + 1 - ν_{A} (x_{i})}{2} - \frac{μ_{B} (x_{i}) + 1 - ν_{B} (x_{i})}{2})} \end{matrix}] .$ (3)

Similarly, the expected amount of information for discrimination of A^c against B^c is given by: $I_{1} (A^{c}, B^{c}) = \sum_{i = 1}^{n} [\begin{matrix} \frac{1}{2} - (\frac{μ_{A} (x_{i}) + 1 - ν_{A} (x_{i})}{2}) \\ e^{(\frac{μ_{B} (x_{i}) + 1 - ν_{B} (x_{i})}{2} - \frac{μ_{A} (x_{i}) + 1 - ν_{A} (x_{i})}{2})} \end{matrix}] .$ (4)

Now, I₁ (A, B) ≠ I₁ (A^c, B^c). Thus we define the discrimination measure of information between IFSs A and B given by $\begin{matrix} I (A, B) = \sum_{i = 1}^{n} \\ [\begin{matrix} 1 - (\frac{ν_{A} (x_{i}) + 1 - μ_{A} (x_{i})}{2}) e^{\frac{((μ_{A} (x_{i}) - μ_{B} (x_{i}) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}} \\ - (\frac{μ_{A} (x_{i}) + 1 - ν_{A} (x_{i})}{2}) e^{\frac{((μ_{B} (x_{i}) - μ_{A} (x_{i}) - (ν_{B} (x_{i}) - ν_{A} (x_{i}))}{2}} \end{matrix}]; \end{matrix}$ (5)

Similarly, we have $\begin{matrix} I (B, A) = \sum_{i = 1}^{n} \\ [\begin{matrix} 1 - (\frac{ν_{B} (x_{i}) + 1 - μ_{B} (x_{i})}{2}) e^{\frac{((μ_{B} (x_{i}) - μ_{A} (x_{i}) - (ν_{B} (x_{i}) - ν_{A} (x_{i}))}{2}} \\ - (\frac{μ_{B} (x_{i}) + 1 - ν_{B} (x_{i})}{2}) e^{\frac{((μ_{A} (x_{i}) - μ_{B} (x_{i}) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}} \end{matrix}] . \end{matrix}$ (6)

Thus, the intuitionistic fuzzy exponential divergence between IFSs A and B is given by $\begin{matrix} D_{IFS}^{E} (A, B) = I (A, B) + I (B, A) \\ = \sum_{i = 1}^{n} [\begin{matrix} 2 - (\frac{ν_{A} (x_{i}) + 1 - μ_{A} (x_{i})}{2} + \frac{μ_{B} (x_{i}) + 1 - ν_{B} (x_{i})}{2}) \\ e^{\frac{((μ_{A} (x_{i}) - μ_{B} (x_{i}) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}} \\ - (\frac{μ_{A} (x_{i}) + 1 - ν_{A} (x_{i})}{2} + \frac{ν_{B} (x_{i}) + 1 - μ_{B} (x_{i})}{2}) \\ e^{\frac{((μ_{B} (x_{i}) - μ_{A} (x_{i}) - (ν_{B} (x_{i}) - ν_{A} (x_{i}))}{2}} \end{matrix}] \\ = \sum_{i = 1}^{n} [\begin{matrix} 2 - (1 - \frac{(μ_{A} (x_{i}) - μ_{B} (x_{i})) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}) \\ e^{\frac{(μ_{A} (x_{i}) - μ_{B} (x_{i})) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{A} (x_{i}) - μ_{B} (x_{i})) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}) \\ e^{\frac{(ν_{A} (x_{i}) - ν_{B} (x_{i})) - (μ_{A} (x_{i}) - μ_{B} (x_{i}))}{2}} \end{matrix}] \end{matrix}$ (7)

It is natural to then ask “Is the defined divergence measure $D_{IFS}^{E} (A, B)$ reasonable?” We answer this question in the following theorem satisfying the properties defined in Hung and Yang [8].

Theorem 1. The defined intuitionistic fuzzy exponential divergence measure $D_{IFS}^{E} (A, B)$ in (7) between IFSs A and B satisfies the following properties:

$0 \leq D_{IFS}^{E} (A, B) \leq 1$

$D_{IFS}^{E} (A, B) = 0$ if and only if A = B.

$D_{IFS}^{E} (A, B) = D_{IFS}^{E} (B, A)$

If A ⊆ B ⊆ C, A, B, C ∈ IFSs (X)

Then $D_{IFS}^{E} (A, B) \leq D_{IFS}^{E} (A, C)$ and $D_{IFS}^{E} (B, C) \leq D_{IFS}^{E} (A, C)$ .

Proof. (i) Since (7) is a convex function, refer theorem 3.1 of Fan and Xie [7]. Then $D_{IFS}^{E} (A, B)$ increases as ∥A - B ∥ ₁ increases where

∥A - B ∥ ₁ = |μ_A (x_i) - μ_B (x_i) | + |ν_A (x_i) - ν_B (x_i) |. Then $D_{IFS}^{E} (A, B)$ attains its maximum value at the following degenerate cases: $A = (1, 0), B = (0, 1) or A = (0, 1), B = (1, 0) .$

It gives us that $0 \leq D_{IFS}^{E} (A, B) \leq 1$ .

Obviously, the properties (ii) and (iii) are satisfied by $D_{IFS}^{E} (A, B)$ .

