On intuitionistic fuzzy order- α divergence and entropy measures with MABAC method for multiple attribute group decision-making

Abstract

The development of information measures associated with fuzzy and intuitionistic fuzzy sets is an important research area from the past few decades. Divergence and entropy are two significant information measures in the intuitionistic fuzzy set (IFS) theory, which have gained wider attention from researchers due to their extensive applications in different areas. In the literature, the existing information measures for IFSs have some drawbacks, which make them irrelevant to use in application areas. In order to obtain more robust and flexible information measures for IFSs, the present work develops and studies some parametric information measures under the intuitionistic fuzzy environment. First, the paper reviews the existing intuitionistic fuzzy divergence measures in detail with their shortcomings and then proposes four new order-α divergence measures between two IFSs. It is worth mentioning that the developed divergence measures satisfy several elegant mathematical properties. Second, we define four new entropy measures called order-α intuitionistic fuzzy entropy measures in order to quantify the fuzziness associated with an IFS. We prove basic and advanced properties of the order-α intuitionistic fuzzy entropy measures for justifying their validity. The paper shows that the introduced measures include various existing fuzzy and intuitionistic fuzzy information measures as their special cases. Further, utilizing the conventional multi-attributive border approximation area comparison (MABAC) model, we develop an intuitionistic fuzzy MABAC method to solve real-life multiple attribute group decision-making problems. Finally, the proposed method is demonstrated by using a practical application of personnel selection.

Keywords

Divergence measure entropy measure intuitionistic fuzzy set MABAC model personnel selection

1 Introduction

Divergence measure provides an important tool for determining the amount of information discrimination between two probability distributions (PDs) or two information sets (ISs). In the past few decades, divergence measures have been extensively studied under different disciplines by several eminent researchers. In 1951, the first measure of divergence was proposed by Kullback and Leibler [1] as an extension of a well-known measure of Shannon’s entropy [2]. This measure is widely known in the literature as KL-divergence. Since then, a wide range of parametric generalized divergence measures has been developed to apply in different application areas [3, 4]. In 1991, Lin [5] pointed out some drawbacks of KL-divergence and defined a modified divergence measure between two probability distributions. In addition, using the concavity property of Shannon’s entropy and Jensen inequality, Lin [5] also introduced the notion of Jenson-Shannon divergence (JSD) to analyze the difference/dissimilarity between two PDs and further generalized it to an arbitrary number of probability distributions. With the use of JSD measure, one can assign different weights to the PDs according to their importance. Later, Kapur [6] made a detailed study on JSD measure and defined various parametric generalizations of JSD by considering different entropy functions in place of Shannon’s entropy.

As an alternative to probability theory, Zadeh [7] introduced the theory of fuzzy sets (FSs) for modeling non-statistical uncertain and vague concepts. Inspired by the idea of KL-divergence, Bhandari and Pal [8] proposed the notion of fuzzy divergence in 1992, which provides a new measure to quantify the amount of information discrimination between two fuzzy sets. Shang and Jiang [9] given a modified fuzzy divergence measure corresponding to Lin [5]. Afterward, several measures of fuzzy divergence have been defined by various researchers, including Bajaj and Hooda [10], Verma [11], Verma and Maheshwari [12], Verma et al. [13].

The FS theory considers the membership degree (MD), and the nonmembership degree (NMD) is the complement of the MD. However, in real life, linguistic negation does not satisfy the logical negation. In practical situations, there may be some kind of hesitation while defining the membership function. Also, the membership function is only a single-valued function, which cannot be used to express the support and objection evidence simultaneously in real-life problems. Due to this reason, the notion of intuitionistic fuzzy sets (IFSs) was introduced by Atanassov [14] in 1986, which is characterized by the degrees of both membership and nonmembership with a degree of hesitancy. In the last 20 years, IFS theory has been widely studied and used to solve many practical problems, including decision-making [15 –19], medical diagnosis [20, 21], pattern recognition [22, 23], market prediction [24], clustering analysis [25, 26], image segmentation [27]. In 2007, Vlachos and Sergiadis [28] proposed an intuitionistic fuzzy divergence measure and studied its applications in pattern recognition, medical diagnosis, and image segmentation. Later, Wei and Ye [29] pointed out the importance of hesitancy degree in a measure and developed a modified divergence measure between two IFSs. In 2012, Verma and Sharma [16] defined a generalized measure of intuitionistic fuzzy divergence and developed a decision-making method to solve multi-criteria decision-making (MCDM) with intuitionistic fuzzy information. Hung and Yang [22] proposed a parametric J-divergence measure between two IFSs. In the sequel, Xia and Xu [17], Verma and Sharma [18], Mao et al. [30], Srivastava and Maheshwari [20], Mishra and Rani [31], Ansari et al. [32], Mishra et al. [33], Song et al. [34] defined some intuitionistic fuzzy divergence measures independently and studied their application in different areas.

Another important research topic in IFS theory is the measure of uncertainty associated with an IFS. In the last 25 years, several attempts have been made to address this problem by various researchers worldwide. Burillo and Bustince firstly [35] defined an entropy measure for IFSs to measure the degree of intuitionism of an IFS. Szmidt and Kacprzyk [36] extended the axioms of [37] and defined an intuitionistic fuzzy entropy measure based on the geometric interpretation of IFSs. Using probability theory, Hung and Yang [22] gave their axiomatic definition of entropy for IFSs. Vlachos and Sergiadis [28] defined an intuitionistic fuzzy entropy measure and proved a mathematical connection between the notions of entropy for FS and IFS. Zhang and Jiang [38] proposed intuitionistic fuzzy entropy by means of intersection and union of the MD and NMD of an IFS. Verma and Sharma [39] defined an exponential intuitionistic fuzzy entropy measure based on the ideal of Pal and Pal’s exponential entropy [40]. Up to now, several intuitionistic fuzzy entropy measures have been proposed by many authors [31 , 41–43].

The above-discussed studies highlight that the development of new intuitionistic fuzzy divergence and entropy measures is an important and interesting research topic because of their broader applications in different areas. In the literature, many existing intuitionistic fuzzy divergence measures have various shortcomings that make them invalid in practical use. It is also worth mentioning that the parametric measures always provide an extra degree of freedom in many real-life applications. Therefore, the main objective of this work is to define some new parametric divergence and entropy measures for IFSs. For doing so, first, the paper briefly reviews the existing intuitionistic fuzzy divergence measures and point out some of their significant drawbacks. Second, in order to cope with these drawbacks, we introduce four new order-α divergence measures between two IFSs and study their properties in detail. Interestingly, the proposed intuitionistic fuzzy divergence measures include several existing fuzzy and intuitionistic fuzzy divergence measures as their special cases. Third, by utilizing the relationship between divergence and entropy measures [28], four new order-α intuitionistic fuzzy entropy functions are defined and prove their validity requirements with limiting and particular cases. Further, we utilize the developed information measures to design a new nonlinear optimization model for calculating the attribute weights in multiple attribute group decision making (MAGDM) problems with completely unknown or partially known information about the attribute weights. Then, we generalize the conventional multi-attributive border approximation area comparison (MABAC) model [44] under an intuitionistic fuzzy environment to solve real-life MAGDM problems with intuitionistic fuzzy information.

The rest of the paper is organized as follows. In Section 1 we briefly review the basic results of intuitionistic fuzzy set theory, intuitionistic fuzzy divergence, and entropy measures. Section 2 defines four new order-α intuitionistic fuzzy divergence measures to quantify the information discrimination between two IFSs. We also prove several important properties associated with them and discuss particular cases. Next, the paper proposes four new order-α intuitionistic fuzzy entropy measures, which satisfy the axiomatic requirements suggested by Szmidt and Kacprzyk [36] and some other properties. We also show that the proposed entropy measures include several existing fuzzy and intuitionistic fuzzy entropy measures as their special and particular cases. In Section 4 we formulate the basic steps of the intuitionistic fuzzy MABAC method with the help of proposed order- α information measures to solve MAGDM. Then, the paper considers a real-life decision-making problem to demonstrate the solution procedure of the developed approach. A comparative study with some existing methods is also carried out to validate the obtained results. Finally, conclusions and some future directions are discussed in Section 5.

2 Preliminaries

In this section, we present some basic results related to intuitionistic fuzzy set theory, intuitionistic fuzzy divergence, and entropy measures, which will be needed in the following analysis.

2.1 Intuitionistic fuzzy Set

Definition 1 [14]: An IFS R in a finite universe of discourse M ={ m₁, m₂, …, m_n } is defined by $R = {〈 m, ξ_{R} (m), ζ_{R} (m) 〉 | m \in M},$ (1) where ξ_R : M → [0, 1] and ζ_R : M → [0, 1] denote the MD and the NMD of the element m to the set R, respectively, satisfying the condition that 0 ⩽ ξ_R (m) + ζ_R (m) ⩽ 1. For an IFS R, the hesitancy degree (HD) of m to the set R is obtained by η_R (m) = 1 - ξ_R (m) - ζ_R (m) . Throughout this paper, we denote the family of all IFSs in M by IFS (M). For a given IFS, the pair (ξ_R (m) , ζ_R (m)) is called an intuitionistic fuzzy number (IFN) [45] and denoted by ℘ = (ξ_℘, ζ_℘).

Definition 2 [14, 46]: Let R, S ∈ IFS (M) be two IFSs, given by $R = {〈 m, ξ_{R} (m), ζ_{R} (m) 〉 | m \in M} and$ $S = {〈 m, ξ_{S} (m), ζ_{S} (m) 〉 | m \in M} .$

Atanassov [14] and He et al. [46] defined the following basic set-theoretic operational laws on IFSs as follows:

R ⊆ S ifandonlyif ξ_R (m) ⩽ ξ_S (m) and ζ_R (m) ⩾ ζ_S (m) ∀ m ∈ M;

R = S ifandonlyif R ⊆ S and R ⊇ S;

R^C ={ 〈 m, ζ_R (m) , ξ_R (m) 〉 |m ∈ M };

min (ζ_R (m) , ζ_S (m))〉 |m ∈ M };

max (ζ_R (m) , ζ_S (m))〉 |m ∈ M };

(1 - ξ_R (m)) (1 - ξ_S (m)) - (1 - ξ_R (m) - ζ_R (m)) (1 - ξ_S (m) - ζ_S (m))〉 |m ∈ M };

- (1 - ξ_R (m) - ζ_R (m)) (1 - ξ_S (m) - ζ_S (m)) , 1 - (1 - ζ_R (m)) (1 - ζ_S (m))〉 |m ∈ M };

- ζ_R (m)) ^λ, 1 - (1 - ζ_R (m)) ^λ〉 |m ∈ M };

- (1 - ξ_R (m) - ζ_R (m)) ^λ〉 |m ∈ M }.

2.2 Intuitionistic fuzzy interactive weighted average operator

To aggregate different IFNs, He et al. [46] proposed the intuitionistic fuzzy interactive weighted average (IFIWA) operator as follows:

Definition 3 [46]: Let ℘_j = (ξ_{℘_j}, ζ_{℘_j}) (j = 1, 2, … , n) be a collection of n IFNs and ω = (ω₁, ω₂, …, ω_n) ^T be the weight vector of ℘_j with ω_j ⩾ 0 (j = 1, 2, …, n) and $\sum_{j = 1}^{n} ω_{j} = 1$ . Then the function IFIWA : Vⁿ → V defined as

$\begin{matrix} IFIWA (℘_{1}, ℘_{2}, \dots, ℘_{n}) \\ = (ω_{1} \overset{IF}{*} ℘_{1}) \overset{IF}{\oplus} (ω_{2} \overset{IF}{*} ℘_{2}) \overset{IF}{\oplus} \dots \overset{IF}{\oplus} (ω_{n} \overset{IF}{*} ℘_{n}) \\ = (1 - \prod_{j = 1}^{n} {(1 - ξ_{℘_{j}})}^{ω_{j}}, \prod_{j = 1}^{n} {(1 - ξ_{℘_{j}})}^{ω_{j}} \\ - \prod_{j = 1}^{n} {(1 - ξ_{℘_{j}} - ζ_{℘_{j}})}^{ω_{j}}), \end{matrix}$ (2)

is called the intuitionistic fuzzy interactive weighted average operator.

2.3 Intuitionistic fuzzy entropy measure

Szmidt and Kacprzyk [36] proposed the following definition of an entropy measure for IFSs.

Definition 4 [36]: An entropy on IFS (M) is a real-valued mapping $E : IFS (M) \to [0, 1]$ , which satisfies the following properties:

EP1 $E (R) = 0$ if and only if R is a crisp set;

EP2 $E (R) = 1$ if and only if ξ_R (m) = ζ_R (m) ∀ m ∈ M;

EP3 $E (R) ⩽ E (S)$ if R is less fuzzy than S, i.e., ξ_R (m) ⩽ ξ_S (m) , ζ_R (m) ⩾ ζ_S (m) for ξ_S (m) ⩽ ζ_S (m) or ξ_R (m) ⩾ ξ_S (m) , ζ_R (m) ⩽ ζ_S (m) for ξ_S (m) ⩾ ζ_S (m) for any m ∈ M;

EP4 $E (R) = E (R^{C})$ .

In the literature, many entropy measures have been introduced to measure the degree uncertainty associated with an IFS. In Table 1, we present some existing intuitionistic fuzzy entropy measures defined by various researchers.

Table 1
Some existing intuitionistic fuzzy entropy measures defined by various researchers

Burillo and Bustince [35] $^{BB} E (R) = \frac{1}{n} \sum_{i = 1}^{n} η_{R} (m_{i}) .$

Zeng and Li [47] $^{ZL} E (R) = 1 - \frac{1}{n} \sum_{i = 1}^{n} | ξ_{R} (m_{i}) - ζ_{R} (m_{i}) | .$

Szmidt and Kacrzyk [36] $^{SK} E (R) = \frac{1}{n} \sum_{i = 1}^{n} (\frac{ξ_{R} (m_{i}) \land ζ_{R} (m_{i}) + η_{R} (m_{i})}{ξ_{R} (m_{i}) \lor ζ_{R} (m_{i}) + η_{R} (m_{i})}) .$

Vlachos and Sergiadis [28] $^{VS} E (R) = - \frac{1}{n} \sum_{j = 1}^{n} [ξ_{R}^{j} {log}_{2} ξ_{R}^{j} + ζ_{R}^{j} {log}_{2} ζ_{R}^{j} - (1 - η_{R}^{j}) {log}_{2} (1 - η_{R}^{j}) - η_{R}^{j}]$

Zhang and Jiang [38] $^{ZJ} E (R) = \frac{1}{n} \sum_{i = 1}^{n} (\frac{ξ_{R} (m_{i}) \land ζ_{R} (m_{i})}{ξ_{R} (m_{i}) \lor ζ_{R} (m_{i})})$

Hung and Yang [48] $^{HY} E_{hc}^{2} (R) = \frac{1}{n} \sum_{j = 1}^{n} [1 - ({(ξ_{R}^{j})}^{2} + {(ζ_{R}^{j})}^{2} + {(η_{R}^{j})}^{2})]$

$^{HY} E_{r}^{1 / 2} (R) = \frac{2}{n} \sum_{j = 1}^{n} {log}_{2} ({(ξ_{R}^{j})}^{\frac{1}{2}} + {(ζ_{R}^{j})}^{\frac{1}{2}} + {(η_{R}^{j})}^{\frac{1}{2}})$

Ye [49] $^{Y} E (R) = \frac{1}{n} \sum_{j = 1}^{n} [{\sqrt{2} cos π (\frac{ξ_{R}^{j} - ζ_{R}^{j}}{4}) - 1} \times \frac{1}{\sqrt{2} - 1}]$

Wei et al. [50] $^{WGG} E (R) = \frac{1}{n} \sum_{j = 1}^{n} cos π (\frac{ξ_{R}^{j} - ζ_{R}^{j}}{4 (1 + η_{R}^{j})}) .$

Verma and Sharma [39] $^{VS 1} E (R) = \frac{1}{n (\sqrt{e} - 1)} \sum_{j = 1}^{n} ((\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2}) e^{(\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2})} + (\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2}) e^{(\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2})} - 1) .$

Verma and Sharma [43] $^{VS 2} E (R) = \frac{1}{n (1 - 2^{\frac{1 - R}{R}})} \sum_{j = 1}^{n} (1 - {({(\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2})}^{R} + {(\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2})}^{R})}^{\frac{1}{R}}), R > 0 and R \neq 1 .$

Wang and Wang [51] $^{WW} E (R) = \frac{1}{n} \sum_{j = 1}^{n} cot π (\frac{π}{4} + \frac{| ξ_{R}^{j} - ζ_{R}^{j} |}{4 (1 + η_{R}^{j})})$

Liu and Ren [41] $^{LR} E (R) = \frac{1}{n} \sum_{j = 1}^{n} cot π (\frac{π}{4} + \frac{| {(ξ_{R}^{j})}^{2} - {(ζ_{R}^{j})}^{2} |}{4}) .$

Mishra et al. [33] $^{MJH} E (R) = \frac{1}{n \sqrt{e} (\sqrt{e} - 1)} \sum_{j = 1}^{n} (e - (\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2}) e^{(\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2})} - (\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2}) e^{(\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2})}) .$

Mishra and Rani [31] $^{MR} E_{1} (R) = \frac{1}{n (1 - e^{- 1 / 2})} \sum_{j = 1}^{n} [{1 - e^{(\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2})}} I_{[ξ_{R}^{j} ⩾ ζ_{R}^{j}]} + {1 - e^{(\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2})}} I_{[ξ_{R}^{j} < ζ_{R}^{j}]}] .$

$^{MR} E_{2} (R) \frac{1}{n (\sqrt{e} - 1)} \sum_{j = 1}^{n} ((\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2}) e^{(\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2})} + (\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2}) e^{(\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2})} - 1)$

$^{MR} E_{3} (R) = 1 - \frac{1}{n} \sum_{j = 1}^{n} [{ξ_{R}^{j} - ζ_{R}^{j}} I_{[ξ_{R}^{j} ⩾ ζ_{R}^{j}]} + {ζ_{R}^{j} - ξ_{R}^{j}} I_{[ξ_{R}^{j} < ζ_{R}^{j}]}] .$

Note: In the following text, we will denote the $ξ_{*} (m_{j}) = ξ_{*}^{j}, ζ_{*} (m_{j}) = ζ_{*}^{j}$ , $η_{*} (m_{j}) = η_{*}^{j}$ and exponential by e.

