# Some Single-Valued Neutrosophic Dombi Weighted Aggregation Operators for Multiple Attribute Decision-Making

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{A}(x), I

_{A}(x) and F

_{A}(x) are represented independently, which lie in real standard or nonstandard subsets of ]

^{−}0, 1

^{+}[, i.e., T

_{A}(x): X → ]

^{−}0, 1

^{+}[, I

_{A}(x): X → ]

^{−}0, 1

^{+}[, and F

_{A}(x): X → ]

^{−}0, 1

^{+}[. Thus, the nonstandard interval ]

^{−}0, 1

^{+}[ may result in the difficulty of actual applications. Based on the real standard interval [0, 1], therefore, the concepts of a single-valued neutrosophic set (SVNS) [5] and an interval neutrosophic set (INS) [6] was presented as subclasses of NS to be easily used for actual applications, and then Ye [7] introduced a simplified neutrosophic set (SNS), including the concepts of SVNS and INS, which are the extension of IFS and IVIFS. Obviously, SNS is a subclass of NS, while SVNS and INS are subclasses of SNS. As mentioned in the literature [4,5,6,7], NS is the generalization of FS, IFS, and IVIFS. Thereby, Figure 1 shows the flow chart extended from FS to NS (SNS, SVNS, INS).

## 2. Some Concepts of SVNSs

**Definition**

**1.**

_{N}(x), an indeterminacy-membership function u

_{N}(x), and a falsity-membership function v

_{N}(x). Then, a SVNS N can be denoted as the following form:

_{N}(x), u

_{N}(x), v

_{N}(x) $\in $ [0, 1] satisfy the condition 0 ≤ t

_{N}(x) + u

_{N}(x) + v

_{N}(x) ≤ 3 for x $\in $ X.

_{N}(x), u

_{N}(x), v

_{N}(x)> in N is denoted by s = <t, u, v>, which is called a SVNN.

**Definition**

**2.**

_{1}= <t

_{1}, u

_{1}, v

_{1}> and s

_{2}= <t

_{2}, u

_{2}, v

_{2}> be two SVNNs. Then the ranking method for s

_{1}and s

_{2}is defined as follows:

- (1)
- If E(s
_{1}) > E(s_{2}), then s_{1}$\succ $ s_{2}, - (2)
- If E(s
_{1}) = E(s_{2}) and H(s_{1}) > H(s_{2}), then s_{1}$\succ $ s_{2}, - (3)
- If E(s
_{1}) = E(s_{2}) and H(s_{1}) = H(s_{2}), then s_{1}= s_{2}.

## 3. Some Single-Valued Neutrosophic Dombi Operations

**Definition**

**3.**

**Definition**

**4.**

_{1}= <t

_{1}, u

_{1}, v

_{1}> and s

_{2}= <t

_{2}, u

_{2}, v

_{2}> be two SVNNs, ρ ≥ 1, and λ > 0. Then, the Dombi T-norm and T-conorm operations of SVNNs are defined below:

## 4. Dombi Weighted Aggregation Operators of SVNNs

**Definition**

**5.**

_{j}= <t

_{j}, u

_{j}, v

_{j}> (j = 1, 2, …, n) be a collection of SVNNs and

**w**= (w

_{1}, w

_{2}, …, w

_{n}) be the weight vector for s

_{j}with w

_{j}$\in $ [0, 1] and ${\sum}_{j=1}^{n}{w}_{j}}=1$. Then, the SVNDWAA and SVNDWGA operators are defined, respectively, as follows:

**Theorem**

**1.**

_{j}= <t

_{j}, u

_{j}, v

_{j}> (j = 1, 2, …, n) be a collection of SVNNs and

**w**= (w

_{1}, w

_{2}, …, w

_{n}) be the weight vector for s

_{j}with w

_{j}$\in $ [0, 1] and ${\sum}_{j=1}^{n}{w}_{j}}=1$. Then, the aggregated value of the SVNDWAA operator is still a SVNN, which is calculated by the following formula:

