Multi-attribute decision making based on novel generalized parametric exponential intuitionistic fuzzy divergence measure

Abstract

In this work, the main theme is to investigate a novel generalized parametric exponential intuitionistic fuzzy divergence measure along with the study of its detailed properties for its authenticity. The applications of this newly developed generalized intuitionistic fuzzy divergence measure have been provided to multi-attribute decision making (MADM). Numerical verification has been illustrated to demonstrate the proposed method for solving multi-attribute decision making problem under fuzzy environment.

Keywords

Fuzzy sets fuzzy divergence measure intuitionistic fuzzy sets generalized exponential intuitionistic fuzzy divergence measure multi-attribute decision making

1. Introduction

In the literature, information theory plays a vital role for representing the uncertainties in the data in the form of measure theory. These measures are valid only when the given data is precisely known. But in day-to-day life, due to the presence of various constraints, the judgement of the decision makers based on uncertain and imprecise in nature. To handle this, the theory of fuzzy sets (FSs) introduced by Zadeh [27,28] gained remarkable importance due to its capability of handling the uncertainties in the form of membership function. The applications of fuzzy set theory in many areas of science and technology including clustering, decision making, signal and image processing, speech recognition, bioinformatics, fuzzy aircraft control, feature selection etc. This theory possesses the capability to model non-statistical imprecision or vague concepts.

Afterwards, the concept of intuitionistic fuzzy set (IFS) proposed by Atanassov [1–3], which is the extension of FS with the addition of degree of non membership along with the membership degree and a hesitancy degree. The generalization of FSs more appropriate and powerful tool to handling uncertainties and vagueness present in the real-life applications such as decision making, medical diagnosis, pattern recognition, artificial intelligence, data analysis etc. Many researchers used the IFS theory in different fields for handling the uncertainties present in the given data and their corresponding analysis is more meaningful than their crisp set analysis.

In classical information theory, the measure of divergence, first introduced by Kullback and Leibler [14] plays an important role because of its applications to a variety of disciplines and it is a measure of the extent to which the assumed probability distribution deviates from the true one. After the introduction of fuzzy sets, a large number of researchers studied the divergence measures for fuzzy distributions in different ways and provided their applications in different areas of research. Bhandari and Pal [4] proposed a measure of fuzzy divergence between two fuzzy sets corresponding to Kullback–Leibler’s [14,15] measure of divergence. Fan and Xie [8] introduced the fuzzy divergence measure based on exponential operation and studied its relation with fuzzy divergence measure introduced by Bhandari and Pal [4]. Montes et al. [18] studied the special classes of divergence measures and used the link between fuzzy and probability uncertainty. Parkash [19] introduced some new symmetric fuzzy divergence measures and studied its detailed properties. Parkash and Kumar [20,21] proposed some new parametric and non-parametric divergence measure for probability distribution and also modified version fuzzy divergence measures along with their detailed properties. In the past few decades, many researchers have paid great attention the theory of intuitionistic fuzzy sets (IFSs) and have been successfully applied to many practical problems such as decision making, expert systems, pattern recognition, medical diagnosis, a clustering analysis [9–13,16,17]. Das et al. [7] studied robust decision making using the intuitionistic fuzzy numbers. Later on, Chen & Chang [5] and Chen et al. [6] present fuzzy multiattribute decision making based on transformation technique of intuitionistic fuzzy values, IF geometric averaging operators and also novel similarity measure between IFSs based on the centroid points of transformed fuzzy numbers with application to pattern recognition. Motivated form the above study, we propose the generalized methodology for measuring the difference between two intuitionistic fuzzy sets (IFSs).

The present study consists of four sections. In Section 2, some basic definitions related to the literature of intuitionistic fuzzy set and directed divergence measure for probability distribution has been provided. In Section 3, we use a flexible approach which provides further leverage of choice to the user, and propose a generalized parametric exponential divergence measure for IFSs analogous to the measure given [23]. It may be remarked that the strength of a measure lies in its properties. The new measure has elegant properties, proved in the paper, to enhance the employability of this measure. In Section 5, an algorithm to solve a multi-attribute decision making problem is discussed and also numerical verification has been presented to illustrate the procedure of proposed algorithm to solve multi-attribute decision making.

2. Preliminary studies

In this section, we provide some basic definitions and notions that are applied to develop the present study.