2.4 Intuitionistic fuzzy divergence measure

Vlachos and Sergiadis [28] given the standard definition of divergence measure for IFSs as follows:

Definition 5: Let R, S ∈ IFS (M), then a real-valued function $D : IFS (M) \times IFS (M) \to [0, 1]$ is called a normalized divergence measure if $D$ satisfies the following properties:

DP1: $D (R | S) ⩾ 0$ with equality if and only if R = S;

DP2: $0 ⩽ D (R | S) ⩽ 1 .$

Here, we review some existing divergence measures for IFSs defined by various researchers as:

Vlachos and Sergiadis [28]:

$\begin{matrix} D_{VS} (R | S) \\ = \frac{1}{n} \sum_{j = 1}^{n} [ξ_{R}^{j} {log}_{2} (\frac{2 ξ_{R}^{j}}{ξ_{R}^{j} + ξ_{S}^{j}}) + ζ_{R}^{j} {log}_{2} (\frac{2 ζ_{R}^{j}}{ζ_{R}^{j} + ζ_{S}^{j}})]; \end{matrix}$ (3)

Hung and Yang [22]:

$\begin{matrix} D_{HY} (R | S) = \frac{1}{n (2^{1 - β} - 1)} \sum_{j = 1}^{n} \\ [\begin{matrix} {(\frac{ξ_{R}^{j} + ξ_{S}^{j}}{2})}^{β} + {(\frac{ζ_{R}^{j} + ζ_{S}^{j}}{2})}^{β} + {(\frac{η_{R}^{j} + η_{S}^{j}}{2})}^{β} \\ - (\frac{{(ξ_{R}^{j})}^{β} + {(ξ_{S}^{j})}^{β}}{2}) - (\frac{{(ζ_{R}^{j})}^{β} + {(ζ_{S}^{j})}^{β}}{2}) - (\frac{{(η_{R}^{j})}^{β} + {(η_{S}^{j})}^{β}}{2}) \end{matrix}], β \neq 1 (β > 0) \end{matrix}$ (4)

Zhang and Jiang [38]:

$\begin{matrix} D_{ZJ} (R | S) = \frac{1}{n} \sum_{i = 1}^{n} \\ [(\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2}) log \frac{2 (ξ_{R}^{j} + 1 - ζ_{R}^{j})}{(ξ_{R}^{j} + 1 - ζ_{R}^{j}) + (ξ_{S}^{j} + 1 - ζ_{S}^{j})} \\ + (\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2}) log \frac{2 (ζ_{R}^{j} + 1 - ξ_{R}^{j})}{(ζ_{R}^{j} + 1 - ξ_{R}^{j}) + (ζ_{S}^{j} + 1 - ξ_{S}^{j})}] \end{matrix}$ (5)

Wei and Ye [29]:

$\begin{matrix} D_{WY} (R | S) = \frac{1}{n} \sum_{j = 1}^{n} [ξ_{R}^{j} {log}_{2} (\frac{2 ξ_{R}^{j}}{ξ_{R}^{j} + ξ_{S}^{j}}) \\ + ζ_{R}^{j} {log}_{2} (\frac{2 ζ_{R}^{j}}{ζ_{R}^{j} + ζ_{S}^{j}}) + η_{R}^{j} {log}_{2} (\frac{2 η_{R}^{j}}{η_{R}^{j} + η_{S}^{j}})]; \end{matrix}$ (6)

Mao et al. [30]:

$\begin{matrix} D_{MYW} (R | S) = \frac{1}{n} \sum_{i = 1}^{n} [| ξ_{R}^{j} - ζ_{R}^{j} | {log}_{2} \\ (\frac{2 | ξ_{R}^{j} - ζ_{R}^{j} |}{| ξ_{R}^{j} - ζ_{R}^{j} | + | ξ_{S}^{j} - ζ_{S}^{j} |}) + η_{R}^{j} {log}_{2} (\frac{2 η_{R}^{j}}{η_{R}^{j} + η_{S}^{j}})]; \end{matrix}$ (7)

Verma and Sharma [18]:

$\begin{matrix} D_{V - S} (R | S) = \frac{1}{n (1 - α)} \sum_{i = 1}^{n} [\begin{matrix} log {{(λ_{1} ξ_{R}^{j} + λ_{2} ξ_{S}^{j})}^{α} + {(λ_{1} ζ_{R}^{j} + λ_{2} ζ_{S}^{j})}^{α} + {(λ_{1} η_{R}^{j} + λ_{2} η_{S}^{j})}^{α}} \\ - λ_{1} log ({(ξ_{R}^{j})}^{α} + {(ζ_{R}^{j})}^{α} + {(η_{R}^{j})}^{α}) - λ_{2} log ({(ξ_{S}^{j})}^{α} + {(ζ_{S}^{j})}^{α} + {(η_{S}^{j})}^{α}) \end{matrix}], \\ where α \in (0, 1), λ_{1}, λ_{2} \in [0, 1], λ_{1} + λ_{2} = 1; \end{matrix}$ (8)

Xia and Xu [17]:

$\begin{matrix} D_{XX} (R | S) \\ = \frac{1}{T} \sum_{i = 1}^{n} [\begin{matrix} \frac{(1 + q ξ_{R}^{j}) {log}_{2} (1 + q ξ_{R}^{j}) + (1 + q ξ_{S}^{j}) {log}_{2} (1 + q ξ_{S}^{j})}{2} + \frac{(1 + q ζ_{R}^{j}) {log}_{2} (1 + q ζ_{R}^{j}) + (1 + q ζ_{S}^{j}) {log}_{2} (1 + q ζ_{S}^{j})}{2} \\ + \frac{(1 + q η_{R}^{j}) {log}_{2} (1 + q η_{R}^{j}) + (1 + q η_{S}^{j}) {log}_{2} (1 + q η_{S}^{j})}{2} - (\frac{1 + q ξ_{R}^{j} + 1 + q ξ_{S}^{j}}{2}) {log}_{2} (\frac{1 + q ξ_{R}^{j} + 1 + q ξ_{S}^{j}}{2}) \\ - (\frac{1 + q ζ_{R}^{j} + 1 + q ζ_{S}^{j}}{2}) {log}_{2} (\frac{1 + q ζ_{R}^{j} + 1 + q ζ_{S}^{j}}{2}) - (\frac{1 + q η_{R}^{j} + 1 + q η_{S}^{j}}{2}) {log}_{2} (\frac{1 + q η_{R}^{j} + 1 + q η_{S}^{j}}{2}) \end{matrix}], \\ where q > 0 and T = 1 / ((1 + q) {log}_{2} (1 + q) - (2 + q) {log}_{2} (2 + q) + 1); \end{matrix}$ (9)

Maheshwari and Srivastava [20]:

$\begin{matrix} D_{MS} (R | S) = - log \\ (\frac{1}{2} (1 + \frac{1}{n} \sum_{i = 1}^{n} (\sqrt{ξ_{R}^{j} ξ_{S}^{j}} + \sqrt{ζ_{R}^{j} ζ_{S}^{j}} + \sqrt{η_{R}^{j} η_{S}^{j}}))); \end{matrix}$ (10)

Ansari et al. [32]:

$\begin{matrix} D_{AMA} (R | S) = \frac{1}{n (2^{α - 1} - 1)} \\ [e^{{(α - 1) \sum_{i = 1}^{n} (ξ_{R}^{j} {log}_{2} (\frac{2 ξ_{R}^{j}}{ξ_{R}^{j} + ξ_{S}^{j}}) + ζ_{R}^{j} {log}_{2} (\frac{2 ζ_{R}^{j}}{ζ_{R}^{j} + ζ_{S}^{j}}))}} - 1], \\ α \neq 1 (α > 0); \end{matrix}$ (11)

Mishra and Rani [31]:

$\begin{matrix} D_{MR}^{1} (R | S) = \frac{1}{n (1 - e^{- 1})} \sum_{i = 1}^{n} \\ [{e^{- ((ξ_{S}^{j} - ξ_{R}^{j}) + (ζ_{R}^{j} - ζ_{S}^{j}) / 2)} - 1} I_{[ξ_{R}^{j} ⩾ ζ_{R}^{j}]} \\ + {e^{- ((ξ_{R}^{j} - ξ_{S}^{j}) + (ζ_{S}^{j} - ζ_{R}^{j}) / 2)} - 1} I_{[ξ_{R}^{j} < ζ_{R}^{j}]}]; \end{matrix}$ (12)

$D_{M R}^{2} (R | S) = \frac{1}{n (e^{- 1} - 1)} \sum_{i = 1}^{n} [1 - (\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2}) e^{(\frac{((ξ_{S}^{j} - ξ_{R}^{j}) + (ζ_{R}^{j} - ζ_{S}^{j}))}{2})} - (\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2}) e^{(\frac{((ξ_{R}^{j} - ξ_{S}^{j}) + (ζ_{S}^{j} - ζ_{R}^{j}))}{2})}]$ (13)

Mishra et al. [33]

$D_{MKS}^{1} (R | S) = \frac{1}{n (\sqrt{e} - 1)} \sum_{i = 1}^{n} [\begin{matrix} {\begin{matrix} (\frac{(ξ_{R}^{j} + ξ_{S}^{j}) + 2 - (ζ_{R}^{j} + ζ_{S}^{j})}{4}) e^{((ζ_{R}^{j} + ζ_{S}^{j}) + 2 - (ξ_{R}^{j} + ξ_{S}^{j}) / 4)} \\ + (\frac{(ζ_{R}^{j} + ζ_{S}^{j}) + 2 - (ξ_{R}^{j} + ξ_{S}^{j})}{4}) e^{((ξ_{R}^{j} + ξ_{S}^{j}) + 2 - ((ζ_{R}^{j} + ζ_{S}^{j})) / 4)} \end{matrix}} \\ - \frac{1}{2} {\begin{matrix} (\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2}) e^{(ζ_{R}^{j} + 1 - ξ_{R}^{j} / 2)} + (\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2}) e^{(ξ_{R}^{j} + 1 - ζ_{R}^{j} / 2)} \\ + (\frac{ξ_{S}^{j} + 1 - ζ_{S}^{j}}{2}) e^{(ζ_{S}^{j} + 1 - ξ_{S}^{j} / 2)} + (\frac{ζ_{S}^{j} + 1 - ξ_{S}^{j}}{2}) e^{(ξ_{S}^{j} + 1 - ζ_{S}^{j} / 2)} \end{matrix}} \end{matrix}];$ (14)

$\begin{matrix} D_{MKS}^{2} (R | S) = \frac{1}{n \sqrt{e} (1 - \sqrt{e})} \sum_{i = 1}^{n} [\begin{matrix} {\begin{matrix} (\frac{(ξ_{R}^{j} + ξ_{S}^{j}) + 2 - (ζ_{R}^{j} + ζ_{S}^{j})}{4}) e^{((ξ_{R}^{j} + ξ_{S}^{j}) + 2 - (ζ_{R}^{j} + ζ_{S}^{j}) / 4)} \\ + (\frac{(ζ_{R}^{j} + ζ_{S}^{j}) + 2 - (ξ_{R}^{j} + ξ_{S}^{j})}{4}) e^{((ζ_{R}^{j} + ζ_{S}^{j}) + 2 - (ξ_{R}^{j} + ξ_{S}^{j}) / 4)} \end{matrix}} \\ - \frac{1}{2} {\begin{matrix} (\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2}) e^{(ξ_{R}^{j} + 1 - ζ_{R}^{j} / 2)} + (\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2}) e^{(ζ_{R}^{j} + 1 - ξ_{R}^{j} / 2)} \\ + (\frac{ξ_{S}^{j} + 1 - ζ_{S}^{j}}{2}) e^{(ξ_{S}^{j} + 1 - ζ_{S}^{j} / 2)} + (\frac{ζ_{S}^{j} + 1 - ξ_{S}^{j}}{2}) e^{(ζ_{S}^{j} + 1 - ξ_{S}^{j} / 2)} \end{matrix}} \end{matrix}]; \end{matrix}$ (15)

Song et al. [34]:

$\begin{matrix} D_{SFWW} (R | S) = \frac{1}{n} \sum_{i = 1}^{n} [(| ξ_{R}^{j} - ζ_{R}^{j} | - | ξ_{S}^{j} - ζ_{S}^{j} |) {log}_{2} \\ (\frac{1 + | ξ_{R}^{j} - ζ_{R}^{j} |}{1 + | ξ_{S}^{j} - ζ_{S}^{j} |}) + (η_{R}^{j} - η_{S}^{j}) {log}_{2} (\frac{1 + η_{R}^{j}}{1 + η_{S}^{j}})] . \end{matrix}$ (16)

We consider some numerical examples to show the shortcomings of the existing divergence measures, as mentioned above.

Example 1: Let R₁, S₁ ∈ IFS (M), given by $R_{1} = {〈 m_{1}, 0.5, 0 〉, 〈 m_{2}, 0, 0 〉, 〈 m_{3}, 0, 0.5 〉},$ $S_{1} = {〈 m_{1}, 0.5, 0.5 〉, 〈 m_{2}, 0.5, 0 〉, 〈 m_{3}, 0.5, 0.5 〉} .$

Here, we get D_VS (R₁|S₁) = 0, but we can easily verify that R₁ and S₁ are not equal. Hence, property DP1 is violated by the measure D_VS (R|S)

Example 2: Suppose that R₂, S₂ ∈ IFS (M), given by

R₂ ={ 〈 m₁, 0.5, 0.5 〉 , 〈 m₂, 0.1, 0.1 〉 } and

S₂ ={ 〈 m₁, 0.3, 0.3 〉 , 〈 m₂, 0.2, 0.2 〉 },

then D_ZJ (R₂|S₂) = 0, $D_{MR}^{1} (R_{2} | S_{2}) = 0$ , $D_{MR}^{2} (R_{2} | S_{2}) = 0$ , $D_{MKS}^{1} (R_{2} | S_{2}) = 0$ and $D_{MKS}^{2} (R_{2} | S_{2}) = 0$ , whenever R₂ ≠ S₂. This is a case of violation of property DP1 by the divergence measures given in Equations (5), (12)–(15).

Example 3: Let R₃, S₃ ∈ IFS (M), given by $\begin{matrix} R_{3} = & {〈 m_{1}, 0.44, 0.39 〉, 〈 m_{2}, 0.43, 0.39 〉, \\ 〈 m_{3}, 0.42, 0.38 〉} and \\ S_{3} = & {〈 m_{1}, 0.34, 0.48 〉, 〈 m_{2}, 0.37, 0.46 〉, \\ 〈 m_{3}, 0.38, 0.45 〉} . \end{matrix}$

Here, we get D_VS (R₃|S₃) = -0.0071, D_MYW (R₃|S₃) = -0.0381,D_AMA (R₃|S₃) = -0.0037 (α = 0.8). Hence, the divergence measures given in Equations (3), (7) and (11) do not satisfy the property DP1.

Example 4: LetR₄, S₄ ∈ IFS (M), given by $\begin{matrix} R = {〈 m_{1}, 0.5, 0 〉, 〈 m_{2}, 0.5, 0 〉} and \\ S = {〈 m_{1}, 0, 0.5 〉, 〈 m_{2}, 0, 0.5 〉} . \end{matrix}$

We obtain D_ZJ (R|S) = 0,D_MYW (R|S) = 0 and D_SFWW (R|S) = 0, again, DP1 is violated.

The above discussed numerical analysis shows that the measures D_VS (R|S), D_ZJ (R|S), $D_{MR}^{1} (R | S)$ , $D_{MR}^{2} (R | S)$ , $D_{MKS}^{1} (R | S)$ , $D_{MKS}^{2} (R | S)$ , D_AMA (R|S), D_MYW (R|S) and D_SFWW (R|S) are not valid intuitionistic fuzzy divergence measures.

To overcome these types of situations, in the next section, we propose some new order-α divergence measures for IFSs and present a detailed study on their important properties and particular cases. Then, we define some new parametric entropy measures for IFSs and prove their basic requirements.

