# A Complementary Dual of Single-Valued Neutrosophic Entropy with Application to MAGDM

^{*}

## Abstract

**:**

## 1. Introduction

- We propose an entropy-like measure in the neutrosophic settings and termed it a single-valued neutrosophic knowledge measure.
- We also discuss certain properties of the neutrosophic knowledge measure and establish its connection with the single-valued neutrosophic similarity and dissimilarity measure.
- An algorithm of MAGDM is proposed and implemented with the help of a numerical example.
- Comparative analysis to check the effectiveness of the proposed knowledge measure has also been presented.

## 2. Preliminaries

**Definition**

**1**.

**([1]).**Let$Y=\left\{{y}_{1},{y}_{2},{y}_{3}\dots \dots {y}_{n}\right\}$be the universal set then a fuzzy set in Y is defined as

**Definition**

**2**.

**([2]).**Intuitionistic fuzzy set B on a universal set Y is defined

**Definition**

**3**.

**([3,4]).**A single-valued neutrosophic set on a universal set B is defined as

**Remark**

**1**:

**Operations on single-valued neutrosophic values (SVNVs)**[4]:

**Definition**

**4**.

**Definition**

**5**.

**Definition**

**6**.

**Definition**

**7**.

**Definition**

**8**.

**([12]).**The similarity measure S between two neutrosophic sets B and C is a function$S:B\times C\to \left[0,1\right]$which satisfies the given condition:

**NSM1:**0$\le $S (B, C)$\le $1;**NSM2:**S (B, C) = 1 if B = C;**NSM3:**S (B, C) = S (C, B);**NSM4:**S (A, C)$\le $S (A, B); S (A, C)$\le $S (B, C), if A$\subseteq $B$\subseteq C$.

**Definition**

**9**.

**([6]).**An entropy E on a single-valued neutrosophic element,$\psi =\left({\psi}_{1},{\psi}_{2,}{\psi}_{3}\right)$is a function$E:N\to \left[0,1\right]$which satisfies the following condition:

**NSE1:**E_{NS}($\psi $) = 0 if$\psi $is a crisp set i.e.,$\psi =\left(1,0,0\right)or\left(0,0,1\right);$**NSE2:**E_{NS}($\psi $)= 1 if (${\psi}_{1},{\psi}_{2},{\psi}_{3}$) = (0.5, 0.5, 0.5);**NSE3:**E_{NS}($\theta $)$\ge $E_{NS}($\psi )$if$\theta $is more uncertain;**NSE4:**E ($\psi $) = E (${\psi}^{C}),$where${\psi}^{C}=\left(1-{\psi}_{1},1-{\psi}_{2},1-{\psi}_{3}\right)$.

**Remark**

**2**:

## 3. A Knowledge Measure on Single-Valued Neutrosophic Set

**NSK1:**K ($\psi $) = 1 if and only if ${\psi}_{t}=0$ or ${\psi}_{t}=1$; t = 1, 2, 3;**NSK2:**K ($\psi )$ = 0 if and only if (${\psi}_{1},{\psi}_{2},{\psi}_{3}$) = (0.5, 0.5, 0.5);**NSK3:**K ($\psi )$ = K (${\psi}^{c})$;**NSK4:**K ($\psi )$ $\ge $ K ($\theta )$ if $\theta =\left({\theta}_{1},{\theta}_{2},{\theta}_{3}\right)$ is more uncertain than $\psi $ i.e., ${\theta}_{t}\ge {\psi}_{t}$ when ${\theta}_{t}-{\theta}_{t}^{c}$ $\le $ 0; t = 1, 2, 3 or ${\theta}_{t}\le {\psi}_{t}$ when ${\theta}_{t}-{\theta}_{t}^{c}\ge 0$.

