Journal of Applied Mathematics

Journal of Applied Mathematics / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 432092 | https://doi.org/10.1155/2014/432092

Huimin Zhang, "Linguistic Intuitionistic Fuzzy Sets and Application in MAGDM", Journal of Applied Mathematics, vol. 2014, Article ID 432092, 11 pages, 2014. https://doi.org/10.1155/2014/432092

Linguistic Intuitionistic Fuzzy Sets and Application in MAGDM

Academic Editor: Reinaldo Martinez Palhares
Received19 Jan 2014
Revised05 Apr 2014
Accepted08 Apr 2014
Published29 Apr 2014

Abstract

To better deal with imprecise and uncertain information in decision making, the definition of linguistic intuitionistic fuzzy sets (LIFSs) is introduced, which is characterized by a linguistic membership degree and a linguistic nonmembership degree, respectively. To compare any two linguistic intuitionistic fuzzy values (LIFVs), the score function and accuracy function are defined. Then, based on -norm and -conorm, several aggregation operators are proposed to aggregate linguistic intuitionistic fuzzy information, which avoid the limitations in exiting linguistic operation. In addition, the desired properties of these linguistic intuitionistic fuzzy aggregation operators are discussed. Finally, a numerical example is provided to illustrate the efficiency of the proposed method in multiple attribute group decision making (MAGDM).

1. Introduction

Intuitionistic fuzzy set (IFS) [1], which is characterized by a degree of membership and a degree of nonmembership, is a very powerful tool to process vague information. After the pioneering study of Atanassov [1], the IFS has captured much attention from researchers in various fields and many achievements have been made, such as entropy measure of IFS [27], distance, or similarity measure between IFSs [813] and aggregation operators of IFS [1421]. In addition, related to IFS, some authors proposed several other tools to handle vague and imprecise information whereby two or more sources of vagueness appear simultaneously [22]. Atanassov and Gargov [23] introduced the notion of interval-valued intuitionistic fuzzy set (IVIFS), which is characterized by a membership function and a nonmembership function with interval values. Torra [24] and Torra and Narukawa [25] gave a definition of hesitant fuzzy set (HFS), which can better deal with the situations where several values are possible to determine the membership of an element. Zhu et al. [26] defined dual hesitant fuzzy set in terms of two functions that return two sets of membership values and nonmembership values, respectively, for each element in the domain.

Although, the foregoing fuzzy tools are suitable for dealing with problems that are defined as quantitative situations [22], uncertainty is often because of the vagueness of meanings that is used by experts in problems whose nature is rather qualitative. For example, for reason of the increasing complexity of the decision making environment, time pressure, and the lack of data or knowledge about the problem domain, in the process of decision making under intuitionistic fuzzy environment, a decision maker may have difficulty in expressing the degree of membership and nonmembership as exact values, whereas he or she may think the use of linguistic values is more straightforward and suitable to express the degree of membership and nonmembership. Similar to IFS, linguistic intuitionistic fuzzy set (LIFS) is characterized by a linguistic membership degree and a linguistic nonmembership degree, respectively. By using the LIFS, decision makers are able to consider a linguistic hesitancy degree in the belongingness of an element to a set, where they cannot easily express their subjective judgment with a single linguistic term.

The outline of the paper is organized as follows. The following section presents a brief introduction to the basic knowledge that will be used in the definition of LIFS. Section 3 gives the concept of LIFS and constructs the score function and accuracy function for LIFS. Section 4 develops several aggregation operators for LIFS. Section 5 proposes a MAGDM method with linguistic intuitionistic fuzzy information. In Section 6, an application of the new approach is presented. Finally, conclusions are provided in Section 7.

2. Preliminaries

In the following, some basic concepts and knowledge related to IFS and linguistic approach are briefly described.

Definition 1 (see [1]). Let be a universal set. An IFS in is given as where the functions , stand for the degree of membership and nonmembership of the element to , respectively. Any meets the condition .

is called intuitionistic index or degree of indeterminacy of to . Obviously, if , IFS is reduced to a fuzzy set.

Some basic definitions and operations on IFS are presented as follows.

Definition 2 (see [14, 15]). If and are two IFSs of the set , then(1) if and only if , , and ;(2), where is the complement of ;(3);(4);(5);(6);(7),  ;(8), , .

In real world, many decision making problems present qualitative aspects that are complex to assess by means of numerical values. In such cases, it may be more suitable to consider them as linguistic variables.