2.1. Intuitionistic fuzzy set [1]

Let X be a non-empty set known as universe of discourse. An intuitionistic fuzzy set (IFS) A in X is defined as an object of the following form $\begin{eqnarray}\displaystyle A=\{\langle x,{\mu}_{A}(x),{\nu}_{A}(x)\rangle |x\in X\} & & \displaystyle\end{eqnarray}$ (1) where 𝜇_A, 𝜈_A: X → [0,1] represent as “membership degree” and “non-membership degree” of the element x ∈ X respectively, with the condition that 0 ≤ 𝜇_A(x) + 𝜈_A(x) ≤ 1. The degree of non-determinacy (uncertainty) for each element x of X in the IFS A, is defined as $\begin{eqnarray}\displaystyle \hspace{-10.0pt}{\pi}_{A}(x)=1-{\mu}_{A}(x)-{\nu}_{A}(x)\quad \text{where }{\pi}_{A}(x)\in [0,1]. & & \displaystyle \nonumber\end{eqnarray}$ Let $A=\{\langle x,{\mu}_{A}(x),{\nu}_{A}(x)\rangle |x\in X\}$ and B = $\{\langle x,{\mu}_{B}(x),{\nu}_{B}(x)\rangle |x\in X\}$ be the two IFSs defined on X, the operations on IFSs are defined as follows:

•

Union: $A\cup B=\big\{\langle \max ({\mu}_{A}(x),{\mu}_{B}(x)),\min ({\nu}_{A}(x)$ , ${\nu}_{B}(x))\rangle \big\}$

•

Intersection: $A\cap B=\big\{\langle \min ({\mu}_{A}(x),{\mu}_{B}(x))$ , $\max ({\nu}_{A}(x),{\nu}_{B}(x))\rangle \big\}$

•

Complement: $A^{C}=\{\langle {\nu}_{A}(x),{\mu}_{A}(x)\rangle \}$

•

Containment: A ⊆ B iff 𝜇_A(x) ≤ 𝜇_B(x), 𝜈_A(x) ≥ 𝜈_B(x) and A ⊇ B iff 𝜇_A(x) ≥ 𝜇_B(x), 𝜈_A(x) ≤ 𝜈_B(x)

2.2. Divergence measure in probability distribution

The divergence measure in probability distribution is the distance formula between two probability distributions. Kullback–Leibler’s [14,15] firstly proposed the measure of directed divergence between the two distributions P = (p₁, p₂, …, p_n), Q = (q₁, q₂, …, q_n) is defined as follows: $\begin{eqnarray}\displaystyle D(P;Q)=\mathop{\sum }_{i=1}^{n}p_{i}\ln \frac{p_{i}}{q_{i}}. & & \displaystyle\end{eqnarray}$ (2) This measure satisfies the following conditions:

D (P; Q) ≥ 0,

D (P; Q) = 0 if and only if P = Q.

D (P; Q) is convex function of both the probability distributions P and Q.

Keeping in view the fundamental properties of directed divergence for fuzzy distributions. Bhandari and Pal [4] extended the concept of divergence measure from probabilistic to fuzzy set theory by giving a fuzzy information measure for discrimination of fuzzy set B relative to some other fuzzy set A and is given as $\begin{eqnarray}\displaystyle I(A,B) & = & \displaystyle \frac{1}{n}\mathop{\sum }_{i=1}^{n}\left[{\mu}_{A}(x_{i})\log \frac{{\mu}_{A}(x_{i})}{{\mu}_{B}(x_{i})}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\left.(1-{\mu}_{A}(x_{i}))\log \frac{(1-{\mu}_{A}(x_{i}))}{(1-{\mu}_{B}(x_{i}))}\right]\end{eqnarray}$ (3) which satisfies the following conditions:

I (A, B) ≥ 0

I (A, B) = 0 if and only if 𝜇_A(x_i) = 𝜇_B(x_i) and

I (A, B) is a convex function.

If 𝜇_B(x_i) approaches either 0 or 1, the above measure tends to infinity and hence they give inaccurate result. To overcome this drawback, Shang and Jiang [22] proposed a modified version of it is given by $\begin{eqnarray}\displaystyle \hspace{-18.0pt}D(A;B) & = & \displaystyle \frac{1}{n}\mathop{\sum }_{i=1}^{n}\left[{\mu}_{A}(x_{i})\log \left(\frac{2{\mu}_{A}(x_{i})}{{\mu}_{A}(x_{i})+{\mu}_{B}(x_{i})}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-34.0pt}\left.+∼(1-{\mu}_{A}(x_{i}))\log \left(\frac{2(1-{\mu}_{A}(x_{i}))}{2-{\mu}_{A}(x_{i})-{\mu}_{B}(x_{i})}\right)\right].\end{eqnarray}$ (4) Afterwards, the notion of divergence measure was extended from fuzzy sets to intuitionistic fuzzy sets by Vlachos and Sergiadis [25]. They defined intuitionistic fuzzy divergence measure corresponding to Shang and Jiang’s fuzzy divergence measure of IFS Brelative IFS A by $\begin{eqnarray}\displaystyle \hspace{-18.0pt}D(A;B) & = & \displaystyle \frac{1}{n}\mathop{\sum }_{i=1}^{n}\left[{\mu}_{A}(x_{i})\log \left(\frac{2{\mu}_{A}(x_{i})}{{\mu}_{A}(x_{i})+{\mu}_{B}(x_{i})}\right)\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \left.+∼{\nu}_{A}(x_{i})\log \left(\frac{2{\nu}_{A}(x_{i})}{{\nu}_{A}(x_{i})+{\nu}_{B}(x_{i})}\right)\right].\end{eqnarray}$ (5) Wei and Ye [26] extended the divergence measure for IFS given by Vlachos and Sergiadis [25]. Later on, Verma and Sharma [24] proposed the improved version of Wei and Ye [26]. Also, Tomar and Ohlan [23] presented a new parametric generalized exponential divergence measure for fuzzy sets. Inspired from the above work, we present the generalized parametric exponential divergence measure for IFSs corresponding to [23].