**Theorem**

**1**.

**Proof**.

**NSK1:**We have $\psi =\left({\psi}_{1},{\psi}_{2},{\psi}_{3}\right)$. Let us suppose that ${\psi}_{t}$= 0 or ${\psi}_{t}$ = 1 then ${\psi}_{t}-{\psi}_{t}^{c}$ = 1 or ${\psi}_{t}-{\psi}_{t}^{c}$ = −1 for t = 1, 2, 3.Using (1), we have K ($\psi $) = 1.On the other hand, we assume that K ($\psi )$ = 1then, ${\psi}_{t}-{\psi}_{t}^{c}$ = $({\psi}_{t}-\left(1-{\psi}_{t}\right)$ = $2{\psi}_{t}$− 1.$\Rightarrow $ ${\psi}_{t}-{\psi}_{t}^{c}$ $\u03f5$ [−1, 1].Therefore, every term in the summation of (1) is positive. As K ($\psi $) =1, then every term in this summation should be equal to one, i.e.,$$-\frac{1}{3}\left[\mathrm{cos}\left(\frac{\left({\psi}_{t}-{\psi}_{t}{}^{c}\right)}{2}\right)\pi -1\right]=1.$$Hence, K ($\psi )$ = 1 if and only if ${\psi}_{t}$ = 0 or ${\psi}_{t}$ = 1 for t = 1, 2, 3.**NSK2:**If (${\psi}_{1},{\psi}_{2},{\psi}_{3}$) = (0.5, 0.5, 0.5), we have ${\psi}_{t}-{\psi}_{t}^{c}$ = 0. Then, from Equation (1), K ($\psi )$ = 0. On the other hand, from the above analysis, we have ${\psi}_{t}-{\psi}_{t}^{c}$ $\u03f5$ [−1, 1], it is obvious that 0 $\le $ K ($\psi $) $\le $ 1.If K ($\psi $) = 0 then ${\psi}_{t}-{\psi}_{t}^{c}$ = 0 for t = 1, 2, 3.It follows that ${\psi}_{t}$ = 0.5, t = 1, 2, 3 i.e., (${\psi}_{1},{\psi}_{2},{\psi}_{3}$) = (0.5, 0.5, 0.5).**NSK3:**Since ${\psi}^{c}$ = (1 − ${\psi}_{1},1-{\psi}_{2},1-{\psi}_{3}$) then ${\left({\psi}^{c}\right)}^{c}$ = $\psi .$ Thus,$$\begin{array}{l}\mathrm{K}\left({\psi}^{c}\right)=-\frac{1}{3}\left[\mathrm{cos}\left(\frac{\left({\psi}_{t}^{c}-{\left({\psi}_{t}{}^{c}\right)}^{c}\right)}{2}\right)\pi -1\right]\\ =-\frac{1}{3}\left[\mathrm{cos}\left(\frac{\left({\psi}_{t}^{c}-{\psi}_{t}\right)}{2}\right)\pi -1\right]\\ =-\frac{1}{3}\left[\mathrm{cos}\left(\frac{\left({\psi}_{t}-{\psi}_{t}^{c}\right)}{2}\right)\pi -1\right].\end{array}$$Therefore, K (${\psi}^{c}$) = K ($\psi $).**NSK4:**Assume that $\theta $ is more uncertain than $\psi $. Therefore, in view of Remark 2, we have two cases: ${\psi}_{t}$ $\le $ ${\theta}_{t}$ for ${\theta}_{t}-{\theta}_{t}^{c}\le $ 0 and ${\psi}_{t}$ $\ge $ ${\theta}_{t}$, when ${\theta}_{t}-{\theta}_{t}^{c}\ge $0; $t=1,2,3$.

**NSK4**. □

**Theorem**

**2**.

**Proof**.

## 4. Single-Valued Neutrosophic Similarity Measure and Distance Measure

#### 4.1. Single-Valued Neutrosophic Similarity Measure

**NS1:**S ($\psi ,\theta $) = 0 if and only if ${\psi}_{t}$ $-$ ${\theta}_{t}$ = 1 or ${\psi}_{t}$ $-$ ${\theta}_{t}$. = −1, t = 1, 2, 3;**NS2:**S ($\psi ,\theta $) = 1 if and only if (${\psi}_{1},{\psi}_{2},{\psi}_{3}$) = (${\theta}_{1},{\theta}_{2},{\theta}_{3}$); t = 1, 2, 3;**NS3:**S ($\psi ,\theta $) = S ($\theta ,\psi $);**NS4:**S ($\psi ,\phi $) $\le $ S ($\psi ,\theta $), S ($\psi ,\phi $) $\le $S ($\theta ,\phi $) if ${\psi}_{t}$ $\le $ ${\theta}_{t}\le $ ${\phi}_{t}$ or ${\psi}_{t}$ $\ge $ ${\theta}_{t}\ge $ ${\phi}_{t}$, t = 1, 2, 3.

**Theorem**

**3**.

**Proof**.

- (a)
- K ($\psi $) = 1 if and only if 1 − S ($\psi ,{\psi}^{c}$) = 1 which implies S ($\psi ,{\psi}^{c}$) = 0 if and only if ${\psi}_{t},-{\psi}_{t}{}^{c}$ = 1 or ${\psi}_{t}-{\psi}_{t}{}^{c}=-1$i.e., K ($\psi $) = 1 if and only if ${\psi}_{t}-{\psi}_{t}{}^{c}$ = 1 or ${\psi}_{t},-{\psi}_{t}{}^{c}=-1$.
- (b)
- K (($\psi $) = 0 which implies that 1 − S ($\psi ,{\psi}^{c}$) = 0 if and only if S ($\psi ,{\psi}^{c}$) = 1 and S ($\psi ,{\psi}^{c}$) = 1 if and only ${\psi}_{t}$ $={\psi}_{t}{}^{c}$ i.e., K ($\psi $) = 0 if and only if (${\psi}_{1},{\psi}_{2},{\psi}_{3}$) =$({\psi}_{1}^{c}$,${\psi}_{2}^{c},{\psi}_{3}^{c})$ i.e., K (($\psi $) = 0 if and only if (${\psi}_{1},{\psi}_{2},{\psi}_{3})=\left(0.5,0.5,0.5\right)$.
- (c)
- K (${\psi}^{c}$) = 1 − S (${\psi}^{c}$,${\left({\psi}^{c}\right)}^{c}$) which implies K (${\psi}^{c}$) = 1 − S (${\psi}^{c}$, $\psi $) = K ($\psi $).
- (d)
- Let $\psi =\left({\psi}_{1},{\psi}_{2},{\psi}_{3}\right)and\theta =\left({\theta}_{1},{\theta}_{2},{\theta}_{3}\right)$ be two SVNEs. Suppose that ${\psi}_{t}\le $ ${\theta}_{t}$when ${\theta}_{t}$− ${\theta}_{t}{}^{c}\le $ 0 then ${\theta}_{t}\le $ 1 − ${\theta}_{t}$ $\le $ 0 i.e., ${\theta}_{t}\le $ 1 − ${\theta}_{t}$ and we have$${\psi}_{t}\le {\theta}_{t}\le 1-{\theta}_{t}\le 1-{\psi}_{t}$$

