Intuitionistic fuzzy sets - Wiley Online Library
Intuitionistic Fuzzy Sets: Spherical Representation and Distances Y. Yang† , F. Chiclana∗ Centre for Computational Intelligence, School of Computing, De Montfort University, Leicester LE1 9BH, UK
Most existing distances between intuitionistic fuzzy sets are defined in linear plane representations in 2D or 3D space. Here, we define a new interpretation of intuitionistic fuzzy sets as a restricted spherical surface in 3D space. A new spherical distance for intuitionistic fuzzy sets is introduced. We prove that the spherical distance is different from those existing distances in that it is nonlinear C 2009 Wiley with respect to the change of the corresponding fuzzy membership degrees. Periodicals, Inc.
1. INTRODUCTION Research in cognition science1 has shown that people are faster at identifying an object that is significantly different from other objects than at identifying an object similar to others. The semantic distance between objects plays a significant role in the performance of these comparisons.2 For the concepts represented by fuzzy sets and intuitionistic fuzzy sets, an element with full membership (nonmembership) is usually much easier to be determined because of its categorical difference from other elements. This requires the distance between intuitionistic fuzzy sets or fuzzy sets to reflect the semantic context of where the membership/nonmembership values are, rather than a simple relative difference between them. Most existing distances based on the linear representation of intuitionistic fuzzy sets are linear in nature, in the sense of being based on the relative difference between membership degrees.3−15 Obviously, in some semantic contexts these distances might not seem to be the most appropriate ones. In such cases, nonlinear distances between intuitionistic fuzzy sets may be more adequate to capture the semantic difference. Here, new nonlinear distances between two intuitionistic fuzzy sets are introduced. We call these distances spherical distances because their definition is based on a spherical representation of intuitionistic fuzzy sets.
∗ †
Author to whom all correspondence should be addressed: e-mail: [email protected]. e-mail: [email protected].
INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 24, 399–420 (2009) C 2009 Wiley Periodicals, Inc. Published online in Wiley InterScience (www.interscience.wiley.com). • DOI 10.1002/int.20342
400
YANG AND CHICLANA
The paper is set out as follows. In the following section, some preliminary definitions and notation on intuitionistic fuzzy sets needed throughout the paper are provided. In Section 3, a review of the existing geometrical interpretations of intuitionistic fuzzy sets and the distance functions proposed and usually used in the literature is given. The spherical interpretation of intuitionistic fuzzy sets and spherical distance functions are introduced in Section 4. Because fuzzy sets are particular cases of intuitionistic fuzzy sets, the corresponding spherical distance functions for fuzzy sets are derived in Section 5. Finally, Section 6 presents our conclusion. 2. INTUITIONISTIC FUZZY SETS: PRELIMINARIES Intuitionistic fuzzy sets were introduced by Atanassov in Ref. 3. The following provides its definition, which will be needed throughout the paper: DEFINITION 1. (Intuitionistic fuzzy set) An intuitionistic fuzzy set A of the universe of discourse U is given by A = {u, μA (u), νA (u) |u ∈ U } where μA : U → [0, 1] , νA : U → [0, 1] and 0 ≤ μA (u) + νA (u) ≤ 1 ∀u ∈ U. For each u, the numbers μA (u) and νA (u) are the degree of membership and degree of nonmembership of u to A, respectively. Another concept related to intuitionistic fuzzy sets is the hesitancy degree, τA (u) = 1 − μA (u) − νA (u), which represents the hesitance to the membership of u to A. We note that the main difference between intuitionistic fuzzy sets and traditional fuzzy sets resides in the use of two parameters for membership degrees instead of a single value. Obviously, μA (u) represents the lowest degree of u belonging to A, and νA (u) gives the lowest degree of u not belonging to A. If we consider μ = v − as membership and ν = 1 − v + as nonmembership, then we come up with the so-called interval-valued fuzzy sets [v − , v + ].16 DEFINITION 2. (Interval-valued fuzzy set) universe of discourse U is given by
An interval-valued fuzzy set A of the
A = {u, MA (u) |u ∈ U } International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS
401
where the function MA : U → P [0, 1] defines the degree of membership of an element u to A, being
P [0, 1] = {[a, b], a, b ∈ [0, 1], a ≤ b} . Interval-valued fuzzy sets16,17 and intuitionistic fuzzy sets3−5 emerged from different grounds and thus they have associated different semantics.18 However, they have been proven to be mathematically equivalent.19−23 and because of this we do not distinguish them throughout this paper. For the sake of convenience when comparing existing distances, we will apply the notation of intuitionistic fuzzy sets. Our conclusions could easily be adapted to interval-valued fuzzy sets. 3. EXISTING GEOMETRICAL REPRESENTATIONS OF AND DISTANCES BETWEEN INTUITIONISTIC FUZZY SETS In contrast to traditional fuzzy sets where only a single number is used to represent membership degree, more parameters are needed for intuitionistic fuzzy sets. Geometrical interpretations have been associated with these parameters,24 which are especially useful when studying the similarity or distance between sets. One of these geometrical interpretations of intuitionistic fuzzy sets was given by Atanassov in Ref. 5, as shown in Figure 1a, where a universe U and subset OST in the Euclidean plane with Cartesian coordinates are represented. According to this interpretation, given an intuitionistic fuzzy set A, a function fA from U to OST can be constructed such that if u ∈ U , then P = fA (u) ∈ OST is the point with coordinates μA (u), νA (u) for which 0 ≤ μA (u) ≤ 1, 0 ≤ νA (u) ≤ 1, 0 ≤ μA (u) + νA (u) ≤ 1. We note that the triangle OST in Figure 1a is an orthogonal projection of the 3D representation proposed by Szmidt and Kacprzyk,10 as shown in Figure 1b. In Figure 1b, in addition to μA (u) and νA (u), a third dimension is present, τA (u) = 1 − μA (u) − νA (u). Because μA (u) + νA (u) + τA (u) = 1, the restricted plane RST can be interpreted as the 3D counterpart of an intuitionistic fuzzy set. Therefore, in a similar way to Atanassov’s procedure, for an intuitionistic fuzzy set A a function fA from U to RST can be constructed in such a way that given u ∈ U , then S = fA (u) ∈ RST has coordinates μA (u), νA (u), τA (u) for which 0 ≤ μA (u) ≤ 1, 0 ≤ νA (u) ≤ 1, 0 ≤ τA (u) ≤ 1 and μA (u) + νA (u) + τA (u) = 1. International Journal of Intelligent Systems
DOI 10.1002/int
402
YANG AND CHICLANA
Figure 1. 2D and 3D representations of intuitionistic fuzzy sets.