Let be a finite linguistic term set with odd cardinality, where represents a possible linguistic term for a linguistic variable. For example, a set of seven terms can be expressed as follows: It is required that the linguistic term set should satisfy the following characteristics [2730].(1)The set is ordered: , if and only if .(2)There is a negation operator: such that .(3)Max operator: , if and only if .(4)Min operator: , if and only if .

To preserve all the given information, Xu [31] extended the discrete term set to a continuous linguistic term set , where, if , then is called the original linguistic term. Otherwise, is called the virtual linguistic term.

Definition 3 (see [31, 32]). Consider any two linguistic terms , , and , the add and multiply operations of linguistic variable are defined as follows: -norm and -conorm have been widely used to construct operations for fuzzy sets and IFSs.

Definition 4 (see [33, 34]). A -norm is a mapping satisfying, for all ,(1);(2);(3);(4) whenever .

The four basic -norms ,  ,  , and are given as follows:, (lattice operation);, (algebraic operation);, (Lukasiewicz operation); (drastic operation).

Definition 5 (see [33, 34]). A -conorm is a mapping satisfying, for all ,  ;  ;  ;   whenever .

The four basic -conorms , , , and are given as follows:, (lattice operation);, (algebraic operation);, (Lukasiewicz operation); (drastic operation).

3. Linguistic Intuitionistic Fuzzy Set

The concept of linguistic intuitionistic fuzzy set (LIFS) is given as follows.

Definition 6. Let be a finite universal set and a continuous linguistic term set. A LIFS in is given as where stand for the linguistic membership degree and linguistic nonmembership of the element to , respectively.

For any , the condition is always satisfied. is called linguistic indeterminacy degree of to . Obviously, if , then LIFS has the minimum linguistic indeterminacy degree, that is, , which means the membership degree of to can be precisely expressed with a single linguistic term and LIFS is reduced to a linguistic variable. Oppositely, if , then LIFS has the maximum linguistic indeterminacy degree; that is, . Similar to IFS, the LIFS can be transformed into an interval linguistic variable , which indicates that the minimum and maximum linguistic membership degrees of the elements to are and , respectively.

For notational simplicity, we suppose both LIFS and contain only one element, which stand for linguistic intuitionistic fuzzy values (LIFVs), that is, the pairs and .

To compare any two LIFVs, the score function and accuracy function are defined as follows.

Definition 7. Let and be two LIFVs, with . The score function of is defined as and the accuracy function is defined as
Thus, and can be ranked by the following procedure:(1)if , then ;(2)if and(a), then ;(b), then ;(c), then .
It is easy to see that and , which means , .

Example 8. Let ,  , and be LIFVs, which are derived from .
Applying formulas (5) and (6), we have
Thus, we obtain .

4. Aggregation Operators for Linguistic Intuitionistic Fuzzy Sets

Since the definition of LIFS is given, it is necessary to introduce the operations and computations between them.

Definition 9. Let and be two LIFVs; then(1) if and only if and ;(2), where is the complement of ;(3)the intersection of and : ;(4)the union of and : .
Motivated by -norm and -conorm, we propose the following operation laws for linguistic variables.

Definition 10. Considering any two linguistic terms , the add and multiply operations of linguistic variable are defined as follows: where and are -conorm and -norm, respectively.

Since , , we have , , which indicate the operation results match the original linguistic term set ; that is, ,  . In addition, it is worth noting that, because of the monotonicity of -conorm and -norm, the values of function and are monotonically increasing with the increasing of and , which means the operation results obtained by (8) and (9) are in accord with our intuition.

If we take the well-known and into (8) and (9), respectively, then they can be rewritten as follows:

Example 11. Let . Applying (10), we have , , , and .
Thus, we obtain and . Such results seem to be intuitive and can be easily accepted.

Alternatively, if we take the operation laws of Definition 3, we have , , where the subscripts of and are bigger than the cardinality of linguistic term set . In addition, if we extend the discrete term set to a continuous term set , [35] where    is a sufficiently large positive integer, there is an unavoidable question on how to define the semantics for and . Obviously, or has different semantics in different linguistic term set with different cardinalities. As a result, it is unrealistic to assign semantics to a given linguistic value derived from linguistic term set with variable cardinality. If we follow the method of and [36] for ,, then we have and . Such results seem to be counter-intuitive and may not be easily accepted. Applying (10), we can overcome the limitations that the subscripts of the linguistic variable are bigger than the cardinality of the corresponding linguistic term set and obtain results agreed with our intuition.

