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Journal of Applied Mathematics

Volume 2013 (2013), Article ID 670285, 33 pages

http://dx.doi.org/10.1155/2013/670285

## Interval-Valued Intuitionistic Hesitant Fuzzy Aggregation Operators and Their Application in Group Decision-Making

College of Mathematics and Computer Science, Hebei University, Baoding, Hebei 071002, China

Received 4 March 2013; Revised 30 May 2013; Accepted 17 June 2013

Academic Editor: Luis Javier Herrera

Copyright © 2013 Zhiming Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Hesitant fuzzy sets, permitting the membership of an element to be a set of several possible values, can be used as an efficient mathematical tool for modelling people’s hesitancy in daily life. In this paper, we extend the hesitant fuzzy set to interval-valued intuitionistic fuzzy environments and propose the concept of interval-valued intuitionistic hesitant fuzzy set, which allows the membership of an element to be a set of several possible interval-valued intuitionistic fuzzy numbers. The aim of this paper is to develop a series of aggregation operators for interval-valued intuitionistic hesitant fuzzy information. Then, some desired properties of the developed operators are studied, and the relationships among these operators are discussed. Furthermore, we apply these aggregation operators to develop an approach to multiple attribute group decision-making with interval-valued intuitionistic hesitant fuzzy information. Finally, a numerical example is provided to illustrate the application of the developed approach.

#### 1. Introduction

In many practical problems, when defining the membership degree of an element, the difficulty of establishing the membership degree is not because we have a margin of error (as in intuitionistic fuzzy sets [1] and interval-valued fuzzy sets [2]) or some possibility distribution (as in type 2 fuzzy sets [3]) on the possibility values, but because we have several possible numerical values. To deal with such cases, Torra [4] introduced the concept of hesitant fuzzy set to permit the membership of an element to be a set of several possible values between 0 and 1, which can depict the human’s hesitance more objectively and precisely.

It should be noted that hesitant fuzzy sets permit the membership of an element to be a set of several possible values. All these possible values are crisp real numbers that belong to . However, in the process of some practical decision-makings, sometimes, due to the time pressure and lack of knowledge or data or the decision makers’ (DMs) limited attention and information processing capacities, the DMs cannot provide their evaluations with a single numerical value, a margin of error, some possibility distribution on the possible values, several possible numerical values, several possible interval numbers, or several possible intuitionistic fuzzy numbers but several possible interval-valued intuitionistic fuzzy numbers. For example, to get a reasonable decision result, a decision organization, which contains a lot of decision makers, is required to estimate the degree that an alternative satisfies an attribute. Suppose there are three cases: some decision makers provide , some assign , and the others provide , and these three parts cannot persuade each other to change their opinions. We can easily see that such cases cannot be dealt with by fuzzy sets [5], hesitant fuzzy sets, and their extensions, such as interval-valued fuzzy sets, intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets, type 2 fuzzy sets, interval-valued hesitant fuzzy sets [6], and generalized hesitant fuzzy sets [7]. Thus, it is very necessary to introduce a new extension of hesitant fuzzy sets to address this issue. The aim of this paper is to present the notion of interval-valued intuitionistic hesitant fuzzy set, which extends the hesitant fuzzy set to interval-valued intuitionistic fuzzy environments and permits the membership of an element to be a set of several possible interval-valued intuitionistic fuzzy numbers. Thus, interval-valued intuitionistic hesitant fuzzy set is a very useful tool to deal with the situations in which the experts hesitate between several possible interval-valued intuitionistic fuzzy numbers to assess the degree to which an alternative satisfies an attribute. In the previous example, the degree to which the alternative satisfies the attribute can be represented by an interval-valued intuitionistic hesitant fuzzy set . Moreover, in many multiple attribute group decision-making (MAGDM) problems, considering that the estimations of the attribute values are interval-valued intuitionistic hesitant fuzzy sets, it therefore is very necessary to give some aggregation techniques to aggregate the interval-valued intuitionistic hesitant fuzzy information. However, we are aware that the existing aggregation techniques have difficulty in coping with group decision-making problems with interval-valued intuitionistic hesitant fuzzy information. Therefore, we in the current paper propose a series of aggregation operators for aggregating the interval-valued intuitionistic hesitant fuzzy information and investigate some properties of these operators. Then, based on these aggregation operators, we develop an approach to MAGDM with interval-valued intuitionistic hesitant fuzzy information. Moreover, we use a numerical example to show the application of the developed approach.

