# Multi-Criteria Decision-Making Method Based on Prioritized Muirhead Mean Aggregation Operator under Neutrosophic Set Environment

^{*}

## Abstract

**:**

## 1. Introduction

- to handle the impact of the some unduly high or unduly low values provided by the decision makers on to the final ranking;
- to present some new aggregation operators to aggregate the preferences of experts element;
- to develop an algorithm to solve the decision-making problems based on proposed operators;
- to present some example in which relevance of the preferences in SVN decision problems is made explicit.

## 2. Preliminaries

**Definition**

**1**

**.**A neutrosophic set (NS) α comprises of three independent degrees in particular truth (${\mu}_{\alpha}$), indeterminacy (${\rho}_{\alpha}$), and falsity (${\nu}_{\alpha}$) which are characterized as

**Definition**

**2**

**.**A single-valued neutrosophic set (SVNS) α in X is defined as

**Definition**

**3**

**.**Let $\alpha =({\mu}_{\alpha},{\rho}_{\alpha},{\nu}_{\alpha})$ be a SVNN. A score function s of α is defined as

**Definition**

**4**

**.**Let $\alpha =(\mu ,\rho ,\nu )$, ${\alpha}_{1}=({\mu}_{1},{\rho}_{1},{\nu}_{1})$ and ${\alpha}_{2}=({\mu}_{2},{\rho}_{2},{\nu}_{2})$ be three SVNNs and $\lambda >0$ be real number. Then, we have

- 1.
- ${\alpha}^{c}=(\nu ,\rho ,\mu )$;
- 2.
- ${\alpha}_{1}\le {\alpha}_{2}$ if ${\mu}_{1}\le {\mu}_{2},{\rho}_{1}\ge {\rho}_{2}$ and ${\nu}_{1}\ge {\nu}_{2}$;
- 3.
- ${\alpha}_{1}={\alpha}_{2}$ if and only if ${\alpha}_{1}\le {\alpha}_{2}$ and ${\alpha}_{2}\le {\alpha}_{1}$;
- 4.
- ${\alpha}_{1}\cap {\alpha}_{2}=(min({\mu}_{1},{\mu}_{2}),max({\rho}_{1},{\rho}_{2}),max({\nu}_{1},{\nu}_{2}))$;
- 5.
- ${\alpha}_{1}\cup {\alpha}_{2}=(max({\mu}_{1},{\mu}_{2}),min({\rho}_{1},{\rho}_{2}),min({\nu}_{1},{\nu}_{2}))$;
- 6.
- ${\alpha}_{1}\oplus {\alpha}_{2}=({\mu}_{1}+{\mu}_{2}-{\mu}_{1}{\mu}_{2},{\rho}_{1}{\rho}_{2},{\nu}_{1}{\nu}_{2})$;
- 7.
- ${\alpha}_{1}\otimes {\alpha}_{2}=({\mu}_{1}{\mu}_{2},{\rho}_{1}+{\rho}_{2}-{\rho}_{1}{\rho}_{2},{\nu}_{1}+{\nu}_{2}-{\nu}_{1}{\nu}_{2})$;
- 8.
- $\lambda {\alpha}_{1}=(1-{(1-{\mu}_{1})}^{\lambda},{\rho}_{1}^{\lambda},{\nu}_{1}^{\lambda})$;
- 9.
- ${\alpha}_{1}^{\lambda}=({\mu}_{1}^{\lambda},1-{(1-{\rho}_{1})}^{\lambda},1-{(1-{\nu}_{1})}^{\lambda})$.