(iv) For A ⊆ B ⊆ C, we have ∥A - B ∥ ₁ ≤ ∥ A - C ∥ ₁ and ∥B - C ∥ ₁ ≤ ∥ A - C ∥ ₁

Thus $D_{IFS}^{E} (A, B) \leq D_{IFS}^{E} (A, C)$ and $D_{IFS}^{E} (B, C) \leq D_{IFS}^{E} (A, C) .$

That is, (iv) holds.

3.2 Properties of intuitionistic fuzzy exponential divergence measure

In this section we provide some more properties of the proposed intuitionistic fuzzy exponential divergence measure (7) in the following theorems. While proving these theorems we consider the separation of X into two parts X₁ and X₂, such that $X_{1} = {x_{i} / x_{i} \in X, A (x_{i}) \subseteq B (x_{i})}$ and $X_{2} = {x_{i} / x_{i} \in X, A (x_{i}) \supseteq B (x_{i})}$ where $A (x_{i}) = {(x_{i}, μ_{A} (x_{i}), ν_{A} (x_{i}))}$ and $B (x_{i}) = {(x_{i}, μ_{B} (x_{i}), ν_{B} (x_{i}))} .$

Thus on using the operations explained above inSection 2, we get

In set X₁, μ_A (x_i) ≤ μ_B (x_i) and ν_A (x_i) ≥ ν_B (x_i).

In set X₂, μ_A (x_i) ≥ μ_B (x_i) and ν_A (x_i) ≤ ν_B (x_i).

Theorem 2. For A, B, C ∈ IFS (X),

$D_{IFS}^{E} (A \cup B, C) \leq D_{IFS}^{E} (A, C) + D_{IFS}^{E} (B, C)$ .

$D_{IFS}^{E} (A \cap B, C) \leq D_{IFS}^{E} (A, C) + D_{IFS}^{E} (B, C)$ .

Proof. 2(i) Let us consider the expression $\begin{matrix} D_{IFS}^{E} (A, C) + D_{IFS}^{E} (B, C) - D_{IFS}^{E} (A \cup B, C) \\ = \sum_{i = 1}^{n} [\begin{matrix} 2 - (1 - \frac{(μ_{A} (x_{i}) - μ_{C} (x_{i})) - (ν_{A} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(μ_{A} (x_{i}) - μ_{C} (x_{i})) - (ν_{A} (x_{i}) - ν_{C} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{A} (x_{i}) - μ_{C} (x_{i})) - (ν_{A} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(ν_{A} (x_{i}) - ν_{C} (x_{i})) - (μ_{A} (x_{i}) - μ_{C} (x_{i}))}{2}} \end{matrix}] \\ + \sum_{i = 1}^{n} [\begin{matrix} 2 - (1 - \frac{(μ_{B} (x_{i}) - μ_{C} (x_{i})) - (ν_{B} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(μ_{B} (x_{i}) - μ_{C} (x_{i})) - (ν_{B} (x_{i}) - ν_{C} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{B} (x_{i}) - μ_{C} (x_{i})) - (ν_{B} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(ν_{B} (x_{i}) - ν_{C} (x_{i})) - (μ_{B} (x_{i}) - μ_{C} (x_{i}))}{2}} \end{matrix}] \\ - \sum_{i = 1}^{n} [\begin{matrix} 2 - (1 - \frac{(μ_{A \cup B} (x_{i}) - μ_{C} (x_{i})) - (ν_{A \cup B} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(μ_{A \cup B} (x_{i}) - μ_{C} (x_{i})) - (ν_{A \cup B} (x_{i}) - ν_{C} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{A \cup B} (x_{i}) - μ_{C} (x_{i})) - (ν_{A \cup B} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(ν_{A \cup B} (x_{i}) - ν_{C} (x_{i})) - (μ_{A \cup B} (x_{i}) - μ_{C} (x_{i}))}{2}} \end{matrix}] \\ = \sum_{X_{1}} [\begin{matrix} 2 - (1 - \frac{(μ_{A} (x_{i}) - μ_{C} (x_{i})) - (ν_{A} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(μ_{A} (x_{i}) - μ_{C} (x_{i})) - (ν_{A} (x_{i}) - ν_{C} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{A} (x_{i}) - μ_{C} (x_{i})) - (ν_{A} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(ν_{A} (x_{i}) - ν_{C} (x_{i})) - (μ_{A} (x_{i}) - μ_{C} (x_{i}))}{2}} \end{matrix}] \\ + \sum_{X_{2}} [\begin{matrix} 2 - (1 - \frac{(μ_{B} (x_{i}) - μ_{C} (x_{i})) - (ν_{B} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(μ_{B} (x_{i}) - μ_{C} (x_{i})) - (ν_{B} (x_{i}) - ν_{C} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{B} (x_{i}) - μ_{C} (x_{i})) - (ν_{B} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(ν_{B} (x_{i}) - ν_{C} (x_{i})) - (μ_{B} (x_{i}) - μ_{C} (x_{i}))}{2}} \end{matrix}] \\ \geq 0 . \end{matrix}$ Thus, the proof of 2(i) holds. Similarly, 2(ii) can also be proved.

