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Articles

Type-2 Hesitant Fuzzy Sets

Liu FengInstitute of Systems Science, Northeastern University, Shenyang, People's Republic of China;Dalian Survey Group of the National Statistics Bureau, Dalian, People's Republic of ChinaCorrespondence[email protected]
,
Fan Chuan-qiangInstitute of Systems Science, Northeastern University, Shenyang, People's Republic of China;School of Science, Liaoning Shihua University, Fushun, People's Republic of China
&
Xie Wei-heDepartment of Naval Gun, Dalian Naval Academy, Dalian, People's Republic of China
Pages 249-259 | Published online: 25 Oct 2018

ABSTRACT

By using type-2 fuzzy sets and hesitant fuzzy sets, type-2 hesitant fuzzy sets are defined and their mathematical structure and characteristics are given. The relations of the structures of type-2 hesitant fuzzy sets and hesitant fuzzy sets are further studied. Consequently, we prove that type-2 hesitant fuzzy sets are the generalization of hesitant fuzzy sets. Type-2 hesitant fuzzy sets may deal with the problem that hesitant fuzzy sets can't have repeated memberships. Otherwise, a part of the special type-2 hesitant fuzzy sets can be changed into discrete type-2 fuzzy sets, but their operations have many differences. On dealing with the fact problems, type-2 hesitant fuzzy sets are the better methods to solve the problems and can be easy to get better results.

1. Introduction

In 1965, fuzzy sets [Citation1–5] were firstly introduced by L. A. Zadeh and have been widely used in many areas of artificial intelligence and control. Since then several extensions have been developed and are applied to practice and got many good results, such as intutionistic fuzzy sets [Citation6–7], type-2 fuzzy sets [Citation8–27], and hesitant fuzzy sets [Citation28–32]. Especially, type-2 fuzzy sets [Citation8–27] allow the membership of a given element as a fuzzy set and have been widely researched and application, for example, interval type-2 fuzzy systems [Citation33–35] etc. By now the structures of type-2 fuzzy sets are very perfectly, and have made a lot of the actual application results. At the same time, hesitant fuzzy sets are new theory now, and also permits their membership having a set of possible values. Hesitant fuzzy sets are very similar to the fuzzy sets. However, hesitant fuzzy sets were firstly put forward in 2010, there are a lot of problems and new theories need to be researched, and they affect their practical application. Such as the new theory of type-2 hesitant fuzzy sets that are parallel to the type-2 fuzzy sets have never been bought up by now. Thus in order to in-depth study hesitant fuzzy sets and expand the application scopes of hesitant fuzzy sets, so according to the theory of type-2 fuzzy sets, the new theory of type-2 hesitant fuzzy sets are firstly proposed in this paper. For the next step of study interval type-2 hesitant fuzzy systems and their application, type-2 hesitant fuzzy sets are the first step and the necessary theory foundation.

Hesitant fuzzy sets have been applied in some decision making problems and received some good practical effects [Citation29–32] now. In order to widely use hesitant fuzzy sets, so it is necessary to further in-depth study the theory of hesitate to fuzzy sets. Firstly, On the basis of the basic theories of type-2 fuzzy sets and hesitant fuzzy sets, type-2 hesitant fuzzy sets are firstly defined and their mathematical structure and characteristics are also given. Then the relations of the structures of type-2 hesitant fuzzy sets and hesitant fuzzy sets are further studied. Because of the discrepancy of sets itself, the elements of hesitant fuzzy sets can't be repeated. But the fact has a lot of repeated possibilities. Type-2 hesitant fuzzy sets solve the above defect of hesitant fuzzy sets, thus the new type-2 hesitant fuzzy sets are a bit of promotion of old hesitant fuzzy sets. The formations of the definitions of type-2 hesitant fuzzy sets and discrete type-2 fuzzy sets are very similar but their basic operations have some different points, their basic operations can't be completely transformed each other except for very special circumstances. On dealing with the fact problems, type-2 hesitant fuzzy sets are closer to the fuzzy reality, and can cover more fuzzy information, and are the new methods to solve the problems and can be easy to get better results, and can open up a new path like type-2 fuzzy sets.