A fuzzy set is a special case of an intuitionistic fuzzy set where τA (x) = 0 holds for all elements. In this case, both OST and RST in Figure 1 converge to the segment ST . Therefore, under this interpretation, the distance between two fuzzy sets is based on the membership functions, which depends on just one parameter (membership). Given any two fuzzy subsets A = {ui , μA (ui ) : ui ∈ U } and B = {ui , μB (ui ) : ui ∈ U } with U = {u1 , u2 , . . . , un }, the following distances have been proposed8 : • Hamming distance d1 (A, B) d1 (A, B) =
n
|μA (ui ) − μB (ui )|
i=1
• Normalized Hamming distance l1 (A, B) l1 (A, B) =
n 1 |μA (ui ) − μB (ui )| n i=1
• Euclidean distance e1 (A, B) n e1 (A, B) = (μA (ui ) − μB (ui ))2 i=1
• Normalized Euclidean distance q1 (A, B) n 1 (μA (ui ) − μB (ui ))2 q1 (A, B) = n i=1 International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS
403
In the case of intuitionistic fuzzy sets, different distances have been defined according to either of the 2D or 3D interpretations. Atanassov in Ref. 5 presents the following distances for any two intuitionistic fuzzy subsets A = {ui , μA (ui ), νA (ui ) : ui ∈ U } and B = {ui , μB (ui ), νB (ui ) : ui ∈ U } using the above 2D interpretation: • Hamming distance d2 (A, B) 1 [|μA (ui ) − μB (ui )| + |νA (ui ) − νB (ui )|] 2 i=1 n
d2 (A, B) =
• Normalized Hamming distance l2 (A, B) l2 (A, B) =
n 1 [|μA (ui ) − μB (ui )| + |νA (ui ) − νB (ui )|] 2n i=1
• Euclidean distance e2 (A, B) n 1 [(μA (ui ) − μB (ui ))2 + (νA (ui ) − νB (ui ))2 ] e2 (A, B) = 2 i=1 • Normalized Euclidean distance q2 (A, B) n 1 [(μA (ui ) − μB (ui ))2 + (νA (ui ) − νB (ui ))2 ] q2 (A, B) = 2n i=1
Based on the 3D representation, Szmidt and Kacprzyk9,10 modified these distances to include the third parameter τA (u): • Hamming distance d3 (A, B) 1 [|μA (ui ) − μB (ui )| + |νA (ui ) − νB (ui )| + |τA (ui ) − τB (ui )|] 2 i=1 n
d3 (A, B) =
• Normalized Hamming distance l3 (A, B) l3 (A, B) =
n 1 [|μA (ui ) − μB (ui )| + |νA (ui ) − νB (ui )| + |τA (ui ) − τB (ui )|] 2n i=1
• Euclidean distance e3 (A, B) n 1 [(μA (ui ) − μB (ui ))2 + (νA (ui ) − νB (ui ))2 + (τA (ui ) − τB (ui ))2 ] e3 (A, B) = 2 i=1 International Journal of Intelligent Systems
DOI 10.1002/int
404
YANG AND CHICLANA • Normalized Euclidean distance q3 (A, B)
n 1 [(μA (ui ) − μB (ui ))2 + (νA (ui ) − νB (ui ))2 + (τA (ui ) − τB (ui ))2 ] q3 (A, B) = 2n i=1
All the above distances clearly adopt the linear plane interpretation, as shown in Figure 1, and therefore they reflect only the relative differences between the memberships, nonmemberships, and hesitancy degrees of intuitionistic fuzzy sets. The following lemma illustrates this characteristic: LEMMA 1. Let A = {ui , μA (ui ), νA (ui ) : ui ∈ U }, B = {ui , μB (ui ), νB (ui ) : ui ∈ U }, C = {ui , μC (ui ), νC (ui ) : ui ∈ U }, and G = {ui , μG (ui ), νG (ui ) : ui ∈ U } be four intuitionistic fuzzy sets of the universe of discourse U = {u1 , u2 , . . . , un }. If the following conditions hold μA (ui ) − μB (ui ) = μC (ui ) − μG (ui ) νA (ui ) − νB (ui ) = νC (ui ) − νG (ui ) then D(A, B) = D(C, G) being D any of the above Atanassov’s 2D or Szmidt and Kacprzyk’s 3D distance functions.
Proof. The proof is obvious for Atanassov’s 2D distances. For Szmidt and Kacprzyk’s 3D distances, the proof follows from the fact that μA (ui ) − μB (ui ) = μC (ui ) − μG (ui ) ∧ νA (ui ) − νB (ui ) = νC (ui ) − νG (ui ) imply τA (ui ) − τB (ui ) = τC (ui ) − τG (ui )
If |τA (ui ) − τB (ui )| = |τC (ui ) − τG (ui )|, then conditions in Lemma 1 can be generalized to |μA (ui ) − μB (ui )| = |μC (ui ) − μG (ui )| |νA (ui ) − νB (ui )| = |νC (ui ) − νG (ui )| International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS
405
This means that if we move both sets in the space shown in Figure 1b with the same changes in membership, nonmembership, and hesitancy degrees, then we obtain exactly the same distance between the two fuzzy sets. This linear feature of the above distances may not be adequate in some cases, because the human perception is not necessarily always linear. For example, we can classify the human behavior as perfect, good, acceptable, poor, and worst. Using fuzzy sets, we can assign their fuzzy membership as 1, 0.75, 0.5, 0.25, and 0. To find out if someone’s behavior is perfect or not, we only need to check if there is anything wrong with him. However, to differentiate good from acceptable, we have to count their positive and negative points. Obviously, the semantic distance between perfect and good should be greater than the semantic distance between good and acceptable. This semantic difference is not shown by the linear distances between their memberships. Therefore, a nonlinear representation of the distance between two intuitionistic fuzzy sets may benefit the representative power of intuitionistic fuzzy sets. Although nonlinearity could be modeled by using many different expressions, we will consider and use a simple one to model it. Here, we propose a new geometrical interpretation of intuitionistic fuzzy sets in 3D space using a restricted spherical surface. This new representation provides a convenient nonlinear measure of the distance between two intuitionistic fuzzy sets.
4. SPHERICAL INTERPRETATION OF INTUITIONISTIC FUZZY SETS: SPHERICAL DISTANCE Let A = {u, μA (u), νA (u) : u ∈ U } be an intuitionistic fuzzy set. We have μA (u) + νA (u) + τA (u) = 1 which can be equivalently transformed to x 2 + y 2 + z2 = 1 with x 2 = μA (u), y 2 = νA (u), z2 = τA (x) It is obvious that this is not the only transformation that could be used. However, as shown in the existing distances, there is no special reason to discriminate μA (u), νA (u), and τA (u). Therefore, a simple nonlinear transformation to unit sphere in a 3D Euclidean space is selected here, as shown in Figure 2. This transformation allows us to interpret an intuitionistic fuzzy set as a restricted spherical surface. An immediate consequence of this interpretation is that the distance between two elements of an intuitionistic fuzzy set can be defined as the spherical distance between their corresponding points on its restricted spherical surface representation. This distance is defined as the shortest path between the two International Journal of Intelligent Systems
DOI 10.1002/int
406
YANG AND CHICLANA
Figure 2. 3D sphere representation of intuitionistic fuzzy sets.