Theorem 3. For A, B, C ∈ IFS (X), $\begin{matrix} D_{IFS}^{E} (A \cup B, C) + D_{IFS}^{E} (A \cap B, C) \\ = D_{IFS}^{E} (A, C) + D_{IFS}^{E} (B, C) . \end{matrix}$

Proof. $\begin{matrix} D_{IFS}^{E} (A \cup B, C) \\ = \sum_{i = 1}^{n} [\begin{matrix} 2 - (1 - \frac{(μ_{A \cup B} (x_{i}) - μ_{C} (x_{i})) - (ν_{A \cup B} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(μ_{A \cup B} (x_{i}) - μ_{C} (x_{i})) - (ν_{A \cup B} (x_{i}) - ν_{C} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{A \cup B} (x_{i}) - μ_{C} (x_{i})) - (ν_{A \cup B} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(ν_{A \cup B} (x_{i}) - ν_{C} (x_{i})) - (μ_{A \cup B} (x_{i}) - μ_{C} (x_{i}))}{2}} \end{matrix}] \\ = \sum_{X_{1}} [\begin{matrix} 2 - (1 - \frac{(μ_{B} (x_{i}) - μ_{C} (x_{i})) - (ν_{B} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(μ_{B} (x_{i}) - μ_{C} (x_{i})) - (ν_{B} (x_{i}) - ν_{C} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{B} (x_{i}) - μ_{C} (x_{i})) - (ν_{B} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(ν_{B} (x_{i}) - ν_{C} (x_{i})) - (μ_{B} (x_{i}) - μ_{C} (x_{i}))}{2}} \end{matrix}] \\ + \sum_{X_{2}} [\begin{matrix} 2 - (1 - \frac{(μ_{A} (x_{i}) - μ_{C} (x_{i})) - (ν_{A} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(μ_{A} (x_{i}) - μ_{C} (x_{i})) - (ν_{A} (x_{i}) - ν_{C} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{A} (x_{i}) - μ_{C} (x_{i})) - (ν_{A} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(ν_{A} (x_{i}) - ν_{C} (x_{i})) - (μ_{A} (x_{i}) - μ_{C} (x_{i}))}{2}} \end{matrix}] \end{matrix}$ (8) Now $\begin{matrix} D_{IFS}^{E} (A \cap B, C) \\ = \sum_{i = 1}^{n} [\begin{matrix} 2 - (1 - \frac{(μ_{A \cap B} (x_{i}) - μ_{C} (x_{i})) - (ν_{A \cap B} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(μ_{A \cap B} (x_{i}) - μ_{C} (x_{i})) - (ν_{A \cap B} (x_{i}) - ν_{C} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{A \cap B} (x_{i}) - μ_{C} (x_{i})) - (ν_{A \cap B} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(ν_{A \cap B} (x_{i}) - ν_{C} (x_{i})) - (μ_{A \cap B} (x_{i}) - μ_{C} (x_{i}))}{2}} \end{matrix}] \\ = \sum_{X_{1}} [\begin{matrix} 2 - (1 - \frac{(μ_{A} (x_{i}) - μ_{C} (x_{i})) - (ν_{A} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(μ_{A} (x_{i}) - μ_{C} (x_{i})) - (ν_{A} (x_{i}) - ν_{C} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{A} (x_{i}) - μ_{C} (x_{i})) - (ν_{A} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(ν_{A} (x_{i}) - ν_{C} (x_{i})) - (μ_{A} (x_{i}) - μ_{C} (x_{i}))}{2}} \end{matrix}] \\ + \sum_{X_{2}} [\begin{matrix} 2 - (1 - \frac{(μ_{B} (x_{i}) - μ_{C} (x_{i})) - (ν_{B} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(μ_{B} (x_{i}) - μ_{C} (x_{i})) - (ν_{B} (x_{i}) - ν_{C} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{B} (x_{i}) - μ_{C} (x_{i})) - (ν_{B} (x_{i}) - ν_{C} (x_{i}))}{2}) \\ e^{\frac{(ν_{B} (x_{i}) - ν_{C} (x_{i})) - (μ_{B} (x_{i}) - μ_{C} (x_{i}))}{2}} \end{matrix}] \end{matrix}$ (9)

On adding (8) and (9), we get the result.

Hence, the proof of theorem holds.

Theorem 4. For A, B, C ∈ IFS (X),

$D_{IFS}^{E} (A \cup B, A \cap B) = D_{IFS}^{E} (A, B)$ .

$D_{IFS}^{E} (A \cap B, A \cup B) = D_{IFS}^{E} (A, B)$ .