**Proof.**

- (1)
- Reducibility: When
**w**= (1/n, 1/n, …, 1/n), it is obvious that there exists$$SVNDWAA({s}_{1},{s}_{2},\dots ,{s}_{n})=\langle 1-\frac{1}{1+{\left\{{\displaystyle \sum _{j=1}^{n}\frac{1}{n}{\left(\frac{{t}_{j}}{1-{t}_{j}}\right)}^{\rho}}\right\}}^{1/\rho}},\frac{1}{1+{\left\{{\displaystyle \sum _{j=1}^{n}\frac{1}{n}{\left(\frac{1-{u}_{j}}{{u}_{j}}\right)}^{\rho}}\right\}}^{1/\rho}},\frac{1}{1+{\left\{{\displaystyle \sum _{j=1}^{n}\frac{1}{n}{\left(\frac{1-{v}_{j}}{{v}_{j}}\right)}^{\rho}}\right\}}^{1/\rho}}\rangle .$$ - (2)
- Idempotency: Let all the SVNNs be s
_{j}= <t_{j}, u_{j}, v_{j}> = s (j = 1, 2, …, n). Then, SVNDWAA(s_{1}, s_{2}, …, s_{n}) = s. - (3)
- Commutativity: Let the SVNS (s
_{1}’, s_{2}’, …, s_{n}’) be any permutation of (s_{1}, s_{2}, …, s_{n}). Then, there is SVNDWAA(s_{1}’, s_{2}’, …, s_{n}’) = SVNDWAA(s_{1}, s_{2}, …, s_{n}). - (4)
- Boundedness: Let s
_{min}= min(s_{1}, s_{2}, …, s_{n}) and s_{max}= max(s_{1}, s_{2}, …, s_{n}). Then, s_{min}≤ SVNDWAA(s_{1}, s_{2}, …, s_{n}) ≤ s_{max}.

**Proof.**

_{j}= <t

_{j}, u

_{j}, v

_{j}> = s (j = 1, 2, …, n). Then, by using Equation (7) we can obtain the following result:

_{1}, s

_{2}, …, s

_{n}) = s holds.

_{min}= min(s

_{1}, s

_{2}, …, s

_{n}) = <t

^{−}, u

^{−}, v

^{−}> and s

_{max}= max(s

_{1}, s

_{2}, …, s

_{n}) = <t

^{+}, u

^{+}, v

^{+}>. Then, we have ${t}^{-}=\underset{j}{\mathrm{min}}({t}_{j})$, ${u}^{-}=\underset{j}{\mathrm{max}}({u}_{j})$, ${v}^{-}=\underset{j}{\mathrm{max}}({v}_{j})$, ${t}^{+}=\underset{j}{\mathrm{max}}({t}_{j})$, ${u}^{+}=\underset{j}{\mathrm{min}}({u}_{j})$, and ${v}^{+}=\underset{j}{\mathrm{min}}({v}_{j})$. Thus, there are the following inequalities:

_{min}≤ SVNDWAA(s

_{1}, s

_{2}, …, s

_{n}) ≤ s

_{max}holds. □

**Theorem**

**2.**

_{j}= <t

_{j}, u

_{j}, v

_{j}> (j = 1, 2, …, n) be a collection of SVNNs and

**w**= (w

_{1}, w

_{2}, …, w

_{n}) be the weight vector for s

_{j}with w

_{j}$\in $ [0, 1] and ${\sum}_{j=1}^{n}{w}_{j}}=1$. Then, the aggregated value of the SVNDWGA operator is still a SVNN, which is calculated by the following formula:

- (1)
- Reducibility: When the weight vector is
**w**= (1/n, 1/n, …, 1/n), it is obvious that there exists the following result:$$SVNDWGA({s}_{1},{s}_{2},\dots ,{s}_{n})=\langle \frac{1}{1+{\left\{{\displaystyle \sum _{j=1}^{n}\frac{1}{n}{\left(\frac{1-{t}_{j}}{{t}_{j}}\right)}^{\rho}}\right\}}^{1/\rho}},1-\frac{1}{1+{\left\{{\displaystyle \sum _{j=1}^{n}\frac{1}{n}{\left(\frac{{u}_{j}}{1-{u}_{j}}\right)}^{\rho}}\right\}}^{1/\rho}},1-\frac{1}{1+{\left\{{\displaystyle \sum _{j=1}^{n}\frac{1}{n}{\left(\frac{{v}_{j}}{1-{v}_{j}}\right)}^{\rho}}\right\}}^{1/\rho}}\rangle .$$ - (2)
- Idempotency: Let all the SVNNs be s
_{j}= <t_{j}, u_{j}, v_{j}> = s (j = 1, 2, …, n). Then, SVNDWGA(s_{1}, s_{2}, …, s_{n}) = s. - (3)
- Commutativity: Let the SVNS (s
_{1}’, s_{2}’, …, s_{n}’) be any permutation of (s_{1}, s_{2}, …, s_{n}). Then, there is SVNDWGA(s_{1}’, s_{2}’, …, s_{n}’) = SVNDWGA(s_{1}, s_{2}, …, s_{n}). - (4)
- Boundedness: Let s
_{min}= min(s_{1}, s_{2}, …, s_{n}) and s_{max}= max(s_{1}, s_{2}, …, s_{n}). Then, s_{min}≤ SVNDWGA(s_{1}, s_{2}, …, s_{n}) ≤ s_{max}.