**Definition**

**5**

**.**For a collection of SVNNs ${\alpha}_{j}=({\mu}_{j},{\rho}_{j},{\nu}_{j})(j=1,2,\dots ,n)$, the prioritized weighted aggregation operators are defined as

- 1.
- SVN prioritized weighted average (SVNPWA) operator$$\begin{array}{c}\hfill \mathit{SVNPWA}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\left(1-\prod _{j=1}^{n}{(1-{\mu}_{j})}^{\frac{{H}_{j}}{{\displaystyle {\displaystyle \sum _{j=1}^{n}}}{H}_{j}}},{\displaystyle \prod _{j=1}^{n}}{\left({\rho}_{j}\right)}^{\frac{{H}_{j}}{{\displaystyle {\displaystyle \sum _{j=1}^{n}}}{H}_{j}}},\prod _{j=1}^{n}{\left({\nu}_{j}\right)}^{\frac{{H}_{j}}{{\displaystyle \sum _{j=1}^{n}}{H}_{j}}}\right),\end{array}$$
- 2.
- SVN prioritized geometric average (SVNPGA) operator$$\begin{array}{c}\hfill \mathit{SVNPGA}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\left(\prod _{j=1}^{n}{\left({\mu}_{j}\right)}^{\frac{{H}_{j}}{{\displaystyle {\displaystyle \sum _{j=1}^{n}}}{H}_{j}}},1-\prod _{j=1}^{n}{(1-{\rho}_{j})}^{\frac{{H}_{j}}{{\displaystyle {\displaystyle \sum _{j=1}^{n}}}{H}_{j}}},1-\prod _{j=1}^{n}{(1-{\nu}_{j})}^{\frac{{H}_{j}}{{\displaystyle {\displaystyle \sum _{j=1}^{n}}}{H}_{j}}}\right),\end{array}$$

**Definition**

**6**

- If $P=(1,0,\dots ,0)$, the MM is reduced to$$\begin{array}{c}\hfill {\mathrm{MM}}^{(1,0,\dots ,0)}({h}_{1},{h}_{2},\dots ,{h}_{n})=\frac{1}{n}\sum _{j=1}^{n}{h}_{j},\end{array}$$
- If $P=(1/n,1/n,\dots ,1/n)$, the MM is reduced to$$\begin{array}{c}\hfill {\mathrm{MM}}^{(1/n,1/n,\dots ,1/n)}({h}_{1},{h}_{2},\dots ,{h}_{n})=\prod _{j=1}^{n}{h}_{j}^{1/n},\end{array}$$
- If $P=(1,1,0,0,\dots ,0)$, then the MM is reduced to$$\begin{array}{c}\hfill {\mathrm{MM}}^{(1,1,0,0,\dots ,0)}({h}_{1},{h}_{2},\dots ,{h}_{n})={\left(\frac{1}{n(n+1)}\sum _{\genfrac{}{}{0pt}{}{i,j=1}{i\ne j}}^{n}{h}_{i}{h}_{j}\right)}^{1/2},\end{array}$$
- If $P=(\stackrel{k}{\overbrace{1,1,\dots ,1}},\stackrel{n-k}{\overbrace{0,0,\dots ,0}})$, then the MM is reduced to$$\begin{array}{c}\hfill {\mathrm{MM}}^{(\stackrel{k}{\overbrace{1,1,\dots ,1}},\stackrel{n-k}{\overbrace{0,0,\dots ,0}})}({h}_{1},{h}_{2},\dots ,{h}_{n})={\left(\frac{1}{{C}_{k}^{n}}\sum _{\genfrac{}{}{0pt}{}{1\le {i}_{1}<}{\dots <{i}_{k}\le n}}\prod _{j=1}^{k}{h}_{{i}_{j}}\right)}^{1/k},\end{array}$$