Proof. 4(i) $\begin{matrix} D_{IFS}^{E} (A \cup B, A \cap B) = \sum_{i = 1}^{n} \\ [\begin{matrix} 2 - (1 - \frac{(μ_{A \cup B} (x_{i}) - μ_{A \cap B} (x_{i})) - (ν_{A \cup B} (x_{i}) - ν_{A \cap B} (x_{i}))}{2}) \\ e^{\frac{(μ_{A \cup B} (x_{i}) - μ_{A \cap B} (x_{i})) - (ν_{A \cup B} (x_{i}) - ν_{A \cap B} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{A \cup B} (x_{i}) - μ_{A \cap B} (x_{i})) - (ν_{A \cup B} (x_{i}) - ν_{A \cap B} (x_{i}))}{2}) \\ e^{\frac{(ν_{A \cup B} (x_{i}) - ν_{A \cap B} (x_{i})) - (μ_{A \cup B} (x_{i}) - μ_{A \cap B} (x_{i}))}{2}} \end{matrix}] \\ = \sum_{X_{1}} [\begin{matrix} 2 - (1 - \frac{(μ_{B} (x_{i}) - μ_{A} (x_{i})) - (ν_{B} (x_{i}) - ν_{A} (x_{i}))}{2}) \\ e^{\frac{(μ_{B} (x_{i}) - μ_{A} (x_{i})) - (ν_{B} (x_{i}) - ν_{A} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{B} (x_{i}) - μ_{A} (x_{i})) - (ν_{B} (x_{i}) - ν_{A} (x_{i}))}{2}) \\ e^{\frac{(ν_{B} (x_{i}) - ν_{A} (x_{i})) - (μ_{B} (x_{i}) - μ_{A} (x_{i}))}{2}} \end{matrix}] \\ + \sum_{X_{2}} [\begin{matrix} 2 - (1 - \frac{(μ_{A} (x_{i}) - μ_{B} (x_{i})) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}) \\ e^{\frac{(μ_{A} (x_{i}) - μ_{B} (x_{i})) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{A} (x_{i}) - μ_{B} (x_{i})) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}) \\ e^{\frac{(ν_{A} (x_{i}) - ν_{B} (x_{i})) - (μ_{A} (x_{i}) - μ_{B} (x_{i}))}{2}} \end{matrix}] \\ = D_{IFS}^{E} (A, B) \end{matrix}$

Similarly, 4(ii) can be proved. Hence, the proof of theorem holds.

Theorem 5. For A, B, C ∈ IFS (X),

$D_{IFS}^{E} (A, A \cup B) + D_{IFS}^{E} (A, A \cap B) = D_{IFS}^{E} (A, B)$ .

$D_{IFS}^{E} (B, A \cup B) + D_{IFS}^{E} (B, A \cap B) = D_{IFS}^{E} (A, B)$ .

Proof. 5(i) $\begin{matrix} D_{IFS}^{E} (A, A \cup B) \\ = \sum_{i = 1}^{n} [\begin{matrix} 2 - (1 - \frac{(μ_{A} (x_{i}) - μ_{A \cup B} (x_{i})) - (ν_{A} (x_{i}) - ν_{A \cup B} (x_{i}))}{2}) \\ e^{\frac{(μ_{A} (x_{i}) - μ_{A \cup B} (x_{i})) - (ν_{A} (x_{i}) - ν_{A \cup B} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{A} (x_{i}) - μ_{A \cup B} (x_{i})) - (ν_{A} (x_{i}) - ν_{A \cup B} (x_{i}))}{2}) \\ e^{\frac{(ν_{A} (x_{i}) - ν_{A \cup B} (x_{i})) - (μ_{A} (x_{i}) - μ_{A \cup B} (x_{i}))}{2}} \end{matrix}] \\ = \sum_{X_{1}} [\begin{matrix} 2 - (1 - \frac{(μ_{A} (x_{i}) - μ_{B} (x_{i})) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}) \\ e^{\frac{(μ_{A} (x_{i}) - μ_{B} (x_{i})) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{A} (x_{i}) - μ_{B} (x_{i})) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}) \\ e^{\frac{(ν_{A} (x_{i}) - ν_{B} (x_{i})) - (μ_{A} (x_{i}) - μ_{B} (x_{i}))}{2}} \end{matrix}] \end{matrix}$ (10) Now, $\begin{matrix} D_{IFS}^{E} (A, A \cap B) = \sum_{X_{2}} \\ [\begin{matrix} 2 - (1 - \frac{(μ_{A} (x_{i}) - μ_{B} (x_{i})) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}) \\ e^{\frac{(μ_{A} (x_{i}) - μ_{B} (x_{i})) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{A} (x_{i}) - μ_{B} (x_{i})) - (ν_{A} (x_{i}) - ν_{B} (x_{i}))}{2}) \\ e^{\frac{(ν_{A} (x_{i}) - ν_{B} (x_{i})) - (μ_{A} (x_{i}) - μ_{B} (x_{i}))}{2}} \end{matrix}] \end{matrix}$ (11)

On adding (10) and (11), the proof of theorem 5(i) holds.

Similarly, 5(ii) can be proved. Hence, the proof of theorem holds.

Theorem 6. For A, B, C ∈ IFS (X),

$D_{IFS}^{E} (A, B) = D_{IFS}^{E} (A^{c}, B^{c})$ .

$D_{IFS}^{E} (A^{c}, B) = D_{IFS}^{E} (A, B^{c})$ .

$\begin{matrix} D_{IFS}^{E} (A, B) + D_{IFS}^{E} (A^{c}, B) \\ = D_{IFS}^{E} (A^{c}, B^{c}) + D_{IFS}^{E} (A^{c}, B) . \end{matrix}$

Proof. Proof of above parts directly follows from the definition of operations provided in Section 2.

4 Numerical example

We now demonstrate the efficiency of the proposed intuitionistic fuzzy exponential divergence measure in the context of pattern recognition by comparing it with the existing intuitionistic fuzzy divergence measures presented in Li [6], Hung and Yang [8], Vlachos and Sergiadis [14] and Zhang and Jiang [26].

