Research Article  Open Access
Weibin Deng, Changlin Xu, Jin Liu, Feng Hu, "A Novel Distance between Vague Sets and Its Applications in Decision Making", Mathematical Problems in Engineering, vol. 2014, Article ID 281095, 10 pages, 2014. https://doi.org/10.1155/2014/281095
A Novel Distance between Vague Sets and Its Applications in Decision Making
Abstract
A novel distance between vague sets (VSs) is presented after the inadequacies of existing distance measures between vague sets are analyzed by artificial vague sets. The proposed method investigates the assignment of degree of hesitation to the membership and nonmembership degree, and the properties are also discussed. The performances of the new method are illustrated by pattern classification problem. Finally, the proposed method is applied into multicriteria fuzzy decision making, where the linear programming method is taken to generate optimal weights for every criterion and the best alternative is obtained by the weighted sum of distance measures between each alternative and the idea alternative with respect to a set of criteria. The experimental results show the effectiveness of the proposed method.
1. Introduction
A fuzzy set (FS) , as proposed by Zadeh [1], is a class of objects along with a degree of membership function, and the fuzzy sets theory has been applied widely in various fields [2]. The membership function , , assigns to each object a degree of membership ranging between 0 and 1; that is, . Obviously, , is a single value between 0 and 1. This single value combines the evidence for and the evidence against , without indicating how much there is of each. The single number tells us nothing about its accuracy. Therefore, as a generalization of fuzzy sets, Atanassov [3] introduced the concept of the intuitionistic fuzzy sets (IFSs) in 1983 and Gau and Buehrer [4] introduced the notion of vague sets (VSs) in 1993. Bustince and Burillo [5] showed that IFSs and VSs are equivalent. The VSs (or IFSs) consider the degree of membership, nonmembership, and hesitation of to , which make the VSs express the true state of uncertain information better than the fuzzy sets (FSs) [6]. The VSs have been successfully applied into edge detection [7], image segmentation [8, 9], fuzzy decision making [10–13], faulttree analysis [14], pattern recognition [15, 16], and so on.
As important contents in fuzzy mathematics, similarity measure and distance measure between VSs, which are involved in fuzzy decision making, pattern recognition, fuzzy reasoning, machine learning, and so forth, have attracted many researchers. At present, there are many distances between VSs, which can be divided into four categories. (1) Distances between VSs based on Hamming distance and Euclidean distance, for example, Atanassov [17] defined the Hamming distances and Euclidean distances between VSs in 1999. In 2000, Szmidt and Kacprzyk [18] considered the degree of hesitation into Atanassov’s VSs distances and redefined the Hamming distances and Euclidean distances of VSs, Liu [19] defined the distances formula of VSs in 2005, and so forth. (2) Distances between VSs based on Hausdorff distance, for example, Hung and Yang [20], Grzegorzewski [21], Chen [22], and Yang and Chiclana [23] proposed the Hausdorff distances of VSs, respectively. (3) Distances between VSs based on fuzzy implications, for example, Hatzimichailidis et al. [24] presented the distances in 2012. (4) The other distances satisfying the axiomatic definition of distance, for example, Wang and Xin [25] constructed some distances and weighted distances of VSs in 2005, and so forth. In the abovementioned methods, some only consider the impacts of the degree of membership and nonmembership on the distances of VSs and some consider the impacts of the degree of membership, nonmembership, and hesitation.
The distances between VSs not only meet the axiomatic definition of distance, but also satisfy the definition of distances between VSs presented by Wang and Xin [25] (see Section 2), since there exists a certain relationship between the degree of membership, nonmembership, and hesitation for vague sets. Therefore, in the paper, the defects of the existing distances between VSs are analyzed and discussed in detail firstly, and a numerical example is used to demonstrate these defects. Thereafter, a new distance measure between VSs is proposed, and some properties of the new method are also discussed and proved. In addition, the effectiveness of the proposed method is illustrated by a pattern classification problem. Finally, we apply the new method into the multicriteria fuzzy decision and select the optimal weight of each criterion through the optimization method so as to obtain the best solution based on the weighted distance between candidate solution and ideal solution in each criterion. The final decisionmaking result shows the effectiveness and the feasibility of the proposed method.