_{4}), it is deduced that

#### 4.2. Single-Valued Neutrosophic Distance Measure

**NSD1:**d ($\psi ,\theta $) = d ($\theta $, $\psi $);

**NSD2:**d ($\psi ,\theta $) = 1 if and only if ${\psi}_{t}$ = 0 or ${\psi}_{t}$= 1 for t = 1, 2, 3;

**NSD3:**d ($\psi ,\theta $) = 0 if and only if $\left({\psi}_{1},{\psi}_{2},{\psi}_{3}\right)=\left({\theta}_{1},{\theta}_{2},{\theta}_{3}\right)$;

**NSD4:**d ($\psi ,\phi $) $\ge $ d ($\psi ,\theta $); d$\left(\psi ,\phi \right)\ge $ $d\left(\theta ,\phi \right)$ if ${\psi}_{t}\le $ ${\theta}_{t}\le $ ${\phi}_{t}$ or ${\psi}_{t}\ge $ ${\theta}_{t}\ge $ ${\phi}_{t}.$

**Theorem**

**4**.

**Proof**.

**NSD1:**As we know K(($\psi $) = 1 which implies d ($\psi ,{\psi}^{c}$) = 1 if and only if ${\psi}_{t}$ − ${\psi}_{t}{}^{c}$ = 1 or ${\psi}_{t}$. − ${\psi}_{t}{}^{c}$ = −1 i.e., ${\psi}_{t}-\left(1-{\psi}_{t}\right)$ = 1 or ${\psi}_{t}-\left(1-{\psi}_{t}\right)$ = −1 and this equation holds if and only ${\psi}_{t}$ = 0 or ${\psi}_{t}$ = 1.

**NSD 2:**K ($\psi $) = 0 which implies that d ($\psi ,{\psi}^{c}$) = 0 if and only if (${\psi}_{1},{\psi}_{2},{\psi}_{3}$) = (${\psi}_{1}{}^{c},{\psi}_{2}{}^{c},{\psi}_{3}{}^{c}$) or (${\psi}_{1},{\psi}_{2},{\psi}_{3}$) = ($1-{\psi}_{1},1-{\psi}_{2},1-{\psi}_{3}$) or ${\psi}_{t}$ = 1 − ${\psi}_{t}$ which implies ${\psi}_{t}$ = 0.5 for t = 1, 2, 3 i.e., (${\psi}_{1},{\psi}_{2},{\psi}_{3}$) = (0.5, 0.5, 0.5).

**NSD 3:**K (${\psi}^{c}$) = d (${\psi}^{c},{\psi}^{c}{)}^{c}$) which implies d (${\psi}^{c},\psi ,$) = K (${\psi}^{c}$) = K ($\psi $).

**NSD 4:**Let $\psi =\left({\psi}_{1},{\psi}_{2},{\psi}_{3}\right)and\theta =\left({\theta}_{1},{\theta}_{2},{\theta}_{3}\right)$ be two SVNEs. Suppose that ${\psi}_{t}\le $ ${\theta}_{t}$when ${\theta}_{t}$− ${\theta}_{t}{}^{c}\le $ 0 then ${\theta}_{t}-\left(1-{\theta}_{t}\right)\le $ 0 or ${\theta}_{t}$ $\le $ 1 − ${\theta}_{t}$ and we have, ${\psi}_{t}\le $ ${\theta}_{t}\le $. $1-{\theta}_{t}\le 1-{\psi}_{t}$ or ${\psi}_{t}\le $ ${\theta}_{t}\le {\theta}_{t}{}^{c}\le {\psi}_{t}{}^{c}$. Therefore, by NSD4

## 5. The MAGDM Problem

- (1)
- Decision matrices/table based on the neutrosophic knowledge-base of each decision maker.
- (2)
- A unified decision table aggregating the opinion of the decision makers with different knowledge and background. The procedure of opinion aggregation essentially needs to consider the level of expertise of each of the decision-makers. Therefore, some level of importance or weight should be assigned to each decision expert. The weight computed in this manner may be considered as the level of expertise. Now question arises how to compute this weight. In such a scenario, the objective weights of decision-makers can be obtained using some mathematical procedure connecting the information base of the decision-makers. The correlation coefficient among the neutrosophic knowledge base of experts gives the linear association or degree of agreement in the opinion of the experts. The normalized correlation efficiency computes the relative agreement level of each of the expert at normalized scale. Thus, normalized correlation efficiency can be perceived as the weight to the expertise of a decision-maker.
- (3)
- The weights of the decision-makers are utilized to obtain the collective decision matrix. The fusion of decision matrices also requires a suitable aggregation operator. In the present scenario, we use a single-valued neutrosophic weighted averaging operator.
- (4)
- Finally, the rating of alternatives can be obtained.