Li [6] introduced the divergence measure for IFSs A and B given by

$\begin{matrix} d (A, B) & = & \sum_{x \in X} (| μ_{A} (x) - μ_{B} (x) | \\ + | ν_{A} (x) + ν_{B} (x) |) / 2 \end{matrix}$ (12)

Hung and Yang [8] defined the distance between IFSs A and B using hausdorff distance as follows: $d_{H} (A, B) = \frac{1}{n} H (I_{A} (x_{i}), I_{B} (x_{i}))$ (13)

where I_A (x_i) and I_B (x_i) be subintervals on [0,1] denoted by I_A (x_i) = [μ_A (x_i) , 1 - ν_A (x_i)] and I_B (x_i) = [μ_B (x_i) , 1 - ν_B (x_i)] and the Hausdorff distance H (A, B) = max{ |a₁ - b₁|, |a₂ - b₂| } is defined for two intervals A = [a₁, a₂] and B = [b₁, b₂].

Vlachos and Sergiadis [14] provided the intuitionistic fuzzy divergence measure given by $D_{IFS} (A, B) = I_{IFS} (A, B) + I_{IFS} (B, A)$ (14) where $\begin{matrix} I_{IFS} (A, B) \\ = \sum_{i = 1}^{n} [\begin{matrix} μ_{A} (x_{i}) In \frac{μ_{A} (x_{i})}{\frac{1}{2} (μ_{A} (x_{i}) + μ_{B} (x_{i}))} \\ + ν_{A} (x_{i}) In \frac{ν_{A} (x_{i})}{\frac{1}{2} (ν_{A} (x_{i}) + ν_{B} (x_{i}))} \end{matrix}] \end{matrix}$ and $I^{μ} (A, B) = In \frac{μ_{A} (x_{i})}{μ_{B} (x_{i})}$ is the amount of discrimination of μ_A (x_i) from μ_B (x_i).

Zhang and Jiang [26] presented a measure of divergence between IFSs/vague sets A and B as $D^{*} (A, B) = D (A, B) + D (B, A)$ (15)

where $\begin{matrix} D (A, B) \\ = \sum_{i = 1}^{n} [\begin{matrix} (\frac{μ_{A} (x_{i}) + 1 - ν_{A} (x_{i})}{2}) \\ {log}_{2} \frac{[μ_{A} (x_{i}) + 1 - ν_{A} (x_{i})] / 2}{\frac{1}{4} {[μ_{A} (x_{i}) + 1 - ν_{A} (x_{i})] + [μ_{B} (x_{i}) + 1 - ν_{B} (x_{i})]}} \\ + (\frac{1 - μ_{A} (x_{i}) + ν_{A} (x_{i})}{2}) \\ {log}_{2} \frac{[1 - μ_{A} (x_{i}) + ν_{A} (x_{i})] / 2}{\frac{1}{4} {[1 - μ_{A} (x_{i}) + ν_{A} (x_{i})] + [1 - μ_{B} (x_{i}) + ν_{B} (x_{i})]}} \end{matrix}] \end{matrix}$

4.1 Procedure to solve pattern recognition problem

Suppose that we are given m known patterns P₁, P₂ , P₃, …, P_m which have classifications C₁, C₂, C₃, …, C_mrespectively. The patterns are represented by the following IFSs in the universe of discourse X = {x₁, x₂, x₃, …, x_n}: $P_{i} = {〈 x_{j}, μ_{P_{i}} (x_{j}), ν_{P_{i}} (x_{j}) 〉 / x_{j} \in X}$

where i = 1, 2, …, m and j = 1, 2, …, n.

Given an unknown pattern Q, represented by IFS $Q_{i} = {〈 x_{j}, μ_{Q_{i}} (x_{j}), ν_{Q_{i}} (x_{j}) 〉 / x_{j} \in X} .$

Our aim is to classify Q to one of the classes C₁, C₂, C₃, …, C_m. According to the principle of minimum divergence degree between IFSs (cf.[34]), the process of assigning Q to C_{k
^*} is described by $k^{*} = arg min_{k} {D (P_{k}, Q)} .$

According to this algorithm, the given pattern can be recognized so that the best class can be selected.

Example 1. Given three known patterns P₁, P₂ and P₃ which have classifications C₁, C₂ and C₃ respectively. These are represented by the following IFSs in the universe of discourse X = {x₁, x₂, x₃}: $\begin{matrix} P_{1} & = & {(x_{1}, 0.2, 0.5), (x_{2}, 0.5, 0.4), (x_{3}, 0.2, 0.4)}; \\ P_{2} & = & {(x_{1}, 0.4, 0.3), (x_{2}, 0.6, 0.1), (x_{3}, 0.5, 0.2)}; \\ P_{3} & = & {(x_{1}, 0.1, 0.4), (x_{2}, 0.3, 0.5), (x_{3}, 0.7, 0.1)} . \end{matrix}$ where for i = 1, 2, 3 $P_{i} = {\begin{matrix} 〈 x_{1}, μ_{P_{i}} (x_{1}), ν_{P_{i}} (x_{1}) 〉, 〈 x_{2}, μ_{P_{i}} (x_{2}), ν_{P_{i}} (x_{2}) 〉, \\ 〈 x_{3}, μ_{P_{i}} (x_{3}), ν_{P_{i}} (x_{3}) 〉 \end{matrix}} .$

we have an unknown pattern Q, represented by IFS $Q = {(x_{1}, 0.4, 0.5), (x_{2}, 0.3, 0.2), (x_{3}, 0.6, 0.3)}$

Our aim is to classify Q to one of the classes C₁, C₂ and C₃. From the formula (7), (12), (13), (14) and (15), we compute the values of different intuitionistic fuzzy divergence measures are presented in Table 1.

