# A Multi-Criteria Group Decision-Making Method with Possibility Degree and Power Aggregation Operators of Single Trapezoidal Neutrosophic Numbers

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## Abstract

**:**

## 1. Introduction

- (1)
- The novel operation laws of SVTNNs are conducted to overcome the lack of operation laws of SVTNNs appeared in previous paper.
- (2)
- Based on the novel operations of SVTNNs, the SVTNPA and SVTNPG operators are developed.
- (3)
- Based on the concept of the possibility degree, the possibility degree of SVTNNs is defined and presented.
- (4)
- Based on possibility degree of SVTNNs, SVTNPA and SVTNPG operators, a novel method for solving MCGDM problems under single trapezoidal neutrosophic environment is developed.

## 2. Preliminaries

#### 2.1. NS and SVNS

**Definition**

**1**

**(**[14]

**).**Let$X$be a space of points (objects), with a generic element in $X$ denoted by $x$. A NS $A$ in $X$ is characterized by three membership functions, namely truth-membership function ${T}_{A}(x)$, indeterminacy-membership function ${I}_{A}(x)$ and falsity-membership function ${F}_{A}(x)$, where ${T}_{A}(x)$, ${I}_{A}(x)$ and ${F}_{A}(x)$ are real standard or nonstandard subsets of ${]}^{-}0,{1}^{+}[$, i.e., ${T}_{A}(x):X\to {]}^{-}0,{1}^{+}[$, ${I}_{A}(x):X\to {]}^{-}0,{1}^{+}[$ and ${F}_{A}(x):X\to {]}^{-}0,{1}^{+}[$. Therefore, it is no restriction on the sum of ${T}_{A}(x)$, ${I}_{A}(x)$ and ${F}_{A}(x)$ and ${}^{-}0\le {T}_{A}(x)+{I}_{A}(x)+{F}_{A}(x)\le {3}^{+}$.

**Definition**

**2**

**(**[13]

**).**Let$X$be a space of points (objects). A SVNS $A$ in $X$ can be expressed as follows:

#### 2.2. The Trapezoidal Fuzzy Number and SVTNNs

**Definition**

**3**

**(**[43,47]

**).**Let$\tilde{a}$be a trapezoidal fuzzy number$\tilde{a}=({a}_{1},{a}_{2},{a}_{3},{a}_{4})$and${a}_{1}\le {a}_{2}\le {a}_{3}\le {a}_{4}$. Then its membership function${\mu}_{\tilde{a}}(x):R\to [0,1]$can be defined as follows:

**Definition**

**4**

**(**[44]

**).**Let$U$be a space of points (objects). Then a SVTNN $\alpha $ can be represented as

**Example**

**1.**

#### 2.3. PA and PG Operators

**Definition**

**5**

**(**[45,46]

**).**Let$\tilde{h}=\{{h}_{1},{h}_{2},\cdot \cdot \cdot ,{h}_{n}\}$a collection of positive real numbers, then PA operator and PG operator can be defined, respectively, as follows:

- (1)
- $Sup({h}_{i},{h}_{j})\in [0,1]$.
- (2)
- $Sup({h}_{i},{h}_{j})=Sup({h}_{j},{h}_{i})$.
- (3)
- If$\left|{h}_{i}-{h}_{j}\right|\le \left|a-b\right|$, then$Sup({h}_{i},{h}_{j})\ge Sup(a,b)$, where$a$and$b$are two positive real numbers.