#### 5.1. Algorithm for MADM Problem in Neutrosophic Set

**Step 1:**There may be lots of attributes in a decision-making problem. Among them, only some of the attributes are appropriate and technically sound. Therefore, appropriate attributes are identified with the help of the domain experts.**Step 2:**Different alternatives may be good in different attributes. On the basis of their performance level, some ratings are given to each alternative with regard to each attribute by decision-makers and these scores are given in the form of linguistic terms. The alternatives with neutrosophic ratings of attributes are shown in the following decision matrix D:$$\begin{array}{ccc}{C}_{1}& {C}_{2}\dots & {C}_{n}\end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{c}{A}_{1}\\ {A}_{\begin{array}{c}2\\ .\\ .\\ .\\ .\end{array}}\\ {A}_{n}\end{array}\left[\begin{array}{ccc}\left({T}_{11},{I}_{11},{F}_{11}\right)& \left({T}_{12},{I}_{12},{F}_{12}\right)\dots & \left({T}_{1m},{I}_{1m},{F}_{1m}\right)\\ \left({T}_{21},{I}_{21},{F}_{21}\right)& \left({T}_{22},{I}_{22},{F}_{22}\right)\dots & \left({T}_{2m},{I}_{2m},{F}_{2m}\right)\\ \left({T}_{n1},{I}_{n1},{F}_{n1}\right)& \left({T}_{n2},{I}_{n2},{F}_{n2}\right)\dots & \left({T}_{nm},{I}_{nm},{F}_{nm}\right)\end{array}\right]$$In the given matrix ${d}_{ij}=\left({T}_{ij},{I}_{ij},{F}_{ij}\right)$ represent degree of truthness, indeterminacy, and falsity respectively.**Step 3:**In the next step, we construct decision matrices for k decision-maker with the help of linguistic term. It is necessary to find out the weights of each decision-makers because each decision-maker has their own significance. For this, the linguistic terms for each decision-maker are rated with the help of neutrosophic number $\left({T}_{ij},{I}_{ij},{F}_{ij}\right)$. Using the correlation coefficient formula given in Definition 5 between each decision-maker helps us to find the correlation measure and correlation coefficient between linguistic opinions of the decision-makers. The correlation efficiency of each decision-maker can be considered as a more realistic weight of the decision-maker as it computes the objective and subjective assessments.**Step 4:**Correlation efficiency concerning the intuitionistic fuzzy sets was computed by Singh et al. [10]. Analogously, we compute the correlation efficiency $\gamma $ and normalized correlation efficiency N$\gamma $ in the neutrosophic environment in Definition 6 and Definition 7. The normalized correlation efficiency of each decision maker was considered as the weights of the decision-maker.**Step 5:**With the help of the decision-maker’s assessment, construct the aggregated single-valued neutrosophic decision matrix was constructed with the help of SVNWA (single-valued neutrosophic weighted averaging) operator (Biswas et al. [11])$${\left({d}_{ij}\right)}_{m\times n}=\left\{\begin{array}{c}\left(1-{\displaystyle \prod _{k=1}^{n}}{\left(1-{T}_{ij}{}^{p}\right)}^{\lambda k}\right),\\ {\displaystyle \prod _{k=1}^{n}}{\left({I}_{ij}{}^{p}\right)}^{\lambda k},{\displaystyle \prod _{k=1}^{n}}{\left({F}_{ij}{}^{p}\right)}^{\lambda k}\end{array}\right\}.$$**Step 6:**The knowledge measure of the selected attribute is calculated using Equation (1). From the knowledge measure of attributes, we can find the weight of an attribute as follows.$${w}_{i}=\frac{K\left({C}_{i}\right)}{{{\displaystyle \sum}}^{}K\left({C}_{i}\right)},\mathrm{where}i=1,2,3,\dots ,n.$$**Step 7:**Aggregation-weighted neutrosophic decision matrix is constructed for each alternative with respect to each attribute with the help of the given formula:$$\left({T}_{ij},{I}_{ij},{F}_{ij}\right)\to \left({w}_{i}{T}_{ij},{w}_{i}{I}_{ij},{w}_{i}{F}_{ij}\right).$$**Step 8:**Obtain relative neutrosophic positive ideal (RNPIS) and relative neutrosophic negative ideal solution (RNNIS) for each attribute from aggregated single neutrosophic decision matrix as follows.$${Q}^{i+}=\left\{max.\left({T}_{ij}\right),min.\left({I}_{ij}\right),min.\left({F}_{ij}\right)\right\}.$$$${Q}^{i-}=\left\{min.\left({T}_{ij}\right),max.\left({I}_{ij}\right),max.\left({F}_{ij}\right)\right\}.$$**Step 9:**The distance measure of each alternative from RNPIS (${Q}^{i+}$) and RNNIS (${Q}^{i-}$) is determined with the help of the given formula:$${d}^{i+}=\sqrt{\frac{1}{3n}{\displaystyle \sum}_{i=1}^{n}\begin{array}{c}{\left({T}_{ij}{}^{wi}\left({x}_{i}\right)-{T}_{i}{}^{w+}\left({x}_{i}\right)\right)}^{2}+\\ {\left({I}_{ij}{}^{wi}\left({x}_{i}\right)-{I}_{i}{}^{w+}\left({x}_{i}\right)\right)}^{2}+\\ {\left({F}_{ij}{}^{wi}\left({x}_{i}\right)-{F}_{i}{}^{w+}\left({x}_{i}\right)\right)}^{2}.\end{array}}$$$${d}^{i-}=\sqrt{\frac{1}{3n}{\displaystyle \sum}_{i=1}^{n}\begin{array}{c}{\left({T}_{ij}{}^{wi}\left({x}_{i}\right)-{T}_{i}{}^{w-}\left({x}_{i}\right)\right)}^{2}+\\ {\left({I}_{ij}{}^{wi}\left({x}_{i}\right)-{I}_{i}{}^{w-}\left({x}_{i}\right)\right)}^{2}+\\ {\left({F}_{ij}{}^{wi}\left({x}_{i}\right)-{F}_{i}{}^{w-}\left({x}_{i}\right)\right)}^{2}.\end{array}}$$**Step 10:**Finally, the relative closeness coefficient to the neutrosophic ideal solution is obtained which is defined as follows:$${C}_{i}{}^{*}=\frac{{d}^{i-}}{{d}^{i+}+{d}^{i-}}.$$The larger relative closeness coefficient value depicts the most suitable and appropriate alternative.