In order to do this, this paper is organized as follows. Section 2 introduces some concepts and properties of interval-valued intuitionistic hesitant fuzzy sets. In Section 3, we present a series of aggregation operators for interval-valued intuitionistic hesitant fuzzy information and examine the relationships among these aggregation operators. Section 4 develops an approach to group decision-makings with interval-valued intuitionistic hesitant fuzzy information. In the sequel, the application of the developed approach in group decision-making problems is shown by an illustrative example in Section 5. The final section offers some concluding remarks.

#### 2. Preliminaries

In this section, we will briefly introduce the basic notions of hesitant fuzzy sets [4], interval-valued intuitionistic fuzzy sets [8], and interval-valued intuitionistic hesitant fuzzy sets.

##### 2.1. Hesitant Fuzzy Sets and Hesitant Fuzzy Elements

*Definition 1 (see [4]). *Let be a fixed set; a hesitant fuzzy set (HFS) on is given in terms of a function that when applied to returns a subset of .

To be easily understood, we express the HFS by a mathematical symbol

where is a set of some values in , denoting the possible membership degrees of the element to the set . For convenience, Xia and Xu [9] called a hesitant fuzzy element (HFE) and the set of all HFEs.

Given three HFEs represented by , , and , Torra [4] defined some operations on them, which can be described as

Furthermore, in order to aggregate hesitant fuzzy information, Xia and Xu [9] defined some new operations on the HFEs , , and :

To compare the HFEs, Xia and Xu [9] defined the following comparison laws.

*Definition 2 (see [9]). *For an HFE , is called the score function of , where is the number of the elements in . For two HFEs and , if , then ; if , then .

##### 2.2. Interval-Valued Intuitionistic Fuzzy Sets and Interval-Valued Intuitionistic Fuzzy Numbers

*Definition 3 (see [8]). *Let be an ordinary nonempty set. An interval-valued intuitionistic fuzzy set in is an object that has the form

where and satisfy for all and are, respectively, called the membership degree and the nonmembership degree of the element to .

Xu [10] called each pair in an interval-valued intuitionistic fuzzy number (IVIFN), and, for convenience, each IVIFN can be simply denoted by , where , , and . Let be the set of all IVIFNs.

Xu [10] introduced the following operational laws for IVIFNs.

*Definition 4 (see [10]). *Let , , and be any three IVIFNs. Then,

Theorem 5 (see [10]). *Let , , and be any three IVIFNs. Then, , , , and are also IVIFNs.*

Xu [10] introduced the score function to get the score of and defined an accuracy function to evaluate the accuracy degree of . Xu [10] gave an order relation between two IVIFNs and .(1)If , then .(2)If , then the following hold.(a)If , then .(b)If , then .(c)If , then .

##### 2.3. Interval-Valued Intuitionistic Hesitant Fuzzy Sets and Interval-Valued Intuitionistic Hesitant Fuzzy Elements

In the following, we propose the concept of interval-valued intuitionistic hesitant fuzzy sets, which permit the membership of an element to be a set of several possible interval-valued intuitionistic fuzzy numbers. The motivation is that when defining the membership degree of an element, the difficulty of establishing the membership degree is not because we have a margin of error (as in intuitionistic fuzzy sets) or some possibility distribution (as in type 2 fuzzy sets) on the possible values, but because we have several possible interval-valued intuitionistic fuzzy numbers.

*Definition 6. *Let be a fixed set; an interval-valued intuitionistic hesitant fuzzy set (IVIHFS) on is given in terms of a function that when applied to returns a subset of .

To be easily understood, we express the IVIHFS by a mathematical symbol

where is a set of some IVIFNs in , denoting the possible membership degree intervals and nonmembership degree intervals of the element to the set . For convenience, we call an interval-valued intuitionistic hesitant fuzzy element (IVIHFE) and the set of all IVIHFEs. If , then is an IVIFN, and it can be denoted by .