points, i.e. the length of the arc of the great circle passing through both points. For points P and Q in Figure 2, their spherical distance is25 . 1 (xP − xQ )2 + (yP − yQ )2 + (zP − zQ )2 . ds (P , Q) = arccos 1 − 2 This expression can be used to obtain the spherical distance between two intuitionistic fuzzy sets, A = {ui , μA (ui ), νA (ui ) : ui ∈ U } and B = {ui , μB (ui ), νB (ui ) : ui ∈ U } of the universe of discourse U = {u1 , u2 , . . . , un }, as follows: n
2 2 1 ds (A, B) = arccos 1 − μA (ui ) − μB (ui ) π i=1 2 +
νA (ui ) −
2
2 νB (ui ) + τA (ui ) − τB (ui )
where the factor π2 is introduced to get distance values in the range [0, 1] instead of [0, π2 ]. Because μA (ui ) + νA (ui ) + τA (ui ) = 1 and μB (ui ) + νB (ui ) + τB (ui ) = 1, we have that ds (A, B) =
n 2 arccos μA (ui )μB (ui ) + νA (ui )νB (ui ) + τA (ui )τB (ui ) π i=1
This is summarized in the following definition: International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS
407
DEFINITION 3. (Spherical distance) Let A = {ui , μA (ui ), νA (ui ) : ui ∈ U } and B = {ui , μB (ui ), νB (ui ) : ui ∈ U } be two intuitionistic fuzzy sets of the universe of discourse U = {u1 , u2 , . . . , un }. The spherical and normalized spherical distances between A and B are • Spherical distance ds (A, B)
ds (A, B) =
n 2 arccos μA (ui )μB (ui ) + νA (ui )νB (ui ) + τA (ui )τB (ui ) π i=1
• Normalized spherical distance dns (A, B)
dns (A, B) =
n 2 arccos μA (ui )μB (ui ) + νA (ui )νB (ui ) + τA (ui )τB (ui ) nπ i=1
Clearly, we have that 0 ≤ ds (A, B) ≤ n and 0 ≤ dns (A, B) ≤ 1. Different from the distances in Section 3, the proposed spherical distances implement in their definition not only the difference between membership, nonmembership, and hesitancy degrees but also their actual values. This is shown in the following result: LEMMA 2. Let A = {ui , μA (ui ), νA (ui ) : ui ∈ U } and B = {ui , μB (ui ), νB (ui ) : ui ∈ U } be two intuitionistic fuzzy subsets of the universe of discourse U = {u1 , u2 , . . . , un }. Let a = {a1 , a2 , . . . , an } and b = {b1 , b2 , . . . , bn } be two sets of real numbers (constants). If the following conditions hold for each ui ∈ U : μB (ui ) = μA (ui ) + ai νB (ui ) = νA (ui ) + bi then the following inequalities hold: n n 2 2 arccos 1 − ei2 ≤ ds (A, B) ≤ arccos (1 − ci )(1 − |ai | − |bi |) π i=1 π i=1 n n 2 2 arccos 1 − ei2 ≤ dns (A, B) ≤ arccos (1 − ci )(1 − |ai | − |bi |) nπ i=1 nπ i=1
International Journal of Intelligent Systems
DOI 10.1002/int
408
YANG AND CHICLANA
where ci = max{|ai |, |bi |} and ei = min{|ai |, |bi |}. The maximum distance between A and B is obtained if and only if one of them is a fuzzy set or their available information supports only opposite membership degree for each one.
Proof. See Appendix A.
Owing to the nonlinear characteristic of the spherical distance, they do not satisfy Lemma 1. However, the following properties hold for the spherical distances: LEMMA 3. Let A = {ui , μA (ui ), νA (ui ) :ui ∈ U }, B = {ui , μB (ui ), νB (ui ) : ui ∈ U } and E = {ui , μE (ui ), νE (ui ) : ui ∈ U } be three intuitionistic fuzzy subsets of the universe of discourse U = {u1 , u2 , . . . , un }. Let a = {a1 , a2 , . . . , an } and b = {b1 , b2 , . . . , bn } two sets of real positive numbers (constants) satisfying the following conditions: |μB (ui ) − μA (ui )| = ai , |νB (ui ) − νA (ui )| = bi |μE (ui ) − μA (ui )| = ai , |νE (ui ) − νA (ui )| = bi If E is one of the two extreme crisp sets with either μE (ui ) = 1 or νE (ui ) = 1 for all ui ∈ U , then the following inequalities hold: ds (A, B) < ds (A, E), dns (A, B) < dns (A, E)
The distance between intuitionistic fuzzy sets A and B is always lower than the distance between A and the extreme crisp sets E under the same difference of their memberships and nonmemberships.
Proof. We provide the proof just for the extreme fuzzy set with full memberships, being the proof for full nonmembership similar. With E being the extreme crisp set with full memberships, we have μE (ui ) = 1, νE (ui ) = 0, τE (ui ) = 0 Because |μE (ui ) − μA (ui )| = ai and |νE (ui ) − νA (ui )| = bi , then μA (ui ) = 1 − ai , νA (ui ) = bi , τA (ui ) = ai − bi From |μB (ui ) − μA (ui )| = ai and |νB (ui ) − νA (ui )| = bi , it is μB (ui ) = 1 − 2ai , νB (ui ) = 2bi , τB (ui ) = 2(ai − bi ). International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS
409
Therefore, ds (A, B) =
n 2 arccos (1 − ai )(1 − 2ai ) + 2bi2 + 2(ai − bi )2 π i=1 =
n √ 2 arccos 2ai + (1 − ai )(1 − 2ai ) π i=1
and ds (A, E) =
n 2 arccos 1 − ai π i=1
Obviously, we have √ 2ai + (1 − ai )(1 − 2ai ) > 1 − ai Thus ds (A, B) < ds (A, E) and dividing by n dns (A, B) < dns (A, E)
Lemma 3 shows that the extreme crisp sets with full memberships or full nonmemberships are categorically different from other intuitionitic fuzzy sets. With the same difference of memberships and nonmemberships, the distance from an extreme crip set is always greater than the distances from other intuitionistic fuzzy sets. This conclusion agrees with our human perception about the quality change against quantity change and captures the semantic difference between extreme situation and intermediate situations. 5. SPHERICAL DISTANCES FOR FUZZY SETS As we have already mentioned, fuzzy sets are particular cases of intuitionistic fuzzy sets. Therefore, the above spherical distances can be applied to fuzzy sets. In the following we provide Lemma 4 for the distance between two fuzzy sets. LEMMA 4. Let A = {ui , μA (ui ) : ui ∈ U } and B = {ui , μB (ui ) : ui ∈ U } be two fuzzy sets in the universe of discourse U = {u1 , u2 , . . . , un }. Let a = {a1 , a2 , . . . , an } be a set of nonnegative real constants. If |μA (ui ) − μB (ui )| = ai holds for each International Journal of Intelligent Systems
DOI 10.1002/int
410
YANG AND CHICLANA
ui ∈ U , then the following inequalities hold: n n 2 2 arccos 1 − ai 2 ≤ ds (A, B) ≤ arccos 1 − ai π i=1 π i=1 n n 2 2 arccos 1 − ai 2 ≤ dns (A, B) ≤ arccos 1 − ai nπ i=1 nπ i=1
The maximum distance between A and B is achieved if and only if one of them is a crisp set.