3 New order-α divergence and entropy measures for IFSs

3.1 Order-α divergence measures for IFSs

Let R and S be two IFSs defined in a one-element universal set M = { m } . For simplicity R and S are denoted by (ξ_R, ζ_R, η_R) and (ξ_S, ζ_S, η_S). From Definition 1, we know $\begin{matrix} ξ_{R} + ζ_{R} + η_{R} = 1, 0 ⩽ ξ_{R}, ζ_{R}, η_{R} ⩽ 1 \\ ξ_{S} + ζ_{S} + η_{S} = 1, 0 ⩽ ξ_{S}, ζ_{S}, η_{S} ⩽ 1 \end{matrix}} .$ (17)

Equation (17) acclaims that (ξ_R, ζ_R, η_R) and (ξ_S, ζ_S, η_S) may be regarded as two probability distributions associated with m. Then, corresponding to the order-α divergence between two probability distributions [3], we propose the following order-α divergence measures between two IFSs R and S given by

$\begin{matrix} ^{1} {\tilde{D}}_{α}^{•} (R (m) | S (m)) = \frac{1}{(2^{(α - 1)} - 1)} \\ [({(ξ_{R})}^{α} {(\frac{ξ_{R} + ξ_{S}}{2})}^{1 - α} + {(ζ_{R})}^{α} {(\frac{ζ_{R} + ζ_{S}}{2})}^{1 - α} \\ + {(η_{R})}^{α} {(\frac{η_{R} + η_{S}}{2})}^{1 - α}) - 1], \end{matrix}$ (18)

$\begin{matrix} ^{2} {\tilde{D}}_{α}^{•} (R (m) | S (m)) = \frac{1}{(α - 1)} {log}_{2} \\ [{(ξ_{R})}^{α} {(\frac{ξ_{R} + ξ_{S}}{2})}^{1 - α} + {(ζ_{R})}^{α} {(\frac{ζ_{R} + ζ_{S}}{2})}^{1 - α} \\ + {(η_{R})}^{α} {(\frac{η_{R} + η_{S}}{2})}^{1 - α}], \end{matrix}$ (19)

$\begin{matrix} ^{3} {\tilde{D}}_{α}^{•} (R (m) | S (m)) = \frac{1}{(e^{2^{α - 1}} - e)} \\ [e^{({(ξ_{R})}^{α} {(\frac{ξ_{R} + ξ_{S}}{2})}^{1 - α} + {(ζ_{R})}^{α} {(\frac{ζ_{R} + ζ_{S}}{2})}^{1 - α} + {(η_{R})}^{α} {(\frac{η_{R} + η_{S}}{2})}^{1 - α})} - e], \end{matrix}$ (20)

$\begin{matrix} ^{4} {\tilde{D}}_{α}^{•} (R (m) | S (m)) = \frac{1}{(2^{(α - 1) / α} - 1)} \\ [({(ξ_{R})}^{α} {(\frac{ξ_{R} + ξ_{S}}{2})}^{1 - α} + {(ζ_{R})}^{α} {(\frac{ζ_{R} + ζ_{S}}{2})}^{1 - α} \\ {+ {(η_{R})}^{α} {(\frac{η_{R} + η_{S}}{2})}^{1 - α})}^{\frac{1}{α}} - 1], \end{matrix}$ (21) $α \in (0, 1) .$

In the following theorem, we prove that the measures $^{τ} {\tilde{D}}_{α}^{•} (R (m) | S (m)) (τ = 1, 2, 3, 4)$ are valid divergence measures between IFSs.

Theorem 1: The divergence measures $^{τ} {\tilde{D}}_{α}^{•} (R (m) | S (m)) (τ = 1, 2, 3, 4)$ satisfy the properties DP1 and DP2, as stated in Definition 5.

Proof: (DP1) Let ψ (x) = x^1-α for every x ∈ [0, ∞). When 0 < α < 1 which implies 1 - α > 0, hence the function ψ is concave. Thus, by applying the Jensen inequality, we obtain:

$\begin{matrix} 1 & = {(\sum_{k = 1}^{t} q_{k})}^{1 - α} = {(\sum_{k = 1}^{t} p_{k} \frac{q_{k}}{p_{k}})}^{1 - α} \\ ⩾ \sum_{k = 1}^{t} p_{k} {(\frac{q_{k}}{p_{k}})}^{1 - α} \Rightarrow \sum_{k = 1}^{t} p_{k}^{α} q_{k}^{1 - α} \\ ⩽ 1, with \sum_{k = 1}^{t} p_{k} = \sum_{k = 1}^{t} q_{k} = 1, \end{matrix}$ (22) and the equality in (22) holds if and only if p_k = q_k ∀ k .

Now, using Equation (17) with the inequality given in Equation (22), we get $\begin{matrix} {(ξ_{R})}^{α} {(\frac{ξ_{R} + ξ_{S}}{2})}^{1 - α} + {(ζ_{R})}^{α} {(\frac{ζ_{R} + ζ_{S}}{2})}^{1 - α} \\ + {(η_{R})}^{α} {(\frac{η_{R} + η_{S}}{2})}^{1 - α} ⩽ 1, \forall α \in (0, 1) . \end{matrix}$

Based on the above inequality, we obtain $^{τ} {\tilde{D}}_{α}^{•} (R (m) | S (m)) ⩾ 0 (τ = 1, 2, 3, 4) .$

This proves DP1.

(DP2) When R = S ⇒ ξ_R = ξ_S and ζ_R = ζ_S, then we get $^{τ} {\tilde{D}}_{α}^{•} (R (m) | S (m)) = 0 (τ = 1, 2, 3, 4) .$ (23)

Next, let us consider $^{τ} {\tilde{D}}_{α}^{•} (R (m) | S (m)) = 0 (τ = 1, 2, 3, 4)$ , which imply

$\begin{matrix} ({(ξ_{R})}^{α} {(\frac{ξ_{R} + ξ_{S}}{2})}^{1 - α} + {(ζ_{R})}^{α} {(\frac{ζ_{R} + ζ_{S}}{2})}^{1 - α} \\ + {(η_{R})}^{α} {(\frac{η_{R} + η_{S}}{2})}^{1 - α}) = 1 . \end{matrix}$ (24)

Since α ∈ (0, 1), then Equation (24) holds only when $ξ_{R} = ξ_{S} and ζ_{R} = ζ_{S} \Rightarrow R = S .$ (25)

Hence, by using the results obtained in Equation (23) to Equation (25), we get $^{τ} {\tilde{D}}_{α}^{•} (R (m) | S (m)) = 0 (τ = 1, 2, 3, 4)$ with equality if and only R = S.

(ii) From DP1, we know that $^{τ} {\tilde{D}}_{α}^{•} (R (m) | S (m)) ⩾ 0$ (τ = 1, 2, 3, 4), so we must show that the maximum value attains by the divergence measures $^{τ} {\tilde{D}}_{α}^{•} (R (m) | S (m)) (τ = 1, 2, 3, 4)$ is 1.

Note that the divergence measures $^{τ} {\tilde{D}}_{α}^{•} (R (m) | S (m)) (τ = 1, 2, 3, 4)$ attain their maximum for the following degenerate cases: (a) R = (1, 0, 0) and S = (0, 1, 0) (b) R = (0, 1, 0) and S = (1, 0, 0) (c) R = (0, 0, 1) and S = (0, 1, 0) (d) R = (0, 0, 1) and S = (1, 0, 0). Therefore, in all these situations, we obtain $^{τ} {\tilde{D}}_{α}^{•} (R (m) | S (m)) = 1$ (τ = 1, 2, 3, 4).

Hence, $0 ⩽^{τ} {\tilde{D}}_{α}^{•} (R (m) | S (m)) ⩽ 1$ . This proves the theorem. ■

Now, we propose the order-α divergence measures $^{τ} {\tilde{D}}_{α}^{•} (R (m) | S (m)) (τ = 1, 2, 3, 4)$ between two IFSs defined in the finite universe of discourse M ={ m₁, m₂, …, m_n } as follows:

Definition 7: Let R and S be two IFSs defined in M ={ m₁, m₂, …, m_n }. Then, the order- α divergence measures between IFSs R and S can be defined as:

$\begin{matrix} ^{1} D_{α} (R | S) = \frac{1}{n} \sum_{j = 1}^{n} (^{1} {\tilde{D}}_{α}^{•} R (m_{j}) | S (m_{j})) \\ = \frac{1}{n (2^{(α - 1)} - 1)} \sum_{j = 1}^{n} [({(ξ_{R}^{j})}^{α} {(\frac{ξ_{R}^{j} + ξ_{S}^{j}}{2})}^{1 - α} \\ + {(ζ_{R}^{j})}^{α} {(\frac{ζ_{R}^{j} + ζ_{S}^{j}}{2})}^{1 - α} + {(η_{R}^{j})}^{α} {(\frac{η_{R}^{j} + η_{S}^{j}}{2})}^{1 - α}) - 1], \end{matrix}$ (26) $\begin{matrix} ^{2} D_{α} (R | S) = \frac{1}{n} \sum_{j = 1}^{n} (^{2} {\tilde{D}}_{α}^{•} R (m_{j}) | S (m_{j})) \\ = \frac{1}{n (α - 1)} \sum_{j = 1}^{n} {log}_{2} [({(ξ_{R}^{j})}^{α} {(\frac{ξ_{R}^{j} + ξ_{S}^{j}}{2})}^{1 - α} \\ + {(ζ_{R}^{j})}^{α} {(\frac{ζ_{R}^{j} + ζ_{S}^{j}}{2})}^{1 - α} + {(η_{R}^{j})}^{α} {(\frac{η_{R}^{j} + η_{S}^{j}}{2})}^{1 - α})], \end{matrix}$ (27)

$\begin{matrix} ^{3} D_{α} (R | S) = \frac{1}{n} \sum_{j = 1}^{n} (^{3} {\tilde{D}}_{α}^{•} R (m_{j}) | S (m_{j})) = \frac{1}{n (e^{2^{α - 1}} - e)} \sum_{j = 1}^{n} \\ [e^{({(ξ_{R}^{j})}^{α} {(\frac{ξ_{R}^{j} + ξ_{S}^{j}}{2})}^{1 - α} + {(ζ_{R}^{j})}^{α} {(\frac{ζ_{R}^{j} + ζ_{S}^{j}}{2})}^{1 - α} + {(η_{R}^{j})}^{α} {(\frac{η_{R}^{j} + η_{S}^{j}}{2})}^{1 - α})} - e], \end{matrix}$ (28)

$\begin{matrix} ^{4} D_{α} (R | S) = \frac{1}{n} \sum_{j = 1}^{n} (^{4} {\tilde{D}}_{α}^{•} R (m_{j}) | S (m_{j})) \\ = \frac{1}{n (2^{(α - 1 / α)} - 1)} \sum_{j = 1}^{n} [({(ξ_{R}^{j})}^{α} {(\frac{ξ_{R}^{j} + ξ_{S}^{j}}{2})}^{1 - α} \\ {+ {(ζ_{R}^{j})}^{α} {(\frac{ζ_{R}^{j} + ζ_{S}^{j}}{2})}^{1 - α} + {(η_{R}^{j})}^{α} {(\frac{η_{R}^{j} + η_{S}^{j}}{2})}^{1 - α})}^{\frac{1}{α}} - 1], \end{matrix}$ (29)

$α \in (0, 1) .$

The symmetric version of divergence measures $^{τ} D_{α} (R | S)$ (τ = 1, 2, 3, 4) can be defined as follows: $^{τ} D_{α} (R ∥ S) = (\frac{^{τ} D_{α} (R | S) +^{τ} D_{α} (S | R)}{2}) .$ (30)

Theorem 2: For all R, S, T ∈ IFS (M), the divergence measures $^{τ} D_{α} (R ∥ S)$ (τ = 1, 2, 3, 4) satisfy the following properties:

$^{τ} D_{α} (R | | R \overset{IF}{\cup} S) =^{τ} D_{α} (S | | R \overset{IF}{\cap} S)$ ;

$^{τ} D_{α} (R | | R \overset{q}{\cap} S) =^{τ} D_{α} (S | | R \overset{IF}{\cup} S)$ ;

$^{τ} D_{α} (R \overset{IF}{\cup} S | | R \overset{IF}{\cap} S) =^{τ} D_{α} (R | | S)$ ;

$^{τ} D_{α} (R | | R \overset{IF}{\cup} S) +^{τ} D_{α} (R | | R \overset{IF}{\cap} S) =^{τ} D_{α} (R | | S)$ ;

$^{τ} D_{α} (R \overset{IF}{\cup} S | | T) ⩽^{τ} D_{α} (R | | T) +^{τ} D_{α} (S | | T)$ ;

$^{τ} D_{α} (R \overset{IF}{\cap} S | | T) ⩽^{τ} D_{α} (R | | T) +^{τ} D_{α} (S | | T)$ ;

$^{τ} D_{α} (R \overset{IF}{\cup} S | | T) +^{τ} D_{α} (R \overset{IF}{\cap} S | | T) =^{τ} D_{α} (R | | T) +^{τ} D_{α} (S | | T)$ ;

$^{τ} D_{α} (R | | S) =^{τ} D_{α} (R^{C} | | S^{C})$ ;

$^{τ} D_{α} (R | | S^{C}) =^{τ} D_{α} (R^{C} | | S)$ ;

$^{τ} D_{α} (R | | S) +^{τ} D_{α} (R | | S^{C}) =^{τ} D_{α} (R^{C} | | S^{C}) +^{τ} D_{α} (R^{C} | | S)$ .

Proof: The proof of these properties can be obtained easily by utilizing a similar methodology, as used in the references [16 , 19]. Therefore, we omit the proof from here. ■

3.1.1 Special and particular of $^{τ} D_{α} (R | S)$ and $^{τ} D_{α} (R | | S)$ (τ = 1, 2, 3, 4):

When α → 1, then $^{τ} D_{α} (R | | S)$ (τ = 1, 2, 3, 4) are reduced to intuitionistic fuzzy J-divergence [22].

When α → 1, then divergence measures $^{τ} D_{α} (R | S)$ (τ = 1, 2, 3, 4) become the divergence measure given in Equation (6).

If α = 1/ 2, then $^{τ} D_{α} (R | | S)$ (τ = 1, 2, 3, 4) reduce to the following divergence measures: $\begin{matrix} ^{1} D_{1 / 2} (R | | S) = \frac{1}{n (\sqrt{2} - 2)} \sum_{j = 1}^{n} \\ [{BC (R (j), S (j)) + (BC (S (j), R (j)))} - 2], \end{matrix}$ (31) $\begin{matrix} ^{2} D_{1 / 2} (R | | S) = - \frac{1}{n} \sum_{j = 1}^{n} \\ [{log}_{2} {BC (R (j), S (j)) \times BC (S (j), R (j))}], \end{matrix}$ (32) $\begin{matrix} ^{3} D_{1 / 2} (R | | S) = \frac{1}{2 n (e^{1 / \sqrt{2}} - e)} \sum_{j = 1}^{n} \\ [{e^{(BC (R (j), S (j)))} + e^{(BC (R (j), S (j)))}} - 2 e], \end{matrix}$ (33)

$^{4} D_{1 / 2} (R | | S) = \frac{1}{2 n} \sum_{j = 1}^{n} [{(B C (R (j), S (j)))}^{2} + {(B C (S (j), R (j)))}^{2} - 2],$ (34)

where $\begin{matrix} BC (R (j), S (j)) = \sqrt{ξ_{R}^{j} (\frac{ξ_{R}^{j} + ξ_{S}^{j}}{2})} + \sqrt{(ζ_{R}^{j}) (\frac{ζ_{R}^{j} + ζ_{S}^{j}}{2})} + \sqrt{(η_{R}^{j}) (\frac{η_{R}^{j} + η_{S}^{j}}{2})}, \\ BC (S (j), R (j)) = \sqrt{ξ_{S}^{j} (\frac{ξ_{S}^{j} + ξ_{R}^{j}}{2})} + \sqrt{(ζ_{S}^{j}) (\frac{ζ_{S}^{j} + ζ_{R}^{j}}{2})} + \sqrt{(η_{S}^{j}) (\frac{η_{S}^{j} + η_{R}^{j}}{2})} \end{matrix}},$ (35) denote the Bhattacharyya coefficient [52] between R (m_j) and S (m_j).

When $η_{R}^{j} = η_{S}^{j} = 0 \forall j$ , then $^{τ} D_{α} (R | | S)$ (τ = 1, 2, 3, 4) becomes the fuzzy divergence measures of order-α.

When α → 1 and $η_{R}^{j} = η_{S}^{j} = 0 \forall j$ , then measures $^{τ} D_{α} (R | | S)$ (τ = 1, 2, 3, 4) are reduced to fuzzy divergence measure proposed by Bhandari and Pal [8].

Now, we solve the numerical Examples 1 to 4 again with our proposed measures, the obtained divergence values based on

^{τ} D_{α} (R | | S)

(τ = 1, 2, 3, 4) are shown in Table 2. The results presented in Table 2 clearly show that the proposed intuitionistic fuzzy divergence measures of order- α are free from all the shortcomings existing in previous intuitionistic fuzzy divergence measures. Hence, the measures

^{τ} D_{α} (R | | S)

(τ = 1, 2, 3, 4) are valid and flexible divergence measures for IFSs, which can be utilized to solve a wide range of problems related to different application areas.