3. A novel generalized parametric exponential intuitionistic fuzzy divergence measure

Let A and B be two fuzzy sets defined in a finite universe of discourse $X=\{x_{1},x_{2},\ldots ,x_{n}\}$ having the membership values 𝜇_A(x_i), i = 1, 2, 3, …, n and 𝜇_B(x_i), i = 1, 2, 3, …, n respectively. Then, we propose a new generalized parametric exponential divergence measure for two Intuitionistic fuzzy sets (IFSs) A and B, analogous to the measure given in [23] as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}D_{E}^{{\alpha}}(A,B)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-30.0pt}\quad =\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right];\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-8.0pt}{\alpha}>0.\end{eqnarray}$ (6) From the definition of $D_{E}^{{\alpha}}(A,B)$ , it has been observed that

$D_{E}^{{\alpha}}(A,B)\geq 0$

$D_{E}^{{\alpha}}(A,B)=0$ if and only if 𝜇_A(x_i) = 𝜇_B(x_i) & 𝜈_A(x_i) = 𝜈_B(x_i)

$D_{E}^{{\alpha}}(A,B)\neq D_{E}^{{\alpha}}(B,A)$ but $K_{E}^{{\alpha}}(A,B)=D_{E}^{{\alpha}}$ $(A,B)+D_{E}^{{\alpha}}(B,A)$ is symmetric.

$D_{E}^{{\alpha}}(A,B)$ is convex function of both A and B.

From (6), it is obvious that

$D_{E}^{{\alpha}}(A,B)\geq 0$

$D_{E}^{{\alpha}}(A,B)=0$ if and only if 𝜇_A(x_i) = 𝜇_B(x_i) & 𝜈_A(x_i) = 𝜈_B(x_i)

Now, to prove convexity, partially differentiate equation (6) w.r.t. 𝜇_A(x_i), we have $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\frac{\partial \{D_{E}^{{\alpha}}(A,B)\}}{\partial {\mu}_{A}(x_{i})}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\quad =\frac{1}{2}\left[e^{\left\{\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right\}}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\qquad \times ∼\left\{1-{\alpha}\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\qquad +∼e^{\left\{\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right\}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\left.\qquad \times ∼\left\{{\alpha}\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-1\right\}\right].\nonumber\end{eqnarray}$ Also, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\frac{\partial ^{2}\{D_{E}^{{\alpha}}(A,B)\}}{\partial {\mu}_{_{A}}^{2}(x_{i})}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\quad =\frac{{\alpha}({\alpha}+1)}{4}\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}-1}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\qquad \times ∼e^{\left\{\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right\}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\qquad +∼\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}-1}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\left.\qquad \times ∼e^{\left\{\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right\}}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\qquad -∼\frac{{\alpha}^{2}}{4}\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{2{\alpha}-1}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\qquad \times ∼e^{\left\{\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right\}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\qquad +∼\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{2{\alpha}-1}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\left.\qquad \times ∼e^{\left\{\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right\}}\right]>0\nonumber\end{eqnarray}$ that is, $\frac{\partial ^{2}\{D_{E}^{{\alpha}}(A,B)\}}{\partial {\mu}_{A}^{2}(x_{i})}>∼0$ , for 𝛼 > 0.

Proceeding on the similar lines, we have prove the following $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\left.\frac{\partial ^{2}\{D_{E}^{{\alpha}}(A,B)\}}{\partial {\mu}_{B_{B}}^{2}(x_{i})},\frac{\partial ^{2}\{D_{E}^{{\alpha}}(A,B)\}}{\partial {\nu}_{A_{B}}^{2}(x_{i})},\frac{\partial ^{2}\{D_{E}^{{\alpha}}(A,B)\}}{\partial {\nu}_{B_{B}}^{2}(x_{i})}\right\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-18.0pt}\quad >0\quad \text{for }{\alpha}>0.\nonumber\end{eqnarray}$ Thus, we have prove that $D_{E}^{{\alpha}}(A,B)$ is convex function of both A and B.