## 3. New Operations and Comparison of SVTNNs

#### 3.1. The New Operations of SVTNNs

**Definition**

**6**

**(**[44]

**).**Let$\alpha =\langle \left[{a}_{1},{a}_{2},{a}_{3},{a}_{4}\right],\left(T(\alpha ),I(\alpha ),F(\alpha )\right)\rangle $and$\beta =\langle \left[{b}_{1},{b}_{2},{b}_{3},{b}_{4}\right],(T(\beta ),I(\beta ),F(\beta ))\rangle $be two positive SVTNNs,$0\le {a}_{1}\le {a}_{2}\le {a}_{3}\le {a}_{4}\le 1$,$0\le {b}_{1}\le {b}_{2}\le {b}_{3}\le {b}_{4}\le 1$,$\zeta \ge 0$. Then the operations of SVTNNs can be defined as follows:

- (1)
- $\alpha +\beta =\langle \left[{a}_{1}+{b}_{1},{a}_{2}+{b}_{2},{a}_{3}+{b}_{3},{a}_{4}+{b}_{4}\right],\left(T(\alpha )+T(\beta )-T(\alpha )T(\beta ),I(\alpha )I(\beta ),F(\alpha )F(\beta )\right)\rangle $;
- (2)
- $\alpha \beta =\langle \left[{a}_{1}{b}_{1},{a}_{2}{b}_{2},{a}_{3}{b}_{3},{a}_{4}{b}_{4}\right],\left(T(\alpha )T(\beta ),I(\alpha )+I(\beta )-I(\alpha )I(\beta ),F(\alpha )+F(\beta )-F(\alpha )F(\beta )\right)\rangle $;
- (3)
- $\zeta \alpha =\langle \left[\zeta {a}_{1},\zeta {a}_{2},\zeta {a}_{3},\zeta {a}_{4}\right],\left(1-{\left(1-T(\alpha )\right)}^{\zeta},{\left(I(\alpha )\right)}^{\zeta},{\left(F(\alpha )\right)}^{\zeta}\right)\rangle $;
- (4)
- ${\alpha}^{\zeta}=\langle \left[{a}_{1}{}^{\zeta},{a}_{2}{}^{\zeta},{a}_{3}{}^{\zeta},{a}_{4}{}^{\zeta}\right],\left({\left(T(\alpha )\right)}^{\zeta},1-{\left(1-I(\alpha )\right)}^{\zeta},1-{\left(1-F(\alpha )\right)}^{\zeta}\right)\rangle $;

- (1)
- The trapezoidal fuzzy numbers and three membership degrees of SVTNNs are considered as two separate parts and operated individually in the operation $\alpha +\beta $, which ignore the correlation among them and cannot reflect the actual results.

**Example**

**2.**

- (2)
- The three membership degrees of SVTNNs are also operated as the trapezoidal fuzzy numbers in the operation $\zeta \alpha $, which can produce the repeat operation and make the result bias.

**Example**

**3.**

**Definition**

**7.**

- (1)
- $neg(\alpha )=\langle \left[1-{a}_{4},1-{a}_{3},1-{a}_{2},1-{a}_{1}\right],\left(T(\alpha ),I(\alpha ),F(\alpha )\right)\rangle $;
- (2)
- $\alpha \oplus \beta =\langle \left[{a}_{1}+{b}_{1},{a}_{2}+{b}_{2},{a}_{3}+{b}_{3},{a}_{4}+{b}_{4}\right],(\frac{\phi (\alpha )T(\alpha )+\phi (\beta )T(\beta )}{\phi (\alpha )+\phi (\beta )},\frac{\phi (\alpha )I(\alpha )+\phi (\beta )I(\beta )}{\phi (\alpha )+\phi (\beta )},$$\frac{\phi (\alpha )F(\alpha )+\phi (\beta )F(\beta )}{\phi (\alpha )+\phi (\beta )})\rangle $, where$\phi (\alpha )=\frac{{a}_{1}+2{a}_{2}+2{a}_{3}+{a}_{4}}{6}$,$\phi (\beta )=\frac{{b}_{1}+2{b}_{2}+2{b}_{3}+{b}_{4}}{6}$;
- (3)
- $\alpha \otimes \beta =\langle \left[{a}_{1}{b}_{1},{a}_{2}{b}_{2},{a}_{3}{b}_{3},{a}_{4}{b}_{4}\right],(T(\alpha )T(\beta ),I(\alpha )+I(\beta )-I(\alpha )I(\beta ),F(\alpha )+F(\beta )-F(\alpha )F(\beta ))\rangle $;
- (4)
- $\zeta \alpha =\langle \left[\zeta {a}_{1},\zeta {a}_{2},\zeta {a}_{3},\zeta {a}_{4}\right],\left(T(\alpha ),I(\alpha ),F(\alpha )\right)\rangle $;
- (5)
- ${\alpha}^{\zeta}=\langle \left[{a}_{1}{}^{\zeta},{a}_{2}{}^{\zeta},{a}_{3}{}^{\zeta},{a}_{4}{}^{\zeta}\right],\left({\left(T(\alpha )\right)}^{\zeta},1-{\left(1-I(\alpha )\right)}^{\zeta},1-{\left(1-F(\alpha )\right)}^{\zeta}\right)\rangle $;