Table 2

The divergence values based on different divergence measures $^{τ} D_{α} (R | | S)$ (τ = 1, 2, 3, 4)

α	$^{1} D_{α} (R_{1} ∥ S_{1})$	$^{2} D_{α} (R_{1} ∥ S_{1})$	$^{3} D_{α} (R_{1} ∥ S_{1})$	$^{4} D_{α} (R_{1} ∥ S_{1})$
0.2	0.4221	0.3633	0.4681	0.6358
0.4	0.4269	0.3813	0.4648	0.4950
0.6	0.4309	0.3997	0.4585	0.4518
0.8	0.4343	0.4183	0.4493	0.4383
$^{1} D_{α} (R_{2} ∥ S_{2})$	$^{2} D_{α} (R_{2} ∥ S_{2})$	$^{3} D_{α} (R_{2} ∥ S_{2})$	$^{4} D_{α} (R_{2} ∥ S_{2})$
0.2	0.1095	0.0909	0.1252	0.1873
0.4	0.1177	0.1016	0.1320	0.1433
0.6	0.1247	0.1127	0.1359	0.1330
0.8	0.1307	0.1241	0.1371	0.1323
$^{1} D_{α} (R_{3} ∥ S_{3})$	$^{2} D_{α} (R_{3} ∥ S_{3})$	$^{3} D_{α} (R_{3} ∥ S_{3})$	$^{4} D_{α} (R_{3} ∥ S_{3})$
0.2	0.0014	0.0010	0.0017	0.0031
0.4	0.0026	0.0021	0.0030	0.0034
0.6	0.0036	0.0031	0.0040	0.0039
0.8	0.0045	0.0042	0.0048	0.0045
$^{1} D_{α} (R_{4} ∥ S_{4})$	$^{2} D_{α} (R_{4} ∥ S_{4})$	$^{3} D_{α} (R_{4} ∥ S_{4})$	$^{4} D_{α} (R_{4} ∥ S_{4})$
0.2	0.5000	0.4316	0.5530	0.7443
0.4	0.5000	0.4484	0.5424	0.5764
0.6	0.5000	0.4655	0.5302	0.5230
0.8	0.5000	0.4827	0.5162	0.5043

It is well known that the intuitionistic fuzzy divergence measures give information of discrimination between two IFSs. In 2007, Vlachos and Sergiadis [28] proved a mathematical relation to obtaining an intuitionistic fuzzy entropy measure from intuitionistic fuzzy divergence measure. In the next subsection, we define four new parametric entropy measures for IFSs based on the developed divergence measures.

3.2 Order-α entropy measures for IFSs

Theorem 3: Let R ∈ IFS (M), then

$\begin{matrix} ^{1} E_{α} (R) = -^{1} D_{α} (R | R^{C}) + 1 = \frac{1}{n (2^{1 - α} - 1)} \sum_{j = 1}^{n} \\ [({(ξ_{R}^{j})}^{α} + {(ζ_{R}^{j})}^{α}) {(ξ_{R}^{j} + ζ_{R}^{j})}^{1 - α} + 2^{1 - α} η_{R}^{j} - 1], \end{matrix}$ (36)

$\begin{matrix} ^{2} E_{α} (R) = -^{2} D_{α} (R | R^{C}) + 1 = \frac{1}{n (1 - α)} \sum_{j = 1}^{n} \\ {log}_{2} [({(ξ_{R}^{j})}^{α} + {(ζ_{R}^{j})}^{α}) {(ξ_{R}^{j} + ζ_{R}^{j})}^{1 - α} + 2^{1 - α} η_{R}^{j}], \end{matrix}$ (37)

$\begin{matrix} ^{3} E_{α} (R) = -^{3} D_{α} (R | R^{C}) + 1 = \frac{1}{n (e - e^{2^{α - 1}})} \sum_{j = 1}^{n} \\ [e^{2^{α - 1} (({(ξ_{R}^{j})}^{α} + {(ζ_{R}^{j})}^{α}) {(ξ_{R}^{j} + ζ_{R}^{j})}^{1 - α} + 2^{1 - α} η_{R}^{j})} - e^{2^{α - 1}}], \end{matrix}$ (38)

$\begin{matrix} ^{4} E_{α} (R) = -^{4} D_{α} (R | R^{C}) + 1 = \frac{1}{n (2^{(1 - α / α)} - 1)} \sum_{j = 1}^{n} \\ [{(({(ξ_{R}^{j})}^{α} + {(ζ_{R}^{j})}^{α}) {(ξ_{R}^{j} + ζ_{R}^{j})}^{1 - α} + 2^{1 - α} η_{R}^{j})}^{1 / α} - 1], \end{matrix}$ (39)

$α \in (0, 1),$ are intuitionistic fuzzy entropy measures.

Proof: In order to prove the validity of the proposed entropy measures $^{τ} E_{α} (R) (τ = 1, 2, 3, 4)$ , it is enough to show that the intuitionistic fuzzy entropy measures given in Equation (36), to Equation (39) hold the properties EP1-EP4 listed in Definition 4.

EP1. Let R be a crisp set having membership values either 0 or 1 for all ∀ m_j ∈ M. Then entropy measures $^{τ} E_{α} (R) (τ = 1, 2, 3, 4)$ become 0.

Next, if $^{τ} E_{α} (R) = 0 (τ = 1, 2, 3, 4)$ , then we obtain

$(ξ_{R}^{j} + ζ_{R}^{j}) (\frac{{(ξ_{R}^{j})}^{α} + {(ζ_{R}^{j})}^{α}}{{(ξ_{R}^{j} + ζ_{R}^{j})}^{α}} - 2^{1 - α}) = 1 - 2^{1 - α} \forall j .$ (40)

Since α ∈ (0, 1), then the Equation (40) will hold only when $ξ_{R}^{j} = 0, ζ_{R}^{j} = 1 or ξ_{R}^{j} = 1, ζ_{R}^{j} = 0 \forall j$ , i.e., R is a crisp set.

EP2. Let $ξ_{R}^{j} = ζ_{R}^{j} \forall j,$ then $^{τ} E_{α} (R) (τ = 1, 2, 3, 4)$ yield 1.

Conversely, let $^{τ} E_{α} (R) = 1 (τ = 1, 2, 3, 4)$ , then we have

$\begin{matrix} ({(ξ_{R}^{j})}^{α} + {(ζ_{R}^{j})}^{α}) {(ξ_{R}^{j} + ζ_{R}^{j})}^{1 - α} + 2^{1 - α} η_{R}^{j} \\ = 2^{1 - α} \Rightarrow \frac{{(ξ_{R}^{j})}^{α} + {(ζ_{R}^{j})}^{α}}{2} = {(\frac{ξ_{R}^{j} + ζ_{R}^{j}}{2})}^{α} . \end{matrix}$ (41)

Let us consider a function $g (z) = z^{α}, where z \in (0, 1], α \in (0, 1) .$ (42)

Differentiating Equation (42) w.r.t. z, then we get

$\frac{d (g (z))}{dz} = α z^{α - 1}$ and $\frac{d^{2} (g (z))}{dz} = α (α - 1) z^{α - 2}$ .

Since $\frac{d^{2} (g (z))}{dz} < 0, when 0 < α < 1,$ so g (z) is a concave function for all α ∈ (0, 1). Thus, for any z₁, z₂ ∈ (0, 1], we have the following inequality $\frac{g (z_{1}) + g (z_{2})}{2} ⩽ g (\frac{z_{1} + z_{2}}{2}), when α \in (0, 1),$ (43) with equality only for z₁ = z₂ .

Hence, based on the inequality obtained in Equation (43) with Equation (41), we conclude that $ξ_{R}^{j} = ζ_{R}^{j} \forall j .$

EP3. Let us assume the following functions as

$\begin{matrix} f_{1} (r, s) = \frac{1}{(2^{1 - α} - 1)} [(r^{α} + s^{α}) {(r + s)}^{1 - α} + 2^{1 - α} (1 - r - s) - 1] \\ f_{2} (r, s) = \frac{1}{(1 - α)} {log}_{2} [(r^{α} + s^{α}) {(r + s)}^{1 - α} + 2^{1 - α} (1 - r - s)] \\ f_{3} (r, s) = \frac{1}{(e - e^{2^{α - 1}})} [e^{2^{1 - α} ((r^{α} + s^{α}) {(r + s)}^{1 - α} + 2^{1 - α} (1 - r - s))} - e^{2^{1 - α}}] \\ f_{4} (r, s) = \frac{1}{(2^{(1 - α / α)} - 1)} [{((r^{α} + s^{α}) {(r + s)}^{1 - α} + 2^{1 - α} (1 - r - s))}^{\frac{1}{α}} - 1] \end{matrix}}, where r, s \in [0, 1] and α \in (0, 1) .$

Taking the partial derivatives of f_τ (r, s) (τ = 1, 2, 3, 4) w.r.t. r and s, we get $\begin{matrix} \frac{\partial f_{1} (r, s)}{\partial r} = \frac{1}{2^{1 - α} - 1} [((1 - α) {(r + s)}^{- α} (r^{α} + s^{α}) + α {(r + s)}^{1 - α} r^{α - 1} - 2^{1 - α})] \\ \frac{\partial f_{1} (r, s)}{\partial s} = \frac{1}{2^{1 - α} - 1} [((1 - α) {(r + s)}^{- α} (r^{α} + s^{α}) + α {(r + s)}^{1 - α} s^{α - 1} - 2^{1 - α})] \end{matrix}},$ $\begin{matrix} \frac{\partial f_{2} (r, s)}{\partial r} = \frac{1}{(1 - α)} [\frac{((1 - α) {(r + s)}^{- α} (r^{α} + s^{α}) + α {(r + s)}^{1 - α} r^{α - 1} - 2^{1 - α})}{((r^{α} + s^{α}) {(r + s)}^{1 - α} + 2^{1 - α} (1 - r - s))}] \\ \frac{\partial f_{2} (r, s)}{\partial s} = \frac{1}{(1 - α)} [\frac{((1 - α) {(r + s)}^{- α} (r^{α} + s^{α}) + α {(r + s)}^{1 - α} s^{α - 1} - 2^{1 - α})}{((r^{α} + s^{α}) {(r + s)}^{1 - α} + 2^{1 - α} (1 - r - s))}] \end{matrix}},$

$\begin{matrix} \frac{\partial f_{3} (r, s)}{\partial r} = \frac{1}{(e - e^{2^{α - 1}})} [2^{1 - α} e^{2^{1 - α} ((r^{α} + s^{α}) {(r + s)}^{1 - α} + 2^{1 - α} (1 - r - s))} ((1 - α) {(r + s)}^{- α} (r^{α} + s^{α}) + α {(r + s)}^{1 - α} r^{α - 1} - 2^{1 - α})] \\ \frac{\partial f_{3} (r, s)}{\partial s} = \frac{1}{(e - e^{2^{α - 1}})} [2^{1 - α} e^{2^{1 - α} ((r^{α} + s^{α}) {(r + s)}^{1 - α} + 2^{1 - α} (1 - r - s))} ((1 - α) {(r + s)}^{- α} (r^{α} + s^{α}) + α {(r + s)}^{1 - α} s^{α - 1} - 2^{1 - α})] \end{matrix}},$

$\begin{matrix} \frac{\partial f_{4} (r, s)}{\partial r} = \frac{1}{(2^{(1 - α / α)} - 1)} [\frac{1}{α} \frac{((1 - α) {(r + s)}^{- α} (r^{α} + s^{α}) + α {(r + s)}^{1 - α} r^{α - 1} - 2^{1 - α})}{{((r^{α} + s^{α}) {(r + s)}^{1 - α} + 2^{1 - α} (1 - r - s))}^{1 - \frac{1}{α}}}] \\ \frac{\partial f_{4} (r, s)}{\partial s} = \frac{1}{(2^{(1 - α / α)} - 1)} [\frac{1}{α} \frac{((1 - α) {(r + s)}^{- α} (r^{α} + s^{α}) + α {(r + s)}^{1 - α} s^{α - 1} - 2^{1 - α})}{{((r^{α} + s^{α}) {(r + s)}^{1 - α} + 2^{1 - α} (1 - r - s))}^{1 - \frac{1}{α}}}] \end{matrix}} .$

For a critical point of f_τ (r, s) (τ = 1, 2, 3, 4), we put $\frac{\partial f_{τ} (r, s)}{\partial r} = \frac{\partial f_{τ} (r, s)}{\partial s} = 0$ (τ = 1, 2, 3, 4) which implies r = s.

Since r, s ∈ [0, 1], we have

$\begin{matrix} \frac{\partial f_{τ} (r, s)}{\partial r} ⩾ 0 (τ = 1, 2, 3, 4), when r ⩽ s, α \in (0, 1) \\ \frac{\partial f_{τ} (r, s)}{\partial r} ⩽ 0 (τ = 1, 2, 3, 4), when r ⩾ s, α \in (0, 1) \end{matrix}},$ $\begin{matrix} \frac{\partial f_{τ} (r, s)}{\partial s} ⩽ 0 (τ = 1, 2, 3, 4), when r ⩽ s, α \in (0, 1) \\ \frac{\partial f_{τ} (r, s)}{\partial s} ⩾ 0 (τ = 1, 2, 3, 4), when r ⩾ s, α \in (0, 1) \end{matrix}} .$

Therefore f_τ (r, s)(τ = 1, 2, 3, 4) are increasing functions of r and decreasing functions of s.

Hence, considering the monotonicity of the functions f_τ (r, s) (τ = 1, 2, 3, 4), with Equations (36) to (39), we get $^{τ} E_{α} (R) ⩽^{τ} E_{α} (S) (τ = 1, 2, 3, 4) .$

EP4. We know that $R^{C} = {〈 m_{j}, ζ_{R} (m_{j}), ξ_{R} (m_{j}) 〉 | m_{j} \in M}$ , then $^{τ} E_{α} (R) =^{τ} E_{α} (R^{C})$ (τ = 1, 2, 3, 4).

This completes the proof. ■

The proposed intuitionistic fuzzy entropy measures, $^{τ} E_{α} (R) (τ = 1, 2, 3, 4)$ , also satisfy the following additional properties.

Theorem 4: Let R ={ 〈m_j, ξ_R (m_j) , ζ_R (m_j) 〉 |m_j ∈ M } and S ={ 〈 m_j, ξ_S (m_j) , ζ_S (m_j) 〉 |m_j ∈ M } be two IFSs defined in M ={ m₁, m₂, …, m_n } such that they satisfy for any m_j ∈ M either R ⊆ S or S ⊆ R, then we have

$\begin{matrix} ^{τ} E_{α} (R \overset{IF}{\cup} S) +^{τ} E_{α} (R \overset{IF}{\cap} S) \\ =^{τ} E_{α} (R) +^{τ} E_{α} (S) (τ = 1, 2, 3, 4) . \end{matrix}$ (44)

Proof. To prove the theorem, we divide M ={ m₁, m₂, …, m_n } into two disjoint sets M₁ and M₂ such that

$\begin{matrix} M_{1} = {m_{j} \in M : R \subseteq S} and \\ M_{2} = {m_{j} \in M : R \supseteq S} . \end{matrix}$ (45)

This implies that ∀ m_j ∈ M₁, $ξ_{R}^{j} ⩽ ξ_{S}^{j}, ζ_{R}^{j} ⩾ ζ_{S}^{j},$ (46) and ∀ m_j ∈ M₂,

$ξ_{R}^{j} ⩾ ξ_{S}^{j}, ζ_{R}^{j} ⩽ ζ_{S}^{j} .$ (47)

Using Equations (45)–(47) with Equations (36)–(39), we get the required results.

Theorem 5: The entropy measures $^{τ} E_{α} (R) (τ = 1, 2, 3, 4)$ attain maximum value when $ξ_{R}^{j} = ζ_{R}^{j} \forall j$ and minimum value when $ξ_{R}^{j} = 1, ζ_{R}^{j} = 0 or ξ_{R}^{j} = 0, ζ_{R}^{j} = 1 \forall j$ . Also, the maximum and minimum values are free from parameter α.

Remark 1: It is worth mentioning that the entropy measures defined in Equation (36) and Equation (37) have also been studied independently in the literature by Joshi & Kumar [53] and Verma & Sharma [42], respectively.

3.2.1 Special and particular cases of

^{τ} E_{α} (R)

(τ = 1, 2, 3, 4)

When α → 1, then entropy measures $^{τ} E_{α} (R)$ (τ = 1, 2, 3, 4) are reduced to intuitionistic fuzzy entropy defined by Vlachos and Sergiadis [28].

When $η_{R}^{j} = η_{R}^{j} = 0 \forall j$ , then $^{1} E_{α} (R)$ becomes the fuzzy entropy of order-α [6].

When $η_{R}^{j} = η_{R}^{j} = 0 \forall j$ , then $^{2} E_{α} (R)$ reduces to the fuzzy entropy of order-α [8].