## 3. Neutrosophic Prioritized Muirhead Mean Operators

#### 3.1. Single-Valued Neutrosophic Prioritized Muirhead Mean (SVNPMM) Operator

**Definition**

**7.**

**Theorem**

**1.**

**Proof.**

**Example**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

- If $P=(1,0,\dots ,0)$, then SVNPMM operator becomes the SVN prioritized weighted average (SVNPWA) operator which is given as$$\begin{array}{lll}\mathrm{SVNPMM}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})& =& {\left(\frac{1}{n!}\underset{\sigma \in {S}_{n}}{\u2a01}\left(n\frac{{H}_{\sigma \left(1\right)}}{{\displaystyle \sum _{j=1}^{n}}{H}_{j}}{\alpha}_{\sigma \left(1\right)}\right)\right)}^{\frac{1}{{\displaystyle \sum _{j=1}^{n}}{p}_{j}}}\\ & =& \underset{j=1}{\overset{n}{\u2a01}}\frac{{H}_{j}}{{\displaystyle \sum _{j=1}^{n}}{H}_{j}}{\alpha}_{j}\hfill \\ & =& \mathrm{SVNPWA}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n}).\hfill \end{array}$$
- When $P=(\lambda ,0,\dots ,0)$, then SVNPMM operator yields to SVN generalized hybrid prioritized weighted average (SVNGHPWA) operator as shown below$$\begin{array}{lll}\mathrm{SVNPMM}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})& =& {\left(\frac{1}{n!}\underset{\sigma \in {S}_{n}}{\u2a01}{\left(n\frac{{H}_{\sigma \left(1\right)}}{{\displaystyle \sum _{j=1}^{n}}{H}_{j}}{\alpha}_{\sigma \left(1\right)}\right)}^{\lambda}\right)}^{\frac{1}{\lambda}}\hfill \\ & =& {\left(\frac{1}{n}\underset{j=1}{\overset{n}{\u2a01}}{\left(n\frac{{H}_{j}}{{\displaystyle \sum _{j=1}^{n}}{H}_{j}}{\alpha}_{j}\right)}^{\lambda}\right)}^{\frac{1}{\lambda}}\hfill \\ & =& \mathrm{SVNGHPWA}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n}).\hfill \end{array}$$
- If $P=(1,1,0,\dots ,0)$, then Equation (11) reduces to SVN prioritized bonferroni mean (SVNPBM) operator as below$$\begin{array}{ccc}\hfill \mathrm{SVNPMM}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})& =& {\left(\frac{1}{n!}\underset{\sigma \in {S}_{n}}{\u2a01}\left(n\frac{{H}_{\sigma \left(1\right)}}{{\displaystyle \sum _{j=1}^{n}}{H}_{j}}{\alpha}_{\sigma \left(1\right)}\right)\left(n\frac{{H}_{\sigma \left(2\right)}}{{\displaystyle \sum _{j=1}^{n}}{H}_{j}}{\alpha}_{\sigma \left(2\right)}\right)\right)}^{\frac{1}{2}}\hfill \\ & =& {\left(\frac{{n}^{2}}{n!}\underset{\genfrac{}{}{0pt}{}{r,s=1}{r\ne s}}{\overset{n}{\u2a01}}\left(\frac{{H}_{r}}{\sum _{r=1}^{n}{H}_{r}}{\alpha}_{r}\right)\left(\frac{{H}_{s}}{\sum _{s=1}^{n}{H}_{s}}{\alpha}_{s}\right)\right)}^{\frac{1}{2}}\hfill \\ & =& \mathrm{SVNPBM}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n}).\hfill \end{array}$$
- If $P=(\stackrel{tterms}{\overbrace{1,1\dots ,1}},\stackrel{n-tterms}{\overbrace{0,0\dots ,0}})$, then SVNPMM operator yields to SVN prioritized Maclaurin symmetric mean (SVNPMSM) operator as follows$$\begin{array}{ccc}\hfill \mathrm{SVNPMM}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})& =& {\left(\frac{2{n}^{t}t}{n!}\underset{\genfrac{}{}{0pt}{}{1<{j}_{1}<\cdots}{<{j}_{t}<n}}{\u2a01}\underset{q=1}{\overset{t}{\u2a02}}\left(\frac{{H}_{{j}_{q}}}{{\displaystyle \sum _{r=1}^{n}}{H}_{r}}{\alpha}_{{j}_{q}}\right)\right)}^{\frac{1}{t}}.\hfill \end{array}$$