## 5. MADM Method Using the SVNDWAA Operator or the SVNDWGA Operator

_{1}, S

_{2}, …, S

_{m}} be a discrete set of alternatives and G = {G

_{1}, G

_{2}, …, G

_{n}} be a discrete set of attributes. Assume that the weight vector of the attributes is given as

**w**= (w

_{1}, w

_{2}, …, w

_{n}) such that w

_{j}$\in $ [0, 1] and ${\sum}_{j=1}^{n}{w}_{j}}=1$. If the decision makers are required to provide their suitability evaluation about the alternative S

_{i}(i = 1, 2, …, m) under the attribute G

_{j}(j = 1, 2, …, n) by the SVNN s

_{ij}= <t

_{ij}, u

_{ij}, v

_{ij}> (i = 1, 2, …, m; j = 1, 2, …, n), then, we can elicit a SVNN decision matrix D = (s

_{ij})

_{m}

_{×n}.

**Step 1.**Derive the collective SVNN s

_{i}(i = 1, 2, …, m) for the alternative S

_{i}(i = 1, 2, …, m) by using the SVNDWAA operator:

**w**= (w

_{1}, w

_{2}, …, w

_{n}) is the weight vector such that w

_{j}$\in $ [0, 1] and ${\sum}_{j=1}^{n}{w}_{j}}=1$.

**Step 2.**Calculate the score values of E(s

_{i}) (the accuracy degrees of H(s

_{i}) if necessary) of the collective SVNN s

_{i}(i = 1, 2, …, m) by using Equations (1) and (2).

**Step 3**. Rank the alternatives and select the best one(s).

**Step 4**. End.

## 6. Illustrative Example

_{1}is a car company; (2) S

_{2}is a food company; (3) S

_{3}is a computer company; (4) S

_{4}is an arms company. The investment company must take a decision corresponding to the requirements of the three attributes: (1) G

_{1}is the risk; (2) G

_{2}is the growth; (3) G

_{3}is the environmental impact. The suitability evaluations of the alternative S

_{i}(i = 1, 2, 3, 4) corresponding to the three attributes of G

_{j}(j = 1, 2, 3) are given by some decision makers or experts and expressed by the form of SVNNs. Thus, when the four possible alternatives corresponding to the above three attributes are evaluated by the decision makers, we can give the single-valued neutrosophic decision matrix D(s

_{ij})

_{m}

_{×n}, where s

_{ij}= <t

_{ij}, u

_{ij}, v

_{ij}> (i = 1, 2, 3, 4; j = 1, 2, 3) is SVNN, as follows:

**w**= (0.35, 0.25, 0.4).

**Step 1.**Derive the collective SVNNs of s

_{i}for the alternative S

_{i}(i = 1, 2, 3, 4) by using Equation (9) for ρ = 1 as follows:

_{1}= <0.6667, 0.2000, 0.3571>, s

_{2}= <0.5652, 0.1250, 0.2857>, s

_{3}= <0.4444, 0.2308, 0.4000>, and s

_{4}= <0.6418, 0, 0.1905>.

**Step 2.**Calculate the score values of E(s

_{i}) of the collective SVNN s

_{i}(i = 1, 2, 3, 4) for the alternatives S

_{i}(i = 1, 2, 3, 4) by using Equation (1) as the following results:

_{1}) = 0.7032, E(s

_{2}) = 0.7182, E(s

_{3}) = 0.6046, and E(s

_{4}) = 0.8171.

**Step 3.**Based on the obtained score values, the ranking order of the alternatives is S

_{4}$\succ $ S

_{2}$\succ $ S

_{1}$\succ $ S

_{3}and the best one is S

_{4}.

**Step 1’.**Derive the collective SVNNs of s

_{i}for the alternative S

_{i}(i = 1, 2, 3, 4) by using Equation (10) for ρ = 1 as follows:

_{1}= <0.5000, 0.2000, 0.3966>, s

_{2}= <0.5556, 0.1429, 0.6364>, s

_{3}= <0.4054, 0.2432, 0.6500>, and s

_{4}= <0.6316, 0.1661, 0.6298>.

**Step 2’.**Calculate the score values of E(s

_{i}) of the collective SVNN s

_{i}(i = 1, 2, 3, 4) for the alternatives S

_{i}(i = 1, 2, 3, 4) by using Equation (1) as the following results:

_{1}) = 0.6345, E(s

_{2}) = 0.5921, E(s

_{3}) = 0.5041, and E(s

_{4}) = 0.6119.

**Step 3’.**Based on the obtained score values, the ranking order of the alternatives is S

_{1}$\succ $ S

_{4}$\succ $ S

_{2}$\succ $ S

_{3}and the best one is S

_{1}.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Flow chart extended from fuzzy set (FS) to neutrosophic set (NS) (simplified neutrosophic set (SNS), single-valued neutrosophic set (SVNS), interval neutrosophic set (INS)). IFS: intuitionistic fuzzy set; IVIFS: interval-valued intuitionistic fuzzy set.