**Example**

**4.**

- (1)
- $neg({\alpha}_{1})=\langle [0.5,0.6,0.8,0.9],(0.4,0.1,0.5)\rangle $;
- (2)
- ${\alpha}_{1}\oplus {\alpha}_{2}=\langle [0.3,0.5,1.0,1.2],(0.64,0.22,0.26)\rangle $;
- (3)
- ${\alpha}_{1}\otimes {\alpha}_{2}=\langle [0.02,0.06,0.24,0.35],(0.32,0.37,0.55)\rangle $;
- (4)
- $2{\alpha}_{1}=\langle [0.2,0.4,0.8,1.0],(0.4,0.1,0.5)\rangle $;
- (5)
- ${\alpha}_{1}{}^{2}=\langle [0.04,0.09,0.25,0.36],(0.16,0.19,0.75)\rangle $.

**Theorem**

**1.**

- (1)
- ${\alpha}_{1}\oplus {\alpha}_{2}={\alpha}_{2}\oplus {\alpha}_{1}$;
- (2)
- $({\alpha}_{1}\oplus {\alpha}_{2})\oplus {\alpha}_{3}={\alpha}_{1}\oplus ({\alpha}_{2}\oplus {\alpha}_{3})$;
- (3)
- ${\alpha}_{1}\otimes {\alpha}_{2}={\alpha}_{2}\otimes {\alpha}_{1}$;
- (4)
- $({\alpha}_{1}\otimes {\alpha}_{2})\otimes {\alpha}_{3}={\alpha}_{1}\otimes ({\alpha}_{2}\otimes {\alpha}_{3})$;
- (5)
- $\zeta {\alpha}_{1}\oplus \zeta {\alpha}_{2}=\zeta ({\alpha}_{2}\oplus {\alpha}_{1})$;
- (6)
- ${({\alpha}_{2}\otimes {\alpha}_{1})}^{\tau}={\alpha}_{1}{}^{\tau}\otimes {\alpha}_{2}{}^{\tau}$.

#### 3.2. The Possibility Degree

**Definition**

**8**

**(**[49,50]

**).**Let$y=[{y}_{1},{y}_{2}]\subseteq [0,1]$and$z=[{z}_{1},{z}_{2}]\subseteq [0,1]$be two real number intervals with uniform probability distribution, the probability$y\ge z$can be represented as$p(y\ge z)$, which exists the following properties:

- (1)
- $0\le p(y\ge z)\le 1$.
- (2)
- $p(y\ge z)+p(z\ge y)=1$.
- (3)
- If$y=z$, then$p(y\ge z)=p(z\ge y)=0.5$.
- (4)
- If$\xi $is an arbitrary interval or number,$p(y\ge z)\ge 0.5$,$p(z\ge \xi )\ge 0.5$, then$p(y\ge \xi )\ge 0.5$.
- (5)
- If$\mathrm{min}(y)>\mathrm{max}(z)$, then$p(y\ge z)=1$.