#### 5.2. Numerical Example Based on MADM (Multiple Attribute Decision-Making)

**Step 1:**Each attribute attains its own significance. Some are very major and some are not. Similarly, each decision-maker has their own importance according to their background knowledge base, power, and position in an organization. The importance of attributes is expressed by linguistic term and these terms are rated as single-valued neutrosophic numbers as shown below in Table 1:Linguistic Term SVNNs **Extremely good**(1.0, 0.0, 0.0) **Very good**(0.95, 0.15, 0.05) **Good**(0.75, 0.25, 0.10) **Medium**(0.50, 0.40, 0.30) **Bad**(0.20, 0.60, 0.60) **Very bad**(0.10, 0.80, 0.95) **Step 2:**The linguistic term taken from Table 2 can be expressed as single-valued neutrosophic number using Table 1 for rating the opinion of each decision-maker. We present these ratings in the form of four decision matrices which subsequently helps to assess the objective weights of the decision-makers. The correlation measures of the neutrosophic values were calculated between each possible pair of decision-makers as shown in Table 3. In Table 4, the correlation coefficient between each pair of decision-makers is obtained with the help of the formula given in Definition 5. Further, we determine the correlation efficiency and normalized correlation efficiency as shown in Table 5 and Table 6 by using the formula given in Definition 6 and Definition 7. We consider the normalize correlation efficiency of each decision-maker as the weights of decision-makers.**Table 2.**Linguistic rating for four attributes for the four available alternatives by decision-makers.Alternative Decision-Maker C _{1}C _{2}C _{3}C _{4}**A**_{1}DM _{1}