#### 3.2. Single-Valued Neutrosophic Prioritized Dual Muirhead Mean Operator

**Definition**

**8.**

**Theorem**

**6.**

**Proof.**

**Example**

**2.**

- (P1)
- Monotonicity: If ${\alpha}_{j}\le {\alpha}_{j}^{\prime}$ for all j, then$$\mathrm{SVNPDMM}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})\le \mathrm{SVNPDMM}({\alpha}_{1}^{\prime},{\alpha}_{2}^{\prime},\dots ,{\alpha}_{n}^{\prime}).$$
- (P2)
- Boundedness: If ${\alpha}^{-}$, and ${\alpha}^{+}$ are lower and upper bound of SVNNs then$${\alpha}^{-}\le \mathrm{SVNPDMM}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})\le {\alpha}^{+}.$$
- (P3)
- Commutativity: For any permutation $(\tilde{{\alpha}_{1}},\tilde{{\alpha}_{2}},\dots ,\tilde{{\alpha}_{n}})$ of the $({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})$, we have$$\mathrm{SVNPDMM}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\mathrm{SVNPDMM}(\tilde{{\alpha}_{1}},\tilde{{\alpha}_{2}},\dots ,\tilde{{\alpha}_{n}}).$$

## 4. Multi-Criteria Decision-Making Approach Based on Proposed Operators

#### 4.1. Proposed Decision-Making Approach

- Step 1:
- If in the considered decision-making problem, there exist two kinds of criteria, namely the benefit and the cost types, then all the cost type criteria should be normalized into the benefit type by using the following equation$$\begin{array}{c}\hfill {r}_{ij}=\left\{\begin{array}{ccc}({\nu}_{ij},{\rho}_{ij},{\mu}_{ij})& ;& \mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{cos}\mathrm{t}\phantom{\rule{4.pt}{0ex}}\mathrm{type}\phantom{\rule{4.pt}{0ex}}\mathrm{criteria},\\ ({\mu}_{ij},{\rho}_{ij},{\nu}_{ij})& ;& \mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{benefit}\phantom{\rule{4.pt}{0ex}}\mathrm{type}\phantom{\rule{4.pt}{0ex}}\mathrm{criteria}.\end{array}\right.\end{array}$$
- Step 2:
- Compute ${H}_{ij}(i=1,2,\dots ,m)$ as$$\begin{array}{c}\hfill {H}_{ij}=\left\{\begin{array}{cc}1\hfill & ;\phantom{\rule{1.em}{0ex}}j=1,\hfill \\ {\displaystyle \prod _{k=1}^{j-1}}s\left({r}_{ik}\right)\hfill & ;\phantom{\rule{1.em}{0ex}}j=2,\dots ,n.\hfill \end{array}\right.\end{array}$$
- Step 3:
- For a given parameter $P=({p}_{1},{p}_{2},\dots ,{p}_{n})$, utilize either SVNPMM or SVNPDMM operator to get the collective values ${r}_{i}=({\mu}_{i},{\rho}_{i},{\nu}_{i})(i=1,2,\dots ,m)$ for each alternative as$$\begin{array}{ccc}\hfill {r}_{i}& =& \mathrm{SVNPMM}({r}_{i1},{r}_{i2},\dots ,{r}_{in})\hfill \\ & =& \left(\begin{array}{c}{\left(1-{\left(\prod _{\sigma \in {S}_{n}}\left(1-{\displaystyle \prod _{j=1}^{n}}{\left(1-{(1-{\mu}_{i\sigma \left(j\right)})}^{n\frac{{H}_{i\sigma \left(j\right)}}{{\displaystyle \sum _{j=1}^{n}}{H}_{ij}}}\right)}^{{p}_{j}}\right)\right)}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle \sum _{j=1}^{n}}{p}_{j}}},\hfill \\ 1-{\left(1-{\left(\prod _{\sigma \in {S}_{n}}\left(1-{\displaystyle \prod _{j=1}^{n}}{\left(1-{{\rho}_{i\sigma \left(j\right)}}^{n\frac{{H}_{i\sigma \left(j\right)}}{{\displaystyle \sum _{j=1}^{n}}{H}_{ij}}}\right)}^{{p}_{j}}\right)\right)}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle \sum _{j=1}^{n}}{p}_{j}}},\hfill \\ 1-{\left(1-{\left(\prod _{\sigma \in {S}_{n}}\left(1-{\displaystyle \prod _{j=1}^{n}}{\left(1-{{\nu}_{i\sigma \left(j\right)}}^{n\frac{{H}_{i\sigma \left(j\right)}}{{\displaystyle \sum _{j=1}^{n}}{H}_{ij}}}\right)}^{{p}_{j}}\right)\right)}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle \sum _{j=1}^{n}}{p}_{j}}}\hfill \end{array}\right)\phantom{\rule{2.em}{0ex}}\hfill \end{array}$$$$\begin{array}{ccc}\hfill {r}_{i}& =& \mathrm{SVNPDMM}({r}_{i1},{r}_{i2},\dots ,{r}_{in})\hfill \\ & =& \left(\begin{array}{c}1-{\left(1-{\left(\prod _{\sigma \in {S}_{n}}\left(1-{\displaystyle \prod _{j=1}^{n}}{\left(1-{{\mu}_{i\sigma \left(j\right)}}^{n\frac{{H}_{i\sigma \left(j\right)}}{{\displaystyle \sum _{j=1}^{n}}{H}_{ij}}}\right)}^{{p}_{j}}\right)\right)}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle \sum _{j=1}^{n}}{p}_{j}}},\hfill \\ {\left(1-{\left(\prod _{\sigma \in {S}_{n}}\left(1-{\displaystyle \prod _{j=1}^{n}}{\left(1-{(1-{\rho}_{i\sigma \left(j\right)})}^{n\frac{{H}_{i\sigma \left(j\right)}}{{\displaystyle \sum _{j=1}^{n}}{H}_{ij}}}\right)}^{{p}_{j}}\right)\right)}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle \sum _{j=1}^{n}}{p}_{j}}},\hfill \\ {\left(1-{\left(\prod _{\sigma \in {S}_{n}}\left(1-\prod _{j=1}^{n}{\left(1-{(1-{\nu}_{i\sigma \left(j\right)})}^{n\frac{{H}_{i\sigma \left(j\right)}}{{\displaystyle \sum _{j=1}^{n}}{H}_{ij}}}\right)}^{{p}_{j}}\right)\right)}^{\frac{1}{n!}}\right)}^{\frac{1}{{\displaystyle \sum _{j=1}^{n}}{p}_{j}}}\hfill \end{array}\right).\phantom{\rule{2.em}{0ex}}\hfill \end{array}$$
- Step 4:
- Calculate score values of the overall aggregated values ${r}_{i}=({\mu}_{i},{\rho}_{i},{\nu}_{i})$ $(i=1,2,\dots ,m)$ by using equation$$\begin{array}{c}\hfill s\left({r}_{i}\right)=\frac{1+({\mu}_{i}-2{\rho}_{i}-{\nu}_{i})(2-{\mu}_{i}-{\nu}_{i})}{2}.\end{array}$$
- Step 5:
- Rank all the feasible alternatives ${A}_{i}(i=1,2,\dots ,m)$ according to Definition 3 and hence select the most desirable alternative(s).