For any , if is a real number in , then reduces to a hesitant fuzzy element (HFE) [9]; if is a closed subinterval of the unit interval, then reduces to an interval-valued hesitant fuzzy element (IVHFE) [6]; if is an intuitionistic fuzzy number (IFN) [11], then reduces to an intuitionistic hesitant fuzzy element (IHFE). Therefore, HFEs, IVHFEs, and IHFEs are special cases of IVIHFEs.

*Definition 7. *Given three IVIHFEs represented by , , and , one defines some operations on them, which can be described as

Theorem 8. *Let , , and be three IVIHFEs, and . Then, , , , and are also IVIHFEs.*

*Proof. *Since , , and are three IVIHFEs, we have , , , and . Then, we can obtain

Thus, , , , and are IVIHFEs.

Theorem 9. *Let , , and be three IVIHFEs, and , and . Then, one has
*

*Proof. *Consider

Thus, we have .

Moreover,

Thus, we have that .

Take

Thus, we have that .

Consider

Thus, we have that .

To compare the IVIHFEs, we define the following comparison laws.

*Definition 10. *For an IVIHFE , is called the score function of , where is the number of the elements in . is called the accuracy function of . For any two IVIHFEs and ,(1)if , then ;(2)if , then the following hold.(a)If , then .(b)If , then .(c)If , then .

#### 3. Aggregation Operators for Interval-Valued Intuitionistic Hesitant Fuzzy Information

In the current section, we propose a series of operators for aggregating the interval-valued intuitionistic hesitant fuzzy information and investigate some desired properties of these operators.

##### 3.1. The IVIHFWA, IVIHFWG, GIVIHFWA, and GIVIHFWG Operators

*Definition 11. *Let () be a collection of IVIHFEs, and let be the weight vector of () with and . An interval-valued intuitionistic hesitant fuzzy weighted averaging (IVIHFWA) operator is a mapping such that

If especially, then the IVIHFWA operator reduces to the interval-valued intuitionistic hesitant fuzzy averaging (IVIHFA) operator:

Theorem 12. *Let be a collection of IVIHFEs. Then, their aggregated value calculated using the IVIHFWA operator is an IVIHFE and
*

*Proof. *The first result follows quickly from Definition 11 and Theorem 8. In the following, we prove the second result by using mathematical induction on . First, we show that (16) holds for .

Because
we have

If (16) holds for , in other words,
then, when , IVIHFE operations yield

In other words, (16) holds for . Equation (16) therefore holds for all .

This completes the proof of Theorem 12.

*Definition 13. *Let be a collection of IVIHFEs, and let be the weight vector of with and . An interval-valued intuitionistic hesitant fuzzy weighted geometric (IVIHFWG) operator is a mapping such that

Especially, if , then the IVIHFWG operator reduces to the interval-valued intuitionistic hesitant fuzzy averaging (IVIHFA) operator:

Theorem 14. *Let be a collection of IVIHFEs. Then, their aggregated value calculated using the IVIHFWG operator is an IVIHFE, and
*

In the following, by combining the IVIHFWA and IVIHFWG operators with the generalized mean [12], we develop the generalized interval-valued intuitionistic hesitant fuzzy weighted averaging (GIVIHFWA) operator and the generalized interval-valued intuitionistic hesitant fuzzy weighted geometric (GIVIHFWG) operator, respectively. The main characteristic of the GIVIHFWA and GIVIHFWG operators is that they have an additional parameter controlling the power to which the argument values are raised. Different from the IVIHFWA and IVIHFWG operators, the GIVIHFWA and GIVIHFWG operators extend them with addition of a parameter controlling the power to which the argument values are raised. When we use different choices of the parameters , we will get some special cases.

*Definition 15. *Let () be a collection of IVIHFEs, and let be the weight vector of () with and .(1) A generalized interval-valued intuitionistic hesitant fuzzy weighted averaging (GIVIHFWA) operator is a mapping , where
with .(2) A generalized interval-valued intuitionistic hesitant fuzzy weighted geometric (GIVIHFWG) operator is a mapping , where
with .

If , then the GIVIHFWA operator reduces to the IVIHFWA operator and the GIVIHFWG operator reduces to the IVIHFWG operator.

Using IVIHFE operations and mathematical induction on , (24) and (25) can be transformed into the following forms:

*Example 16. *Suppose that , , and are three IVIHFEs, and is their weight vector. Then, by Definition 15, we can obtain