The remainder of this paper is organized as follows. The basic concepts on vague sets are shown in Section 2. The analysis process of the existing distances between vague sets is presented in Section 3. Section 4 proposes a new distance between vague sets, and the performances of the method are evaluated by pattern classification and multicriteria fuzzy decision making in Section 5. The paper is concluded in Section 6.
2. Vague Set and Its Operations
Definition 1 (see [4]). A vague set in is characterized by a truthmembership function and a falsemembership function ; that is, with condition , , where is a lower bound on the degree of membership of derived from the evidence for and is a lower bound on the negation of derived from the evidence against . and both associate a real number in the interval [] with each point in .
This approach bounds the degree of membership of x to a subinterval of . In other words, the exact degree of membership of x may be unknown but is bounded by , where .
For each vague set in , the uncertainty of our knowledge about for is characterized by . If is small, our knowledge about is relatively precise; if is large, we know correspondingly little. If , our knowledge about is exact, and the vague sets degenerate into the fuzzy sets. If and or and , then the vague sets will revert back to the ordinary sets. For convenience, we denote all the vague sets in as .
For example, let ; if and , then vague set A can be interpreted as “the vote for a resolution is 5 in favor, 2 against, and 3 abstentions .”
Let ; if the universe of discourse is discrete, then If the universe of discourse is continuous, then
Let and be two vague values, where and ; then the operations of two vague values are as follows:
Let , where and ; the operations and relations of and are as follows:
3. Distances between Vague Sets
3.1. Definition of Distances between Vague Sets
Distance measure is a function that characterizes the difference between VSs and can be considered as a dual concept of similarity measure. For VSs, the axiomatic definitions of a distance are as follows.
Definition 2. A distance d in a nonempty set is a real value function , which satisfies the following conditions, :(1) if and only if ;(2);
(3).
The distance measure must satisfy the three conditions presented in Definition 2. But in vague sets, expresses a support degree of to and expresses a degree of opposition of to , while expresses a degree of neutrality of to , which is a degree of decision that cannot be made currently. So the distances between VSs should also meet the definition presented by Wang and Xin [25].
Definition 3 (see [25]). Let be a mapping: . , if satisfies the following properties:(1);
(2) if and only if ;(3);
(4)If , then and .
Then is a distance measure between vague sets and .
From Definition 3, the inclusion relations between VSs can reflect the distance relations between VSs, so the fourth condition must be satisfied. The existing distances between VSs will be analyzed according to Definitions 2 and 3 in Section 3.2.
3.2. Analysis of the Existing Distances between Vague Sets
In order to simplify the description, the following notations are used, :
3.2.1. Distances between Fuzzy Sets
Distances between vague sets are expanded by Kacprzyk according to the distances between fuzzy sets [26]. Therefore, the distances between fuzzy sets are firstly introduced as follows for two fuzzy sets in .
Hamming distance and normalized Hamming distance
Euclidean distance and normalized Euclidean distance where .
Formulas (7) and (8) just consider the membership function . However, there exists the linear relationship between the membership function and the nonmembership function for a fuzzy set ; that is, , where . Therefore, if the nonmembership function is also introduced into the distances between fuzzy sets, then formulas (7) and (8) can be represented as follows, respectively.
Hamming distance and normalized Hamming distance
Euclidean distance and normalized Euclidean distance
Based on expressions (9) and (10), the distances between fuzzy sets are enlarged if fuzzy sets are expressed as the form of vague sets, but there is no substantial influence on the results. Therefore, Atanassov obtained the Hamming distances and Euclidean distances through extending formulas (7) and (8) into vague sets.