From the computed numerical values of different existing measures and the proposed measure $D_{IFS}^{E}$ presented in Table 1, it is observed that the pattern Q should be classified to C₂.

Thus, the proposed intuitionistic fuzzy exponential divergence measure is consistent for the application point of view in the context of pattern recognition.

5 Application of intuitionistic fuzzy exponential divergence in multi-attribute decision making (MADM)

In this Section we present the application of the proposed intuitionistic fuzzy exponential divergence measure in the field of MADM. It is widely known that the decision-making problem is the process of finding the best option from all of the feasible alternatives. For this let us assume that there exist a set A = {A₁, A₂, A₃, …, A_m} of m alternatives and another set of n attributes given by M = {M₁, M₂, M₃, . . . , M_n}. The decision maker has to find the best alternative from the set A corresponding to the set M of n attributes. Further, suppose that D = (d_ij) _n×m is the intuitionistic fuzzy decision matrix, where d_ij = (μ_ij, ν_ij) is an attribute provided by the decision maker.

μ_ij= degree for which attribute M_i is satisfied by the alternative A_j,

ν_ij= degree for which attribute M_i is not satisfied by the alternative A_j.

The computational procedure to solve the intuitionistic fuzzy MADM problem is as follows:

Step 1: Construct the normalized decision matrix

This step converts the various dimensional attributes into non-dimensional attributes, as per the method provided by Xu and Hu [18]. In general all of the attributes may be of the same type or different type. If the attributes are of different type, it is required to make them of the same type. For example, it is assumed that there are two types of attributes, say (i) benefit type and the (ii) cost type. Depending on the nature of attributes, we convert the cost attribute into the benefit attribute. So, we transform the intuitionistic fuzzy decision matrix D = (d_ij) _n*m into the normalized intuitionistic fuzzy decision matrix say R = (r_ij) _m×n. An element r_ij of the normalized decision matrix R is obtained as follows:

$\begin{matrix} r_{ij} = (μ_{ij}, ν_{ij}) = {\begin{matrix} d_{ij}, for benefit attribute, M_{i} \\ d_{ij}^{c}, for cos t attribute, M_{i} \end{matrix}; \\ where i = 1, 2, \dots, n; j = 1, 2, \dots, m . \end{matrix}$ (16) and $d_{ij}^{c}$ is the complement of d_ij with $d_{ij}^{c} = (ν_{ij}, μ_{ij})$ .

Step 2: Specify the options by the characteristic sets

With the normalized matrix R = (r_ij) _m×n, we specify the option A_j, by the characteristic sets given by

$\begin{matrix} A_{j} & = & {〈 M_{i}, μ_{ij}, ν_{ij} 〉 / M_{i} \in M}, \\ i & = & 1, 2, \dots, n; j = 1, 2, \dots, m . \end{matrix}$ (17)

Step 3: Determine the ideal solution A^* $A^{*} = {(μ_{1}^{*}, ν_{1}^{*}), (μ_{2}^{*}, ν_{2}^{*}), (μ_{3}^{*}, ν_{3}^{*}) \dots, (μ_{n}^{*}, ν_{n}^{*})}$ (18) where $(μ_{i}^{*}, ν_{i}^{*}) = (max_{j} μ_{ij}, min_{j} ν_{ij})$ (19) with i = 1, 2, …, n.

Step 4: Calculate the divergence using $D_{IFS}^{E} (A, B)$ given in (7)

Calculate the divergence by using the following expression $\begin{matrix} D_{IFS}^{E} (A_{j}, A^{*}) = \sum_{i = 1}^{n} \\ [\begin{matrix} 2 - (1 - \frac{(μ_{A_{j}} (x_{i}) - μ_{A^{*}} (x_{i})) - (ν_{A_{j}} (x_{i}) - ν_{A^{*}} (x_{i}))}{2}) \\ e^{\frac{(μ_{A_{j}} (x_{i}) - μ_{A^{*}} (x_{i})) - (ν_{A_{j}} (x_{i}) - ν_{A^{*}} (x_{i}))}{2}} \\ - (1 + \frac{(μ_{A_{j}} (x_{i}) - μ_{A^{*}} (x_{i})) - (ν_{A_{j}} (x_{i}) - ν_{A^{*}} (x_{i}))}{2}) \\ e^{\frac{(ν_{A_{j}} (x_{i}) - ν_{A^{*}} (x_{i})) - (μ_{A_{j}} (x_{i}) - μ_{A^{*}} (x_{i}))}{2}} \end{matrix}], \\ \forall j = 1, 2, \dots m \end{matrix}$ (20)

Step 5: Rank the Order of Preference

The best alternative A_k is now obtained corresponding to the smallest degree of divergence measure by $D_{IFS}^{E} (A_{j}, A^{*})$ , ∀ j = 1, 2, … m.