3.1. Generalized parametric symmetric exponential divergence measure for IFSs

TheoremA .
A generalized parametric symmetric exponential divergence for two IFSs A and B based on is denoted as $K_{E}^{{\alpha}}(A,B)$ is defined as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}L_{E}^{{\alpha}}(A;B)=D_{E}^{{\alpha}}(A,B)+D_{E}^{{\alpha}}(B,A)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}\quad =\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}\quad +∼\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right].\nonumber\end{eqnarray}$

From the definition of $D_{E}^{{\alpha}}(A,B)$ , it has also been observed that $L_{E}^{{\alpha}}(A;B)\geq 0$ .
3.2. Some properties of generalized parametric exponential symmetric divergence measure for IFSs

We provide some more properties of the new fuzzy divergence measure for IFSs, while proving these properties; divide the universe X into two parts X₁ and X₂ as: $\begin{eqnarray}\displaystyle X_{1} & = & \displaystyle \big\{x_{i}/x_{i}\in X,{\mu}_{A}(x_{i})\geq {\mu}_{B}(x_{i}),\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\nu}_{A}(x_{i})\leq {\nu}_{B}(x_{i}),\forall x_{i}\in X_{1}\big\}\nonumber\end{eqnarray}$ and $\begin{eqnarray}\displaystyle X_{2} & = & \displaystyle \big\{x_{i}/x_{i}\in X,{\mu}_{A}(x_{i})\leq {\mu}_{B}(x_{i}),\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\nu}_{A}(x_{i})\geq {\nu}_{B}(x_{i}),\forall x_{i}\in X_{2}\big\}.\nonumber\end{eqnarray}$ In set X₁, A ∪ B = Union of A and $B\Leftrightarrow {\mu}_{A\cup B}(x)=\max \{{\mu}_{A}(x),{\mu}_{B}(x)\}={\mu}_{A}(x)$ ; $\begin{eqnarray}\displaystyle A\cap B & = & \displaystyle \text{Intersection of }A\text{ and }B\Leftrightarrow {\mu}_{A\cap B}(x)\nonumber\\ \displaystyle & = & \displaystyle \min \{{\mu}_{A}(x),{\mu}_{B}(x)\}={\mu}_{B}(x).\nonumber\end{eqnarray}$ In set X₂, A ∪ B = Union of A and $B\Leftrightarrow {\mu}_{A\cup B}(x)=\max \{{\mu}_{A}(x),{\mu}_{B}(x)\}={\mu}_{B}(x)$ ; $\begin{eqnarray}\displaystyle A\cap B & = & \displaystyle \text{Intersection of }A\text{ and }B\Leftrightarrow {\mu}_{A\cap B}(x)\nonumber\\ \displaystyle & = & \displaystyle \min \{{\mu}_{A}(x),{\mu}_{B}(x)\}={\mu}_{A}(x).\nonumber\end{eqnarray}$ Similarly, for 𝜈_A(x_i) and 𝜈_B(x_i).

TheoremB .
If A and B be the two IFSs defined on universal set X, such that they satisfy for any x_i ∈ X either A ⊆ B or A ⊇ B, then $\begin{eqnarray}\displaystyle D_{E}^{{\alpha}}(A\cup B;A\cap B)=D_{E}^{{\alpha}}(A;B). & & \displaystyle \nonumber\end{eqnarray}$

TheoremC .
It is clear that $\begin{eqnarray}\displaystyle D_{E}^{{\alpha}}(A\cup B;A\cap B) & = & \displaystyle D_{E}^{{\alpha}}(A\cup B|A\cap B)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼D_{E}^{{\alpha}}(A\cap B|A\cup B).\nonumber\end{eqnarray}$ Now, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}D_{E}^{{\alpha}}(A\cup B|A\cap B)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-28.0pt}\quad =\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A\cup B}(x_{i})+{\nu}_{A\cup B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A\cap B}(x_{i})+{\nu}_{A\cap B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A\cup B}(x_{i})+{\nu}_{A\cup B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A\cup B}(x_{i})+1-{\nu}_{A\cup B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A\cap B}(x_{i})+1-{\nu}_{A\cap B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A\cup B}(x_{i})+1-{\nu}_{A\cup B}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-28.0pt}\quad =\mathop{\sum }_{x\in X_{1}}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}}\right)\\[7.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[1.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}}\right)\\[7.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-28.0pt}\qquad +∼\mathop{\sum }_{x\in X_{2}}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}}\right)\\[7.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[1.0pt] \quad -\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)\\[7.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-28.0pt}D_{E}^{{\alpha}}(A\cap B|A\cup B)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-28.0pt}\quad =\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A\cap B}(x_{i})+{\nu}_{A\cap B}(x_{i})}{2}}\right)\\[7.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A\cup B}(x_{i})+{\nu}_{A\cup B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A\cap B}(x_{i})+{\nu}_{A\cap B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[1.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A\cap B}(x_{i})+1-{\nu}_{A\cap B}(x_{i})}{2}}\right)\\[7.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A\cup B}(x_{i})+1-{\nu}_{A\cup B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A\cap B}(x_{i})+1-{\nu}_{A\cap B}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-28.0pt}\quad =\mathop{\sum }_{x\in X_{1}}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}}\right)\\[7.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[1.0pt] \quad -\left({\displaystyle \frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}}\right)\\[7.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-28.0pt}\qquad +∼\mathop{\sum }_{x\in X_{2}}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}}\right)\\[7.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[1.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}}\right)\\[7.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}.\end{array}\right]\nonumber\end{eqnarray}$ By adding the above equations, we get $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}D_{E}^{{\alpha}}(A\cup B|A\cap B)+D_{E}^{{\alpha}}(A\cap B|A\cup B)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}\quad =\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}}\right)\\[7.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[1.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}}\right)\\[7.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}\qquad +∼\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}\quad =D_{E}^{{\alpha}}(A|B)+D_{E}^{{\alpha}}(B|A)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}\quad =D_{E}^{{\alpha}}(A;B).\nonumber\end{eqnarray}$ Hence the result holds.