When $η_{R}^{j} = η_{R}^{j} = 0 \forall j$ , then $^{4} E_{α} (R)$ is reduced to the parametric fuzzy entropy [54].

When α → 1, $η_{R}^{j} = η_{R}^{j} = 0 \forall j$ , then entropy measures $^{τ} E_{α} (R)$ (τ = 1, 2, 3, 4) become fuzzy entropy proposed in [37].

3.2.2 Comparative study with existing IF entropy measures

In this subsection, we compare the performance of the proposed entropy measures $^{τ} E_{α} (R)$ (τ = 1, 2, 3, 4) with some existing intuitionistic fuzzy entropy measures listed in Table 1.

Example 5: Let $R = {〈 m_{j}, ξ_{R} (m_{j}), ζ_{R} (m_{j}) 〉 | m_{j} \in M}$ be an IFSs defined in M ={ m₁, m₂, …, m_n }. For any positive real number n, De et al. [6] described an IFS Rⁿ as follows

$R^{n} = {〈 m_{j}, {(ξ_{R} (m_{j}))}^{n}, 1 - {(1 - ζ_{R} (m_{j}))}^{n} 〉 | m_{j} \in M} .$ (48)

We assume that the IFS R is defined by

$\begin{matrix} R = {〈 6, 0.1, 0.8 〉, 〈 7, 0.3, 0.5 〉, 〈 8, 0.5, 0.4 〉, \\ 〈 9, 0.9, 0 〉, 〈 10, 1.0, 0 〉} . \end{matrix}$ (49)

Considering R as a linguistic variable “Good” on M ={ m₁, m₂, …, m_n }. Based on Equation (48), we obtain

R^1/2 : resembled as “FAIR”,

R²: resembled as “Very GOOD”,

R³: resembled as “EXCELLENT”

R⁴: resembled as “EXCEPTIONAL.”

We use these IFSs to compare the performance of the proposed entropies with the entropy measures summarized in Table 1. As per the logical consideration, the entropy measures of these IFSs should be obeyed the following order: $E (R^{1 / 2}) > E (R) > E (R^{2}) > E (R^{3}) > E (R^{4}) .$ (50)

The values of different entropy measure for R^1/2, R,R², R³, and R⁴ are shown in Table 3.

Table 3

Numerical values of different entropy measures for R^1/2, R, R², R³, and R⁴

$^{BB} E (R)$		0.092	0.120	0.132	0.134	0.136
$^{ZL} E (R)$		0.379	0.360	0.316	0.232	0.191
$^{SK} E (R)$		0.345	0.374	0.197	0.131	0.109
$^{VS} E (R)$		0.408	0.392	0.272	0.192	0.148
$^{ZJ} E (R)$		0.285	0.305	0.104	0.038	0.0161
$^{HY} E_{hc}^{2} (R)$		0.328	0.340	0.290	0.253	0.226
$^{HY} E_{r}^{1 / 2} (R)$		0.661	0.681	0.605	0.522	0.457
$^{Y} E (R)$		0.545	0.525	0.364	0.259	0.205
$^{WGG} E (R)$		0.866	0.869	0.844	0.826	0.815
$^{VS 1} E (R)$		0.553	0.534	0.377	0.273	0.217
$^{VS 2} E (R)$	0.5	0.626	0.609	0.514	0.420	0.354
	0.8	0.597	0.572	0.462	0.363	0.299
	2	0.535	0.480	0.347	0.243	0.192
	5	0.479	0.415	0.279	0.182	0.143
	10	0.447	0.388	0.257	0.167	0.131
$^{WW} E (R)$		0.391	0.414	0.246	0.173	0.143
$^{LR} E (R)$		0.394	0.419	0.253	0.182	0.155
$^{MJH} E (R)$		0.552	0.533	0.376	0.2719	0.2149
$^{MR} E_{1} (R)$		0.448	0.437	0.272	0.184	0.145
$^{MR} E_{2} (R)$		0.553	0.534	0.377	0.273	0.217
$^{MR} E_{3} (R)$		0.420	0.416	0.238	0.155	0.122
$^{1} E_{α} (R)$	0.2	0.594	0.542	0.513	0.438	0.377
	0.4	0.581	0.506	0.441	0.339	0.269
	0.6	0.569	0.464	0.397	0.286	0.218
	0.8	0.560	0.378	0.368	0.255	0.190
$^{2} E_{α} (R)$	0.2	0.601	0.537	0.544	0.479	0.422
	0.4	0.588	0.502	0.467	0.369	0.299
	0.6	0.575	0.457	0.414	0.306	0.236
	0.8	0.563	0.364	0.377	0.265	0.198
$^{3} E_{α} (R)$	0.2	0.589	0.541	0.490	0.406	0.342
	0.4	0.574	0.506	0.420	0.314	0.244
	0.6	0.563	0.467	0.382	0.270	0.202
	0.8	0.556	0.388	0.360	0.247	0.182
$^{4} E_{α} (R)$	0.2	0.567	0.531	0.401	0.292	0.220
	0.4	0.569	0.506	0.403	0.294	0.225
	0.6	0.565	0.467	0.385	0.273	0.206
	0.8	0.559	0.381	0.366	0.253	0.188

From Table 3, we find that the entropy measures except $^{BB} E (R)$ , $^{SK} E (R)$ , $^{ZJ} E (R)$ , $^{HY} E_{hc}^{2} (R)$ , $^{HY} E_{r}^{1 / 2} (R)$ , $^{WGG} E (R)$ , $^{WW} E (R)$ , $^{LR} E (R)$ follow the order given in Equation (50). Therefore, the performance of the proposed intuitionistic fuzzy entropy measures $^{τ} E_{α} (R)$ (τ = 1, 2, 3, 4) is satisfactory for the viewpoint of structured linguistic variables. Also, note that our proposed intuitionistic fuzzy entropy measures of order-α have a flexible parameter to accommodate different situations in practical applications. Hence, we can say that the proposed intuitionistic fuzzy entropy measures of order- α provide more degree of freedom to the users for measuring the uncertainty associated with an IFS under different complex situations.

3.3 Weighted order-α divergence and entropy measures for IFSs

Let − λ = ( − λ₁, − λ₂, …, − λ_n) ^T be the weight vector of m_j ∈ M such that − λ_j ⩾ 0 ∀ j and $\sum_{j = 1}^{n} {- λ}_{j} = 1$ . Then, corresponding to measures given in Equations (26)–(29) and (36)–(39), we define the following weighted order-α divergence and entropy measures for IFSs:

$\begin{matrix} \overset{1}{\bar{λ}} D_{α} (R | S) = \frac{1}{(2^{(α - 1)} - 1)} \sum_{j = 1}^{n} {- λ}_{j} [({(ξ_{R}^{j})}^{α} {(\frac{ξ_{R}^{j} + ξ_{S}^{j}}{2})}^{1 - α} \\ + {(ζ_{R}^{j})}^{α} {(\frac{ζ_{R}^{j} + ζ_{S}^{j}}{2})}^{1 - α} + {(η_{R}^{j})}^{α} {(\frac{η_{R}^{j} + η_{S}^{j}}{2})}^{1 - α}) - 1], \\ \overset{2}{\bar{λ}} D_{α} (R | S) = \frac{1}{(α - 1)} \sum_{j = 1}^{n} {- λ}_{j} {log}_{2} [{(ξ_{R}^{j})}^{α} {(\frac{ξ_{R}^{j} + ξ_{S}^{j}}{2})}^{1 - α} \\ + {(ζ_{R}^{j})}^{α} {(\frac{ζ_{R}^{j} + ζ_{S}^{j}}{2})}^{1 - α} + {(η_{R}^{j})}^{α} {(\frac{η_{R}^{j} + η_{S}^{j}}{2})}^{1 - α}], \end{matrix}$

$\begin{matrix} \overset{3}{\bar{λ}} D_{α} (R | S) = \frac{1}{(e^{2^{α - 1}} - e)} \sum_{j = 1}^{n} {- λ}_{j} \\ [e^{{(ξ_{R}^{j})}^{α} {(\frac{ξ_{R}^{j} + ξ_{S}^{j}}{2})}^{1 - α} + {(ζ_{R}^{j})}^{α} {(\frac{ζ_{R}^{j} + ζ_{S}^{j}}{2})}^{1 - α} + {(η_{R}^{j})}^{α} {(\frac{η_{R}^{j} + η_{S}^{j}}{2})}^{1 - α}} - e], \end{matrix}$

$\begin{matrix} \overset{4}{\bar{λ}} D_{α} (R | S) = \frac{1}{(2^{(α - 1 / α)} - 1)} \sum_{j = 1}^{n} {- λ}_{j} \\ [({(ξ_{R}^{j})}^{α} {(\frac{ξ_{R}^{j} + ξ_{S}^{j}}{2})}^{1 - α} + {(ζ_{R}^{j})}^{α} {(\frac{ζ_{R}^{j} + ζ_{S}^{j}}{2})}^{1 - α} \\ {+ {(η_{R}^{j})}^{α} {(\frac{η_{R}^{j} + η_{S}^{j}}{2})}^{1 - α})}^{1 / α} - 1], \end{matrix}$

$\begin{matrix} \overset{1}{\bar{λ}} E_{α} (R) = \frac{1}{(2^{1 - α} - 1)} \sum_{j = 1}^{n} {- λ}_{j} \\ [({(ξ_{R}^{j})}^{α} + {(ζ_{R}^{j})}^{α}) {(ξ_{R}^{j} + ζ_{R}^{j})}^{1 - α} + 2^{1 - α} η_{R}^{j} - 1], \\ \overset{2}{\bar{λ}} E_{α} (R) = \frac{1}{(1 - α)} \sum_{j = 1}^{n} \\ {- λ}_{j} {log}_{2} [({(ξ_{R}^{j})}^{α} + {(ζ_{R}^{j})}^{α}) {(ξ_{R}^{j} + ζ_{R}^{j})}^{1 - α} + 2^{1 - α} η_{R}^{j}], \\ \overset{3}{\bar{λ}} E_{α} (R) = \frac{1}{(e - e^{2^{α - 1}})} \sum_{j = 1}^{n} {- λ}_{j} \\ [e^{2^{α - 1} (({(ξ_{R}^{j})}^{α} + {(ζ_{R}^{j})}^{α}) {(ξ_{R}^{j} + ζ_{R}^{j})}^{1 - α} + 2^{1 - α} η_{R}^{j})} - e^{2^{α - 1}}], \\ \overset{4}{\bar{λ}} E_{α} (R) = \frac{1}{n (2^{(1 - α / α)} - 1)} \sum_{j = 1}^{n} {- λ}_{j} \\ [{(({(ξ_{R}^{j})}^{α} + {(ζ_{R}^{j})}^{α}) {(ξ_{R}^{j} + ζ_{R}^{j})}^{1 - α} + 2^{1 - α} η_{R}^{j})}^{1 / α} - 1] . \end{matrix}$

In the next section, by utilizing proposed order-α divergence and entropy measures for IFSs, we propose an intuitionistic fuzzy version of the well-known MABAC model [44] for solving real-life MAGDM problems with unknown/partially known attribute weights.

4 A new multiple-attribute group decision-making approach

Decision-making is one of the most significant activities in our daily life actions. MAGDM, a part of DM theory, has become a very popular research area in recent years. In MAGDM problems, the best alternative is selected by a group of decision-makers (DMs) under the several known and unknown attributes. Now, we formulate an intuitionistic fuzzy version of the well-known MABAC method by using the proposed order-α divergence and entropy measures to solve MAGDM problems with intuitionistic fuzzy information.

4.1 Intuitionistic fuzzy MABAC method

In the literature, a wide range of MADM decision-making methods have been investigated by numerous scholars, including the VIKOR approach, the MULTIMOORA approach, the TOPSIS approach, the PROMETHEE approach, the GRA approach, the TODIM approach, and the ELECTRE approach. In 2015, Pamučar and Ćirović developed a novel MADM approach called MABAC (multi-attributive border approximation area comparison) method, which can take the conflicting attributes into consideration. This decision-making method is focused on the distance of the alternatives from the border approximation area. It has a simple computational procedure and systematic process that makes it capable and competent in solving real-world decision-making problems. In the last years, some extensions of the MABAC method have been formulated by several researchers under different information environments and applied in many practical areas [55 –57].

For a MAGDM problem with intuitionistic fuzzy information, let R = (R₁, R₂, …, R_m) be a set of m alternatives and A = (A₁, A₂, …, A_n) be a set of n attributes. Further, assume that the attribute weighting vector is denoted by − λ = ( − λ₁, − λ₂, …, − λ_n) ^T such that − λ_j ⩾ 0, j = 1, 2, …, n and $\sum_{j = 1}^{n} {- λ}_{j} = 1$ . To make the decision result more conclusive, a panel of ^′p′ experts is invited to evaluate the available alternatives, and the panel of experts is denoted as $E = (E^{(1)}, E^{(2)}, \dots, E^{(p)})$ . The weight vector associated with a panel of experts is given by ω = (ω₁, ω₂, …, ω_p) ^T such that ω_κ ⩾ 0, κ = 1, 2, …, p and $\sum_{κ = 1}^{p} ω_{κ} = 1$ . The experts evaluate the available alternatives under the given set of attributes and provide their assessment values in terms of linguistic variables (LVs) $ℓ_{ij}^{(κ)}$ . Let $L^{(κ)} = {[ℓ_{ij}^{(κ)}]}_{m \times n}$ be the linguistic matrix provided by the expert $E^{(κ)}$ , where $ℓ_{ij}^{(κ)}$ represents the evaluation information provided by the expert $E^{(κ)}$ for alternative R_i concerning attribute A_j

Now, for solving the above MAGDM problem, the intuitionistic fuzzy-MABAC method contains the following decision steps:

Step 1: Construct the intuitionistic fuzzy decision matrices

First, transform LVs $ℓ_{ij}^{(κ)}$ into IFNs $℘_{ij}^{(κ)} = 〈 ξ_{℘_{ij}}^{(κ)}, ζ_{℘_{ij}}^{(κ)} 〉$ and construct the intuitionistic fuzzy decision matrices $B^{(κ)}$ (κ = 1, 2, …, p) corresponding to $L^{(κ)} = {[ℓ_{ij}^{(κ)}]}_{m \times n}$ (κ = 1, 2, …, p), represented by: $\begin{matrix} \begin{matrix} A_{1} & A_{2} & \dots & A_{n} \end{matrix} \\ B^{(κ)} = \begin{matrix} R_{1} \\ R_{2} \\ ⋮ \\ R_{m} \end{matrix} (\begin{matrix} 〈 ξ_{℘_{11}}^{(κ)}, ζ_{℘_{11}}^{(κ)} 〉 & 〈 ξ_{℘_{12}}^{(κ)}, ζ_{℘_{12}}^{(κ)} 〉 & \dots & 〈 ξ_{℘_{1 n}}^{(κ)}, ζ_{℘_{1 n}}^{(κ)} 〉 \\ 〈 ξ_{℘_{21}}^{(κ)}, ζ_{℘_{21}}^{(κ)} 〉 & 〈 ξ_{℘_{22}}^{(κ)}, ζ_{℘_{22}}^{(κ)} 〉 & \dots & 〈 ξ_{℘_{2 n}}^{(κ)}, ζ_{℘_{2 n}}^{(κ)} 〉 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 〈 ξ_{℘_{m 1}}^{(κ)}, ζ_{℘_{m 1}}^{(κ)} 〉 & 〈 ξ_{℘_{m 2}}^{(κ)}, ζ_{℘_{m 2}}^{(κ)} 〉 & \dots & 〈 ξ_{℘_{mn}}^{(κ)}, ζ_{℘_{mn}}^{(κ)} 〉 \end{matrix}) \end{matrix},$ (51)

Step 2: Normalization of decision matrices Convert all the cost attributes into benefit attributes and form the normalized intuitionistic decision matrices ${\tilde{B}}^{(κ)} = {({\tilde{℘}}_{ij}^{(κ)})}_{m \times n}$ , where ${\tilde{℘}}_{ij}^{(κ)} = {\begin{matrix} 〈 ξ_{℘_{ij}}^{(κ)}, ζ_{℘_{ij}}^{(κ)} 〉 for benefit type attributes, \\ 〈 ζ_{℘_{ij}}^{(κ)}, ξ_{℘_{ij}}^{(κ)} 〉 for cost type attributes . \end{matrix}$ (52)

Step 3: Determine the weight of experts During the group decision-making process, it is very important to assign suitable weights to the different experts according to their expertise and experience. Suppose the weight values ω_κ (κ = 1, 2, …, p) of the experts are given in terms of LVs. After converting the LVs into IFNs, we use the following formula to determine the experts’ weight, as suggested by Boran et al. [58]: $ω_{κ} = \frac{ξ_{℘_{κ}} + η_{℘_{κ}} (\frac{ξ_{℘_{κ}}}{ξ_{℘_{κ}} + ζ_{℘_{κ}}})}{\sum_{κ = 1}^{p} (ξ_{℘_{κ}} + η_{℘_{κ}} (\frac{ξ_{℘_{κ}}}{ξ_{℘_{κ}} + ζ_{℘_{κ}}}))}, κ = 1, 2, \dots, p,$ (53) where ω_κ ⩾ 0, κ = 1, 2, …, p and $\sum_{κ = 1}^{p} ω_{κ} = 1$ .