Proof. According to Definition 3, we have n 2 ds (A, B) = arccos μA (ui )μB (ui ) + νA (ui )νB (ui ) + τA (ui )τB (ui ) π i=1
For fuzzy sets, we have τA (ui ) = 0 and νA (ui ) = 1 − μA (ui ) τB (ui ) = 0 and νB (ui ) = 1 − μB (ui ) Because |μA (ui ) − μB (ui )| = ai and μA (ui ) ≥ 0, we have μB (ui ) = μA (ui ) ± ai If μB (ui ) = μA (ui ) + ai then ds (A, B) =
=
n 2 arccos μA (ui )μB (ui ) + (1 − μA (ui ))(1 − μB (ui ) π i=1 n 2 arccos μA (ui )(μA (ui ) + ai ) π i=1
+ (1 − μA (ui ))(1 − μA (ui ) − ai ) International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS
411
This can be rewritten as ds (A, B) =
n 2 arccos fi (μA (ui )) π i=1
√ √ with fi (t) = t(t + ai ) + (1 − t)(1 − t − ai ), t ∈ [0, 1 − ai ]. The extremes of function fi (t) will be among the solution of fi (t) = 0 t ∈ (0, 1 − ai ) and the values 0 and 1 − ai , i.e, among (1 − ai )/2, 0 and 1 − ai . The maximum √ value 1 − ai 2 is obtained when t = (1 − ai )/2, whereas the minimum value 1 − ai is obtained in both 0 and 1 − ai . We conclude that n n 2 2 arccos 1 − ai 2 ≤ ds (A, B) ≤ arccos 1 − ai π i=1 π i=1
When t = 0 or t = 1 − ai , we have, respectively, μA (ui ) = 0 and μB (ui ) = 1 − ai + ai = 1, which implies that one set among A and B has to be crisp in order to reach the maximum value under the given difference in their membership degrees. Following a similar reasoning, it is easy to prove that the same conclusion is obtained in the case μB (ui ) = μA (ui ) − ai . If μB (ui ) = μA (ui ) − ai for some i, and μB (uj ) = μA (uj ) + aj for some j , then we could separate the elements into two different groups, each of them satisfying the inequalities, and therefore their summation obviously satisfying it too. The normalized inequality is obtained just by dividing the first one by n. Because spherical distances are quite different from the traditional distances, the semantics associated with them also differ. For the same relative difference in membership degrees, the spherical distance varies with the locations of its two relevant sets in the membership degree space, 2D for fuzzy sets and 3D for intuitionistic fuzzy sets. The spherical distance achieves its maximum when one of the fuzzy sets is an extreme crisp set. The following example illustrates this effect: Example 1. Consider our previous example on human behavior. We can classify our behavior as perfect, good, acceptable, poor, and worst, with corresponding fuzzy membership of 1, 0.75, 0.5, 0.25, and 0, respectively. Let A = {u, 0.75 : u ∈ U }, B = {u, 0.5 : u ∈ U }, and E = {u, 1 : u ∈ U } be three fuzzy subsets, and U = {u} is a universe of discourse with one element only. From Section 3, we have d1 (A, B) = l1 (A, B) = e1 (A, B) = q1 (A, B) = 0.25 d1 (A, E) = l1 (A, E) = e1 (A, E) = q1 (A, E) = 0.25 International Journal of Intelligent Systems
DOI 10.1002/int
YANG AND CHICLANA
412
Obviously, we have d1 (A, B) = d1 (A, E),
l1 (A, B) = l1 (A, E)
e1 (A, B) = e1 (A, E),
q1 (A, B) = q1 (A, E)
From Definition 3, we have dns (A, B) = ds (A, B) =
n √ 2 arccos 0.75 ∗ 0.5 + (1 − 0.75)(1 − 0.5) = 0.17 π i=1
dns (A, E) = ds (A, E) =
n √ 2 arccos 0.75 ∗ 1 = 0.55 π i=1
We note that the traditional linear distance of fuzzy sets does not differentiate the semantic difference of a crisp set from a fuzzy set. However, ds (A, E) and dns (A, E) are much greater than ds (A, B) and dns (A, B). This demonstrates that the crisp set E is much more different from A than B although their membership difference are the same. Hence, the proposed spherical distance does show the semantic difference between a crisp set and a fuzzy set. This is useful when this kind of semantic difference is significant in the consideration. Figure 3 shows four comparisons between the spherical distance and the Hamming distance for two fuzzy subsets A, B with a universe of discourse with one element U = {u}. The curves represent the spherical distance, and straight lines denote Hamming distances. Figure 3a displays how the distance changes with respect to μB (x) when μA (x) = 0; Figure 3b uses the value μA (x) = 1; in Figure 3c the value μA (x) = 0.2 is used; finally μA (x) = 0.5 is used in Figure 3d. Clearly, the spherical distance changes sharply for values close to the two lower and upper memberships values, but slightly for values close to the middle membership value. In the case of the Hamming distance, the same rate of change is always obtained. Figure 4 displays how the spherical distance and Hamming distance changes with respect to μB (x) for all possible values of μA (x). Figure 4a shows that the spherical distance forms a curve surface, whereas a plane surface produced by the Hamming distance is shown in Figure 4b. Their contours in the bottom clearly show their differences. The contours for spherical distances are ellipses coming from (0, 0) and (1, 1) with curvatures increasing sharply near (0, 1) and (1, 0). Compared with these ellipses, the contours of Hamming distance are a set of parallel lines. These figures prove our conclusions in Lemma 4: The spherical distances do not remain constant as Hamming distances do when both sets experiment a same change in their membership degrees. International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS
413
Figure 3. Comparison between spherical and Hamming distances for fuzzy sets.
Figure 4. Grid of spherical and Hamming distances for fuzzy sets.
6. CONCLUSIONS An important issue related with the representation of intuitionistic fuzzy sets is that of measuring distances. Most existing distances are based on linear plane International Journal of Intelligent Systems
DOI 10.1002/int
414
YANG AND CHICLANA
representation of intuitionistic fuzzy sets, and therefore are also linear in nature, in the sense of being based on the relative difference between membership degrees. In this paper, we have looked at the issue of 3D representation of intuitionistic fuzzy sets. We have introduced a new spherical representation, which allowed us to define a new distance function between intuitionistic fuzzy sets: the spherical distance. We have shown that the spherical distance is different from those existing distances in that it is nonlinear with respect to the change of the corresponding fuzzy membership degrees, and thus it seems more appropriate than usual linear distances for non linear contexts in 3D spaces.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
19. 20.
Chiaoa JY, Bordeauxa AR, Ambady N. Mental representations of social status. Cognition 2004;93:B49–B57. Viglioccoa G, Vinsona DP, Damianb MF, Levelt W. Semantic distance effects on object and action naming. Cognition 2002;85:B61–B69. Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets Syst 1986;20:87–96. Atanassov KT. More on intuitionistic fuzzy sets. Fuzzy Sets Syst 1989;33:37–46. Atanassov KT. Intuitionistic fuzzy sets. Heidelberg west, Germany: Physica-Verlag; 1999. Hung W, Yang M. Fuzzy entropy on intuitionistic fuzzy sets. Int J Intell Syst 2006;21(4):443– 451. Hung W, Yang M. On similarity measures between intuitionistic fuzzy sets. Int J Intell Syst 2008;23(3):364–383. Kacprzyk J. Multistage fuzzy control. Chichester, UK: Wiley; 1997. Szmidt E, Kacprzyk J. On measuring distances between intuitionistic fuzzy sets. Notes IFS 1997;3:1–13. Szmidt E, Kacprzyk J. Distance between intuitionistic fuzzy sets. Fuzzy Sets Syst 2000;114(3):505–518. Szmidt E, Kacprzyk J. Similarity of intuitionistic fuzzy sets and the jaccard coefficient. In: IPMU’2004 Proceedings, Perugia, Italy, 2004; pp 1405–1412. Szmidt E, Baldwin J. Entropy for intuitionistic fuzzy set theory and mass assignment theory. Notes on IFS 2004;10(3):15–286. Szmidt E, Kacprzyk J. A new concept of a similarity measure for intuitionistic fuzzy sets and its use in group decision making. In: Torra V, Narukawa Y, Miyamoto S, editors. Modelling decisions for artificial intelligence. LNAI 3558, Berlin: Springer; 2005. pp 272–282. Szmidt E, Baldwin J. Assigning the parameters for intuitionistic fuzzy sets. Notes on IFS 2005;11(6):1–12. Szmidt E, Kacprzyk J. A new similarity measure for intuitionistic fuzzy sets: straightforward approaches may not work. In: Proceedings of Int. IEEE Conf: Fuzzy Systems, 2007. pp 1–6. Sambuc R. Functions φ -floues. application i’aide au diagnostic en patholologie thyroidienne, Ph.D. thesis, University of Marseille, France, 1975. Turksen B. Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst 1986;20:191– 210. Cornelis C, Atanassov KT, Kerre EE. Intuitionistic fuzzy sets and interval-valued fuzzy sets: a critical comparison. In: Wagenknecht M, Hampel R, editors. In: Third EUSFLAT Proceedings, European Society for Fuzzy Logic and Technology, Zittau, Germany, 2003. pp 159–163. Bustince H, Burillo P. Vague sets are intuitionistic fuzzy sets. Fuzzy Sets Syst 1996;79:403– 405. Deschrijver G, Kerre E. On the relationship between some extensions of fuzzy set theory. Fuzzy Sets Syst 2003;133(2):227–235. International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS 21.