In order to do that, the remainders of the paper are organized as follows. In Section 2, the basic theories of hesitant fuzzy sets and fuzzy sets are introduced. In Section 3, type-2 hesitant fuzzy sets are defined and their mathematical structure and characteristics are also given. In section 4, the relations of the structures of type-2 hesitant fuzzy sets and hesitant fuzzy sets are further studied. Section 5 prove that a part of the special type-2 hesitant fuzzy sets can be changed into type-2 hesitant fuzzy sets. Section 6, type-n hesitant fuzzy sets are defined and their mathematical structure and characteristics are also given. Section 7 gives some concluding remarks.

2. Hesitant Fuzzy Sets and Discrete Type-2 Fuzzy Sets

In this section, we review the basic conceptions and operations of fuzzy sets and hesitant fuzzy sets, because they will be used later to discuss type-2 hesitant fuzzy sets.

Definition 2.1:

[Citation28] Given a reference set X, a fuzzy set A on X is represented in terms of a function X→[0,1], for ∈X.

Definition 2.2:

[Citation28] Let X be a reference set, then hesitant fuzzy sets are defined on X in terms of a function h that when applied to X returns a subset of [0,1].

Definition 2.3:

[Citation28] Let M = {,,, … ,} be a set of N membership functions, then hesitant fuzzy sets associated with M, that is , is defined as .

Note: in this paper, we only discuss the kind of hesitant fuzzy sets that are given by definition 3.

Given a hesitant fuzzy set represented by its membership function h,, its lower, upper,α-lower andα-upper bound are defined as follows:

  1. lower bound: = min ;

  2. upper bound: = max

  3. α -lower bound: = {h∈};

  4. α -upper bound: = {h∈}.

Definition 2.4:

[Citation28] Given a hesitant fuzzy set represented by its membership function h, its complement as follows:

Definition 2.5

[Citation28] Given two hesitant fuzzy sets represented by their membership function h1 and h2, their union and intersection that are respectively represented by h1 ∪ h2 and h1 ∩ h2 are defined as follows:

  1. = {}, or equivalently,

  2. = {, for ;

  3. = {}, or equivalently,

  4. = {, for .

Definition 2.6

[Citation21, Citation22] A discrete type-2 fuzzy set is one whose primary varable and secondary membership functions are discrete (sampled), and it is represented as:

In this equation, + also denotes union. Note that has been discretized into M values.

3. Type-2 Hesitant Fuzzy Sets

On the basis of the basic theories of type-2 fuzzy sets and hesitant fuzzy sets, type-2 hesitant fuzzy sets are firstly defined, then their mathematical structure and characteristics are also given.

In order to facilitate the following comparison and unify mathematical symbol, the following the size comparison of two-dimensional coordinate points are defined as follows:

Definition 3.1:

Suppose (a,b), (c,d), (e, f) and (l,t) ∈ [0,1] × [0,1], then

  1. (a,b) ≤ (c,d) ⇔ a ≤ c and b ≤ d;

  2. (a,b) ≥ (c,d) ⇔ a ≥ c and b ≥ d;

  3. (a,b) = (c,d) ⇔ a = c and b = d;

  4. (e,f) = max{(a,b), (c,d)} = (max{a, c}, max{b,d));

  5. (l,t) = min{(a,b), (c,d)} = (min{a, c}, min{b,d)).

Example 3.1:

(3,4) ≤ (3,5), (5,7) ≥ (4,4), max{(3,4), (4,3)} = (4,4), min{(3,4), (4,3)} = (3,3).

Then imitating the definitions and operations of hesitant fuzzy sets, the corresponding conceptions of type-2 hesitate fuzzy sets are step by step given.

Definition 3.2:

Let X be a reference set, then type-2 hesitant fuzzy sets are defined on X in terms of a function h that when applied to X returns a subset of [0,1] × [0,1].

Definition 3.3:

Let M2 = {, ,, … , } be a set of N dual membership functions, then type-2 hesitant fuzzy sets associated with M2, that is defined as .

Note: (1) empty set: = {(0,0)} for all ;

(2) full set: ={(1,1)}for all .

Example 3.2:

Let X be a reference set, , are two the membership functions of type-2 hesitant fuzzy sets. Then are two type-2 hesitant fuzzy sets on X.

Given a type-2 hesitant fuzzy set represented by its membership function , and M2 = {,, … , },then its lower, upper,α-lower and α-upper bound are defined as follows:

  1. lower bound:  = (min , min );

  2. upper bound:  = (max , max ).

