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# Novel Distance Measures on Hesitant Fuzzy Sets Based on Equal-Probability Transformation and Their Application in Decision Making on Intersection Traffic Control

1 Naval Architecture and Port Engineering College, Shandong Jiaotong University, Weihai, 264209, China

2 College of Transport and Communications, Shanghai Maritime University, Shanghai, 201306, China

3 College of Automotive and Traffic Engineering, Nanjing Forestry University, Nanjing, 210037, China

4 School of Civil and Environmental Engineering, Ningbo University, Ningbo, 315211, China

5 School of Business, National University of Singapore, 118414, Singapore

6 Durham Law School, Durham University, Durham, DH1 3LE, UK

* Corresponding Author: Yi Zhao. Email:

(This article belongs to the Special Issue: Decision making Modeling, Methods and Applications of Advanced Fuzzy Theory in Engineering and Science)

*Computer Modeling in Engineering & Sciences* **2023**, *135*(2), 1589-1602. https://doi.org/10.32604/cmes.2022.022431

**Received** 09 March 2022; **Accepted** 07 June 2022; **Issue published** 27 October 2022

## Abstract

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.## Keywords

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. [16]. To describe the idea clearly, the remainder of this study is arranged as follows. Section 2 introduces some basic concepts on HFSs, and introduces a series of classical distance measures. Section 3 introduces the concept of equal-probability transformation on HFSs, gives three properties of the transformation, and proposes a series of improved distance measures. Section 4 introduces a traffic control mode decision making problem, and solves it by the proposed improved distance measures on HFSs. Finally, the main innovation points are concluded in Section 5.

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

Definition 1 (Torra [2]) Let X be a given set, an HFS E on X is demonstrated as a function that when applied to X returns a subset of

Definition 2 (Xu et al. [4]) Let M and N be two HFSs on X, then the distance measure on M and N is described as

Definition 3 (Xu et al. [4]) Let M and N be two HFSs on X. For any

where

Definition 4 (Xu et al. [4]) Let M and N be two HFSs on

where

In classical calculation process on HFSs, to satisfy Eqs. (1)–(4), part of the information on HFSs has to be artificially added when the cardinality of HFEs is different. In an environment characterized by uncertainty, this process further increases the uncertainty of computing problems and weakens support for decision-makers. To reduce the uncertainty in the calculation for HFSs, for any given HFE

3.1 Probability Density Function for HFE

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.

Definition 5 Suppose that there is an HFE

where

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

Property 1 By Eq. (5), for any given HFE

Property 2 If any given HFE h and its probability density function

Property 3 If any given HFE h and its probability density function

3.2 Equal-Probability Function and Their Properties

For any given HFE, the occurrence interval of the true value of its membership degree can be calculated by Eq. (5). On the premise of keeping the probability density function of true value at any point in the interval unchanged, the expression form of HFE can be changed by specific skills. Specifically, the equal-probability function for HFSs is proposed in this subsection.

Definition 6 Suppose that there is an HFE

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.

Property 4 Suppose that there are two HFEs

Proof By Eq. (5), it gets that

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

Case 1 Suppose that there is an HFE

Firstly, the probability density function for

Secondly, by Eq. (6),

Therefore, it gets

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

Definition 7 Suppose that there are K HFSs

Definition 8 Suppose that there are K HFSs

3.3 Improved Distance Measures on HFSs

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

Definition 9 Suppose that there are K HFSs

and

where

Definition 10 When one takes the weight

Analogously, for any

where

In terms of cardinality, Eqs. (9)–(15) are consistent with Eqs. (1)–(4). To illustrate the performance of the proposed distance measures, an example is given in the following section.

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” [17]. For the sake of understanding, these three kinds of traffic control modes are illustrated by the picture (Figs. 1–3), respectively.

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 [18]. For the sake of convenience, these three kinds of control models are denoted by

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

Figs. 4 and 5 illustrate that the suitable intersection Traffic Control mode varies as the parameter

In this subsection, the given problem can is solved by using classical methods. For example, by using Definition (3) proposed by Xu et al. [4], E0 can be transferred as

Then, for

Table 2 illustrates that the suitable intersection traffic control mode varies as the parameter. Specifically, when

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:

(i) 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.

(ii) 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.

(iii) 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.

Ethical Approval: This article does not contain any studies with human participants performed by any of the authors.

Funding Statement: The Fangwei Zhang’s work is partially supported by Shanghai Pujiang Program (No. 2019PJC062), the Natural Science Foundation of Shandong Province (No. ZR2021MG003), the Research Project on Undergraduate Teaching Reform of Higher Education in Shandong Province (No. Z2021046).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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*Computer Modeling in Engineering & Sciences*,

*135*

*(2)*, 1589-1602. https://doi.org/10.32604/cmes.2022.022431

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*Comput. Model. Eng. Sci.*, vol. 135, no. 2, pp. 1589-1602. 2023. https://doi.org/10.32604/cmes.2022.022431

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