TheoremD .
For any two IFSs A and B, we have

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

(b) $D_{E}^{{\alpha}}(A\cup B,C)+D_{E}^{{\alpha}}(A\cap B,C)=D_{E}^{{\alpha}}(A,C)+D_{E}^{{\alpha}}(B,C)$

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

TheoremE .
We prove only the first part, all other can analogously prove.

(a) First, let $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-27.0pt}D_{E}^{{\alpha}}(A\cup B,A)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-30.0pt}\quad =\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A\cup B}(x_{i})+{\nu}_{A\cup B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A\cup B}(x_{i})+{\nu}_{A\cup B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A\cup B}(x_{i})+1-{\nu}_{A\cup B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A\cup B}(x_{i})+1-{\nu}_{A\cup B}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-30.0pt}\quad =\mathop{\sum }_{x\in X_{1}}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-30.0pt}\qquad +∼\mathop{\sum }_{x\in X_{2}}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-30.0pt}\quad =\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-27.0pt}D_{E}^{{\alpha}}(A\cap B,A)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-30.0pt}\quad =\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A\cap B}(x_{i})+{\nu}_{A\cap B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A\cap B}(x_{i})+{\nu}_{A\cap B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A\cap B}(x_{i})+1-{\nu}_{A\cap B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A\cap B}(x_{i})+1-{\nu}_{A\cap B}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-30.0pt}\quad =\mathop{\sum }_{x\in X_{1}}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-30.0pt}\qquad +\mathop{\sum }_{x\in X_{2}}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-30.0pt}\quad =\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-27.0pt}D_{E}^{{\alpha}}(A\cup B,A)+D_{E}^{{\alpha}}(A\cap B,A)=D_{E}^{{\alpha}}(B,A).\nonumber\end{eqnarray}$ This completes the proof.

TheoremF .
(a) $D_{E}^{{\alpha}}(A,\overline{A})=D_{E}^{{\alpha}}(\overline{A},A)$

(b) $D_{E}^{{\alpha}}(\overline{A},\overline{B})=D_{E}^{{\alpha}}(A,B)$

(c) $D_{E}^{{\alpha}}(A,\overline{B})=D_{E}^{{\alpha}}(\overline{A},B)$

(d) $D_{E}^{{\alpha}}(A,B)+D_{E}^{{\alpha}}(\overline{A},B)=D_{E}^{{\alpha}}(\overline{A},\overline{B})+D_{E}^{{\alpha}}$ (A, B ).

TheoremG .
By using Eq. (6) and proceeding on the similar lines as above, we have proved part (a)–(d).

TheoremH .
If A and B be two IFSs defined on the universal set X, then

(a) $D_{E}^{{\alpha}}(A;C)+D_{E}^{{\alpha}}(B;C)-D_{E}^{{\alpha}}(A\cup B;C)\geq 0$ .

(b) $D_{E}^{{\alpha}}(A;C)+D_{E}^{{\alpha}}(B;C)-D_{E}^{{\alpha}}(A\cap B;C)\geq 0$ .