Now the application of introducing intuitionistic fuzzy exponential divergence measure $D_{IFS}^{E} (A, B)$ is demonstrated with the help of a numerical example below:

Example 2. Let us assume that a car company wants to select a suitable material supplier among five alternative suppliers A₁, A₂, A₃, A₄ and A₅ while considering six attributes: (i) Quality of Product (M₁), (ii) Price (M₂), (iii) Technical Capability (M₃), (iv) Delivery (M₄), (v) Service (M₅), and (vi) Flexibility (M₆).

Table 2 shows the intuitionistic fuzzy decision matrix D = (d_ij) _6×5 having the characteristics of the alternatives A_j (j = 1, 2, 3, 4, 5).

Step 1. Here M₂ is the cost attribute while other five are benefit attributes. We transform the cost attribute M₂ into benefit attribute $M_{2}^{c}$ , we get $\begin{matrix} M_{2}^{c} & = & {(0 . 1, 0 . 8), (0 . 3, 0 . 5), (0 . 3, 0 . 7), \\ (0.4, 0 . 3), (0 . 1, 0 . 7)} \end{matrix}$

Table 3 presents the normalized intuitionistic fuzzy decision matrix R.

Step 2. Characteristic sets presenting the options A_j are given by $\begin{matrix} A_{1} & = & {(M_{1}, 0.4, 0.5), (M_{2}, 0.1, 0.8), \\ (M_{3}, 0.7, 0.3), (M_{4}, 0.6, 0.2), \\ (M_{5}, 0.5, 0.4), (M_{6}, 0.3, 0.4)} \\ A_{2} & = & {(M_{1}, 0.7, 0.2), (M_{2}, 0.3, 0.5), \\ (M_{3}, 0.3, 0.4), (M_{4}, 0.8, 0.1), \\ (M_{5}, 0.2, 0.6), (M_{6}, 0.4, 0.5)} \\ A_{3} & = & {(M_{1}, 0.6, 0.1), (M_{2}, 0.3, 0.7), \\ (M_{3}, 0.6, 0.2), (M_{4}, 0.4, 0.1), \\ (M_{5}, 0.3, 0.4), (M_{6}, 0.8, 0.2)} \\ A_{4} & = & {(M_{1}, 0.5, 0.4), (M_{2}, 0.4, 0.3), \\ (M_{3}, 0.8, 0.1), (M_{4}, 0.7, 0.2), \\ (M_{5}, 0.6, 0.1), (M_{6}, 0.7, 0.1)} \\ A_{5} & = & {(M_{1}, 0.4, 0.3), (M_{2}, 0.1, 0.7), \\ (M_{3}, 0.5, 0.2), (M_{4}, 0.9, 0.1), \\ (M_{5}, 0.8, 0.0), (M_{6}, 0.6, 0.4)} . \end{matrix}$

Step 3. The ideal solution A^* obtained using (18) and (19) is given by $\begin{matrix} A^{*} & = & {(M_{1}, 0.7, 0.1), (M_{2}, 0.4, 0.3), \\ (M_{3}, 0.8, 0.1), (M_{4}, 0.9, 0.1), \\ (M_{5}, 0.8, 0.0), (M_{6}, 0.8, 0.1)} \end{matrix}$

Step 4. Table 4 gives the values of $D_{IFS}^{E} (A_{k}, A^{*})$ using the measure (20).

Step 5. The best alternative is A₄ obtained corresponding to the smallest degree of divergence measure by $D_{IFS}^{E} (A_{j}, A^{*})$ for j = 4.

Hence, the numerical example shows that the proposed intuitionistic fuzzy exponential divergence measure is a very suitable measure to solve the MADM problems.

6 Application of intuitionistic fuzzy TOPSIS and MOORA methods for MADM: A comparative analysis

We now present the application of proposed intuitionistic fuzzy exponential divergence measure in context of MADM using TOPSIS (Hwang and Yoon [29]) and MOORA (Brauers and Zavadskas [30], Stanujkic et al. [31]) in intuitionistic fuzzy environment.

6.1 Intuitionistic fuzzy TOPSIS method

Let A = {A₁, A₂, A₃, . . . , A_m} be a set of m alternatives and decision maker will choose the best one from A according to an attribute set given by M = {M₁, M₂, M₃, …, M_n}.

Various computational steps in the Intuitionistic fuzzy TOPSIS method are as follows:

1. Construction of intuitionistic fuzzy decision matrix

In this step intuitionistic fuzzy decision matrix D = (d_ij) _n×m of intuitionistic fuzzy value d_ij = (μ_ij, ν_ij) is constructed.

2. Construct the normalized intuitionistic fuzzy decision matrix R = (r_ij) _n×m

The normalized value r_ij is calculated as $\begin{matrix} r_{ij} = (μ_{ij}, ν_{ij}) = (μ_{ij} / \sqrt{\sum_{j = 1}^{m} μ_{ij}^{2}}, ν_{ij} / \sqrt{\sum_{j = 1}^{m} ν_{ij}^{2}}), \\ i = 1, 2, \dots, n; j = 1, 2, \dots, m . \end{matrix}$ (21)

3. Construct the weighted normalized intuitionistic fuzzy decision matrix

The weighted normalized value $v_{ij} = w_{i} r_{ij}, i = 1, 2, \dots, n; j = 1, 2, \dots, m,$ (22)

where the weight matrix for each attribute is as follows: W = [1, 1, 1, 1, 1] and w_i is weight or preference value of ith attribute.

