The purpose of this study is to reduce the uncertainty in the calculation process on hesitant fuzzy sets (HFSs). The innovation of this study is to unify the cardinal numbers of hesitant fuzzy elements (HFEs) in a special way. Firstly, a probability density function is assigned for any given HFE. Thereafter, equal-probability transformation is introduced to transform HFEs with different cardinal numbers on the condition into the same probability density function. The characteristic of this transformation is that the higher the consistency of the membership degrees in HFEs, the higher the credibility of the mentioned membership degrees is, then, the bigger the probability density values for them are. According to this transformation technique, a set of novel distance measures on HFSs is provided. Finally, an illustrative example of intersection traffic control is introduced to show the usefulness of the given distance measures. The example also shows that this study is a good complement to operation theories on HFSs.

As an important tool of group decision making, hesitant fuzzy set (HFS) assigns the membership degree of an element to a set with a set of possible values

The characteristic of the newly proposed method is to make full use of the existing decision making information, and not to add artificial one, so as to keep the objectivity of decision making process. The technique adopted in the proposed method is equal probability transformation, which guarantees the constant probability of the theoretical truth value appearing at each point before and after the transformation. Besides, this technique is also suitable to be used in aggregation operators on HFSs. For more details on this issue, please refer to Xia et al. [

In this section, some basic definitions and some classical distance measures on HFSs are reviewed. For convenience’s sake

In classical calculation process on HFSs, to satisfy

From the viewpoint of probability, the truth value of membership function of any given HFE can appear at any point between the smallest and the largest occurred membership degree, but the probability of occurrence is different for different values. Logically, the probability that a certain point is the true value of membership degree of HFE is related to the occurred value of membership degree near this point. When the occurred value of membership degree function is far away from the point, it is thought that the probability of the true value in this point is low; otherwise, the probability is high. Guided by this idea, a probability density function for HFE is introduced as follows.

On the probability density function for HFSs, some properties are summarized as follows.

For any given HFE, the occurrence interval of the true value of its membership degree can be calculated by

By using Definition 6, an HFE is transferred to another new HFE under the condition that the two variables share the same probability distribution function. An important property on equal-probability function for HFSs is introduced as follows.

To illustrate Definition 6 clearly, a case is given as follows.

Firstly, the probability density function for

Secondly, by

Therefore, it gets

By further study, Def. 7 is obtained in the following.

By using equal-probability equations, a series of improved distance measures on HFSs are obtained as follows.

Analogously, for any

In terms of cardinality,

At present, the at-grade intersection is an important kind of complex node in urban road. In traffic engineering fields, there are three basic methods of traffic control which could be implemented at an intersection, i.e., “method 1-uncontrolled intersection”; “method 2-intersection with right assignment using Yield or Stop signs”; and “method 3-signalized intersection” [

In China, the most frequently used traffic control modes at intersections are “roundabout control”, “Yield or Stop signs control”, and “traffic signal control”, where they also belong to the above three control methods, respectively [_{0} needs to be designed and constructed. Suppose that the traffic control modes

Intersection number | Mode | _{1} |
_{2} |
_{3} |
_{4} |
---|---|---|---|---|---|

Intersection 1 | _{1} |
0.75 | 0.81 | 0.82, 0.88 | 0.76, 0.79 |

Intersection 2 | _{2} |
0.81 | 0.78 | 0.76, 0.78 | 0.85, 0.86 |

Intersection 3 | _{3} |
0.72 | 0.79 | 0.82, 0.88 | 0.80, 0.82 |

Intersection 4 | _{1} |
0.78, 0.82 | 0.77, 0.84 | 0.79, 0.84, 0.86 | 0.78, 0.81, 0.82 |

Intersection 5 | _{2} |
0.81, 0.86 | 0.76, 0.84 | 0.81, 0.88, 0.89 | 0.76, 0.78, 0.81 |

Intersection 6 | _{3} |
0.74, 0.78 | 0.78, 0.81 | 0.85, 0.87, 0.89 | 0.75, 0.84, 0.88 |

Intersection 7 | _{1} |
0.79, 0.81 | 0.79, 0.85 | 0.85, 0.87 | 0.84, 0.85 |

Intersection 8 | _{2} |
0.76, 0.78 | 0.78, 0.81 | 0.78, 0.79 | 0.79, 0.83 |

Intersection 9 | _{3} |
0.75, 0.78 | 0.75, 0.85 | 0.86, 0.87 | 0.77, 0.83 |

Studied one | _{0} |
0.70, 0.89 | 0.75, 0.80 | 0.72, 0.74 | 0.75, 0.81 |

By information aggregation,

Similarly, the studied intersection could be expressed as a hesitant fuzzy set

In this subsection, the decision making problem is solved by using the novel distance measures. Firstly, sort the elements of each

Thereafter, for every

Then, the intervals corresponding to

The following, by

Analogously,

For

_{2}; when

In this subsection, the given problem can is solved by using classical methods. For example, by using Definition (3) proposed by Xu et al. [_{0} can be transferred as

Then, for

Distance | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

0.063 | 0.074 | 0.082 | 0.088 | 0.093 | 0.097 | 0.100 | 0.103 | 0.105 | 0.107 | |

0.048 | 0.060 | 0.072 | 0.082 | 0.090 | 0.096 | 0.101 | 0.106 | 0.109 | 0.112 | |

0.076 | 0.092 | 0.101 | 0.107 | 0.112 | 0.115 | 0.118 | 0.120 | 0.122 | 0.124 |

In calculating the distance between two HFEs with different cardinal numbers, the cardinalities of them should be unified. To reduce the uncertainty in the unifying process, equal-probability transformation is used. Specially, a series of improved distance measures on HFSs are proposed. Since the essence of equal-probability transformation is a kind of dimensional transformation of the same information in the way of expression, the proposed method retains the decision information completely. By contrast, it is difficult to achieve this effect using classical methods. Moreover, the main innovations of this study are concluded as follows:

The theoretical basis of this study is to deal with the membership function value of HFSs from the viewpoint of probability. Moreover, an equal-probability transformation technique is proposed to transform any given HFE into a new one with specified cardinal number.

To express the enhancement effect for the same membership function value of HFE occurring more than once, impulse function is introduced into hesitant fuzzy fields. This is consistent with people’s production experience.

This study is a multidisciplinary combination of the theories of the cardinality, equal-probability mapping, and impulse function.

In general, the innovation of this study mainly lies in the unification of multi-source heterogeneous data. This innovation can be applied not only to HFSs, but also to HFSs or NFSs, etc.