Based on (10), we can get the following operation laws for LIFVs.

Definition 12. Let and be two LIFVs, where , , , with ; then Some special cases of and are obtained as follows.
If , then
If , then
If , then
If , then
If , then

Theorem 13. Let and be two LIFVs, where with . Then, one has(1),(2),(3),(4).

Proof. By (11), we have . Thus, based on (13), we have
Similarly, since and , then
Hence, we obtain .
By (13), we have and ; thus, we obtain
By (14), we get and ; thus, based on (12), we have
Since , then, by (14), we have
Hence, we obtain .
By (14), we have and ; thus, we have which completes the proof of Theorem 13.

Motivated by the intuitionistic fuzzy aggregation operators [14, 15], in what follows, we define some aggregation operators for LIFVs.

Definition 14. Let   () be a set of LIFVs. Then, the linguistic intuitionistic fuzzy weighted averaging (LIFWA) operator is defined as where is the weight vector of   (), with and .

In Particular, if , then the LIFWA operator is reduced to a linguistic intuitionistic fuzzy averaging (LIFA) operator; that is,

Based on Definition 14, we get some properties of the LIFWA operator.

Theorem 15. Let   () be a set of LIFVs and the weight vector of   (), with and  ; then one has the following.(1)Idempotency. If all   () are equal, that is, , for any , then (2)Monotonicity. Let   () be a set of LIFVs. If and , for any , then for any .(3)Boundary. Consider

Proof. Since , for any , then
If , that is, , for any , then we have
Similarly, when , for any , we can get .
According to Definition 7, we obtain that is,
Since and , for any , then, based on the monotonicity of Theorem 15, we derive

Definition 16. Let   () be a set of LIFVs. Then, the linguistic intuitionistic fuzzy ordered weighted averaging (LIFOWA) operator is defined as where is the th largest of and is the associated weight vector of , with and .

Similar to Theorem 15, we have some properties of the LIFOWA operator.

Theorem 17. Let    be a set of linguistic intuitionistic fuzzy values and the associated weight vector of   , with and ; then one has the following.(1)Idempotency. If all    are equal, that is, , for any , then (2)Monotonicity. Let    be a set of linguistic intuitionistic fuzzy values. If and , for any , then for any .(3)Boundary. Consider (4)Commutativity. Let    be a set of linguistic intuitionistic fuzzy values, then for any where is any permutation of .

Definition 18. Let    be a set of LIFVs. Then, the linguistic intuitionistic fuzzy weighted geometric (LIFWG) operator is defined as where is the weight vector of   , with and .

The LIFWG operator has the following properties.

Theorem 19. Let    be a set of LIFVs and the weight vector of   , with and ; then one has the following.(1)Idempotency. If all    are equal, that is, , for any , then (2)Monotonicity. Let    be a set of linguistic intuitionistic fuzzy values. If and , for any , then for any .(3)Boundary. Consider

Definition 20. Let    be a set of LIFVs. Then, the linguistic intuitionistic fuzzy ordered weighted geometric (LIFOWG) operator is defined as where is the th largest of and is the associated weight vector of   ), with and .

Similar to Theorem 15, we have some properties of the LIFOWG operator.

Theorem 21. Let    be a set of LIFVs and the associated weight vector of   , with and ; then one has the following.(1)Idempotency. If all    are equal, that is, , for any , then (2)Monotonicity. Let    be a set of linguistic intuitionistic fuzzy values. If and , for any , then for any .(3)Boundary. Consider (4)Commutativity. Let    be a set of linguistic intuitionistic fuzzy values; then for any where is any permutation of .

Lemma 22 (see [37, 38]). Let ,  ,  , and ; then with equality if and only if .

Based on Lemma 22, we have the following theorem.

Theorem 23. Let be a set of LIFVs; then one has with equality if and only if .

Proof. Let be the weight vector of   , with and ; then, by Lemma 22, we have , with equality if and only if ; that is, , with equality if and only if ; and , with equality if and only if ; that is, , with equality if and only if .
Consequently, by Definition 7, we obtain , with equality if and only if and ; that is, with equality if and only if.
Similarly, we can also prove , with equality if and only if .
Besides the above properties, we can derive the following desirable results of the LIFOWA and LIFOWG operators.

Theorem 24. Let    be a set of LIFVs and the associated weight vector of    , with and . Then one has the following.(1)If , then (2)If , then