A Novel Distance between Vague Sets and Its Applications in Decision Making

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.


Introduction
A fuzzy set (FS) , as proposed by Zadeh [1], is a class of objects  = { 1 ,  2 , . . .,   } 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,   :  → [0, 1].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][11][12][13], fault-tree 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 above-mentioned 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 decision-making 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.

Vague Set and Its Operations
Definition 1 (see [4]).A vague set  in  is characterized by a truth-membership function   () and a false-membership function   (); that is, with condition 0 ≤   () +   () ≤ 1, ∀ ∈ , 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 [0, 1] with each point in .
For each vague set  in , the uncertainty of our knowledge about  for  is characterized by   () = 1 −   () −   ().If   () is small, our knowledge about  is relatively precise; if   () is large, we know correspondingly little.If   () = 0, our knowledge about  is exact, and the vague sets degenerate into the fuzzy sets.If   () and   () = 1 or   () = 1 and   () = 0, then the vague sets will revert back to the ordinary sets.For convenience, we denote all the vague sets in  as VSs().
For example, let  ∈ VSs(); if   () = 0.5 and   () = 0.2, then vague set A can be interpreted as "the vote for a resolution is 5 in favor, 2 against, and 3 abstentions (  () = 0.3)." Let  ∈ VSs(); if the universe of discourse  is discrete, then If the universe of discourse  is continuous, then (5)

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.
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].
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.

Analysis of the Existing Distances between Vague Sets.
In order to simplify the description, the following notations are used, ∀,  ∈ VSs():

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 .
(1) Hamming distance   and normalized Hamming distance (2) Euclidean distance   and normalized Euclidean distance where Δ  () =   (  ) −   (  ).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,   (  ) = 1 −   (  ), 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. ( (4) 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.

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 , Δ  (), Δ  (), and Δ  () have the following relations: From ( 12), the distance |Δ  ()| between the degree of hesitation   (  ) and the degree of hesitation   (  ) is not strictly linear relations with the distances |Δ  ()| and |Δ  ()| 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.
If  is continuous, the distance between  and  is If  is discrete, the distance between  and  is In ( 16) and ( 17), 1 <  < +∞.
If  = 2, Liu's method is Szmidt and Kacprzyk's normalized Euclidean distance.Thus, Liu's method may not satisfy the fourth condition in Definition 3.
Based on Hausdorff measure, Hung and Yang [20] and Grzegorzewski [21] defined the Hausdorff distances between vague sets, which are revised by Chen [22] 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.

Other Distances. According to Definition 3, Wang and
Xin [25] defined the distances between vague sets; that is, If  = 1, (, ).Thus, Wang and Xin's method also has the same deficiencies as Atanassov's distances between vague sets.
The deficiencies of the above-mentioned methods will be illustrated by Example 4.
Example 4. Let  = [1, 1],  = [0, 0], and  = [0, 1] 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 above-mentioned distances between vague sets , , and  are shown in Table 1 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.
Proof.Similar to the proof of Theorem 6 (omitted).

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: (1) pattern classification and (2) multicriteria fuzzy decision.

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  1 ,  2 ,  3 , and  4 in the feature space  = { 1 ,  2 , . . .,  12 }; 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   ( = 1, 2, 3, 4).The  belongs to the class   when the distance between  and each   is minimal among the distances between  and each   ( = 1, 2, 3, 4).
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 DOC () 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.

Presentation of Multicriteria Fuzzy Decision Based on the
Distance Measures between VSs.Let A = { 1 ,  2 , . . .,   } where 0 ≤   +   ≤ 1, 1 ≤  ≤ , and 1 ≤  ≤ .  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   (  ) ( = 1, 2, . . ., ), where   (  ) represents the weighted sum of the distance between   and   ; namely, where the vague set   = [  , 1 −   ] represents the characteristic of the alternative   about the criterion   and   represents an idea alternative about the criterion   ( = 1, 2, . . ., ;  = 1, 2, . . ., ) and is determined by the following expression: where the operators "∨" and "∧" depend on the criterion   .
With regard to the choice of weights  1 ,  2 , . . .,   , many researchers give a certain number between 0 and 1 [13,[30][31][32].In the paper, we use the optimal method to determine the weights  1 ,  2 , . . .,   based on [11,33].Namely, the optimal weights  1 ,  2 , . . .,   satisfy min where    =   ,    = 1 −   ,   and   indicate the degree of membership and nonmembership with respect to the criterion   to the fuzzy concept "importance, " respectively, and 0 ≤   ,   ≤ 1 as well as 0 ≤   +   ≤ 1; that is, the weight   is also expressed by a vague set [  , 1 −   ], and   = 1 −   −   indicates the degree of hesitation.In this way, the weight   of the criterion   is expressed by a closed interval [   ,
Therefore, the minimum value is   ( 1 ) = 0.0850.Namely, the best alternative is  1 .The optimal ranking order of the alternatives is given by  1 ≻  3 ≻  2 , which is consistent with the results of Xu and Wei [11] and Li [33].However, the method is more concise than Li's method.

Conclusions
In this paper, we give a novel distance between vague sets, which considers the assignment of hesitancy degree () to the membership () and nonmembership degree (), after analyzing the existing method, and the properties are also discussed.The performances of the proposed method are illustrated by pattern classification.Finally, our method is applied into multicriteria fuzzy decision making, where we take the linear programming method to generate optimal weights for every criteria 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 demonstrate the effectiveness of the proposed method.
as following: Chiclana's Distances between Vague Sets:

Table 1 :
The results of existing distances between vague sets , , and .
2, which means   (  ) = 0.2.If we persuade that he should vote for, then the best result is  +final (  ) =   (  ) +   (  ) = 0.8 with  +final (  ) = 0.2.On the contrary, if he votes against, then the best result that the opponents can achieve is  +final (  ) =   (  ) +   (  ) = 0.4 with  +final (  ) = 0.6.It may happen that  +final (  ) could be any number from [0.6, 0.8] and  +final (  ) may be any number from [0.2, 0.4].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.