415
Dubois D, Ostasiewicz W, Prade H. Fuzzy set: history and basic notions. In: Dubois D, Prade H, editors. Fundamentals of fuzzy sets. Boston, MA: Kluwer; 2000. pp 21–124. Dubois D, Prade H. Fuzzy sets and systems: theory and applications. New York: Academic Press; 1980. Wang G, He Y. Intuitionistic fuzzy sets and l-fuzzy sets. Fuzzy Sets Syst 2000;110:271–274. Xu Z, Yager RR. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 2006;35(4):417–433. Abramowitz M, Stegum IS, editors. Handbook of mathematical functions with formulas, graphs and mathematical tables. New York: Dover; 1972. Szmidt E, Baldwin J. New similarity measures for intuitionistic fuzzy set theory and mass assignment theory. Notes on IFS 2003;9(3):60–76.
22. 23. 24. 25. 26.
APPENDIX: PROOF FOR LEMMA 2 According to Definition 3, we have n 2 ds (A, B) = arccos μA (ui )μB (ui ) + νA (ui )νB (ui ) + τA (ui )τB (ui ) π i=1
Because A and B satisfy μB (ui ) = μA (ui ) + ai νB (ui ) = νA (ui ) + bi then τB (ui ) = 1 − μA (ui ) − νA (ui ) − ai − bi and n 2 ds (A, B) = arccos μA (ui )(μA (ui ) + ai ) + νA (ui )(νA (ui ) + bi ) π i=1
√ + (1 − μA (ui ) − νA (ui ))(1 − μA (ui ) − νA (ui ) − ai − bi ) Let f (u, v) =
u(u + ai ) + v(v + bi ) + (1 − u − v)(1 − u − v − ai − bi ).
Denoting a = |ai | and b = |bi |, then we distinguish 4 possible cases: Case 1: ai ≥ 0 and bi ≥ 0. In this case, f (u, v) =
u(u + a) + v(v + b) + (1 − u − v)(1 − u − v − a − b). International Journal of Intelligent Systems
DOI 10.1002/int
416
YANG AND CHICLANA
Let f (u, v)|u = 0, we have u=
a(1 − a − b − v) a(v − 1) and v = . b 2a + b
Let f (u, v)|v = 0, then v=
b(u − 1) b(1 − a − b − u) and v = . a 2b + a
u=
b(u − 1) a(v − 1) < 0 and v = < 0, b a
Because
the valid solutions are u=
b(1 − a − b − u) a(1 − a − b − v) and v = . 2a + b 2b + a
Solving these equations, we have u=
b(1 − a − b) a(1 − a − b) and v = , 2(a + b) 2(a + b)
and therefore f0 = f
a(1 − a − b) b(1 − a − b) , 2(a + b) 2(a + b)
=
1 − (a + b)2 .
Obviously, the third square root in f (u, v) must be defined so 0 ≤ u ≤ 1 − a − b and 0 ≤ v ≤ 1 − a − b. The boundary points are reached when u and v get their minimum or maximum values. Denoting t = τA (ui ), the following needs to be verified u + v + t = 1. If u = v = 1 − a − b then t = 2a + 2b − 1, and (1 − u − v)(1 − u − v − a − b) = (2a + 2b − 1)(a + b − 1) ≤ 0, which implies that the third square root in f (u, v) is not defined. Therefore, we have three boundary points for A u = 0, v = 1 − a − b, t = a + b u = 1 − a − b, v = 0, t = a + b International Journal of Intelligent Systems
DOI 10.1002/int
417
INTUITIONISTIC FUZZY SETS
u = 0, v = 0, t = 1, and therefore f1 = f (0, 1 − a − b) = f2 = f (1 − a − b, 0) =
(1 − a)(1 − a − b)
(1 − b)(1 − a − b)
f3 = f (0, 0) =
√ 1 − a − b.
If ai ≥ 0 and bi ≥ 0, then a + b ≤ 1, and we have f1 ≤ f3 ≤ f0 f2 ≤ f3 ≤ f0 . The relationship between f1 and f2 depends on the relationship between a and b. Let c = max{a, b} and e = min{a, b}, then we have
(1 − c)(1 − a − b) ≤ f (u, v) ≤
1 − e2 .
Case 2: ai ≤ 0 and bi ≤ 0. In this case, following a similar reasoning to the above one, the same conclusion is obtained. Case 3: ai ≤ 0 and bi ≥ 0. In this case f (u, v) =
u(u − a) + v(v + b) + (1 − u − v)(1 − u − v + a − b).
Let f (u, v)|u = 0, then u=
a(1 − v) a(1 + a − b − v) and v = . b 2a − b
Let f (u, v)|v = 0, then v=
b(1 − u) −b(1 + a − b − u) and v = . a a − 2b
For u = a(1 − v)/b and v = b(1 − u)/a, we get a = b, hence u + v = 1, and then both A and B are fuzzy sets. According to Lemma 4, we know that √ 1 − a ≤ f (u, v) ≤ 1 − a 2 . Note that the following also holds: (1 − a)(1 − a − b) ≤ (1 − a). International Journal of Intelligent Systems
DOI 10.1002/int
418
YANG AND CHICLANA
For u = [a(1 + a − b − v)]/(2a − b) and v = [−b(1 + a − b − u)]/(a − 2b), we have u=
a(1 + a − b) b(1 + a − b) and v = − . 2(a − b) 2(a − b)
If a > b then u ≥ 0 and v ≤ 0, and therefore the only possibility here is v = 0, i.e b = 0 in which case it is 1+a f0 = f , 0 = 1 − a2. 2 If a < b then u ≤ 0 and v ≥ 0, and therefore a = 0, in which case it is 1−b f1 = f 0, = 1 − b2 . 2 For u = a(1 − v)/b and v = [−b(1 + a − b − u)]/(a − 2b), we have u=
1−b a(1 + b) and v = 2b 2
and we obtain f
a(1 + b) 1 − b , 2b 2
=
1 − b2 = f1 .
Also, because u + v ≤ 1 we have that (a − b) · (1 + b) ≤ 0 and, therefore, it is a ≤ b. For u = [a(1 + a − b − v)]/(2a − b) and v = b(1 − u)/a, we have u=
b(1 − a) 1+a and v = 2 2a
and f
1 + a b(1 − a) , 2 2a
=
1 − a 2 = f0 .
In this case it is a ≥ b. The ranges for u and v are a ≤ u ≤ 1 and 0 ≤ v ≤ 1 − max{a, b} therefore, we have the following boundary points for A u = a, v = 1 − a, t = 0 International Journal of Intelligent Systems
DOI 10.1002/int
419
INTUITIONISTIC FUZZY SETS
u = a, v = 1 − b, t = b − a u = 1, v = 0, t = 0 u = a, v = 0, t = 1 − a Thus f2 = f (a, 1 − a) =
(1 − a)(1 − a + b)
f3 = f (a, 1 − b) =
√ 1−b
f4 = f (1, 0) =
√ 1−a
f5 = f (a, 0) =
(1 − a)(1 − b)
The following inequalities hold f2 ≤ f4 ≤ f0 f5 ≤ f4 ≤ f0 f5 ≤ f3 ≤ f1 Denoting fmin =
(1 − c)(1 − a − b) and fmax =
1 − e2
with e = min{a, b} and c = max{a, b}, then we have
1 − e2 ≥ f (u, v) ≥
(1 − c)(1 − a − b).