Step 4: Calculate the attribute weights In MAGDM problems, the attribute weights have a significant role in the ranking of alternatives. To differentiate the significance of different attributes, we need to calculate the attribute weights. In the literature, some approaches are available to compute the attribute weights, including the entropy approach [59], TOPSIS approach [60], and maximizing deviation method [15], divergence based method [17]. To calculate the attribute weights, we formulate an optimization model based on proposed information measures as follows:

For the expert $E^{(κ)}$ , the total divergence measure among all the alternatives under an attribute A_j can be obtained as $D_{j}^{(κ)} = \frac{1}{(m - 1)} \sum_{i = 1}^{m} \sum_{k = 1}^{m} D (℘_{ij}^{(κ)} | | ℘_{kj}^{(κ)}) .$ (54)

Also, if the weights of all the experts are taken into account, then the overall divergence measure among all the alternatives for an attribute A_j is represented as

$\begin{matrix} D_{j} & = \sum_{κ = 1}^{p} ω_{κ} D_{j}^{(κ)} \\ = \sum_{κ = 1}^{p} ω_{κ} \frac{1}{(m - 1)} \sum_{i = 1}^{m} \sum_{k = 1}^{m} D (℘_{ij}^{(κ)} | | ℘_{kj}^{(κ)}) . \end{matrix}$ (55)

The overall entropy value of an attribute A_j can be calculated by the following expression $E_{j} = \sum_{κ = 1}^{p} ω_{κ} \sum_{i = 1}^{m} E (℘_{ij}^{(κ)}),$ (56) and the dispersion measure [61] corresponding to the attribute weighting vector − λ = ( − λ₁, − λ₂, …, − λ_n) ^T is given by $D i s p (- λ) = - \sum_{j = 1}^{n} {- λ}_{j} {log}_{2} {- λ}_{j}$ (57)

The ideal weighting vector should maximize the dispersion and divergence but minimize the entropy of the total IF decision matrices. Therefore, we define the following function

$\begin{matrix} \hat{F} = - \sum_{j = 1}^{n} {- λ}_{j} {log}_{2} {- λ}_{j} + \sum_{j = 1}^{n} \\ ({- λ}_{j} \sum_{κ = 1}^{p} ω_{κ} \frac{1}{(m - 1)} \sum_{i = 1}^{m} \sum_{k = 1}^{m} D (℘_{ij}^{(κ)} | | ℘_{kj}^{(κ)})) \\ - \sum_{j = 1}^{n} ({- λ}_{j} \sum_{κ = 1}^{p} ω_{κ} \sum_{i = 1}^{m} E (℘_{ij}^{(κ)})) \\ = \sum_{j = 1}^{n} {- λ}_{j} (\sum_{κ = 1}^{p} \sum_{i = 1}^{m} ω_{κ} (\frac{1}{(m - 1)} \sum_{k = 1}^{m} D (℘_{ij}^{(κ)} | | ℘_{kj}^{(κ)}) \\ - E (℘_{ij}^{(κ)})) - {log}_{2} {- λ}_{j}) \\ = \sum_{j = 1}^{n} {- λ}_{j} (D_{j} - E_{j} - {log}_{2} {- λ}_{j}) . \end{matrix}$ (58)

Hence, the following optimization model is designed to calculate the weighting vector − λ = ( − λ₁, − λ₂, …, − λ_n) ^T:

(MOD 1) $Max \hat{F} = \sum_{j = 1}^{n} {- λ}_{j} (D_{j} - E_{j} - {log}_{2} {- λ}_{j})$ $s . t . {\begin{matrix} - λ \in T, \\ \sum_{j = 1}^{n} {- λ}_{j} = 1, \\ {- λ}_{j} ⩾ 0, j = 1, 2, \dots, n \end{matrix}$ where T is the set of all incomplete information about the attribute weights.

If we use the developed order-α divergence and entropy measures, then MOD 1 can be written as:

(MOD 2) $Max \hat{F} = \sum_{j = 1}^{n} {- λ}_{j} (\sum_{κ = 1}^{p} \sum_{i = 1}^{m} ω_{κ} (\frac{1}{(m - 1)} \sum_{k = 1}^{m}^{τ} D_{α} (℘_{ij}^{(κ)} | | ℘_{kj}^{(κ)}) -^{τ} E_{α} (℘_{ij}^{(κ)})) - {log}_{2} {- λ}_{j})$ , where τ = 1, 2, 3, 4 $s . t . {\begin{matrix} - λ \in T, \\ \sum_{j = 1}^{n} {- λ}_{j} = 1, \\ {- λ}_{j} ⩾ 0, j = 1, 2, \dots, n . \end{matrix}$

Step 5: Construction of normalized aggregated intuitionistic fuzzy decision matrix

To combine all the individual normalized intuitionistic fuzzy decision matrices ${\tilde{B}}^{(κ)} = {({\tilde{℘}}_{ij}^{(κ)})}_{m \times n} (κ = 1, 2, \dots, p)$ , a combined intuitionistic fuzzy decision matrix is constructed based on IFIWA operator as

$\begin{matrix} \begin{matrix} A_{1} & A_{2} & \dots & A_{n} \end{matrix} \\ \tilde{B} = {({\tilde{℘}}_{ij})}_{m \times n} = \begin{matrix} R_{1} \\ R_{2} \\ ⋮ \\ R_{m} \end{matrix} (\begin{matrix} 〈 ξ_{{\tilde{℘}}_{11}}, ζ_{{\tilde{℘}}_{11}} 〉 & 〈 ξ_{{\tilde{℘}}_{12}}, ζ_{{\tilde{℘}}_{12}} 〉 & \dots & 〈 ξ_{{\tilde{℘}}_{1 n}}, ζ_{{\tilde{℘}}_{1 n}} 〉 \\ 〈 ξ_{{\tilde{℘}}_{21}}, ζ_{{\tilde{℘}}_{21}} 〉 & 〈 ξ_{{\tilde{℘}}_{22}}, ζ_{{\tilde{℘}}_{22}} 〉 & \dots & 〈 ξ_{{\tilde{℘}}_{2 n}}, ζ_{{\tilde{℘}}_{2 n}} 〉 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 〈 ξ_{{\tilde{℘}}_{m 1}}, ζ_{{\tilde{℘}}_{m 1}} 〉 & 〈 ξ_{{\tilde{℘}}_{m 2}}, ζ_{{\tilde{℘}}_{m 2}} 〉 & \dots & 〈 ξ_{{\tilde{℘}}_{mn}}, ζ_{{\tilde{℘}}_{mn}} 〉 \end{matrix}) \end{matrix},$ (59) where $\begin{matrix} {\tilde{℘}}_{ij} = 〈 ξ_{{\tilde{℘}}_{ij}}, ζ_{{\tilde{℘}}_{ij}} 〉 = IFIWA ({\tilde{℘}}_{ij}^{(1)}, {\tilde{℘}}_{ij}^{(2)}, \dots, {\tilde{℘}}_{ij}^{(p)}) \\ = 〈 1 - \prod_{κ = 1}^{p} {(1 - ξ_{{\tilde{℘}}_{ij}}^{(κ)})}^{ω_{κ}}, \prod_{κ = 1}^{p} {(1 - ξ_{{\tilde{℘}}_{ij}}^{(κ)})}^{ω_{κ}} \\ - \prod_{κ = 1}^{p} {(1 - ξ_{{\tilde{℘}}_{ij}}^{(κ)} - ζ_{{\tilde{℘}}_{ij}}^{(κ)})}^{ω_{κ}} 〉, i = 1, 2, \dots, \\ m; j = 1, 2, \dots, n . \end{matrix}$ (60)

Step 6: Calculate the weighted normalized aggregated intuitionistic fuzzy decision matrix

Utilizing the decision matrix $\tilde{B} = {({\tilde{℘}}_{ij})}_{m \times n}$ and the attributes weighting vector − λ = ( − λ₁, − λ₂, …, − λ_n) ^T obtained in Step 3, we can compute the weighted normalized aggregated intuitionistic fuzzy decision matrix as follows: $\hat{\tilde{}} B = {(^{\hat{˜}} ℘_{ij})}_{m \times n} = {({- λ}_{j} \overset{IF}{*} {\tilde{℘}}_{ij})}_{m \times n} .$ (61)

Step 7: Construct the border approximation area (BAA) matrix

The BAA matrix is obtained from the following expression $G = {[g_{j}]}_{1 \times n} = {[{({\overset{IF}{\otimes}}_{i = 1}^{m}^{\hat{˜}} ℘_{ij})}^{\frac{1}{m}}]}_{1 \times n} .$ (62)

Step 8: Divergence computation

Compute the divergence between alternatives and the BAA using the following equation

$^{τ} D_{α}^{ij} = {\begin{matrix} ^{τ} D_{α} (^{\hat{˜}} ℘_{ij} | | g_{j}), if^{\hat{˜}} ℘_{ij} > g_{j} \\ 0, if^{\hat{˜}} ℘_{ij} = g_{j} \\ -^{τ} D_{α} (^{\hat{˜}} ℘_{ij} | | g_{j}), if^{\hat{˜}} ℘_{ij} < g_{j} \end{matrix} (τ = 1, 2, 3, 4) .$ (63)

Step 9: Obtain total divergence from BAA

The total divergence measure of each alternative from the BAA is determined as follows $^{τ} D_{α}^{i} = \sum_{j = 1}^{n}^{τ} D_{α}^{ij} .$ (64)

Step 10: Ranking of alternatives

Finally, we rank all the alternatives based on the total divergence values $^{τ} D_{α}^{i}$ ’s in descending order and select the best alternative with the highest divergence value.

4.2 Numerical example

Example 6: In many enterprises and organizations, one of the most important and challenging activities executed by human resource management (HRM) is personnel selection. To make better personnel selections, several studies have been conducted based on different MAGDM methods. Nowadays, excellent talents are of great importance for any organization to survive in a highly competitive environment.

Let us suppose a manufacturing company intends to recruit a sales manager for his new branch office. In doing so, the company released an opening recruitment advertisement in leading newspapers and on the company’s official website. A total of 15 candidates were applied from different cities for this post. After making a strict screening process, five candidates R ={ R₁, R₂, R₃, R₄, R₅ } have been short-listed for the final interview. To select the most suitable candidate for this post, a panel of four experts $E = (E^{(1)}, E^{(2)}, E^{(3)}, E^{(4)})$ from the human resources department of the company is invited to have a face-to-face interview with the five candidates. The importance of the experts in a selection committee may not be equal to each other and is considered as linguistic terms (LTs) expressed in IFNs in Table 4. The company management decides the importance of each expert according to his/her expertise and is presented in Table 5. The decision is made based on the following six attributes: (1) Oral communication skill (A₁) (2) Self-confidence (A₂) (3) Past experience (A₃) (4) Leadership (A₄) (5) General aptitude (A₅) (6) First impression (A₆). Table 6 represents the LTs in the form of IFNs to estimate the relative importance of the considered attributes. The experts $E = (E^{(1)}, E^{(2)}, E^{(3)}, E^{(4)})$ evaluate the five candidates based on the attributes A_j (j = 1, 2, …, 6) and use Table 6 to provide his/her assessment information in the form of LTs. Tables 7 –10 present the assessment information of five candidates provided by the experts concerning the attributes A_j (j = 1, 2, …, 6).

Table 4
Linguistic terms for the rating of the experts

LTs Very Important Important Medium Unimportant Very unimportant

(VI) (I) (M) (UI) (VUI)

IFNs 〈0.80, 0.10〉〈0.55, 0.25〉〈0.50, 0.50〉〈0.30, 0.50〉〈0.20, 0.70〉

LTs	Very Important	Important	Medium	Unimportant	Very unimportant
IFNs	〈0.80, 0.10〉	〈0.55, 0.25〉	〈0.50, 0.50〉	〈0.30, 0.50〉	〈0.20, 0.70〉

Table 5

Importance weight of experts as LTs

	$E^{(1)}$	$E^{(2)}$	$E^{(3)}$	$E^{(4)}$
LTs	VI	M	I	I

Table 6

Linguistic terms for the rating of the candidates

LTs	IFNs
Extremely good (EG)	〈1.00, 0.00〉
Very good (VG)	〈0.75, 0.10〉
Good (G)	〈0.60, 0.25〉
Slightly good (SG)	〈0.50, 0.40〉
Fair (F)	〈0.50, 0.50〉
Slightly poor (SP)	〈0.40, 0.50〉
Poor (P)	〈0.25, 0.60〉
Very poor (VP)	〈0.10, 0.75〉
Extremely poor (EP)	〈0.00, 0.90〉

Table 7

Linguistic assessment matrix $L^{(1)}$ provided by the expert $E^{(1)}$

	A ₁	A ₂	A ₃	A ₄	A ₅	A ₆
R ₁	SG	VG	F	G	F	VG
R ₂	G	EG	SG	EG	VG	SP
R ₃	F	G	G	SP	F	G
R ₄	VG	VG	F	F	SP	EG
R ₅	G	G	F	F	VG	G

Table 8

Linguistic assessment matrix $L^{(2)}$ provided by the expert $E^{(2)}$

	A ₁	A ₂	A ₃	A ₄	A ₅	A ₆
R ₁	G	VG	F	F	F	SP
R ₂	EG	EG	G	SP	F	G
R ₃	SP	SG	G	P	G	G
R ₄	G	VG	SP	VP	G	F
R ₅	VG	G	SG	SP	F	G

Table 9

Linguistic assessment matrix $L^{(3)}$ provided by the expert $E^{(3)}$

	A ₁	A ₂	A ₃	A ₄	A ₅	A ₆
R ₁	G	VG	SP	G	F	SP
R ₂	EG	EG	G	SP	G	G
R ₃	SP	G	F	F	SG	F
R ₄	G	F	F	SG	P	VG
R ₅	SP	G	VG	SG	SG	F

Table 10

Linguistic assessment matrix $L^{(4)}$ provided by the expert $E^{(4)}$

	A ₁	A ₂	A ₃	A ₄	A ₅	A ₆
R ₁	G	G	SG	F	P	SG
R ₂	SG	EG	EG	G	VG	F
R ₃	SP	SP	G	F	G	SG
R ₄	VG	G	SP	G	F	G
R ₅	VG	G	VG	VG	SP	G

The stepwise decision procedure as follows:

Step 1: Utilizing Table 6, we construct the intuitionistic fuzzy decision matrices $B^{(κ)}$ corresponding to the linguistic assessment matrices $L^{(κ)}$ as summarized in Tables 11-14.

Step 2: Since all the attributes are of benefit type, so ${\tilde{B}}^{(κ)} = B^{(κ)} (κ = 1, 2, 3, 4) .$

Step 3: The weight of each expert is calculated based on Equation (53) as follows:

$ω_{1} = \frac{0.80 + 0.10 \times (\frac{0.80}{0.80 + 0.10})}{0.80 + 0.10 \times (\frac{0.80}{0.80 + 0.10}) + 0.50 + 0.0 \times (\frac{0.50}{0.50 + 0.50}) + 0.55 + 0.20 \times (\frac{0.55}{0.55 + 0.25}) + 0.55 + 0.20 \times (\frac{0.55}{0.55 + 0.25})} = 0.3216,$ $ω_{2} = \frac{0.50 + 0.0 \times (\frac{0.50}{0.50 + 0.50})}{0.80 + 0.10 \times (\frac{0.80}{0.80 + 0.10}) + 0.50 + 0.0 \times (\frac{0.50}{0.50 + 0.50}) + 0.55 + 0.20 \times (\frac{0.55}{0.55 + 0.25}) + 0.55 + 0.20 \times (\frac{0.55}{0.55 + 0.25})} = 0.1810,$ $ω_{3} = \frac{0.55 + 0.20 \times (\frac{0.55}{0.55 + 0.25})}{0.80 + 0.10 \times (\frac{0.80}{0.80 + 0.10}) + 0.50 + 0.0 \times (\frac{0.50}{0.50 + 0.50}) + 0.55 + 0.20 \times (\frac{0.55}{0.55 + 0.25}) + 0.55 + 0.20 \times (\frac{0.55}{0.55 + 0.25})} = 0.2487,$ $ω_{4} = \frac{0.55 + 0.20 \times (\frac{0.55}{0.55 + 0.25})}{0.80 + 0.10 \times (\frac{0.80}{0.80 + 0.10}) + 0.50 + 0.0 \times (\frac{0.50}{0.50 + 0.50}) + 0.55 + 0.20 \times (\frac{0.55}{0.55 + 0.25}) + 0.55 + 0.20 \times (\frac{0.55}{0.55 + 0.25})} = 0.2487 .$ Now, we consider the following two cases:

Case (i) When the information about the attribute weights is completely unknown.

