# New Distance Measures for Dual Hesitant Fuzzy Sets and Their Application to Multiple Attribute Decision Making

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Dual Hesitant Fuzzy Sets

**Definition**

**1.**

**Definition**

**2.**

- (1)
- if $s({e}_{1})<s({e}_{2})$, then ${e}_{1}$ is inferior than ${e}_{2}$, denoted by ${e}_{1}\prec {e}_{2}$;
- (2)
- if $s({e}_{1})=s({e}_{2})$, then
- (i)
- if $p\left({e}_{1}\right)=p\left({e}_{2}\right)$, then ${e}_{1}$ is equal to ${e}_{2}$, denoted by ${e}_{1}\sim {e}_{2}$;
- (ii)
- if $p\left({e}_{1}\right)<p\left({e}_{2}\right)$, then ${e}_{1}$ is inferior than ${e}_{2}$, denoted by ${e}_{1}\prec {e}_{2}$;

**Definition**

**3.**

#### 2.2. Existing Distance Measures for DHFSs

**Definition**

**4.**

- $(P1)0\le d(A,\text{}B)\le 1$;
- $(P2)\text{}d(A,\text{}B)=0$if and only if$A=B$;
- $(P3)\text{}d(A,\text{}B)=d(B,\text{}A)$;

**Definition**

**5.**

**Definition**

**6.**

## 3. New Distance Measure for DHFSs

#### 3.1. Analysis of Deficiencies of Existing Distance Measures

**Definition**

**7.**

- $(P1)0\le d(A,\text{}B)\le 1$;
- $(P2)\text{}d(A,\text{}B)=0$if and only if$A=B$;
- $(P3)\text{}d(A,\text{}B)=d(B,\text{}A)$;
- $(P4)\text{}$Let$C$be any DHFS, then$d\left(A,B\right)\le d\left(A,C\right)+d\left(B,C\right)$;

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

#### 3.2. Construction of New Distance Measures for DHFSs

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

**Theorem**

**1.**

**Lemma**

**1.**

**Proof.**

- The new distance measures for DHFSs are developed in terms of the mean, variance and number of elements of DHFEs, which efficiently considers the characteristics of DHFS. The new distance measures not only focus on the difference among the value of DHFEs but also pay much attention to the volatility of DHF information.
- The new distance measures between DHFSs are developed without the consideration of two potential assumptions adopted in aforementioned distance measures. We do not need to extend the shorter DHFEs to the same length by adding any value in it, which can avoid inaccuracy by adding values to DHFEs artificially.
- As a basic property of the distance measures, the triangle inequality is an essential part of the axioms of distance measures. We improve the Definition 4 by adding the property of triangle inequality, which makes the axioms of distance measures more complete. The new distance measures for DHFSs not only meet all properties in Definition 4 but also the property of triangular inequality.

## 4. MADM Method Based on New Distance Measure for DHFSs

#### 4.1. Determination of Completely Unknown Attribute Weights

#### 4.2. Algorithm for MADM Problem with DHF Assessment

## 5. Application of the Proposed Method in MADM

#### 5.1. Description of the Airline Service Quality Evaluation Problem

#### 5.2. Solution Procedure of Airline Service Quality Evaluation Problem

_{i}(i = 1, …, 4) can be calculated and we have CC

_{1}= 0.6052, CC

_{2}= 0.3289, CC

_{3}= 0.3979, CC

_{4}= 0.5279.

_{i}(i = 1, …, 4), we have ${A}_{1}\succ {A}_{4}\succ {A}_{3}\succ {A}_{2}$. Thus, Northern Airlines ${A}_{1}$ is the optional solution for the airline service quality evaluation problem.

#### 5.3. Comparison Analysis with the Existing Distance Measures

#### 5.4. Sensitivity Analysis on the Parameter of the Proposed Distance Measure

_{i}(i = 1, 2, 3, 4) with a variation in $\lambda $ is shown in Figure 1, Figure 2 and Figure 3.

_{i}(i = 1, …, 4) are affected as values of the parameter $\lambda $ changes according to Figure 1, Figure 2 and Figure 3. As can be seen from Figure 1, the values of CC

_{i}(i = 1, …, 4) are monotonically decreasing but the values of CC

_{i}(i = 1, …, 4) in Figure 2 and Figure 3 are not all decreasing. For instance, as the value of parameter $\lambda $ continues to increase, when α = 0.2, CC

_{2}is monotonically decreasing, while when α = 0.8, CC

_{2}is increasing. The reason for this result is that the preference value α (or β) of the mean and volatility have a considerable effect on ${d}_{wpg}$ and ${d}_{wpg}$ directly affect the value of the CC

_{i}(i = 1, …, 4).