Case 4: ai ≥ 0 and bi ≤ 0. In this case, following a similar reasoning to the above one, the same conclusion is obtained. In the four cases, we conclude that n n 2 2 2 arccos 1 − ei ≤ ds (A, B) ≤ arccos (1 − ci )(1 − |ai | − |bi |) π i=1 π i=1 n n 2 2 2 arccos 1 − ei ≤ ds (A, B) ≤ arccos (1 − ci )(1 − |ai | − |bi |) nπ i=1 nπ i=1
where ci = max{|ai |, |bi |} and ei = min{|ai |, |bi |}. International Journal of Intelligent Systems
DOI 10.1002/int
420
YANG AND CHICLANA
According to the boundary points discussed in the above four cases, if ai bi ≥ 0 holds for each ui , we have μA (ui ) = 0, νA (ui ) = 1 − a − b, τA (ui ) = a + b or μA (ui ) = 1 − a − b, νA (ui ) = 0, τA (ui ) = a + b hence, set B has to satisfy μB (ui ) = a, νB (ui ) = 1 − a, τB (ui ) = 0 or μB (ui ) = 1 − b, νB (ui ) = b, τB (ui ) = 0 Obviously, B is a√fuzzy set. A similar conclusion can be drawn if ai bi < 0 holds √ for each ui and (1 − c)(1√ − c + e) ≤ (1 − a)(1 √ − b). However, if ai < 0 and bi > 0 hold for each ui and (1 − a)(1 − a + b) > (1 − a)(1 − b), we have μA (ui ) = a, νA (ui ) = 0, τA (ui ) = 1 − a hence, the set B satisfies μB (ui ) = 0, νB (ui ) = b, τB (ui ) = 1 − b Obviously, A is an intuitionistic fuzzy set with available information supporting only membership, whereas B is an intuitionistic fuzzy set with available information supporting only nonmembership.
International Journal of Intelligent Systems
DOI 10.1002/int
Most existing distances between intuitionistic fuzzy sets are defined in linear plane representations in 2D or 3D space. Here, we define a new interpretation of intuitionistic fuzzy sets as a restricted spherical surface in 3D space. A new spherical distance for intuitionistic fuzzy sets is introduced. We prove that the spherical distance is different from those existing distances in that it is nonlinear C 2009 Wiley with respect to the change of the corresponding fuzzy membership degrees. Periodicals, Inc.
1. INTRODUCTION Research in cognition science1 has shown that people are faster at identifying an object that is significantly different from other objects than at identifying an object similar to others. The semantic distance between objects plays a significant role in the performance of these comparisons.2 For the concepts represented by fuzzy sets and intuitionistic fuzzy sets, an element with full membership (nonmembership) is usually much easier to be determined because of its categorical difference from other elements. This requires the distance between intuitionistic fuzzy sets or fuzzy sets to reflect the semantic context of where the membership/nonmembership values are, rather than a simple relative difference between them. Most existing distances based on the linear representation of intuitionistic fuzzy sets are linear in nature, in the sense of being based on the relative difference between membership degrees.3−15 Obviously, in some semantic contexts these distances might not seem to be the most appropriate ones. In such cases, nonlinear distances between intuitionistic fuzzy sets may be more adequate to capture the semantic difference. Here, new nonlinear distances between two intuitionistic fuzzy sets are introduced. We call these distances spherical distances because their definition is based on a spherical representation of intuitionistic fuzzy sets.
∗ †
Author to whom all correspondence should be addressed: e-mail: [email protected]. e-mail: [email protected].
INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 24, 399–420 (2009) C 2009 Wiley Periodicals, Inc. Published online in Wiley InterScience (www.interscience.wiley.com). • DOI 10.1002/int.20342
400
YANG AND CHICLANA
The paper is set out as follows. In the following section, some preliminary definitions and notation on intuitionistic fuzzy sets needed throughout the paper are provided. In Section 3, a review of the existing geometrical interpretations of intuitionistic fuzzy sets and the distance functions proposed and usually used in the literature is given. The spherical interpretation of intuitionistic fuzzy sets and spherical distance functions are introduced in Section 4. Because fuzzy sets are particular cases of intuitionistic fuzzy sets, the corresponding spherical distance functions for fuzzy sets are derived in Section 5. Finally, Section 6 presents our conclusion. 2. INTUITIONISTIC FUZZY SETS: PRELIMINARIES Intuitionistic fuzzy sets were introduced by Atanassov in Ref. 3. The following provides its definition, which will be needed throughout the paper: DEFINITION 1. (Intuitionistic fuzzy set) An intuitionistic fuzzy set A of the universe of discourse U is given by A = {u, μA (u), νA (u) |u ∈ U } where μA : U → [0, 1] , νA : U → [0, 1] and 0 ≤ μA (u) + νA (u) ≤ 1 ∀u ∈ U. For each u, the numbers μA (u) and νA (u) are the degree of membership and degree of nonmembership of u to A, respectively. Another concept related to intuitionistic fuzzy sets is the hesitancy degree, τA (u) = 1 − μA (u) − νA (u), which represents the hesitance to the membership of u to A. We note that the main difference between intuitionistic fuzzy sets and traditional fuzzy sets resides in the use of two parameters for membership degrees instead of a single value. Obviously, μA (u) represents the lowest degree of u belonging to A, and νA (u) gives the lowest degree of u not belonging to A. If we consider μ = v − as membership and ν = 1 − v + as nonmembership, then we come up with the so-called interval-valued fuzzy sets [v − , v + ].16 DEFINITION 2. (Interval-valued fuzzy set) universe of discourse U is given by
An interval-valued fuzzy set A of the
A = {u, MA (u) |u ∈ U } International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS
401
where the function MA : U → P [0, 1] defines the degree of membership of an element u to A, being
P [0, 1] = {[a, b], a, b ∈ [0, 1], a ≤ b} . Interval-valued fuzzy sets16,17 and intuitionistic fuzzy sets3−5 emerged from different grounds and thus they have associated different semantics.18 However, they have been proven to be mathematically equivalent.19−23 and because of this we do not distinguish them throughout this paper. For the sake of convenience when comparing existing distances, we will apply the notation of intuitionistic fuzzy sets. Our conclusions could easily be adapted to interval-valued fuzzy sets. 3. EXISTING GEOMETRICAL REPRESENTATIONS OF AND DISTANCES BETWEEN INTUITIONISTIC FUZZY SETS In contrast to traditional fuzzy sets where only a single number is used to represent membership degree, more parameters are needed for intuitionistic fuzzy sets. Geometrical interpretations have been associated with these parameters,24 which are especially useful when studying the similarity or distance between sets. One of these geometrical interpretations of intuitionistic fuzzy sets was given by Atanassov in Ref. 5, as shown in Figure 1a, where a universe U and subset OST in the Euclidean plane with Cartesian coordinates are represented. According to this interpretation, given an intuitionistic fuzzy set A, a function fA from U to OST can be constructed such that if u ∈ U , then P = fA (u) ∈ OST is the point with coordinates μA (u), νA (u) for which 0 ≤ μA (u) ≤ 1, 0 ≤ νA (u) ≤ 1, 0 ≤ μA (u) + νA (u) ≤ 1. We note that the triangle OST in Figure 1a is an orthogonal projection of the 3D representation proposed by Szmidt and Kacprzyk,10 as shown in Figure 1b. In Figure 1b, in addition to μA (u) and νA (u), a third dimension is present, τA (u) = 1 − μA (u) − νA (u). Because μA (u) + νA (u) + τA (u) = 1, the restricted plane RST can be interpreted as the 3D counterpart of an intuitionistic fuzzy set. Therefore, in a similar way to Atanassov’s procedure, for an intuitionistic fuzzy set A a function fA from U to RST can be constructed in such a way that given u ∈ U , then S = fA (u) ∈ RST has coordinates μA (u), νA (u), τA (u) for which 0 ≤ μA (u) ≤ 1, 0 ≤ νA (u) ≤ 1, 0 ≤ τA (u) ≤ 1 and μA (u) + νA (u) + τA (u) = 1. International Journal of Intelligent Systems
DOI 10.1002/int
402
YANG AND CHICLANA
Figure 1. 2D and 3D representations of intuitionistic fuzzy sets.