DM_{2}

DM_{3}

DM_{4}G

VG

G

VGG

G

VG

GG

A

VG

AG

G

A

VG**A**_{2}DM _{1}

DM_{2}

DM_{3}

DM_{4}VG

VG

VG

VGG

A

VG

AA

G

G

GA

G

G

G**A**_{3}DM _{1}

DM_{2}

DM_{3}

DM_{4}VG

G

A

GVG

G

A

VGVG

G

A

VGVG

G

A

G**A**_{4}DM _{1}

DM_{2}

DM_{3}

DM_{4}G

A

VG

GG

A

G

GG

G

G

GG

G

G

GFor decision-maker DM_{1}, DM_{2,}DM_{3,}DM_{4}linguistic term are given as below:For decision-maker DM_{1}$$\begin{array}{c}{A}_{1}\\ \begin{array}{c}{A}_{2}\\ \begin{array}{c}{A}_{3}\\ {A}_{4}\end{array}\end{array}\end{array}{\left(\begin{array}{c}\begin{array}{c}\begin{array}{ccc}G& G& GG\\ VG& G& AA\\ VG& VG& VGVG\\ G& G& GG\end{array}\end{array}\end{array}\right)}_{}$$For decision-maker DM_{2}$$\begin{array}{c}{A}_{1}\\ \begin{array}{c}{A}_{2}\\ \begin{array}{c}{A}_{3}\\ {A}_{4}\end{array}\end{array}\end{array}\left(\begin{array}{c}\begin{array}{c}\begin{array}{ccc}VG& G& AG\\ VG& A& GA\\ G& G& GG\\ A& G& GG\end{array}\end{array}\end{array}\right)$$For decision-maker DM_{3}$$\begin{array}{c}{A}_{1}\\ \begin{array}{c}{A}_{2}\\ \begin{array}{c}{A}_{3}\\ {A}_{4}\end{array}\end{array}\end{array}\left(\begin{array}{c}\begin{array}{c}\begin{array}{ccc}G& VG& VGA\\ VG& VG& GG\\ A& A& AA\\ VG& G& GG\end{array}\end{array}\end{array}\right)$$For decision-maker DM_{4}$$\begin{array}{c}{A}_{1}\\ \begin{array}{c}{A}_{2}\\ \begin{array}{c}{A}_{3}\\ {A}_{4}\end{array}\end{array}\end{array}\left(\begin{array}{c}\begin{array}{c}\begin{array}{ccc}VG& G& AVG\\ VG& A& GG\\ G& VG& VGG\\ G& G& GG\end{array}\end{array}\end{array}\right)$$For decision-maker DM_{1}, DM_{2,}DM_{3}, DM_{4}single-valued neutrosophic values corresponding to linguistic terms are$$\left(\begin{array}{cccc}<\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)>\\ <\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.50,0.40,0.30\right)& \hspace{1em}\left(0.50,0.40,0.30\right)>\\ <\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.95,0.15,0.05\right)>\\ <\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)>\end{array}\right)$$For decision maker DM_{2}, the single-valued neutrosophic value corresponding to linguistic terms is$$\left(\begin{array}{cccc}<\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.50,0.40,0.30\right)& \hspace{1em}\left(0.75,0.25,0.10\right)>\\ <\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.50,0.40,0.30\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.50,0.40,0.30\right)>\\ <\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)>\\ <\left(0.50,0.40,0.30\right)& \hspace{1em}\left(0.50,0.40,0.30\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)>\end{array}\right)$$For decision maker DM_{3}, the single-valued neutrosophic value corresponding to linguistic terms is$$\left(\begin{array}{cccc}<\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.50,0.40,0.30\right)>\\ <\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)>\\ <\left(0.50,0.40,0.30\right)& \hspace{1em}\left(0.50,0.40,0.30\right)& \hspace{1em}\left(0.50,0.40,0.30\right)& \hspace{1em}\left(0.50,0.40,0.30\right)>\\ <\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)>\end{array}\right)$$For decision maker DM_{4}, the single-valued neutrosophic value corresponding to linguistic terms is$$\left(\begin{array}{cccc}<\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.50,0.40,0.30\right)& \hspace{1em}\left(0.95,0.15,0.05\right)>\\ <\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.50,0.40,0.30\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)>\\ <\left(0.50,0.40,0.30\right)& \hspace{1em}\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.95,0.15,0.05\right)& \hspace{1em}\left(0.75,0.25,0.10\right)>\\ <\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)& \hspace{1em}\left(0.75,0.25,0.10\right)>\end{array}\right)$$$\mathbf{D}{\mathbf{M}}_{\mathbf{1}}$ $\mathbf{D}{\mathbf{M}}_{\mathbf{2}}$ $\mathbf{D}{\mathbf{M}}_{\mathbf{3}}$ $\mathbf{D}{\mathbf{M}}_{\mathbf{4}}$ $D{M}_{\mathbf{1}}$ 11.336 10.262 10.193 10.984 $D{M}_{\mathbf{2}}$ 10.262 10.194 9.65 10.562 $D{M}_{\mathbf{3}}$ 10.193 9.65 10.939 10.118 $D{M}_{\mathbf{4}}$ 10.984 10.562 10.118 11.341 $\mathbf{D}{\mathbf{M}}_{\mathbf{1}}$ $\mathbf{D}{\mathbf{M}}_{\mathbf{2}}$ $\mathbf{D}{\mathbf{M}}_{\mathbf{3}}$ $\mathbf{D}{\mathbf{M}}_{\mathbf{4}}$ $D{M}_{\mathbf{1}}$ 1 0.954 0.915 0.968 $D{M}_{\mathbf{2}}$ 0.954 1 0.914 0.982 $D{M}_{\mathbf{3}}$ 0.915 0.914 1 0.908 $D{M}_{\mathbf{4}}$ 0.968 0.982 0.908 1 Decision-Maker Correlation Efficiency DM _{1}0.945 DM _{2}0.95 DM _{3}0.912 DM _{4}0.952 Decision-Maker Normalized Correlation Efficiency **DM**_{1}0.251 **DM**_{2}0.252 **DM**_{3}0.242 **DM**_{4}0.253 Where 0.251, 0.252, 0.242, and 0.253 are weights of decision-makers. Take $\lambda $_{1}= $0.251,$ $\lambda $_{2}= $0.252,\lambda $_{3}= 0.242, and $\lambda $_{4}= 0.253.**Step 3:**We construct aggregated neutrosophic decision matrix as given in Table 7 with the help of SVNWA (single-valued neutrosophic weighted averaging aggregation operator).C _{1}C _{2}C _{3}C _{4}**A**_{1}(0.889, 0.192, 0.070) (0.830, 0.221, 0.084) (0.759, 0.280, 0.147) (0.803, 0.246, 0.109) **A**_{2}(0.949, 0.149, 0.050) (0.759, 0.280, 0.147) (0.702, 0.281, 0.131) (0.702, 0.281, 0.131) **A**_{3}(0.802, 0.246, 0.109) (0.868, 0.216, 0.092) (0.764, 0.278, 0.144) (0.803, 0.246, 0.109) **A**_{4}(0.798, 0.248, 0.111) (0.702, 0.281, 0.132) (0.750, 0.250, 0.100) (0.750, 0.250, 0.100) The weights of the attributes are determined with the help of Equation (8), i.e., ${w}_{1}=0.313$, ${w}_{2}=0.246,{w}_{3}=0.207,\mathrm{and}{w}_{4}=0.232$.**Step 4:**We construct an aggregated weighted neutrosophic decision matrix using Equation (9) and present in Table 8.C _{1}C _{2}C _{3}C _{4}**A**_{1}(0.278, 0.060, 0.021) (0.204, 0.054, 0.020) (0.157, 0.057, 0.030) (0.186, 0.057, 0.025) **A**_{2}(0.297, 0.046, 0.015) (0.186, 0.068, 0.036) (0.145, 0.058, 0.027) (0.162, 0.065, 0.030) **A**_{3}(0.251, 0.076, 0.034) (0.213, 0.053, 0.022) (0.158, 0.057, 0.029) (0.186, 0.057, 0.025) **A**_{4}(0.249, 0.077, 0.034) (0.172, 0.069, 0.032) (0.155, 0.051, 0.020) (0.174, 0.058, 0.023) **Step 5:**The neutrosophic relative positive ideal solution and relative negative solution is obtained from aggregated weighted neutrosophic decision matrix given in Equations (10) and (11).Neutrosophic relative positive ideal solution (Q^{i+}):$${Q}^{i+}=\left\{\begin{array}{ccc}<0.297,& 0.046,& 0.015>,\\ <0.213,& 0.053,& 0.020>,\\ <0.158,& 0.051,& 0.020>,\\ <0.186,& 0.057,& 0.023>,\end{array}\right\}.$$Neutrosophic relative positive ideal solution (Q^{i−}):$${Q}^{i-}=\left\{\begin{array}{ccc}<0.249,& 0.077,& 0.034>,\\ <0.172,& 0.069,& 0.036>,\\ <0.145,& 0.058,& 0.030>,\\ <0.162,& 0.065,& 0.030>,\end{array}\right\}.$$**Step 6:**The distance measures (${d}^{i+}and{d}^{i-}$) of each alternative from RNPIS and RNNIS are determined with the help of normalized Euclidean distance measure given in Equations (12) and (13). Finally, we obtain the relative closeness coefficient (**C**) with the help of formula given in Equation (14)._{i}^{*}The largest relative closeness value indicates the most suitable and appropriate alternative. From Table 9, it can be seen that the largest value of relative closeness coefficient is corresponding to ${A}_{1}$. Hence, A_{1}is the most suitable alternative.Alternatives ${\mathit{d}}^{\mathit{i}+}$ ${\mathit{d}}^{\mathit{i}-}$ C _{i}^{*}A _{1}0.0079 0.0173 0.6865 A _{2}0.0134 0.0178 0.5705 A _{3}0.0170 0.0157 0.4801 A _{4}0.0220 0.0065 0.2280