3.2.2. Distance between Vague Sets Based on Hamming Distance and Euclidean Distance
(1) Atanassov’s Distances between Vague Sets. Based on the Hamming distances and Euclidean distances between FSs, the distances between vague sets derived by Atanassov are as follows:
However, in vague sets and , , , and have the following relations: From (12), the distance between the degree of hesitation and the degree of hesitation is not strictly linear relations with the distances and when we use (11). Thus, the degree of hesitation and the degree of hesitation should not be omitted in distances between VSs. As a result, Szmidt and Kacprzyk [18] improved the Atanassov’s distances between VSs.
(2) Szmidt and Kacprzyk’s Distances between Vague Sets:
In fact, formulas (13) are, respectively, the Hamming distances and the Euclidean distances between the vectors and . Therefore, they satisfy the conditions of Definition 2. However, formulas (13) could not strictly satisfy the fourth condition in Definition 3, since and satisfy the condition (see Example 4). To this end, we conduct a detailed analysis as follows.
Let vague values , , and . If , then and , we get where , . But for and , since So, from formulas (14) and (15), we cannot clear assert that or . Thus, may not be true; that is, the distance may be improper. Similarly, , , and are also improper; see Example 4.
The results of the above analysis show that there must be the relationship between and not determined if we introduce into distances between VSs directly. This is because and have no clear relationships when or .
(3) Liu’s Distances between Vague Sets. Liu proposed new distances between VSs based on the geometric interpretation of vague sets [13].
If is continuous, the distance between and is If is discrete, the distance between and is In (16) and (17), .
If , Liu’s method is Szmidt and Kacprzyk’s normalized Euclidean distance. Thus, Liu’s method may not satisfy the fourth condition in Definition 3.
3.2.3. Distances between Vague Sets Based on Hausdorff Measure
(1) HungGrzegorzewskiChen’s Distance between Vague Sets. Based on Hausdorff measure, Hung and Yang [20] and Grzegorzewski [21] defined the Hausdorff distances between vague sets, which are revised by Chen [22] as following:
Although formulas (18) satisfy the conditions of Definitions 2 and 3, they all neglect the degree of hesitation and the degree of hesitation . Yang and Chiclana [23] analyzed that it will obtain the inconsistent results when two of three objects , and are only considered. For this reason, they proposed several distances between vague sets.
(2) Yang and Chiclana’s Distances between Vague Sets:
In 2013, Luo and Xiao [27] introduced the weights into formula (19) and obtained where , . Particularly, if , then formula (23) will be formula (20). Luo and Xiao proved in detail that formula (23) satisfies all the conditions of Definition 3. But formulas (21) and (22) may not satisfy the fourth condition of Definition 3.
3.2.4. Other Distances
According to Definition 3, Wang and Xin [25] defined the distances between vague sets; that is,
If , is ; if , is . Thus, Wang and Xin’s method also has the same deficiencies as Atanassov’s distances between vague sets.
The deficiencies of the abovementioned methods will be illustrated by Example 4.
Example 4. Let , , and be the three vague values. Since , the distance between and is difference from the distance between and . The geometrical interpretation of vague sets , , and is presented in Figure 1. The results of the abovementioned distances between vague sets , , and are shown in Table 1.

From Table 1, the methods , , , and can get , but other methods do not. However, the degree of hesitation is not considered in the methods , , , and .
Based on the analysis in Table 1, we think that the degree of membership, nonmembership, and hesitation should be introduced into the distance measures between vague sets, and the distances between vague sets must satisfy Definitions 2 and 3. For this reason, we propose a new distance which will be a detailed introduction in Section 4.
4. A New Distance between Vague Sets and Its Analysis
Let be objects set of experts who vote for or against a given decision. The degree of each expert voting in favor is , voting against is , and the hesitation is . For example, suppose that the expert at the beginning of negotiations voting in favor is and against is , which means . If we persuade that he should vote for, then the best result is with . On the contrary, if he votes against, then the best result that the opponents can achieve is with . It may happen that could be any number from [] and may be any number from []. Therefore, we should consider the assignment of to and separately if is introduced into the distances between VSs; that is, we investigate the influence of hesitation degree to the distances between VSs indirectly.
Let be a vague value; the assignment of to and is defined as and , respectively, where and represent the weights. Thus, for each vague value , there will be four parts: , , , and . A new distance between vague values, denoted as , can be defined as follows.