TheoremI .
(a) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}D_{E}^{{\alpha}}(A;C)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}\quad =\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}\qquad +∼\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}D_{E}^{{\alpha}}(B;C)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}\quad =\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}\qquad +∼\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\end{eqnarray}$ and $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-27.0pt}D_{E}^{{\alpha}}(A\cup B;C)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-32.0pt}\quad =\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A\cup B}(x_{i})+{\nu}_{A\cup B}(x_{i})}{2}}\right)\\[8.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A\cup B}(x_{i})+{\nu}_{A\cup B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[1.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A\cup B}(x_{i})+1-{\nu}_{A\cup B}(x_{i})}{2}}\right)\\[8.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A\cup B}(x_{i})+1-{\nu}_{A\cup B}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-38.0pt}\qquad +\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}}\right)\\[8.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A\cup B}(x_{i})+{\nu}_{A\cup B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[1.0pt] \quad -\left({\displaystyle \frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}}\right)\\[8.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A\cup B}(x_{i})+1-{\nu}_{A\cup B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-32.0pt}\quad =\mathop{\sum }_{x\in X_{1}}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}}\right)\\[8.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[1.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}}\right)\\[8.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[1.0pt] \quad +1-\left({\displaystyle \frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}}\right)\\[8.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[1.0pt] \quad -\left({\displaystyle \frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}}\right)\\[8.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-38.0pt}\qquad +\mathop{\sum }_{x\in X_{2}}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}}\right)\\[8.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[1.0pt] \quad -\left({\displaystyle \frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}}\right)\\[8.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[1.0pt] \quad +1-\left({\displaystyle \frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}}\right)\\[8.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[1.0pt] \quad -\left({\displaystyle \frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}}\right)\\[8.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right].\nonumber\end{eqnarray}$ Then, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-27.0pt}D_{E}^{{\alpha}}(A;C)+D_{E}^{{\alpha}}(B;C)-D_{E}^{{\alpha}}(A\cup B;C)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-30.0pt}\quad =\mathop{\sum }_{x\in X_{1}}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad +1-\left({\displaystyle \frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A}(x_{i})+{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A}(x_{i})+1-{\nu}_{A}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-30.0pt}\qquad +\hspace{-3.0pt}\mathop{\sum }_{x\in X_{2}}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad +1-\left({\displaystyle \frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{B}(x_{i})+{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{C}(x_{i})+{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{B}(x_{i})+1-{\nu}_{B}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{C}(x_{i})+1-{\nu}_{C}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right].\nonumber\end{eqnarray}$ Since, 𝜇(x_i), 𝜈(x_i) ∈ [0,1], ∀x_i ∈ X. This completes the proof.

(b) Proceeding on the similar line as in above, we also prove the same.
4. Multi-attribute decision making (MADM) based on generalized parametric exponential intuitionistic fuzzy divergence measure

In this section, we present an MADM method based Parametric Generalized Exponential Intuitionistic fuzzy Divergence Measure.

Let us assume that there exists a set of m alternatives $A=\{A_{1},A_{2},\ldots ,A_{m}\}$ and another set of n attributes given by $M=\{M_{1},M_{2},\ldots ,M_{n}\}$ . The decision maker has to find the best alternatives from the set A corresponding to the set M of n attributes.

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

Step 1: Construction of the normalized decision-making matrix: The method consists that the various dimensional attributes converts into non dimensional attributes. Let us suppose that D = (d_ij)_n×m is the intuitionistic fuzzy decision matrix, where d_ij = (𝜇_ij, 𝜈_ij), 𝜇_ij represents the degree that the alternative A_i satisfies the attributes M_j and 𝜈_ij defined as the degree that the attribute M_i does not satisfied by the attributes M_j given by the decision maker such that 𝜇_ij ∈ [0,1], 𝜈_ij ∈ [0,1] such that 𝜇_ij + 𝜈_ij ≤ 1, i = 1, 2, …, m; j = 1, 2, …, n.

Generally, all of the attributes may be of the same type or different type. Firstly, if the attributes are of different type, it is required to make them of the same type. It is assumed that there are two types of attributes, say (i) benefit type, and (ii) the cost type

We convert the cost attribute into the benefit attributes based on the nature of attributes. So, we transform the intuitionistic fuzzy decision matrix say R = (r_ij)_n×m.

An element r_ij of the normalized decision matrix R is represented as follows: $\begin{eqnarray}\displaystyle r_{ij}=({\mu}_{ij},{\nu}_{ij})=\left\{\begin{array}{@{}l@{}}d_{ij},\quad \text{for benefit attribute},M_{i}\\[3.0pt] d_{ij}^{c},\quad \text{for }\cos t\text{ attribute},M_{i}\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$ where i = 1,2, …, n; j = 1,2, …, m. and $d_{ij}^{c}$ is the compliment of d_ij with $d_{ij}^{c}=({\nu}_{ij},{\mu}_{ij})$ .