## 6. Comparative Study

#### 6.1. Effectiveness of the Proposed Knowledge Measure against Different Existing Entropies

#### 6.2. Comparison between the Existing Entropies of the Neutrosophic Sets and the Proposed Knowledge Measure of Neutrosophic Sets on the Basis of Linguistic Hedges

_{1}(A

^{1/2}) > EY

_{1}(A) < EY

_{1}(A

^{2}) < EY

_{1}(A

^{3})

_{2}(A

^{1/2}) > EY

_{2}(A) > EY

_{2}(A

^{2}) > EY

_{2}(A

^{3})

_{3}(A

^{1/2}) > EY

_{3}(A) < EY

_{3}(A

^{2}) < EY

_{3}(A

^{3})

_{4}(A

^{1/2}) > EY

_{4}(A) > EY

_{4}(A

^{2}) > EY

_{4}(A

^{3})

_{5}(A

^{1/2}) > EY

_{5}(A) > EY

_{5}(A

^{2}) > EY

_{5}(A

^{3})

_{6}(A

^{1/2}) > EY

_{6}(A) > EY

_{6}(A

^{2}) < EY

_{6}(A

^{3})

_{7}(A

^{1/2}) > EY

_{7}(A) > EY

_{7}(A

^{2}) > EY

_{7}(A

^{3})

_{8}(A

^{1/2}) > EY

_{8}(A) > EY

_{8}(A

^{2}) > EY

_{8}(A

^{3})

^{1/2}) < K(A) < K(A

^{2}) < K(A

^{3}).