Step 4: Using the MOD 2 with measures $^{1} D_{α} (R | S)$ and $^{1} E_{α} (R)$ , we formulate the following optimal model to determine the weighting vector − λ = ( − λ₁, − λ₂, …, − λ_n) ^T (here, we have taken α = 0.5): $Max . \hat{F} = (\begin{matrix} - 3.7882 {- λ}_{1} - {- λ}_{1} log {- λ}_{1} - 3.3446 {- λ}_{2} - {- λ}_{2} log {- λ}_{2} - 4.1821 {- λ}_{3} - {- λ}_{3} log {- λ}_{3} \\ - 4.0611 {- λ}_{4} - {- λ}_{4} log {- λ}_{4} - 4.2909 {- λ}_{5} - {- λ}_{5} log {- λ}_{5} - 4.0384 {- λ}_{6} - {- λ}_{6} log {- λ}_{6} \end{matrix}),$ $s . t . {\begin{matrix} {- λ}_{1} + {- λ}_{2} + {- λ}_{3} + {- λ}_{4} + {- λ}_{5} + {- λ}_{6} = 1, \\ {- λ}_{j} ⩾ 0, j = 1, 2, 3, 4, 5, 6 . \end{matrix}$ (65) After solving the optimization model given in Equation (65), we get

${- λ}^{1} = {(0.1820, 0.2472, 0.1385, 0.1507, 0.1285, 0.1531)}^{T} .$

Step 5: We use the IFIWA operator given in Equation (60) to get the collective decision matrix $\tilde{B} = {({\tilde{℘}}_{ij})}_{m \times n}$ . Table 15 presents the normalized collective intuitionistic fuzzy decision matrix $\tilde{B}$ :

Step 6: Using the normalized collective intuitionistic fuzzy decision matrix $\tilde{B} = {({\tilde{℘}}_{ij})}_{m \times n}$ and the attributes weighting vector − λ = (0.1820, 0.2472, 0.1385, 0.1507, 0.1285, 0.1531) ^T, we obtain the weighted normalized collective intuitionistic fuzzy decision matrix based on Equation (61) as given in Table 16.

Table 11

Intuitionistic fuzzy assessment matrix $B^{(1)}$

	A ₁	A ₂	A ₃	A ₄	A ₅	A ₆
R ₁	〈0.50, 0.40〉	〈0.75, 0.10〉	〈0.50, 0.50〉	〈0.60, 0.25〉	〈0.50, 0.50〉	〈0.75, 0.10〉
R ₂	〈0.60, 0.25〉	〈1.00, 0.00〉	〈0.50, 0.40〉	〈1.00, 0.00〉	〈0.75, 0.10〉	〈0.40, 0.50〉
R ₃	〈0.50, 0.50〉	〈0.60, 0.25〉	〈0.60, 0.25〉	〈0.40, 0.50〉	〈0.50, 0.50〉	〈0.60, 0.25〉
R ₄	〈0.75, 0.10〉	〈0.75, 0.10〉	〈0.50, 0.50〉	〈0.50, 0.50〉	〈0.40, 0.50〉	〈1.00, 0.00〉
R ₅	〈0.60, 0.25〉	〈0.60, 0.25〉	〈0.50, 0.50〉	〈0.50, 0.50〉	〈0.75, 0.10〉	〈0.60, 0.25〉

Table 12

Intuitionistic fuzzy assessment matrix $B^{(2)}$

	A ₁	A ₂	A ₃	A ₄	A ₅	A ₆
R ₁	〈0.60, 0.25〉	〈0.75, 0.10〉	〈0.50, 0.50〉	〈0.50, 0.50〉	〈0.50, 0.50〉	〈0.40, 0.50〉
R ₂	〈1.00, 0.00〉	〈1.00, 0.00〉	〈0.60, 0.25〉	〈0.40, 0.50〉	〈0.50, 0.50〉	〈0.60, 0.25〉
R ₃	〈0.40, 0.50〉	〈0.50, 0.40〉	〈0.60, 0.25〉	〈0.25, 0.60〉	〈0.60, 0.25〉	〈0.60, 0.25〉
R ₄	〈0.60, 0.25〉	〈0.75, 0.10〉	〈0.40, 0.50〉	〈0.10, 0.75〉	〈0.60, 0.25〉	〈0.50, 0.50〉
R ₅	〈0.75, 0.10〉	〈0.60, 0.25〉	〈0.50, 0.40〉	〈0.40, 0.50〉	〈0.50, 0.50〉	〈0.60, 0.25〉

Table 13

Intuitionistic fuzzy assessment matrix $B^{(3)}$

	A ₁	A ₂	A ₃	A ₄	A ₅	A ₆
R ₁	〈0.60, 0.25〉	〈0.75, 0.10〉	〈0.40, 0.50〉	〈0.60, 0.25〉	〈0.50, 0.50〉	〈0.40, 0.50〉
R ₂	〈1.00, 0.00〉	〈0.60, 0.25〉	〈0.60, 0.25〉	〈0.40, 0.50〉	〈0.60, 0.25〉	〈0.60, 0.25〉
R ₃	〈0.40, 0.50〉	〈0.60, 0.25〉	〈0.50, 0.50〉	〈0.50, 0.50〉	〈0.50, 0.40〉	〈0.50, 0.50〉
R ₄	〈0.60, 0.25〉	〈0.50, 0.50〉	〈0.50, 0.50〉	〈0.50, 0.40〉	〈0.25, 0.60〉	〈0.75, 0.10〉
R ₅	〈0.40, 0.50〉	〈0.60, 0.25〉	〈0.75, 0.10〉	〈0.50, 0.40〉	〈0.50, 0.40〉	〈0.50, 0.50〉

Table 14

Intuitionistic fuzzy assessment matrix $B^{(4)}$

	A ₁	A ₂	A ₃	A ₄	A ₅	A ₆
R ₁	〈0.60, 0.25〉	〈0.60, 0.25〉	〈0.50, 0.40〉	〈0.50, 0.50〉	〈0.25, 0.60〉	〈0.50, 0.40〉
R ₂	〈0.50, 0.40〉	〈1.00, 0.00〉	〈1.00, 0.00〉	〈0.60, 0.25〉	〈0.75, 0.10〉	〈0.50, 0.50〉
R ₃	〈0.40, 0.50〉	〈0.40, 0.50〉	〈0.60, 0.25〉	〈0.50, 0.50〉	〈0.60, 0.25〉	〈0.50, 0.40〉
R ₄	〈0.75, 0.10〉	〈0.60, 0.25〉	〈0.40, 0.50〉	〈0.60, 0.25〉	〈0.50, 0.50〉	〈0.60, 0.25〉
R ₅	〈0.75, 0.10〉	〈0.60, 0.25〉	〈0.75, 0.10〉	〈0.75, 0.10〉	〈0.40, 0.50〉	〈0.60, 0.25〉

Table 15

Normalized collective IF decision matrix $\tilde{B}$

	A ₁	A ₂	A ₃	A ₄	A ₅	A ₆
R ₁	〈0.5702, 0.2981〉	〈0.7190, 0.1310〉	〈0.4768, 0.5232〉	〈0.5597, 0.4403〉	〈0.4470, 0.5530〉	〈0.5673, 0.3188〉
R ₂	〈1.0000, 0.0000〉	〈1.0000, 0.0000〉	〈1.0000, 0.0000〉	〈1.0000, 0.0000〉	〈0.6814, 0.3186〉	〈0.5183, 0.4817〉
R ₃	〈0.4342, 0.5658〉	〈0.5393, 0.3347〉	〈0.5772, 0.4228〉	〈0.4294, 0.5706〉	〈0.5457, 0.4543〉	〈0.5530, 0.4470〉
R ₄	〈0.6941, 0.1559〉	〈0.6661, 0.3339〉	〈0.4593, 0.5407〉	〈0.4739, 0.5261〉	〈0.4368, 0.5632〉	〈1.0000, 0.0000〉
R ₅	〈0.6385, 0.2259〉	〈0.6000, 0.2500〉	〈0.6458, 0.3542〉	〈0.5651, 0.4349〉	〈0.5813, 0.4187〉	〈0.5772, 0.4228〉

Step 7: The BAA matrix is obtained using the Equation (62), as shown in Table 17.

Table 16

Weighted normalized collective intuitionistic fuzzy decision matrix $\hat{\tilde{B}}$

	A ₁	A ₂	A ₃	A ₄	A ₅	A ₆
R ₁	〈0.1425, 0.1661〉	〈0.2693, 0.1050〉	〈0.0858, 0.9142〉	〈0.1163, 0.8837〉	〈0.0733, 0.9185〉	〈0.1204, 0.1626〉
R ₂	〈1.0000, 0.0000〉	〈1.0000, 0.0000〉	〈1.0000, 0.0000〉	〈1.0000, 0.0000〉	〈0.1367, 0.8633〉	〈0.1058, 0.8942〉
R ₃	〈0.0985, 0.9015〉	〈0.1743, 0.2264〉	〈0.1124, 0.8876〉	〈0.0811, 0.9189〉	〈0.0964, 0.9036〉	〈0.1160, 0.8840〉
R ₄	〈0.1939, 0.0980〉	〈0.2375, 0.7625〉	〈0.0816, 0.9184〉	〈0.0923, 0.9077〉	〈0.0711, 0.9289〉	〈1.0000, 0.0000〉
R ₅	〈0.1690, 0.1358〉	〈0.2027, 0.1717〉	〈0.1339, 0.8661〉	〈0.1179, 0.8821〉	〈0.1058, 0.8942〉	〈0.1235, 0.8765〉

Table 17

The BAA matrix $G$

$g_{1}$	$g_{2}$	$g_{3}$	$g_{4}$	$g_{5}$	$g_{6}$
〈0.5771, 0.4229〉	〈0.6712, 0.3288〉	〈0.1602, 0.8398〉	〈0.1593, 0.8407〉	〈0.0938, 0.9062〉	〈0.2634, 0.7366〉

Step 8: The divergence between alternatives and the BAA is calculated based on Equation (63), and the obtained results are summarized in Table 18.

Table 18

The divergence between alternatives and the BAA matrix $G$

	A ₁	A ₂	A ₃	A ₄	A ₅	A ₆
R ₁	–0.4352	–0.3788	–0.0056	–0.0017	–0.0006	0.4615
R ₂	0.2362	0.1784	0.5799	0.5810	0.0019	–0.0185
R ₃	–0.1267	–0.3743	–0.0021	–0.0064	0.0000	–0.0157
R ₄	–0.4483	–0.0865	–0.0064	–0.0044	–0.0007	0.4719
R ₅	–0.4351	–0.3830	–0.0006	–0.0015	0.00002	–0.0139

Step 9: Using Equation (64), we calculate the total divergence measure of the alternatives from BAA. Table 19 presents the results.

Table 19

Total divergence measure of alternatives from the BAA matrix $G$

$^{1} D_{α}^{1}$	$^{1} D_{α}^{2}$	$^{1} D_{α}^{3}$	$^{1} D_{α}^{4}$	$^{1} D_{α}^{5}$
–0.3604	1.5589	–0.5252	–0.0744	–0.8339

Step 10: According to the computed values of $^{1} D_{α}^{i}$ ’s as summarized in Table 19, we can rank all the candidates. The ranking of the candidates is R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅, and the best candidate among the possible five candidates is R₂.

Case (ii) When the information about the attribute weights is partially known.

Let the partially known information about the attribute weights is given as:

$T = {\begin{matrix} {- λ}_{1} ⩾ 0.10, 0.10 ⩽ {- λ}_{2} ⩽ 0.20, 0.25 ⩽ {- λ}_{3} ⩽ 0.30, \\ {- λ}_{4} ⩾ 0.20, 0.20 ⩽ {- λ}_{5} ⩽ 0.30, 0.10 ⩽ {- λ}_{6} ⩽ 0.15 . \end{matrix}$ Then the following optimization model is constructed to calculate the weighting vector − λ = ( − λ₁, − λ₂, …, − λ_n) ^T (here, we have taken α = 0.5): $Max . \hat{F} = (\begin{matrix} - 3.7882 {- λ}_{1} - {- λ}_{1} log {- λ}_{1} - 3.3446 {- λ}_{2} - {- λ}_{2} log {- λ}_{2} - 4.1821 {- λ}_{3} - {- λ}_{3} log {- λ}_{3} \\ - 4.0611 {- λ}_{4} - {- λ}_{4} log {- λ}_{4} - 4.2909 {- λ}_{5} - {- λ}_{5} log {- λ}_{5} - 4.0384 {- λ}_{6} - {- λ}_{6} log {- λ}_{6} \end{matrix}),$ $s . t . {\begin{matrix} {- λ}_{1} ⩾ 0.10, \\ 0.10 ⩽ {- λ}_{2} ⩽ 0.20, \\ 0.25 ⩽ {- λ}_{3} ⩽ 0.35, \\ {- λ}_{4} ⩾ 0.20, \\ 0.20 ⩽ {- λ}_{5} ⩽ 0.30, \\ 0.10 ⩽ {- λ}_{6} ⩽ 0.15, \\ {- λ}_{1} + {- λ}_{2} + {- λ}_{3} + {- λ}_{4} + {- λ}_{5} + {- λ}_{6} = 1, \\ {- λ}_{j} ⩾ 0, j = 1, 2, 3, 4, 5, 6 . \end{matrix}$ After solving it, we get the optimal weight vector associated with attributes given as: ${- λ}^{2} = {(0.10600.14400.25000.20000.20000.1000)}^{T} .$ By repeating steps 5 to 8 with − λ², the obtained divergence measure values between alternatives and the BAA matrix $G$ are summarized in Tables 20.

Table 20

The divergence measure values between alternatives and the BAA matrix $G$

	A ₁	A ₂	A ₃	A ₄	A ₅	A ₆
R ₁	–0.5440	–0.4962	–0.0066	–0.0016	–0.0009	0.5506
R ₂	0.2623	0.2013	0.4862	0.5381	0.0029	–0.0183
R ₃	–0.1446	–0.4895	–0.0019	–0.0070	0.0000	–0.0160
R ₄	–0.5571	–0.1188	–0.0077	–0.0047	–0.0011	0.5254
R ₅	–0.5448	–0.4984	–0.0002	–0.0014	0.00003	–0.0145

Therefore, the total divergence measure $^{1} D_{α}^{i}$ of each alternative from BAA as $\begin{matrix} ^{1} D_{α}^{1} = - 0.4987,^{1} D_{α}^{2} = 1.4725,^{1} D_{α}^{3} = - 0.6590, \\ ^{1} D_{α}^{4} = - 0.1640,^{1} D_{α}^{5} = - 1.0159 . \end{matrix}$ According to $^{1} D_{α}^{2} ≻^{1} D_{α}^{4} ≻^{1} D_{α}^{1} ≻^{1} D_{α}^{3} ≻^{1} D_{α}^{5}$ , the ranking of the candidates is R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅. Hence R₂ is still the best candidate for this post.

Further, to examine the impact of the parameter α on the obtained ranking order of the candidates, we consider different values of the parameter α in the proposed MAGDM approach. The obtained consequences are presented in Table 21. The ranking orders are the same R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅ for all considered values of the parameter α. To select a suitable value of the parameter α, a decision-maker can decide based on his/her past experience and the present evaluation information.

Table 21

The ranking orders based on $^{1} D_{α} (R | S)$ and $^{1} E_{α} (R)$ with different values of α

α	− λ¹	$^{1} D_{α}^{1}$	$^{1} D_{α}^{2}$	$^{1} D_{α}^{3}$	$^{1} D_{α}^{4}$	$^{1} D_{α}^{5}$	Ranking orders
Case (i)
0.2	(0.17730.23680.14570.15950.12410.1566) ^T	-0.3427	1.3698	-0.4124	-0.0332	-0.7513	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.5	(0.1820, 0.2472, 0.1385, 0.1507, 0.1285, 0.1531) ^T	-0.3604	1.5589	-0.5252	0.0744	-0.8339	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.8	(0.18610.25540.13420.14530.12780.1513) ^T	-0.3701	1.6920	-0.6041	-0.1019	-0.8909	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
→1	(0.18050.24820.17460.12780.12310.1458) ^T	-0.3902	1.7288	-0.6570	-0.0381	-0.9424	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
Case (ii)
0.2	(0.10700.14300.25000.20000.20000.1000) ^T	-0.5468	1.5814	-0.7634	-0.2228	-1.1627	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.5	(0.10600.14400.25000.20000.20000.1000) ^T	-0.4987	1.4725	-0.6590	-0.1640	-1.0590	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.8	(0.10540.14460.25000.20000.20000.1000) ^T	-0.5468	1.5814	-0.7634	-0.2228	-1.1627	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
→1	(0.10520.14480.25000.20000.20000.1000) ^T	-0.5710	1.6362	-0.8140	-0.1152	-1.2140	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅

Further, if we use another divergence and entropy measures of order α in the proposed MAGDM approach and solve the above-considered personnel selection problem, the results for them are summarized in Tables 22 –24. The obtained results clearly indicate that R₂ is the best candidate for this post.