#### 4.2. Illustrative Example

- Step 1:
- As all the criteria values are of the same types, the original decision matrix need not be normalized.
- Step 2:
- Compute ${H}_{ij}(j=1,2,3,4)$ by using Equation (16), we get$$\begin{array}{ccc}\hfill H& =& \left[\begin{array}{c}\hfill \begin{array}{cccc}\hfill 1& 0.6650& 0.4921\hfill & 0.3642\\ \hfill 1& 0.9000& 0.7200\hfill & 0.4464\\ \hfill 1& 0.6650& 0.5320\hfill & 0.4575\\ \hfill 1& 0.6650& 0.5154\hfill & 0.1134\\ \hfill 1& 0.8250& 0.6806\hfill & 0.6024\end{array}\end{array}\right].\hfill \end{array}$$
- Step 3:
- Without loss of generality, we take $P=(0.25,0.25,0.25,0.25)$ and use SVNPMM operator given in Equation (17) to aggregate ${r}_{ij}(j=1,2,3,4)$ and hence we get ${r}_{1}=(0.9026$, $0.0004$, $0.0118)$; ${r}_{2}=(0.9963$, $0.0008$, $0.0007)$; ${r}_{3}=(0.9858$, $0.0001$, $0.0029)$; ${r}_{4}=(0.9877$, $0.0021$, $0.0002)$ and ${r}_{5}=(0.9474$, $0.0000$, $0.0093).$
- Step 4:
- By Equation (19), we get $s\left({r}_{1}\right)=0.9959$, $s\left({r}_{2}\right)=0.9992$, $s\left({r}_{3}\right)=0.9998$, $s\left({r}_{4}\right)=0.9978$ and $s\left({r}_{5}\right)=0.9990$.
- Step 5:
- Since $s\left({r}_{3}\right)>s\left({r}_{2}\right)>s\left({r}_{5}\right)>s\left({r}_{4}\right)>s\left({r}_{1}\right)$ and thus ranking order of their corresponding alternatives is ${A}_{3}\succ {A}_{2}\succ {A}_{5}\succ {A}_{4}\succ {A}_{1}$. Here ≻ refers “preferred to”. Therefore, ${A}_{3}$ is the best one according to the requirement of the travel agency.

- Step 1:
- Similar to above Step 1.
- Step 2:
- Similar to above Step 2.
- Step 3:
- For a parameter $P=(0.25,0.25,0.25,0.25)$, use SVNPDMM operator given in Equation (18) we get ${r}_{1}=(0.0069$, $0.7379$, $0.9413)$; ${r}_{2}=(0.1034$, $0.7423$, $0.7782)$; ${r}_{3}=(0.0428$, $0.6021$, $0.8672)$; ${r}_{4}=(0.0625$, $0.8271$, $0.6966)$ and ${r}_{5}=(0.0109$, $0.5340$, $0.9125)$.
- Step 4:
- The evaluated score values by using Equation (19) are $s\left({r}_{1}\right)=0.2226$, $s\left({r}_{2}\right)=0.1628$, $s\left({r}_{3}\right)=0.3396$, $s\left({r}_{4}\right)=-0.0554$ and $s\left({r}_{5}\right)=0.4222$.
- Step 5:
- The ranking order of the alternatives, based on the score values, is ${A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{4}$ and hence ${A}_{5}$ as the best alternative among the others.

#### 4.3. Comparison Study

- Step 1:
- Use SVNPWA operator as given in Equation (4) to calculate the aggregated values ${\beta}_{i}(i=1,2,3,4,5)$ of each alternative ${A}_{i}$ are ${\beta}_{1}=(0.4392,0.2407$, $0.3981)$, ${\beta}_{2}=(0.6681,0.1864,0.2602)$, ${\beta}_{3}=(0.5461,0.1929,0.3414)$, ${\beta}_{4}=(0.6294,0.2844$, $0.2000)$ and ${\beta}_{5}=(0.4291,0.1141,0.3232)$.
- Step 2:
- Compute the cross entropy E for each ${\beta}_{i}$ from ${A}^{+}=(1,0,0)$ and ${A}^{-}=(0,0,1)$ based on the equation $E({\alpha}_{1},{\alpha}_{2})=(sin{\mu}_{1}-sin{\mu}_{2})\times (sin({\mu}_{1}-{\mu}_{2}))+(sin{\rho}_{1}-sin{\rho}_{2})\times (sin({\rho}_{1}-{\rho}_{2}))+(sin{\nu}_{1}-sin{\nu}_{2})\times (sin({\nu}_{1}-{\nu}_{2}))$ and then evaluate ${S}_{{\beta}_{i}}$ by using equation ${S}_{{\beta}_{i}}=\frac{E({\beta}_{i},{A}^{+})}{E({\beta}_{i},{A}^{+})+E({\beta}_{i},{A}^{-})}$. The values corresponding to it are: ${S}_{{\beta}_{1}}=0.4642$, ${S}_{{\beta}_{2}}=0.1755$, ${S}_{{\beta}_{3}}=0.3199$, ${S}_{{\beta}_{4}}=0.1914$ and ${S}_{{\beta}_{5}}=0.4007$.
- Step 3:
- The final ranking of alternative, according to the values of ${S}_{{\beta}_{i}}$, is ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{5}\succ {A}_{1}$.