A fuzzy set is a special case of an intuitionistic fuzzy set where τA (x) = 0 holds for all elements. In this case, both OST and RST in Figure 1 converge to the segment ST . Therefore, under this interpretation, the distance between two fuzzy sets is based on the membership functions, which depends on just one parameter (membership). Given any two fuzzy subsets A = {ui , μA (ui ) : ui ∈ U } and B = {ui , μB (ui ) : ui ∈ U } with U = {u1 , u2 , . . . , un }, the following distances have been proposed8 : • Hamming distance d1 (A, B) d1 (A, B) =
n
|μA (ui ) − μB (ui )|
i=1
• Normalized Hamming distance l1 (A, B) l1 (A, B) =
n 1 |μA (ui ) − μB (ui )| n i=1
• Euclidean distance e1 (A, B) n e1 (A, B) = (μA (ui ) − μB (ui ))2 i=1
• Normalized Euclidean distance q1 (A, B) n 1 (μA (ui ) − μB (ui ))2 q1 (A, B) = n i=1 International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS
403
In the case of intuitionistic fuzzy sets, different distances have been defined according to either of the 2D or 3D interpretations. Atanassov in Ref. 5 presents the following distances for any two intuitionistic fuzzy subsets A = {ui , μA (ui ), νA (ui ) : ui ∈ U } and B = {ui , μB (ui ), νB (ui ) : ui ∈ U } using the above 2D interpretation: • Hamming distance d2 (A, B) 1 [|μA (ui ) − μB (ui )| + |νA (ui ) − νB (ui )|] 2 i=1 n
d2 (A, B) =
• Normalized Hamming distance l2 (A, B) l2 (A, B) =
n 1 [|μA (ui ) − μB (ui )| + |νA (ui ) − νB (ui )|] 2n i=1
• Euclidean distance e2 (A, B) n 1 [(μA (ui ) − μB (ui ))2 + (νA (ui ) − νB (ui ))2 ] e2 (A, B) = 2 i=1 • Normalized Euclidean distance q2 (A, B) n 1 [(μA (ui ) − μB (ui ))2 + (νA (ui ) − νB (ui ))2 ] q2 (A, B) = 2n i=1
Based on the 3D representation, Szmidt and Kacprzyk9,10 modified these distances to include the third parameter τA (u): • Hamming distance d3 (A, B) 1 [|μA (ui ) − μB (ui )| + |νA (ui ) − νB (ui )| + |τA (ui ) − τB (ui )|] 2 i=1 n
d3 (A, B) =
• Normalized Hamming distance l3 (A, B) l3 (A, B) =
n 1 [|μA (ui ) − μB (ui )| + |νA (ui ) − νB (ui )| + |τA (ui ) − τB (ui )|] 2n i=1
• Euclidean distance e3 (A, B) n 1 [(μA (ui ) − μB (ui ))2 + (νA (ui ) − νB (ui ))2 + (τA (ui ) − τB (ui ))2 ] e3 (A, B) = 2 i=1 International Journal of Intelligent Systems
DOI 10.1002/int
404
YANG AND CHICLANA • Normalized Euclidean distance q3 (A, B)
n 1 [(μA (ui ) − μB (ui ))2 + (νA (ui ) − νB (ui ))2 + (τA (ui ) − τB (ui ))2 ] q3 (A, B) = 2n i=1
All the above distances clearly adopt the linear plane interpretation, as shown in Figure 1, and therefore they reflect only the relative differences between the memberships, nonmemberships, and hesitancy degrees of intuitionistic fuzzy sets. The following lemma illustrates this characteristic: LEMMA 1. Let A = {ui , μA (ui ), νA (ui ) : ui ∈ U }, B = {ui , μB (ui ), νB (ui ) : ui ∈ U }, C = {ui , μC (ui ), νC (ui ) : ui ∈ U }, and G = {ui , μG (ui ), νG (ui ) : ui ∈ U } be four intuitionistic fuzzy sets of the universe of discourse U = {u1 , u2 , . . . , un }. If the following conditions hold μA (ui ) − μB (ui ) = μC (ui ) − μG (ui ) νA (ui ) − νB (ui ) = νC (ui ) − νG (ui ) then D(A, B) = D(C, G) being D any of the above Atanassov’s 2D or Szmidt and Kacprzyk’s 3D distance functions.
Proof. The proof is obvious for Atanassov’s 2D distances. For Szmidt and Kacprzyk’s 3D distances, the proof follows from the fact that μA (ui ) − μB (ui ) = μC (ui ) − μG (ui ) ∧ νA (ui ) − νB (ui ) = νC (ui ) − νG (ui ) imply τA (ui ) − τB (ui ) = τC (ui ) − τG (ui )
If |τA (ui ) − τB (ui )| = |τC (ui ) − τG (ui )|, then conditions in Lemma 1 can be generalized to |μA (ui ) − μB (ui )| = |μC (ui ) − μG (ui )| |νA (ui ) − νB (ui )| = |νC (ui ) − νG (ui )| International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS
405
This means that if we move both sets in the space shown in Figure 1b with the same changes in membership, nonmembership, and hesitancy degrees, then we obtain exactly the same distance between the two fuzzy sets. This linear feature of the above distances may not be adequate in some cases, because the human perception is not necessarily always linear. For example, we can classify the human behavior as perfect, good, acceptable, poor, and worst. Using fuzzy sets, we can assign their fuzzy membership as 1, 0.75, 0.5, 0.25, and 0. To find out if someone’s behavior is perfect or not, we only need to check if there is anything wrong with him. However, to differentiate good from acceptable, we have to count their positive and negative points. Obviously, the semantic distance between perfect and good should be greater than the semantic distance between good and acceptable. This semantic difference is not shown by the linear distances between their memberships. Therefore, a nonlinear representation of the distance between two intuitionistic fuzzy sets may benefit the representative power of intuitionistic fuzzy sets. Although nonlinearity could be modeled by using many different expressions, we will consider and use a simple one to model it. Here, we propose a new geometrical interpretation of intuitionistic fuzzy sets in 3D space using a restricted spherical surface. This new representation provides a convenient nonlinear measure of the distance between two intuitionistic fuzzy sets.
4. SPHERICAL INTERPRETATION OF INTUITIONISTIC FUZZY SETS: SPHERICAL DISTANCE Let A = {u, μA (u), νA (u) : u ∈ U } be an intuitionistic fuzzy set. We have μA (u) + νA (u) + τA (u) = 1 which can be equivalently transformed to x 2 + y 2 + z2 = 1 with x 2 = μA (u), y 2 = νA (u), z2 = τA (x) It is obvious that this is not the only transformation that could be used. However, as shown in the existing distances, there is no special reason to discriminate μA (u), νA (u), and τA (u). Therefore, a simple nonlinear transformation to unit sphere in a 3D Euclidean space is selected here, as shown in Figure 2. This transformation allows us to interpret an intuitionistic fuzzy set as a restricted spherical surface. An immediate consequence of this interpretation is that the distance between two elements of an intuitionistic fuzzy set can be defined as the spherical distance between their corresponding points on its restricted spherical surface representation. This distance is defined as the shortest path between the two International Journal of Intelligent Systems
DOI 10.1002/int
406
YANG AND CHICLANA
Figure 2. 3D sphere representation of intuitionistic fuzzy sets.