**Definition**

**9.**

**Example**

**5.**

**Theorem**

**2.**

- (1)
- $0\le p(\alpha \succ \beta )\le 1$.
- (2)
- $p(\alpha \succ \beta )+p(\beta \succ \alpha )=1$.
- (3)
- If${a}_{i}={b}_{i}$,$i=1,2,3,4$,$T(\alpha )=T(\beta )$,$I(\alpha )=I(\beta )$and$F(\alpha )=F(\beta )$, then$p(\alpha \succ \beta )=p(\beta \succ \alpha )=0.5$.
- (4)
- If$\xi $is an arbitrary positive SVTNN,$p(\alpha \succ \beta )\ge 0.5$,$p(\beta \succ \xi )\ge 0.5$, then$p(\alpha \ge \xi )\ge 0.5$.
- (5)
- If${a}_{1}\ge {b}_{4}$,$T(\alpha )\ge T(\beta )$,$I(\alpha )\ge I(\beta )$and$F(\alpha )\le F(\beta )$, then$p(\alpha \succ \beta )=1$.

**Proof.**

#### 3.3. The Comparison Method of SVTNNs

**Definition**

**10**

**.**Let $\alpha =\langle \left[{a}_{1},{a}_{2},{a}_{3},{a}_{4}\right],\left(T(\alpha ),I(\alpha ),F(\alpha )\right)\rangle $ and $\beta =\langle \left[{b}_{1},{b}_{2},{b}_{3},{b}_{4}\right],\left(T(\beta ),I(\beta ),F(\beta )\right)\rangle $ be two SVTNNs. Then the score degree of $\alpha $ $S(\alpha )$ can be defined as follows:

**Example**

**6.**

**Example**

**7.**

**Definition**

**11.**

- (1)
- If$p(\alpha \succ \gamma )>p(\beta \succ \gamma )$, then$\alpha \succ \beta $, i.e.,$\alpha $is superior to$\beta $.
- (2)
- If$p(\alpha \succ \gamma )=p(\beta \succ \gamma )$, then$\alpha \sim \beta $, i.e.,$\alpha $is equal to$\beta $.
- (3)
- If$p(\alpha \succ \gamma )<p(\beta \succ \gamma )$, then$\alpha \prec \beta $, i.e.,$\beta $is superior to$\alpha $.

**Example**

**8.**

## 4. Single Valued Trapezoidal Neutrosophic Power Aggregation Operators

**Definition**

**12.**

- (1)
- $Sup({\alpha}_{i},{\alpha}_{j})\in [0,1]$.
- (2)
- $Sup({\alpha}_{i},{\alpha}_{j})=Sup({\alpha}_{j},{\alpha}_{i})$.
- (3)
- If$\left|p({\alpha}_{i}\succ {\alpha}_{j})-p({\alpha}_{j}\succ {\alpha}_{i})\right|<\left|p(\pi \succ \nu )-p(\nu \succ \pi )\right|$, then$Sup({\alpha}_{i},{\alpha}_{j})>Sup(\pi ,\nu )$, where$\pi $and$\nu $are two positive SVTNNs,$p({\alpha}_{i}\succ {\alpha}_{j})$,$p({\alpha}_{j}\succ {\alpha}_{i})$,$p(\pi \succ \nu )$and$p(\nu \succ \pi )$are the possibility degree of${\alpha}_{i}\succ {\alpha}_{j}$,${\alpha}_{j}\succ {\alpha}_{i}$,$\pi \succ \nu $and$\nu \succ \pi $.

**Theorem**

**3.**

**Proof.**

**Example**

**9.**

**Theorem**

**4.**

**Proof.**

**Definition**

**13.**

**Theorem**

**5.**

**Example**

**10.**

**Theorem**

**6.**

## 5. A MCGDM Method Based on Possibility Degree and Power Aggregation Operators under Single Valued Trapezoidal Neutrosophic Environment

**Step 1.**Normalize the decision matrices.

**Step 2.**Aggregate the values of alternatives on each criterion to get the collective SVTNNs.

**Step 3.**Aggregate the values of alternative on each decision-maker to get the overall SVTNNs.

**Step 4.**Calculate the possibility degrees of the assessment values of each alternative superior than other alternatives’ values.