4. Determine the intuitionistic fuzzy positive ideal solution (IFPIS) and intuitionistic fuzzy negative ideal solution (IFNIS) $\begin{matrix} IFPIS = A^{+} = {v_{1}^{+}, v_{2}^{+}, \dots, v_{n}^{+}} \\ = {\underset{j}{(max} v_{ij} / i \in J), (min_{j} v_{ij} / i \in J^{'})} \end{matrix}$ (23) $\begin{matrix} IFNIS = A^{-} = {v_{1}^{-}, v_{2}^{-}, \dots, v_{n}^{-}} \\ = {\underset{j}{(min} v_{ij} / i \in J), (max_{j} v_{ij} / i \in J^{'})} \end{matrix}$ (24)

where J is associated with the benefit attribute and J′ is associated with the cost attribute.

5. Calculate the separation measures $D_{IFS}^{E}^{+}$ and $D_{IFS}^{E}^{-}$ using the divergence measure in (7).

6. Calculate the relative closeness of the idealsolution.

The relative closeness of alternative A_j with respect to IFPIS is defined by $C_{j} = \frac{D_{IFS}^{E}^{-}}{D_{IFS}^{E}^{-} + D_{IFS}^{E}^{+}}, j = 1, 2, \dots, m .$ (25)

7. Rank the preference order of all alternatives according to the closeness coefficient.

Now the application of proposed divergence measure $D_{IFS}^{E} (A, B)$ with TOPSIS technique is demonstrated using the intuitionistic fuzzy decision matrix considered in Table 2.

Table 5 presents the normalized/weighted normalized intuitionistic fuzzy decision matrix corresponding to the intuitionistic fuzzy decision matrix given in Table 2 using the formulas (21) and (22).

Table 6 shows the intuitionistic fuzzy positive and negative ideal solutions A⁺ and A^- using formulas (23) and (24).

The calculated numerical values of separation measures of each alternative from positive ideal solution and negative solution using measure (7) are given in Table 7.

The calculated values of relative closeness of each alternative to positive ideal solution using the formula (25) and their corresponding ranks are presented in Table 8.

According to the closeness coefficient and ranking of alternative it is obtained that A₅ is the best alternative.

6.2 Intuitionistic fuzzy MOORA method

Intuitionistic fuzzy MOORA method for solving MADM is as follows.

The computational procedure in intuitionistic fuzzy MOORA method up to step 3 is same as discussed in TOPSIS method above.

Step 4. Compute the overall rating M⁺ and M^- of benefit and cost attribute for each alternative from Table 5. $\begin{matrix} M^{+} = {v_{1}^{+}, v_{2}^{+}, \dots, v_{n}^{+}} \\ = {\underset{i}{(max} v_{ij}, min_{i} v_{ij}) / j \in J} \end{matrix}$ (26) $\begin{matrix} M^{-} = {v_{1}^{-}, v_{2}^{-}, \dots, v_{n}^{-}} \\ = {\underset{i}{(min} v_{ij}, max_{i} v_{ij}) / j \in J} \end{matrix}$ (27) where J = {j = 1, 2, 3, …, m/j is associated with the alternative}.

The calculated overall rating M⁺ and M^- of each of alternative are shown in Table 9.

Step 5. Calculate the overall performance index $D_{IFS}^{E} (M^{+}, M^{-})$ for each alternative using the formula (7) and the computed values in Table 9.

Step 6. Ranking alternatives and/ or selecting the most efficient one based on the value of $D_{IFS}^{E} (M^{+}, M^{-})$ . Table 10 presents the overall performance index $D_{IFS}^{E} (M^{+}, M^{-})$ for each alternatives and ranking order of them.

According to the calculated results, ranking order of the alternatives it is obtained that A₄ is the most preferable alternative.

6.3 A comparative analysis of the proposed method and the existing methods of MADM

We now compare the proposed method of MADM with the existing methods while using the proposed divergence measure (7). Form the proposed method in Section 5 it is obtained that the alternative A₄ is the most preferable alternative for a car company among five. However, we above examine from the TOPSIS method that A₅ is the most preferable alternative and from MOORA method we obtained that A₄ is the most preferable alternative which is exactly same as the result obtained by our new method. It is noticed that the proposed method is a very simple, consistent and efficient method than the existing methods compared with us.

7 Conclusion

Despite the fact that many information-theoretic divergence measures between IFSs have been developed in past years, still there is a possibility that the better divergence measure can be developed, which will find applications in the variety of fields. In this paper, we have proposed an information-theoretic exponential framework for IFSs and established some of the properties of the proposed intuitionistic fuzzy exponential divergence measure. A numerical example is given to present that the proposed divergence measure is feasible and efficient in point of view of pattern recognition. In addition, a method to solve MADM problems based on the proposed divergence measure under intuitionistic fuzzy environment is developed. Furthermore, the application of the proposed divergence measure in two existing TOPSIS and MOORA methods of MADM in an intuitionistic fuzzy environment is presented. Finally, a comparative study between the proposed method and the existing methods of MADM presents consistency of the results of the proposed method. We note that the proposed divergence measure is a very appropriate measure to solve the real-world problems related to MADM.

Footnotes

Acknowledgments

The author is thankful to the editor and two anonymous referees of this journal for their comments and suggestions which have resulted in substantial improvement to both the content and presentation of the paper. Any errors are my very own.

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