**Table 1.**Ranking results for different operational parameters of the single-valued neutrosophic Dombi weighted arithmetic average (SVNDWAA) operator.

ρ | E(s_{1}), E(s_{2}), E(s_{3}), E(s_{4}) | Ranking Order |
---|---|---|

1 | 0.7032, 0.7182, 0.6046, 0.8171 | S_{4} $\succ $ S_{2} $\succ $ S_{1} $\succ $ S_{3} |

2 | 0.7259, 0.7356, 0.6257, 0.8326 | S_{4} $\succ $ S_{2} $\succ $ S_{1} $\succ $ S_{3} |

3 | 0.7380, 0.7434, 0.6364, 0.8396 | S_{4} $\succ $ S_{2} $\succ $ S_{1} $\succ $ S_{3} |

4 | 0.7449, 0.7480, 0.6429, 0.8441 | S_{4} $\succ $ S_{2} $\succ $ S_{1} $\succ $ S_{3} |

5 | 0.7492, 0.7511, 0.6472, 0.8474 | S_{4} $\succ $ S_{2} $\succ $ S_{1} $\succ $ S_{3} |

6 | 0.7521, 0.7533, 0.6503, 0.8499 | S_{4} $\succ $ S_{2} $\succ $ S_{1} $\succ $ S_{3} |

7 | 0.7542, 0.7550, 0.6525, 0.8520 | S_{4} $\succ $ S_{2} $\succ $ S_{1} $\succ $ S_{3} |

8 | 0.7558, 0.7564, 0.6543, 0.8536 | S_{4} $\succ $ S_{2} $\succ $ S_{1} $\succ $ S_{3} |

9 | 0.7571, 0.7574, 0.6556, 0.8549 | S_{4} $\succ $ S_{2} $\succ $ S_{1} $\succ $ S_{3} |

10 | 0.7580, 0.7583, 0.6567, 0.8560 | S_{4} $\succ $ S_{2} $\succ $ S_{1} $\succ $ S_{3} |

**Table 2.**Ranking results for different operational parameters of the single-valued neutrosophic Dombi weighted geometric average (SVNDWGA) operator.

ρ | E(s_{1}), E(s_{2}), E(s_{3}), E(s_{4}) | Ranking Order |
---|---|---|

1 | 0.6345, 0.5921, 0.5041, 0.6119 | S_{1} $\succ $ S_{4} $\succ $ S_{2} $\succ $ S_{3} |

2 | 0.6145, 0.5602, 0.4722, 0.5645 | S_{1} $\succ $ S_{4} $\succ $ S_{2} $\succ $ S_{3} |

3 | 0.6026, 0.5460, 0.4549, 0.5454 | S_{1} $\succ $ S_{2} $\succ $ S_{4} $\succ $ S_{3} |

4 | 0.5950, 0.5374, 0.4439, 0.5351 | S_{1} $\succ $ S_{2} $\succ $ S_{4} $\succ $ S_{3} |

5 | 0.5898, 0.5316, 0.4363, 0.5286 | S_{1} $\succ $ S_{2} $\succ $ S_{4} $\succ $ S_{3} |

6 | 0.5861, 0.5272, 0.4308, 0.5241 | S_{1} $\succ $ S_{2} $\succ $ S_{4} $\succ $ S_{3} |

7 | 0.5834, 0.5238, 0.4266, 0.5208 | S_{1} $\succ $ S_{2} $\succ $ S_{4} $\succ $ S_{3} |

8 | 0.5813, 0.5211, 0.4234, 0.5183 | S_{1} $\succ $ S_{2} $\succ $ S_{4} $\succ $ S_{3} |

9 | 0.5797, 0.5190, 0.4208, 0.5163 | S_{1} $\succ $ S_{2} $\succ $ S_{4} $\succ $ S_{3} |

10 | 0.5784, 0.5172, 0.4188, 0.5147 | S_{1} $\succ $ S_{2} $\succ $ S_{4} $\succ $ S_{3} |

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## Share and Cite

**MDPI and ACS Style**

Chen, J.; Ye, J.
Some Single-Valued Neutrosophic Dombi Weighted Aggregation Operators for Multiple Attribute Decision-Making. *Symmetry* **2017**, *9*, 82.
https://doi.org/10.3390/sym9060082

**AMA Style**

Chen J, Ye J.
Some Single-Valued Neutrosophic Dombi Weighted Aggregation Operators for Multiple Attribute Decision-Making. *Symmetry*. 2017; 9(6):82.
https://doi.org/10.3390/sym9060082

**Chicago/Turabian Style**

Chen, Jiqian, and Jun Ye.
2017. "Some Single-Valued Neutrosophic Dombi Weighted Aggregation Operators for Multiple Attribute Decision-Making" *Symmetry* 9, no. 6: 82.
https://doi.org/10.3390/sym9060082