  3. (α,β)-lower bound:  = { ∈ };

  4. (α,β)-upper bound:  = { ∈ }.

Example 3.3:

Let X be a reference set, , then

Definition 3.4:

Given an hesitant fuzzy set represented by its membership function , and M2 = {,, … , } its complement as follows:

Proposition 3.1

The complement is involutive, i.e.

Proof

It is easy to be proved, so omitted.

Definition 3.5

Given two type-2 hesitant fuzzy sets represented by their membership function and , their union and intersection that are respectively represented by ∪ and ∩ are defined as follows:

Example 3.4:

Let X be a reference set, , are two the membership functions of type-2 hesitant fuzzy sets. Then = {(0.4,0.7), (0.8,1.0), (1.0,0.9)}, (0.4,0.5), (0.6,0.3)}.

4. Type-2 Hesitant Fuzzy Sets and Hesitant Fuzzy Sets

Type-2 hesitant fuzzy sets are the generalization of hesitant fuzzy sets. At the same time, type-2 hesitant fuzzy sets keep the operation rules of original hesitant fuzzy sets unchanged.

Proposition 4.1:

Hesitant fuzzy sets are the special cases of type-2 hesitant fuzzy sets.

Proof:

Suppose the hesitant fuzzy sets associated with M, that is , let M2 = {, , , … ,} be a set of N dual membership functions, then , then the hesitant fuzzy sets are the special case of type-2 hesitant fuzzy sets.

Proposition 4.2:

Hesitant fuzzy sets are changed into type-2 hesitant fuzzy sets, then new type-2 hesitant fuzzy sets and the original hesitant fuzzy sets have the following operational formulas.

  1. lower bound:  = (min , 1) = (, 1);

  2. upper bound:  = (max , 1) = (, 1).

  3. (α,1)-lower bound:  = {∈} = ;

  4. (α,1)-upper bound:  = {∈} = 

  5. When , then ;

  6. When = (, then

  7. When  = (, then

Proof:

It is easy to be proved, so omitted.

Proposition 4.3

Type-2 hesitant fuzzy sets can be changed into hesitant fuzzy sets .

Proof

As M2 = {, , , … ,} is a set of N dual membership functions, let M = {, , , … ,}, then let g :M2 M, , k = 1,2, … ,N, and then .

So type-2 hesitant fuzzy sets can be changed into hesitant fuzzy sets.

Example 4.1

In some event, please six experts to appraise the qualities of the three articles, the symbols on behalf of the experts and their quantification of the position in the industry are denoted as , , , , , . In order to facilitate, three papers are recorded as 1,2,3, the back corresponding numerical values representative of the expert's overall evaluation scores of articles. Then six functions defined as follows: And which is the domain of the functions = {1,2,3}, then establishing a hesitant fuzzy set: then Then the above type-2 hesitant fuzzy set can be changed into the hesitant fuzzy set is From the above type-2 hesitant fuzzy set and hesitant fuzzy set, it is obviously that paper 2 is better than paper 1 and paper1 is better than paper 3.

5. Type-2 Hesitant Fuzzy Sets and Discrete Type-2 Fuzzy Sets

The formations of the definitions of type-2 hesitant fuzzy sets and discrete type-2 fuzzy sets are very similarly, but their basic operations are almost completely different, their basic operations can't be completely transformed each other except for very special circumstances.

Proposition 5.1:

Type-2 hesitant fuzzy sets can be changed into discrete type-2 fuzzy sets in the form of the definitions.

Proof:

Suppose the membership function of the type-2 hesitant fuzzy set is as follows: then type-2 hesitant fuzzy set is Then let ∪ is replaced with ∑ and is replaced with ,

So type-2 hesitant fuzzy set is changed into the form: Thus type-2 hesitant fuzzy sets can be changed into discrete type-2 fuzzy sets in the form of the definitions.

Example 5.1:

Let X be a reference set, , are two the membership functions of type-2 hesitant fuzzy sets. Then = {(0.4,0.7), (0.8,1.0), (1.0,0.9)}, (0.4,0.5), (0.6,0.3)}. But the generated two discrete type - 2 fuzzy sets and , then = + + = + + , = + + = + + . Then and are converted into discrete type -2 fuzzy sets, = + + , = + + , it is clearly that ≠ , ≠ .

Only when = , = .

So their basic operations are almost completely different, their basic operations can't be completely transformed each other except for very much special circumstances.