#### 4.4. Influence of Parameter P on the Decision-Making Process

#### 4.5. Further Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | |
---|---|---|---|---|

${A}_{1}$ | $(0.5,0.3,0.4)$ | $(0.5,0.2,0.3)$ | $(0.2,0.2,0.6)$ | $(0.3,0.2,0.4)$ |

${A}_{2}$ | $(0.7,0.1,0.3)$ | $(0.7,0.2,0.3)$ | $(0.6,0.3,0.2)$ | $(0.6,0.4,0.2)$ |

${A}_{3}$ | $(0.5,0.3,0.4)$ | $(0.6,0.2,0.4)$ | $(0.6,0.1,0.2)$ | $(0.5,0.1,0.3)$ |

${A}_{4}$ | $(0.7,0.3,0.2)$ | $(0.7,0.2,0.2)$ | $(0.4,0.5,0.2)$ | $(0.5,0.2,0.2)$ |

${A}_{5}$ | $(0.4,0.1,0.3)$ | $(0.5,0.1,0.2)$ | $(0.4,0.1,0.5)$ | $(0.4,0.3,0.6)$ |

Parameter Vector P | Operator | Score Values of Alternatives | Ranking Results | ||||
---|---|---|---|---|---|---|---|

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | |||

(1, 0, 0, 0) | SVNPMM | 0.9975 | 0.9997 | 0.9999 | 0.9989 | 0.9990 | ${A}_{3}\succ {A}_{2}\succ {A}_{5}\succ {A}_{4}\succ {A}_{1}$ |

SVNPDMM | 0.2184 | 0.0876 | 0.2942 | -0.1233 | 0.3632 | ${A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{4}$ | |

(1, 1, 0, 0) | SVNPMM | 0.9844 | 0.9969 | 0.9988 | 0.9920 | 0.9940 | ${A}_{3}\succ {A}_{2}\succ {A}_{5}\succ {A}_{4}\succ {A}_{1}$ |

SVNPDMM | 0.3638 | 0.2891 | 0.4851 | 0.0162 | 0.5597 | ${A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{4}$ | |

(1, 1, 1, 0) | SVNPMM | 0.9723 | 0.9926 | 0.9968 | 0.9809 | 0.9887 | ${A}_{3}\succ {A}_{2}\succ {A}_{5}\succ {A}_{4}\succ {A}_{1}$ |

SVNPDMM | 0.4268 | 0.3846 | 0.5529 | 0.1219 | 0.6053 | ${A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{4}$ | |

(1, 1, 1, 1) | SVNPMM | 0.9624 | 0.9868 | 0.9942 | 0.9659 | 0.9851 | ${A}_{3}\succ {A}_{2}\succ {A}_{5}\succ {A}_{4}\succ {A}_{1}$ |

SVNPDMM | 0.4617 | 0.4507 | 0.5955 | 0.2079 | 0.6341 | ${A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{4}$ | |

(2, 2, 2, 2) | SVNPMM | 0.9443 | 0.9633 | 0.9836 | 0.9189 | 0.9767 | ${A}_{3}\succ {A}_{5}\succ {A}_{2}\succ {A}_{1}\succ {A}_{4}$ |

SVNPDMM | 0.5165 | 0.5024 | 0.640 | 0.3016 | 0.6698 | ${A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{4}$ | |

(3, 3, 3, 3) | SVNPMM | 0.9322 | 0.9440 | 0.9744 | 0.8896 | 0.9715 | ${A}_{3}\succ {A}_{5}\succ {A}_{2}\succ {A}_{1}\succ {A}_{4}$ |