^{1/2}) < K(A) < K(A

^{2}) < K(A

^{3}), while SVN entropies produce unreasonable results in different instances. Therefore, the performance of our knowledge measure is better than the conventional entropy measures in the neutrosophic settings.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Existing Entropies | Ranking |
---|---|

$E{Y}_{1}$ = $\frac{1}{3\left(\sqrt{2}-1\right)}{\displaystyle \sum}_{t=1}^{3}\left(\left(\sqrt{2}\mathrm{cos}\frac{{\alpha}_{t}-{\alpha}_{t}{}^{c}}{4}\times \pi \right)-1\right)$ (Wu et al. [6]) | ${A}_{1}{A}_{3}{A}_{2}{A}_{4}$ |

$\mathit{E}{\mathit{Y}}_{\mathbf{2}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}\mathbf{\left(}\sqrt{\mathbf{2}}\mathbf{\right)}\mathbf{-}\mathbf{1}}{\displaystyle {\displaystyle \mathbf{\sum}}_{\mathit{t}\mathbf{=}\mathbf{1}}^{\mathbf{3}}}\mathbf{\left(}\frac{\mathbf{sin}\mathbf{\pi}\mathbf{\left(}{\mathbf{\alpha}}_{\mathit{t}}\mathbf{-}{\mathbf{\alpha}}_{\mathit{t}}{}^{\mathit{c}}\mathbf{+}\mathbf{1}\mathbf{\right)}}{\mathbf{4}}\mathbf{\right)}\mathbf{+}$ $\left(\frac{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\pi}\mathbf{\left(}{\mathbf{\alpha}}_{\mathit{t}}\mathbf{-}{\mathbf{\alpha}}_{\mathit{t}}{}^{\mathit{c}}\mathbf{+}\mathbf{1}\mathbf{\right)}}{\mathbf{4}}\mathbf{\right)}\mathbf{-}\mathbf{1}$ (Jin et al. [12]) | ${\mathit{A}}_{\mathbf{1}}\mathbf{>}{\mathit{A}}_{\mathbf{3}}\mathbf{>}{\mathit{A}}_{\mathbf{2}}\mathbf{>}{\mathit{A}}_{\mathbf{4}}$ |

$E{Y}_{3}=1-\frac{1}{n}$ $\sum}_{t=1}^{m}\left({\alpha}_{t}+{\gamma}_{t}\right)\left|2{\beta}_{t}-1\right|$ (Elshabshery and Fattouh [18]) | ${A}_{1}{A}_{3}{A}_{2}{A}_{4}$ |

$E{Y}_{4}=\frac{1}{n}{{\displaystyle \sum}}^{}\left(1-\frac{1}{b-a}\underset{a}{\overset{b}{{\displaystyle \int}}}\left|{\alpha}_{t}-{\gamma}_{t}\right|\left|{\beta}_{t}-{\beta}_{t}{}^{c}\right|\right)dx$ (Aydogdu [15]) | ${A}_{1}>{A}_{3}>{A}_{2}>{A}_{4}$ |

K (Proposed Knowledge measure) | ${A}_{1}{A}_{2}{A}_{3}{A}_{4}$ |

NS | EY_{1} | EY_{2} | EY_{3} | EY_{4} | EY_{5} | ${\mathit{E}}_{\mathit{Y}\mathbf{6}}$ | ${\mathit{E}}_{\mathit{Y}\mathbf{7}}$ | ${\mathit{E}}_{\mathit{Y}\mathbf{8}}$ | K(A) |
---|---|---|---|---|---|---|---|---|---|

A^{1/2} | 0.948 | 2.966 | 0.884 | 2.965 | 0.247 | 0.833 | 0.663 | 0.581 | 0.270 |

A | 0.845 | 2.171 | 0.705 | 2.169 | 0.187 | 0.825 | 0.387 | 0.394 | 0.482 |

A^{2} | 0.850 | 1.009 | 0.802 | 1.006 | 0.108 | 0.802 | 0.153 | 0.301 | 0.762 |

A^{3} | 0.899 | 0.547 | 0.885 | 0.544 | 0.058 | 0.885 | 0.069 | 0.299 | 0.873 |

NS | EY_{1} | EY_{4} | EY_{5} | $\mathit{E}{\mathit{Y}}_{\mathbf{8}}$ | K(A) |
---|---|---|---|---|---|

A^{1/2} | 0.660 | 0.660 | 0.195 | 0.471 | 0.358 |

A | 0.697 | 0.561 | 0.2 | 0.405 | 0.468 |

A^{2} | 0.455 | 0.365 | 0.207 | 0.143 | 0.613 |

A^{3} | 0.430 | 0.345 | 0.2102 | 0.148 | 0.663 |

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Sharma, S.; Singh, S.
A Complementary Dual of Single-Valued Neutrosophic Entropy with Application to MAGDM. *Mathematics* **2022**, *10*, 3726.
https://doi.org/10.3390/math10203726

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Sharma S, Singh S.
A Complementary Dual of Single-Valued Neutrosophic Entropy with Application to MAGDM. *Mathematics*. 2022; 10(20):3726.
https://doi.org/10.3390/math10203726

**Chicago/Turabian Style**

Sharma, Sonam, and Surender Singh.
2022. "A Complementary Dual of Single-Valued Neutrosophic Entropy with Application to MAGDM" *Mathematics* 10, no. 20: 3726.
https://doi.org/10.3390/math10203726