6. Type-n Hesitant Fuzzy Sets

Because of the discrepancy of set itself, the elements of hesitant fuzzy sets can't be repeated. But the fact has a lot of repeated possibilities. Type-2 hesitant fuzzy sets solve the mainly defect of hesitant fuzzy sets, thus the new type-2 hesitant fuzzy sets are the generalization of hesitant fuzzy sets. But for the same reason, the elements of type-2 hesitant fuzzy sets can't be repeated. So it needs to define a higher layer of the hesitant fuzzy set to solve this problem. Thus it is necessary to type-hesitant fuzzy sets are defined below.

Example 6.1

Let h is a hesitant fuzzy set, h = , it is obviously that membership functions of each element haven't duplicate values. If 0.8 and 0.6 all appear twice, let

, then the above problem is solved.

In the same way, the duplicate membership function value of type-2 hesitate fuzzy sets question can be solved through the type-3 hesitant fuzzy sets. Similarly, repeated problems can be solved through the way of upgrade the dimension of hesitate fuzzy sets.

In order to facilitate the following comparison and unify mathematical symbol, the following the size comparison of two-dimensional coordinate points are defined as follows:

Definition 6.1

Suppose (a1,a2, … ,an), (b1,b2, … ,bn), (d1,d2, … ,dn) and (c1,c2, … ,cn)∈[0,1]n, then

  1. (a1,a2, … ,an)≤(b1,b2, … ,bn) ⇔ ai ≤ bi,for i = 1,2, … , n;

  2. (a1,a2, … ,an)≥(b1,b2, … ,bn) ⇔ ai ≥ bi, for i = 1,2, … , n;

  3. (a1,a2, … ,an) = (b1,b2, … ,bn) ⇔ ai = bi, for i = 1,2, … , n;

  4. (d1,d2, … ,dn) = max{(a1,a2, … ,an), (b1,b2, … ,bn)}⇔ (max{a1,b1}, max{a1,b1}, … ,max{an, bn}), for i = 1,2, … , n;

  5. (c1,c2, … ,cn) = min{(a1,a2, … ,an), (b1,b2, … ,bn)}⇔ (min{a1,b1}, min{a1,b1}, … , min{an, bn}), for i = 1,2, … , n;

Definition 6.2

Let X be a reference set, then type-n hesitant fuzzy sets are defined on X in terms of a function h that when applied to X returns a subset of [0,1]n.

Definition 6.3

Let Mn = {, , } be a set of N n-dimensions membership functions, then type-n hesitant fuzzy sets associated with Mn, that is defined as .

Note: (1) empty set: = {} for all ;

(2) full set: = for all .

Given a type-n hesitant fuzzy set represented by its membership function and Mn, then its lower, upper,α-lower and α-upper bound are defined as follows:

  1. lower bound: = (min , min min

  2. upper bound: = (max, max max

  3. (α1,α2,α3, … ,αn)-lower bound:

  4. (α1,α2,α3, … ,αn)-upper bound:

Definition 6.4

Given a hesitant fuzzy set represented by its membership function and Mn, its complement as follows:

Proposition 6.1:

The complement is involutive, i.e.

Definition 6.5

Given two type-n hesitant fuzzy sets represented by their membership function and , their union and intersection that are respectively represented by ∪ and ∩ are defined as follows: According to the need of solving practical problems, determine the level of the type of hesitant fuzzy sets. In general, type-2 hesitant fuzzy sets are enough to solve the problem.

7. Conclusions and Future Work

In this paper, type-2 hesitant fuzzy sets and their basic operations are firstly introduced. Then the relations of type-2 hesitant fuzzy sets and hesitant fuzzy sets are further studied. The relations of type-2 hesitant fuzzy sets and discrete type-2 fuzzy sets are also studied in the form. Because of the discrepancy of set itself, the elements of hesitant fuzzy sets can't be repeated, type-2 hesitant fuzzy sets solve the mainly defect, thus the new type-2 hesitant fuzzy sets are the generalization of hesitant fuzzy sets. In addition, type-n hesitant fuzzy sets and their basic operations are given.

In the future, the practical application and standard measures and distance of type-2 hesitate fuzzy sets will be an in-depth study in order to solve more and more practical problems. At last, the new interval type-2 hesitant fuzzy systems and their application will be further studied at once.

Disclosure statement

No potential conflict of interest was reported by the authors.

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