Table 22

The ranking orders based on $^{2} D_{α} (R | S)$ and $^{2} E_{α} (R)$ with different values of α

α	− λ¹	$^{2} D_{α}^{1}$	$^{2} D_{α}^{2}$	$^{2} D_{α}^{3}$	$^{2} D_{α}^{4}$	$^{2} D_{α}^{5}$	Ranking orders
Case (i)
0.2	(0.17720.23340.14720.16030.12440.1575) ^T	-0.3037	1.2250	-0.3558	-0.0237	-0.6677	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.5	(0.18180.24430.14060.15250.12640.1543) ^T	-0.3315	1.4440	-0.4731	-0.0620	-0.7673	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.8	(0.18440.25530.13560.14650.12600.1521) ^T	-0.3584	1.6409	-0.5778	-0.0966	-0.8614	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
→1	(0.18050.24820.17460.12780.12310.1458) ^T	-0.3902	1.7288	-0.6570	-0.0381	-0.9424	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
Case (ii)
0.2	(0.10790.14210.25000.20000.20000.1000) ^T	-0.3930	1.1710	-0.4487	-0.0682	-0.8298	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.5	(0.10670.14330.25000.20000.20000.1000) ^T	-0.4635	1.3557	-0.6017	-0.1445	-0.9887	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.8	(0.10490.14510.25000.20000.20000.1000) ^T	-0.5307	1.5295	-0.7343	-0.2136	-1.1313	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
→1	(0.10520.14480.25000.20000.20000.1000) ^T	-0.5710	1.6362	-0.8140	-0.1152	-1.2140	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅

Table 23

The ranking orders based on $^{3} D_{α} (R | S)$ and $^{3} E_{α} (R)$ with different values of α

α	− λ¹	$^{3} D_{α}^{1}$	$^{3} D_{α}^{2}$	$^{3} D_{α}^{3}$	$^{3} D_{α}^{4}$	$^{3} D_{α}^{5}$	Ranking orders
Case (i)
0.2	(0.17730.23980.14450.15870.12390.1558) ^T	-0.3732	1.4847	-0.4591	-0.0430	-0.8160	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.5	(0.18290.25110.13730.14970.12640.1526) ^T	-0.3843	1.6564	-0.5710	-0.0855	-0.8893	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.8	(0.18660.25720.13340.14470.12710.1510) ^T	-0.3817	1.7384	-0.6284	-0.1082	-0.9184	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
→1	(0.18660.25720.13340.14470.12710.1510) ^T	-0.3902	1.7288	-0.6570	-0.0381	-0.9424	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
Case (ii)
0.2	(0.10630.14370.25000.20000.20000.1000) ^T	-0.4660	1.4412	-0.5569	-0.0958	-0.9762	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.5	(0.10540.14460.25000.20000.20000.1000) ^T	-0.5286	1.5738	-0.7100	-0.1824	-1.1185	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.8	(0.10510.14490.25000.20000.20000.1000) ^T	-0.5620	1.6307	-0.7902	-0.2332	-1.1926	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
→1	(0.10520.14480.25000.20000.20000.1000) ^T	-0.5710	1.6362	-0.8140	-0.1152	-1.2140	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅

Table 24

The ranking orders based on $^{4} D_{α} (R | S)$ and $^{4} E_{α} (R)$ with different values of α

α	− λ¹	$^{4} D_{α}^{1}$	$^{4} D_{α}^{2}$	$^{4} D_{α}^{3}$	$^{4} D_{α}^{4}$	$^{4} D_{α}^{5}$	Ranking orders
Case (i)
0.2	(0.17690.25190.13920.15520.12420.1526) ^T	-0.4847	1.9204	-0.6429	-0.0844	-1.0523	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.5	(0.18310.25150.13700.14950.12640.1524) ^T	-0.3883	1.6725	-0.5779	-0.0866	-0.8984	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.8	(0.18610.25580.13390.14510.12780.1512) ^T	-0.3735	1.7046	-0.6107	-0.1035	-0.8987	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
→1	(0.18050.24820.17460.12780.12310.1458) ^T	-0.3902	1.7288	-0.6570	-0.0381	-0.9424	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
Case (ii)
0.2	(0.10310.14690.25000.20000.20000.1000) ^T	-0.5468	1.5814	-0.7634	-0.2228	-1.1627	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.5	(0.10530.14470.25000.20000.20000.1000) ^T	-0.5812	1.8879	-0.7479	-0.1560	-1.2076	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
0.8	(0.10530.14470.25000.20000.20000.1000) ^T	-0.5507	1.5946	-0.7704	-0.2256	-1.1707	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅
→1	(0.10520.14480.25000.20000.20000.1000) ^T	-0.5710	1.6362	-0.8140	-0.1152	-1.2140	R₂ ≻ R₄ ≻ R₁ ≻ R₃ ≻ R₅

4.3 Comparative analysis

To compare the performance of the proposed MAGDM approach with some existing decision-making methods under an intuitionistic fuzzy environment, we have been used six methods, as developed Boran et al. [58], Xu [45], Wang and Liu [62], Huang [63], He et al. [46] and Liu and Wang [64] to solve above-considered personnel selection problem by taking Table 15 as the decision matrix and − λ = (0.1820, 0.2472, 0.1385, 0.1507, 0.1285, 0.1531) ^T as the attribute weighting vector. The obtained results corresponding to these methods are recorded in Table 25.

The results presented in Tables 25 indicate that the ranking order of the candidates obtained by using the Boran et al. [58] approach is similar as obtained by our developed approach, and the best candidate by both methods is R₂. On the other hand, when we apply others methods as given by Xu [45], Wang and Liu [62], Huang [63], He et al. [46] and Liu and Wang [64], then the ranking order of the candidates is slightly different from our obtained ranking order. In these methods, we cannot select the best alternative because of the score values of the candidates R₂ and R₄ are equal. It is also worth mentioning that all these methods have no flexible parameters for accommodating different situations. Therefore, the proposed intuitionistic fuzzy MABAC method provides an efficient and flexible approach for solving MAGDM problems with intuitionistic fuzzy information.

Table 25
The ranking order of the candidates based on different methods

Relative Closeness Coefficient Ranking orders

$\overset{⌢}{C} (R_{1})$ $\overset{⌢}{C} (R_{2})$ $\overset{⌢}{C} (R_{3})$ $\overset{⌢}{C} (R_{4})$ $\overset{⌢}{C} (R_{5})$

Boran et al. [58] 0.2501 0.8516 0.1098 0.3685 0.3397 R₂ ≻ R₄ ≻ R₅ ≻ R₁ ≻ R₃

Score Values

$\hat{S} (R_{1})$ $\hat{S} (R_{2})$ $\hat{S} (R_{3})$ $\hat{S} (R_{4})$ $\hat{S} (R_{5})$

Xu [45] (IFWA operator) 0.6411 1.0000 0.5334 1.0000 0.6393 R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃

Wang and Liu [62] (IFEWA operator) 0.6360 1.0000 0.5316 1.0000 0.6380 R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃

Huang [63] (γ = 3)(IFHWA operator) 0.6338 1.0000 0.5307 1.0000 0.6374 R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃

He et al. [46] (IFIWA operator) 0.5874 1.0000 0.5150 1.0000 0.6028 R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃

Liu and Wang [64] (IFIEWA operator) 0.5845 1.0000 0.5140 1.0000 0.6025 R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃

	Relative Closeness Coefficient	Ranking orders
Boran et al. [58]	0.2501	0.8516	0.1098	0.3685	0.3397	R₂ ≻ R₄ ≻ R₅ ≻ R₁ ≻ R₃
	Score Values
	$\hat{S} (R_{1})$	$\hat{S} (R_{2})$	$\hat{S} (R_{3})$	$\hat{S} (R_{4})$	$\hat{S} (R_{5})$
Xu [45] (IFWA operator)	0.6411	1.0000	0.5334	1.0000	0.6393	R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃
Wang and Liu [62] (IFEWA operator)	0.6360	1.0000	0.5316	1.0000	0.6380	R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃
Huang [63] (γ = 3)(IFHWA operator)	0.6338	1.0000	0.5307	1.0000	0.6374	R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃
He et al. [46] (IFIWA operator)	0.5874	1.0000	0.5150	1.0000	0.6028	R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃
Liu and Wang [64] (IFIEWA operator)	0.5845	1.0000	0.5140	1.0000	0.6025	R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃

IFWA: Intuitionistic fuzzy weighted average, IFEWA: Intuitionistic fuzzy Einstein weighted average, IFHWA: Intuitionistic fuzzy Hamacher weighted average, IFIWA: Intuitionistic fuzzy interactive weighted average, IFIEWA: Intuitionistic fuzzy interactive Einstein weighted average

5 Conclusions

In this paper, we have presented a valuable study on parametric information measures under the intuitionistic fuzzy environment. First, we have defined four new order-α divergence measures between two IFSs and proved several basic and important mathematical properties connected with them. The proposed divergence measures provide a generalized approach to quantify information discrimination between IFSs. With the help of some numerical examples, it has been shown that the order-α intuitionistic fuzzy divergence measures have some advantages over the existing intuitionistic fuzzy divergence measures. Further, to measure the degree of fuzziness associated with an IFS, the paper has proposed four new order-α intuitionistic fuzzy entropy measures. The limiting and particular cases of the proposed order-α entropy and divergence measures have been examined in detail. Besides, we have developed the intuitionistic fuzzy MABAC method to solve MAGDM problems in which the information about the attribute weights is completely unknown or partially known. By utilizing the proposed information measures, we have also formulated a nonlinear optimization model to calculate the attribute weighting vector. Finally, a numerical example concerning the personnel selection has been given for illustrating the decision-making steps. A comparative study with some existing methods has been conducted to demonstrate the flexibility, reasonability, and superiority of the developed method.

In future work, we shall study these information measures under different information environments, including interval-valued intuitionistic fuzzy, Pythagorean fuzzy, cubic intuitionistic fuzzy, complex intuitionistic fuzzy, and q-rung orthopair fuzzy. We shall also explore the applications in some other fields such as supply chain management, green supplier selection, and the university faculty recruitment.

Conflict of interest

The author declares that he has no conflict of interest.

Footnotes

Acknowledgments

The work is supported by the Chilean Government (Conicyt) through the Fondecyt Postdoctoral Program (Project Number 3170556).

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Burillo and Bustince [35]	$^{BB} E (R) = \frac{1}{n} \sum_{i = 1}^{n} η_{R} (m_{i}) .$
Zeng and Li [47]	$^{ZL} E (R) = 1 - \frac{1}{n} \sum_{i = 1}^{n} \| ξ_{R} (m_{i}) - ζ_{R} (m_{i}) \| .$
Szmidt and Kacrzyk [36]	$^{SK} E (R) = \frac{1}{n} \sum_{i = 1}^{n} (\frac{ξ_{R} (m_{i}) \land ζ_{R} (m_{i}) + η_{R} (m_{i})}{ξ_{R} (m_{i}) \lor ζ_{R} (m_{i}) + η_{R} (m_{i})}) .$
Vlachos and Sergiadis [28]	$^{VS} E (R) = - \frac{1}{n} \sum_{j = 1}^{n} [ξ_{R}^{j} {log}_{2} ξ_{R}^{j} + ζ_{R}^{j} {log}_{2} ζ_{R}^{j} - (1 - η_{R}^{j}) {log}_{2} (1 - η_{R}^{j}) - η_{R}^{j}]$
Zhang and Jiang [38]	$^{ZJ} E (R) = \frac{1}{n} \sum_{i = 1}^{n} (\frac{ξ_{R} (m_{i}) \land ζ_{R} (m_{i})}{ξ_{R} (m_{i}) \lor ζ_{R} (m_{i})})$
Hung and Yang [48]	$^{HY} E_{hc}^{2} (R) = \frac{1}{n} \sum_{j = 1}^{n} [1 - ({(ξ_{R}^{j})}^{2} + {(ζ_{R}^{j})}^{2} + {(η_{R}^{j})}^{2})]$
	$^{HY} E_{r}^{1 / 2} (R) = \frac{2}{n} \sum_{j = 1}^{n} {log}_{2} ({(ξ_{R}^{j})}^{\frac{1}{2}} + {(ζ_{R}^{j})}^{\frac{1}{2}} + {(η_{R}^{j})}^{\frac{1}{2}})$
Ye [49]	$^{Y} E (R) = \frac{1}{n} \sum_{j = 1}^{n} [{\sqrt{2} cos π (\frac{ξ_{R}^{j} - ζ_{R}^{j}}{4}) - 1} \times \frac{1}{\sqrt{2} - 1}]$
Wei et al. [50]	$^{WGG} E (R) = \frac{1}{n} \sum_{j = 1}^{n} cos π (\frac{ξ_{R}^{j} - ζ_{R}^{j}}{4 (1 + η_{R}^{j})}) .$
Verma and Sharma [39]	$^{VS 1} E (R) = \frac{1}{n (\sqrt{e} - 1)} \sum_{j = 1}^{n} ((\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2}) e^{(\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2})} + (\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2}) e^{(\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2})} - 1) .$
Verma and Sharma [43]	$^{VS 2} E (R) = \frac{1}{n (1 - 2^{\frac{1 - R}{R}})} \sum_{j = 1}^{n} (1 - {({(\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2})}^{R} + {(\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2})}^{R})}^{\frac{1}{R}}), R > 0 and R \neq 1 .$
Wang and Wang [51]	$^{WW} E (R) = \frac{1}{n} \sum_{j = 1}^{n} cot π (\frac{π}{4} + \frac{\| ξ_{R}^{j} - ζ_{R}^{j} \|}{4 (1 + η_{R}^{j})})$
Liu and Ren [41]	$^{LR} E (R) = \frac{1}{n} \sum_{j = 1}^{n} cot π (\frac{π}{4} + \frac{\| {(ξ_{R}^{j})}^{2} - {(ζ_{R}^{j})}^{2} \|}{4}) .$
Mishra et al. [33]	$^{MJH} E (R) = \frac{1}{n \sqrt{e} (\sqrt{e} - 1)} \sum_{j = 1}^{n} (e - (\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2}) e^{(\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2})} - (\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2}) e^{(\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2})}) .$
Mishra and Rani [31]	$^{MR} E_{1} (R) = \frac{1}{n (1 - e^{- 1 / 2})} \sum_{j = 1}^{n} [{1 - e^{(\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2})}} I_{[ξ_{R}^{j} ⩾ ζ_{R}^{j}]} + {1 - e^{(\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2})}} I_{[ξ_{R}^{j} < ζ_{R}^{j}]}] .$
	$^{MR} E_{2} (R) \frac{1}{n (\sqrt{e} - 1)} \sum_{j = 1}^{n} ((\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2}) e^{(\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2})} + (\frac{ζ_{R}^{j} + 1 - ξ_{R}^{j}}{2}) e^{(\frac{ξ_{R}^{j} + 1 - ζ_{R}^{j}}{2})} - 1)$
	$^{MR} E_{3} (R) = 1 - \frac{1}{n} \sum_{j = 1}^{n} [{ξ_{R}^{j} - ζ_{R}^{j}} I_{[ξ_{R}^{j} ⩾ ζ_{R}^{j}]} + {ζ_{R}^{j} - ξ_{R}^{j}} I_{[ξ_{R}^{j} < ζ_{R}^{j}]}] .$

	Relative Closeness Coefficient					Ranking orders
	$\overset{⌢}{C} (R_{1})$	$\overset{⌢}{C} (R_{2})$	$\overset{⌢}{C} (R_{3})$	$\overset{⌢}{C} (R_{4})$	$\overset{⌢}{C} (R_{5})$
Boran et al. [58]	0.2501	0.8516	0.1098	0.3685	0.3397	R₂ ≻ R₄ ≻ R₅ ≻ R₁ ≻ R₃
	Score Values
	$\hat{S} (R_{1})$	$\hat{S} (R_{2})$	$\hat{S} (R_{3})$	$\hat{S} (R_{4})$	$\hat{S} (R_{5})$
Xu [45] (IFWA operator)	0.6411	1.0000	0.5334	1.0000	0.6393	R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃
Wang and Liu [62] (IFEWA operator)	0.6360	1.0000	0.5316	1.0000	0.6380	R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃
Huang [63] (γ = 3)(IFHWA operator)	0.6338	1.0000	0.5307	1.0000	0.6374	R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃
He et al. [46] (IFIWA operator)	0.5874	1.0000	0.5150	1.0000	0.6028	R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃
Liu and Wang [64] (IFIEWA operator)	0.5845	1.0000	0.5140	1.0000	0.6025	R₂ = R₄ ≻ R₅ ≻ R₁ ≻ R₃

On intuitionistic fuzzy order- α divergence and entropy measures with MABAC method for multiple attribute group decision-making

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Intuitionistic fuzzy Set

3.1 Order-α divergence measures for IFSs

3.2.2 Comparative study with existing IF entropy measures

4 A new multiple-attribute group decision-making approach

4.1 Intuitionistic fuzzy MABAC method

Table 4 Linguistic terms for the rating of the experts LTs Very Important Important Medium Unimportant Very unimportant (VI) (I) (M) (UI) (VUI) IFNs 〈0.80, 0.10〉 〈0.55, 0.25〉 〈0.50, 0.50〉 〈0.30, 0.50〉 〈0.20, 0.70〉

Conflict of interest

Footnotes

Acknowledgments

References

Table 4
Linguistic terms for the rating of the experts

LTs Very Important Important Medium Unimportant Very unimportant

(VI) (I) (M) (UI) (VUI)

IFNs 〈0.80, 0.10〉〈0.55, 0.25〉〈0.50, 0.50〉〈0.30, 0.50〉〈0.20, 0.70〉