_{i}(i = 1, …, 4) tend to be stable with the increase of the value of parameter $\lambda $ and ${A}_{1}$ is always the optimal solution. That is to say, the result of the optimal solution is not sensitive to the value of parameter $\lambda $. Through the sensitivity analysis on the parameters of the new distance measure, it can be concluded that the result of the optimal solution has strong stability and further proves that the proposed distance-based method is rational and effective.

- From the perspective of DHF theory, the proposed new distance measures between DHFSs do not depend on two assumptions to deal with DHF information, which makes the calculation results more objective. Moreover, the construction of new distance measures takes into account not only the difference among the value of DHFEs but also the volatility of dual hesitant fuzzy information. The proposed new distance measures further reflect the characteristics of the DHF information. Besides, the preference coefficients can be determined according to the decision maker’s psychological preference, having strong practicability and availability.
- From the perspective of the determination of completely unknown attribute weights, the weights of attributes can be obtained objectively by constructing optimization model with the help of the new distance measure for DHFSs. The decision information is fully utilized and the attribute weight determination model is flexible and easy to operate.
- From the perspective of practical application, DHFS can better characterize the preferences and cognitions of decision makers in MADM problem. The proposed distance measures for DHFSs can be effectively applied to solve the MADM problem of airline service quality evaluation and help the Civil Aviation bureau to find the best service quality airline.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Attribute | Description of the Attribute |
---|---|

Booking and ticketing services ${C}_{1}$ | Booking and ticketing services mainly include the convenience, rapidity and courtesy in the process of purchasing air tickets, etc. |

Security and boarding services ${C}_{2}$ | Security and boarding services mainly include the convenience and efficiency of security inspection, the courtesy of security personnel, the clarity of notices and announcements, etc. |

Cabin Services ${C}_{3}$ | Cabin services can generally be divided into cabin security demonstration services, courtesy and helpfulness of flight attendants, cleanliness and comfort in the cabin, etc. |

Responsive Services ${C}_{4}$ | Responsive services mainly include the appropriateness of call waiting time, the satisfaction degree of complaint handling and baggage parcel loss processing, etc. |

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

${A}_{1}$ | ({0.6,0.4}, {0.4,0.2,0.1}) | ({0.7,0.6}, {0.3,0.2,0.1}) | ({0.9,0.7,0.5}, {0.1}) | ({0.6,0.4}, {0.3,0.2}) | |

${A}_{2}$ | ({0.4,0.3,0.2}, {0.4}) | ({0.6,0.5,0.4}, {0.2,0.1}) | ({0.6,0.5,0.4,0.2}, {0.2,0.1}) | ({0.8,0.5}, {0.2}) | |

${A}_{3}$ | ({0.6,0.4}, {0.3,0.2}) | ({0.8,0.4}, {0.2, 0.1}) | ({0.5,0.3}, {0.4,0.2}) | ({0.6,0.4}, {0.3,0.2}) | |

${A}_{4}$ | ({0.8,0.4}, {0.2,0.1}) | ({0.8,0.5}, {0.2,0.1,0}) | ({0.6,0.4}, {0.3,0.2,0.1}) | ({0.7}, {0.2}) |

**Table 3.**Ranking order of four airlines derived from three dual hesitant fuzzy (DHF) distance measures.

Airlines | CC_{i} Using ${\mathit{d}}_{\mathit{w}\mathit{p}\mathit{g}}$ | Ranking Order | CC_{i} Using ${\mathit{d}}_{\mathbf{2}\mathit{w}}{}^{\mathit{\lambda}}$ | Ranking Order | CC_{i} Using ${\mathit{d}}_{\mathbf{3}\mathit{w}}{}^{\mathit{\lambda}}$ | Ranking Order |
---|---|---|---|---|---|---|

A_{1} | 0.6052 | 1 | 0.5412 | 1 | 0.5832 | 1 |

A_{2} | 0.3289 | 4 | 0.4068 | 4 | 0.4830 | 2 |

A_{3} | 0.3979 | 3 | 0.4255 | 3 | 0.3821 | 4 |

A_{4} | 0.5279 | 2 | 0.4726 | 2 | 0.4328 | 3 |

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**MDPI and ACS Style**

Wang, R.; Li, W.; Zhang, T.; Han, Q.
New Distance Measures for Dual Hesitant Fuzzy Sets and Their Application to Multiple Attribute Decision Making. *Symmetry* **2020**, *12*, 191.
https://doi.org/10.3390/sym12020191

**AMA Style**

Wang R, Li W, Zhang T, Han Q.
New Distance Measures for Dual Hesitant Fuzzy Sets and Their Application to Multiple Attribute Decision Making. *Symmetry*. 2020; 12(2):191.
https://doi.org/10.3390/sym12020191

**Chicago/Turabian Style**

Wang, Rugen, Weimin Li, Tao Zhang, and Qi Han.
2020. "New Distance Measures for Dual Hesitant Fuzzy Sets and Their Application to Multiple Attribute Decision Making" *Symmetry* 12, no. 2: 191.
https://doi.org/10.3390/sym12020191