Step 2: Specify the options by the characteristic sets with the normalized matrix R = (r_ij)_n×m we specify the option A_j, by the characteristic sets given by $\begin{eqnarray}\displaystyle A_{j} & = & \displaystyle \{\langle M_{i},{\mu}_{ij},{\nu}_{ij}\rangle |M_{i}\in M\},\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle i=1,2,\ldots ,n;j=1,2,\ldots ,m.\nonumber\end{eqnarray}$ Step 3: Determine the ideal solution A^∗ $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}A^{\ast }=\big\{({\mu}_{1}^{\ast },{\nu}_{1}^{\ast }),({\mu}_{2}^{\ast },{\nu}_{2}^{\ast }),\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad ({\mu}_{3}^{\ast },{\nu}_{3}^{\ast }),\ldots ,({\mu}_{n}^{\ast },{\nu}_{n}^{\ast })\big\}\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-15.0pt}\text{where }({\mu}_{1}^{\ast },{\nu}_{1}^{\ast })=(\max _{j}{\mu}_{ij},\min _{j}{\nu}_{ij})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \text{with }i=1,2,\ldots ,n.\end{eqnarray}$ (8) Step 4: In this step, using the definition of $D_{E}^{{\alpha}}(A,B)$ & $D_{E}^{{\alpha}}(B,A)$ calculate the divergence measure for IFSs as $\begin{eqnarray}\displaystyle \because L_{E}^{{\alpha}}(A_{j};A^{\ast })=D_{E}^{{\alpha}}\left(A_{j},A^{\ast }\right)+D_{E}^{{\alpha}}(A^{\ast },A_{j}). & & \displaystyle \nonumber\end{eqnarray}$ Now $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-26.0pt}D_{E}^{{\alpha}}(A_{j},A^{\ast })\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-30.0pt}\quad =\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A_{j}}(x_{i})+{\nu}_{A_{j}}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A^{\ast }}(x_{i})+{\nu}_{A^{\ast }}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A_{j}}(x_{i})+{\nu}_{A_{j}}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A_{j}}(x_{i})+1-{\nu}_{A_{j}}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A^{\ast }}(x_{i})+1-{\nu}_{A^{\ast }}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A_{j}}(x_{i})+1-{\nu}_{A_{j}}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right]\nonumber\end{eqnarray}$ and $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-27.0pt}D_{E}^{{\alpha}}(A^{\ast },A_{j})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-32.0pt}\quad =\mathop{\sum }_{i=1}^{n}\left[\begin{array}{@{}l@{}}1-\left({\displaystyle \frac{1-{\mu}_{A^{\ast }}(x_{i})+{\nu}_{A^{\ast }}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{1-{\mu}_{A_{j}}(x_{i})+{\nu}_{A_{j}}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{1-{\mu}_{A^{\ast }}(x_{i})+{\nu}_{A^{\ast }}(x_{i})}{2}\right)^{{\alpha}}\right]}\\[2.0pt] \quad -\left({\displaystyle \frac{{\mu}_{A^{\ast }}(x_{i})+1-{\nu}_{A^{\ast }}(x_{i})}{2}}\right)\\[10.0pt] \quad \times ∼e^{\left[\left(\frac{{\mu}_{A_{j}}(x_{i})+1-{\nu}_{A_{j}}(x_{i})}{2}\right)^{{\alpha}}-\left(\frac{{\mu}_{A^{\ast }}(x_{i})+1-{\nu}_{A^{\ast }}(x_{i})}{2}\right)^{{\alpha}}\right]}\end{array}\right].\nonumber\end{eqnarray}$ Then, calculate, $L_{E}^{{\alpha}}(A_{j};A^{\ast })$ .

Step 5: In this step, ranking the order of preferences.

The most appropriate alternative A_K is now obtained corresponding to smallest value of the divergence measure $D_{E}^{{\alpha}}(A_{j},A^{\ast }),\forall j=∼1,2,\ldots ,m$ .

Now, the application of introducing parametric generalized exponential divergence measure for IFSs is demonstrated with the help of a numerical example below:

Table 1

Intuitionistic fuzzy decision matrix

	C ₁	C ₂	C ₃	C ₄	C ₅
M ₁	(0.4, 0.5)	(0.7, 0.2)	(0.6, 0.1)	(0.5, 0.4)	(0.4, 0.3)
M ₂	(0.1, 0.8)	(0.5, 0.3)	(0.4, 0.3)	(0.3, 0.4)	(0.5, 0.2)
M ₃	(0.7, 0.3)	(0.3, 0.4)	(0.6, 0.2)	(0.8, 0.1)	(0.5, 0.2)
M ₄	(0.6, 0.2)	(0.8, 0.1)	(0.4, 0.0)	(0.7, 0.2)	(0.9, 0.1)
M ₅	(0.5, 0.4)	(0.2, 0.6)	(0.3, 0.4)	(0.6, 0.1)	(0.3, 0.0)
M ₆	(0.2, 0.5)	(0.4, 0.1)	(0.5, 0.4)	(0.3, 0.7)	(0.5, 0.1)

4.1. Illustrative example

Let us suppose that an investment company wants to invest certain amount of money in the best option out of five options: A software company A₁, a pharmaceutical company A₂, a textile company A₃, an automobile company A₄ and a air conditioner company A₅. The investment company needs to take a decision according to the following six criteria: (1) M₁ is the risk analysis (2) M₂ is the growth analysis (3) M₃ is the social-political impact analysis (4) M₄ is the environmental impact analysis (5) M₅ is the level of technology (6) M₆ is service.