Definition 5. Let and be two vague values; then the distance between and is defined as where and represent the assignments of to and , respectively. and represent the assigned weights of to and , respectively, and satisfy .
For Example 4, we can get and , where , same applies hereinafter, and we can prove that is a distance measure that satisfied Definitions 2 and 3.
Theorem 6. Let and be two vague values; then is a distance measure between and and satisfies Definitions 2 and 3.
Proof. From formula (25), . In addition, and . Similarly, . Thus, . That is, satisfies condition in Definition 3.
If , then ; thereby, .
Conversely, if , then ; accordingly, ; that is, .
Thus, satisfies condition in Definition 2 and condition in Definition 3.
From formula (25), . So, satisfies condition in Definition 2 and condition in Definition 3.
If , we have
Thus, , which shows that condition in Definition 2 is satisfied.
If , then and , we have
Thus, .
Similarly, . This result shows that condition in Definition 3 is satisfied.
Therefore, the above proof shows that is a distance measure between and and satisfies Definitions 2 and 3.
Definition 7. Let be two VSs in ; then the distance measure between vague sets and , denoted as , is defined as where , , . and represent the assigned weights of to and , respectively.
Theorem 8. is a distance measure between and in and satisfies Definitions 2 and 3.
Proof. Similar to the proof of Theorem 6 (omitted).
5. Comparative Analysis of the Experiment and Its Application in Multicriteria Fuzzy Decision Making
To study the ability of the proposed metric to count the distance between two VSs, two experiments have been conducted: pattern classification and multicriteria fuzzy decision.
5.1. A Numerical Example for Pattern Classification
Assume that the question which relates to classification is given using VSs. Liang and Shi [28] use the principle of the maximum degree of similarity between VSs to solve the problem of pattern classification. Similarly, we use the principle of minimum distance between VSs to solve the problem. We use the classification data about building materials given by Wang and Xin [25]. Given four classes of building material, each is represented by the vague sets , , , and in the feature space ; see Table 2. Now, given another kind of unknown building material , we justify which class the belongs to through computing the distance between and each . The belongs to the class when the distance between and each is minimal among the distances between and each .

The classification performance is illustrated by the degree of confidence (DOC) proposed by Hatzimichailidis et al. [24]. This factor measures the confidence of each distance metric in recognizing a specific sample that belongs to the class () and has the following form: Obviously, the greater is, the more confident the result of the specific distance metric is.
Table 3 summarizes the distance measures’ results along with the degree of confidence of each one among the above distances between VSs that we have introduced. In Table 3, the minimum distance and the best results with the highest degree of confidence have been denoted in bold. All the distances can classify the test sample correctly. However, the existing distance measures fail to introduce the degree of hesitation into the distance measure between VSs which causes the DOC to be lower, while the proposed method has higher DOC. Thus, the results of classification indicate that our method is effective.

5.2. Presentation of Multicriteria Fuzzy Decision Based on the Distance Measures between VSs
Let be the set of alternatives and let be the corresponding weights of the criteria , where and . The characteristic of the alternative is represented by a vague set: where , , and . indicates the degree of the alternative which satisfies the criterion and indicates the degree of the alternative which does not satisfy the criterion given by the decision maker.
The same as the TOPSIS method proposed by Hwang and Yoon [29], the best alternative is obtained by the minimum value , where represents the weighted sum of the distance between and ; namely, where the vague set represents the characteristic of the alternative about the criterion and represents an idea alternative about the criterion and is determined by the following expression: where the operators “” and “” depend on the criterion .
With regard to the choice of weights , many researchers give a certain number between 0 and 1 [13, 30–32]. In the paper, we use the optimal method to determine the weights based on [11, 33]. Namely, the optimal weights satisfy where , , and indicate the degree of membership and nonmembership with respect to the criterion to the fuzzy concept “importance,” respectively, and , as well as ; that is, the weight is also expressed by a vague set , and indicates the degree of hesitation. In this way, the weight of the criterion is expressed by a closed interval