SVNPDMM | 0.5369 | 0.5018 | 0.6490 | 0.3142 | 0.6853 | ${A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{4}$ | |

$\left({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}}\right)$ | SVNPMM | 0.9824 | 0.9965 | 0.9987 | 0.9903 | 0.9943 | ${A}_{3}\succ {A}_{2}\succ {A}_{5}\succ {A}_{4}\succ {A}_{1}$ |

SVNPDMM | 0.3652 | 0.3217 | 0.4982 | 0.0490 | 0.5661 | ${A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{4}$ | |

$\left({\textstyle \frac{1}{4}},{\textstyle \frac{1}{4}},{\textstyle \frac{1}{4}},{\textstyle \frac{1}{4}}\right)$ | SVNPMM | 0.9959 | 0.9992 | 0.9998 | 0.9978 | 0.9990 | ${A}_{3}\succ {A}_{2}\succ {A}_{5}\succ {A}_{4}\succ {A}_{1}$ |

SVNPDMM | 0.2226 | 0.1628 | 0.3396 | -0.0554 | 0.4222 | ${A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{4}$ | |

(2, 0, 0, 0) | SVNPMM | 0.9890 | 0.9984 | 0.9990 | 0.9953 | 0.9931 | ${A}_{3}\succ {A}_{2}\succ {A}_{4}\succ {A}_{5}\succ {A}_{1}$ |

SVNPDMM | 0.3571 | 0.1886 | 0.4228 | -0.1009 | 0.4781 | ${A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{4}$ | |

(3, 0, 0, 0) | SVNPMM | 0.9814 | 0.9964 | 0.9974 | 0.9898 | 0.9860 | ${A}_{3}\succ {A}_{2}\succ {A}_{4}\succ {A}_{5}\succ {A}_{1}$ |

SVNPDMM | 0.4139 | 0.2426 | 0.4645 | -0.0595 | 0.5008 | ${A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{2}\succ {A}_{4}$ |

Approaches | Whether the Interrelationship of Two Attributes Is Captured | Whether the Interrelationship of Three Attributes Is Captured | Whether the Relationship of Multiple Attributes Is Captured | Whether the Bad Effects of the Unduly High Unduly Low Arguments Can Be Reduced | Whether It Makes the Method Flexible by the Parameter Vector |
---|---|---|---|---|---|

NWA [21] | × | × | × | × | × |

SVNWA [22] | × | × | × | × | × |

SVNOWA [22] | × | × | × | × | × |

SVNWG [22] | × | × | × | × | × |

SVNOWG [22] | × | × | × | × | × |

SVNHWA [25] | × | × | × | × | × |

SVNHWG [25] | × | × | × | × | × |

NWG [21] | × | × | × | × | × |

SVNFWG [24] | × | × | × | × | |

SVNFWA [24] | × | × | × | × | |

SVNFNPBM [37] | × | × | × | ||

WSVNLMSM [34] | × | ||||

SVNNWBM [33] | × | × | × | ||

SVNIGWHM [20] | × | ||||

GNNHWA [25] | × | × | × | × | |

The proposed method |

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## Share and Cite

**MDPI and ACS Style**

Garg, H.; Nancy.
Multi-Criteria Decision-Making Method Based on Prioritized Muirhead Mean Aggregation Operator under Neutrosophic Set Environment. *Symmetry* **2018**, *10*, 280.
https://doi.org/10.3390/sym10070280

**AMA Style**

Garg H, Nancy.
Multi-Criteria Decision-Making Method Based on Prioritized Muirhead Mean Aggregation Operator under Neutrosophic Set Environment. *Symmetry*. 2018; 10(7):280.
https://doi.org/10.3390/sym10070280

**Chicago/Turabian Style**

Garg, Harish, and Nancy.
2018. "Multi-Criteria Decision-Making Method Based on Prioritized Muirhead Mean Aggregation Operator under Neutrosophic Set Environment" *Symmetry* 10, no. 7: 280.
https://doi.org/10.3390/sym10070280