Table 1 shows the intuitionistic fuzzy decision matrix.

Step 2: Characteristic sets presenting the options A_j are given by $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hspace{-27.0pt}A_{1}=\left\{(M_{1},0.4,0.5),(M_{2},0.1,0.8),(M_{3},0.7,0.3),\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-2.0pt}\left.(M_{4},0.6,0.2),(M_{5},0.5,0.4),(M_{6},0.2,0.5)\right\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-27.0pt}A_{2}=\left\{(M_{1},0.7,0.2),(M_{2},0.5,0.3),(M_{3},0.3,0.4),\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-2.0pt}\left.(M_{4},0.8,0.1),(M_{5},0.2,0.6),(M_{6},0.4,0.1)\right\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-27.0pt}A_{3}=\left\{(M_{1},0.6,0.1),(M_{2},0.7,0.3),(M_{3},0.6,0.2),\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-2.0pt}\left.(M_{4},0.4,0.0),(M_{5},0.3,0.4),(M_{6},0.5,0.4)\right\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-27.0pt}A_{4}=\left\{(M_{1},0.5,0.4),(M_{2},0.3,0.4),(M_{3},0.8,0.1),\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-2.0pt}\left.(M_{4},0.7,0.2),(M_{5},0.6,0.1),(M_{6},0.3,0.7)\right\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-27.0pt}A_{5}=\left\{(M_{1},0.4,0.3),(M_{2},0.5,0.2),(M_{3},0.5,0.2),\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{-2.0pt}\left.(M_{4},0.9,0.1),(M_{5},0.8,0.0),(M_{6},0.5,0.1)\right\}.\nonumber\end{eqnarray}$

Step 3: The ideal solution A^∗ obtained using equations A & B is given $\begin{eqnarray}\displaystyle A^{\ast } & = & \displaystyle \big\{(0.7,0.1),(0.5,0.2),(0.8,0.1),\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle (0.9,.0.0),(0.6,0.0),(0.5,0.1)\big\}\nonumber\end{eqnarray}$ Step 4: $L_{E}^{{\alpha}}(A_{j};A^{\ast })=D_{E}^{{\alpha}}(A_{j},A^{\ast })+D_{E}^{{\alpha}}(A^{\ast },A_{j})$ .

Table 2 gives the values of $L_{E}^{{\alpha}}(A_{j},A^{\ast })$ using measure (6).

Step 5: In this step, ranking order of preference as: A₅ ≻ A₃ ≻ A₄ ≻ A₂ ≻ A₁ for 𝛼 = 0.5.

The most appropriate alternative is A₅ obtained corresponding to the smallest value of divergence measure $L_{E}^{{\alpha}}(A_{j},A^{\ast })$ for j = 5. The propose measure in (6) more flexible and effective due to the presence of parameter 𝛼. For different values of the parameter, decision maker applies it to the different situations and give their appropriate decision.

Table 2
Values of $L_{E}^{{\alpha}}(A_{j},A^{\ast })$ for A_j(j = 1, 2, 3, 4, 5)

$L_{E}^{{\alpha}}(A_{1},A^{\ast })=∼0.6513$ , $L_{E}^{{\alpha}}(A_{2},A^{\ast })=∼0.4254$ , $L_{E}^{{\alpha}}(A_{3},A^{\ast })=∼0.2613$ , $L_{E}^{{\alpha}}(A_{4},A^{\ast })=∼0.3089$ , and $L_{E}^{{\alpha}}(A_{5},A^{\ast })=∼0.1347$ ;

5. Conclusions

In this paper, we have proposed a new generalized parametric exponential fuzzy divergence measure for intuitionistic fuzzy sets (IFSs). To prove the authenticity of the above divergence measure for IFSs, we have studied in detail its essential and desirable properties. This intuitionistic fuzzy divergence measure more flexible, easy approach in the application point of view due to the presence of the parameter. Finally, the application of this newly developed generalized intuitionistic fuzzy divergence measure in multi-attribute decision making and also numerical example is given for illustration.

Footnotes

Acknowledgements

The authors are thankful to University Grants Commission (UGC), New Delhi for providing the financial assistance for the preparation of the manuscript.

Conflict of interest

None to report.

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Multi-attribute decision making based on novel generalized parametric exponential intuitionistic fuzzy divergence measure

Abstract

Keywords

1. Introduction

2. Preliminary studies

2.1. Intuitionistic fuzzy set [1]

Table 2 Values of L E α ( A j , A ∗ ) for A j (j = 1, 2, 3, 4, 5) L E α ( A 1 , A ∗ ) = 0.6513 , L E α ( A 2 , A ∗ ) = 0.4254 , L E α ( A 3 , A ∗ ) = 0.2613 , L E α ( A 4 , A ∗ ) = 0.3089 , and L E α ( A 5 , A ∗ ) = 0.1347 ;

Footnotes

Acknowledgements

Conflict of interest

References