**Step 5.**Calculate the collective possibility degree index of each alternative to derive the overall values of the alternatives.

**Step 6.**Rank the alternatives and select the best one.

## 6. Illustrative Example

#### 6.1. Background

#### 6.2. The Procedures of Single Valued Trapezoidal Neutrosophic MCGDM Method

**Step 1.**Normalize the decision matrices.

**Step 2.**Aggregate the values of the four alternatives on each criterion to get the collective SVTNNs.

**Step 3.**Aggregate the values of the four alternatives on each green supplier to get the overall SVTNNs by using the SVTNPA or SVTNPG operator.

**Step 4.**Calculate the possibility degrees of the assessment values of each alternative superior than other alternatives’ values to get the possibility degrees matrix $U$ or $\tilde{U}$.

**Step 5.**Calculate the collective possibility degree index of each alternative to derive the overall values of the alternatives.

**Step 6.**Rank the green suppliers and select the best one.

#### 6.3. Comparison Analysis and Discussion

- (a)
- The new operations of SVTNNs defined in this paper, which take the conservative and reliable principle, can take account of the correlation between trapezoidal fuzzy numbers and three membership degrees of SVTNNs. However, the operations in Reference [44] divide the trapezoidal fuzzy numbers and three membership degrees of SVTNNs into two parts and calculate them separately, which make aggregating results deviate from the reality.
- (b)
- The new comparison of SVTNNs proposed in this paper has some crucial advantages over comparison of SVTNNs based on the score degree function in Reference [44], which can take the preference of decision-makers into consideration.
- (c)
- The relationship among the aggregation information, which exists in the aggregation process of in practical MCDM problems, is ignored [44]. Whereas, the SVTNPA and SVTNPG operators, which can effectively take the relationship among the assessment information being aggregated into consideration and in this paper, the advantages of the possibility degree of SVTNNs are combined to rank the uncertain information reasonably and accurately from the probability viewpoint. Hence, the ranking result of this paper is more objective and reasonable than that obtained by using the operators in Reference [44].

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Methods | Operators | Ranking of Alternatives |
---|---|---|

The method in Reference [44] | NNTWA operator | ${B}_{1}\succ {B}_{2}\succ {B}_{4}\succ {B}_{3}$ |

NNTWG operator | ${B}_{4}\succ {B}_{2}\succ {B}_{1}\succ {B}_{3}$ | |

The proposed method | SVTNPA operator and the possibility degrees SVTNNs | ${B}_{2}\succ {B}_{1}\succ {B}_{3}\succ {B}_{4}$ |

SVTNPG operator and the possibility degrees SVTNNs | ${B}_{2}\succ {B}_{1}\succ {B}_{4}\succ {B}_{3}$ |

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## Share and Cite

**MDPI and ACS Style**

Wu, X.; Qian, J.; Peng, J.; Xue, C.
A Multi-Criteria Group Decision-Making Method with Possibility Degree and Power Aggregation Operators of Single Trapezoidal Neutrosophic Numbers. *Symmetry* **2018**, *10*, 590.
https://doi.org/10.3390/sym10110590

**AMA Style**

Wu X, Qian J, Peng J, Xue C.
A Multi-Criteria Group Decision-Making Method with Possibility Degree and Power Aggregation Operators of Single Trapezoidal Neutrosophic Numbers. *Symmetry*. 2018; 10(11):590.
https://doi.org/10.3390/sym10110590

**Chicago/Turabian Style**

Wu, Xiaohui, Jie Qian, Juanjuan Peng, and Changchun Xue.
2018. "A Multi-Criteria Group Decision-Making Method with Possibility Degree and Power Aggregation Operators of Single Trapezoidal Neutrosophic Numbers" *Symmetry* 10, no. 11: 590.
https://doi.org/10.3390/sym10110590