points, i.e. the length of the arc of the great circle passing through both points. For points P and Q in Figure 2, their spherical distance is25 . 1 (xP − xQ )2 + (yP − yQ )2 + (zP − zQ )2 . ds (P , Q) = arccos 1 − 2 This expression can be used to obtain the spherical distance between two intuitionistic fuzzy sets, A = {ui , μA (ui ), νA (ui ) : ui ∈ U } and B = {ui , μB (ui ), νB (ui ) : ui ∈ U } of the universe of discourse U = {u1 , u2 , . . . , un }, as follows: n
2 2 1 ds (A, B) = arccos 1 − μA (ui ) − μB (ui ) π i=1 2 +
νA (ui ) −
2
2 νB (ui ) + τA (ui ) − τB (ui )
where the factor π2 is introduced to get distance values in the range [0, 1] instead of [0, π2 ]. Because μA (ui ) + νA (ui ) + τA (ui ) = 1 and μB (ui ) + νB (ui ) + τB (ui ) = 1, we have that ds (A, B) =
n 2 arccos μA (ui )μB (ui ) + νA (ui )νB (ui ) + τA (ui )τB (ui ) π i=1
This is summarized in the following definition: International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS
407
DEFINITION 3. (Spherical distance) Let A = {ui , μA (ui ), νA (ui ) : ui ∈ U } and B = {ui , μB (ui ), νB (ui ) : ui ∈ U } be two intuitionistic fuzzy sets of the universe of discourse U = {u1 , u2 , . . . , un }. The spherical and normalized spherical distances between A and B are • Spherical distance ds (A, B)
ds (A, B) =
n 2 arccos μA (ui )μB (ui ) + νA (ui )νB (ui ) + τA (ui )τB (ui ) π i=1
• Normalized spherical distance dns (A, B)
dns (A, B) =
n 2 arccos μA (ui )μB (ui ) + νA (ui )νB (ui ) + τA (ui )τB (ui ) nπ i=1
Clearly, we have that 0 ≤ ds (A, B) ≤ n and 0 ≤ dns (A, B) ≤ 1. Different from the distances in Section 3, the proposed spherical distances implement in their definition not only the difference between membership, nonmembership, and hesitancy degrees but also their actual values. This is shown in the following result: LEMMA 2. Let A = {ui , μA (ui ), νA (ui ) : ui ∈ U } and B = {ui , μB (ui ), νB (ui ) : ui ∈ U } be two intuitionistic fuzzy subsets of the universe of discourse U = {u1 , u2 , . . . , un }. Let a = {a1 , a2 , . . . , an } and b = {b1 , b2 , . . . , bn } be two sets of real numbers (constants). If the following conditions hold for each ui ∈ U : μB (ui ) = μA (ui ) + ai νB (ui ) = νA (ui ) + bi then the following inequalities hold: n n 2 2 arccos 1 − ei2 ≤ ds (A, B) ≤ arccos (1 − ci )(1 − |ai | − |bi |) π i=1 π i=1 n n 2 2 arccos 1 − ei2 ≤ dns (A, B) ≤ arccos (1 − ci )(1 − |ai | − |bi |) nπ i=1 nπ i=1
International Journal of Intelligent Systems
DOI 10.1002/int
408
YANG AND CHICLANA
where ci = max{|ai |, |bi |} and ei = min{|ai |, |bi |}. The maximum distance between A and B is obtained if and only if one of them is a fuzzy set or their available information supports only opposite membership degree for each one.
Proof. See Appendix A.
Owing to the nonlinear characteristic of the spherical distance, they do not satisfy Lemma 1. However, the following properties hold for the spherical distances: LEMMA 3. Let A = {ui , μA (ui ), νA (ui ) :ui ∈ U }, B = {ui , μB (ui ), νB (ui ) : ui ∈ U } and E = {ui , μE (ui ), νE (ui ) : ui ∈ U } be three intuitionistic fuzzy subsets of the universe of discourse U = {u1 , u2 , . . . , un }. Let a = {a1 , a2 , . . . , an } and b = {b1 , b2 , . . . , bn } two sets of real positive numbers (constants) satisfying the following conditions: |μB (ui ) − μA (ui )| = ai , |νB (ui ) − νA (ui )| = bi |μE (ui ) − μA (ui )| = ai , |νE (ui ) − νA (ui )| = bi If E is one of the two extreme crisp sets with either μE (ui ) = 1 or νE (ui ) = 1 for all ui ∈ U , then the following inequalities hold: ds (A, B) < ds (A, E), dns (A, B) < dns (A, E)
The distance between intuitionistic fuzzy sets A and B is always lower than the distance between A and the extreme crisp sets E under the same difference of their memberships and nonmemberships.
Proof. We provide the proof just for the extreme fuzzy set with full memberships, being the proof for full nonmembership similar. With E being the extreme crisp set with full memberships, we have μE (ui ) = 1, νE (ui ) = 0, τE (ui ) = 0 Because |μE (ui ) − μA (ui )| = ai and |νE (ui ) − νA (ui )| = bi , then μA (ui ) = 1 − ai , νA (ui ) = bi , τA (ui ) = ai − bi From |μB (ui ) − μA (ui )| = ai and |νB (ui ) − νA (ui )| = bi , it is μB (ui ) = 1 − 2ai , νB (ui ) = 2bi , τB (ui ) = 2(ai − bi ). International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS
409
Therefore, ds (A, B) =
n 2 arccos (1 − ai )(1 − 2ai ) + 2bi2 + 2(ai − bi )2 π i=1 =
n √ 2 arccos 2ai + (1 − ai )(1 − 2ai ) π i=1
and ds (A, E) =
n 2 arccos 1 − ai π i=1
Obviously, we have √ 2ai + (1 − ai )(1 − 2ai ) > 1 − ai Thus ds (A, B) < ds (A, E) and dividing by n dns (A, B) < dns (A, E)
Lemma 3 shows that the extreme crisp sets with full memberships or full nonmemberships are categorically different from other intuitionitic fuzzy sets. With the same difference of memberships and nonmemberships, the distance from an extreme crip set is always greater than the distances from other intuitionistic fuzzy sets. This conclusion agrees with our human perception about the quality change against quantity change and captures the semantic difference between extreme situation and intermediate situations. 5. SPHERICAL DISTANCES FOR FUZZY SETS As we have already mentioned, fuzzy sets are particular cases of intuitionistic fuzzy sets. Therefore, the above spherical distances can be applied to fuzzy sets. In the following we provide Lemma 4 for the distance between two fuzzy sets. LEMMA 4. Let A = {ui , μA (ui ) : ui ∈ U } and B = {ui , μB (ui ) : ui ∈ U } be two fuzzy sets in the universe of discourse U = {u1 , u2 , . . . , un }. Let a = {a1 , a2 , . . . , an } be a set of nonnegative real constants. If |μA (ui ) − μB (ui )| = ai holds for each International Journal of Intelligent Systems
DOI 10.1002/int
410
YANG AND CHICLANA
ui ∈ U , then the following inequalities hold: n n 2 2 arccos 1 − ai 2 ≤ ds (A, B) ≤ arccos 1 − ai π i=1 π i=1 n n 2 2 arccos 1 − ai 2 ≤ dns (A, B) ≤ arccos 1 − ai nπ i=1 nπ i=1
The maximum distance between A and B is achieved if and only if one of them is a crisp set.
Proof. According to Definition 3, we have n 2 ds (A, B) = arccos μA (ui )μB (ui ) + νA (ui )νB (ui ) + τA (ui )τB (ui ) π i=1
For fuzzy sets, we have τA (ui ) = 0 and νA (ui ) = 1 − μA (ui ) τB (ui ) = 0 and νB (ui ) = 1 − μB (ui ) Because |μA (ui ) − μB (ui )| = ai and μA (ui ) ≥ 0, we have μB (ui ) = μA (ui ) ± ai If μB (ui ) = μA (ui ) + ai then ds (A, B) =
=
n 2 arccos μA (ui )μB (ui ) + (1 − μA (ui ))(1 − μB (ui ) π i=1 n 2 arccos μA (ui )(μA (ui ) + ai ) π i=1
+ (1 − μA (ui ))(1 − μA (ui ) − ai ) International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS
411
This can be rewritten as ds (A, B) =
n 2 arccos fi (μA (ui )) π i=1
√ √ with fi (t) = t(t + ai ) + (1 − t)(1 − t − ai ), t ∈ [0, 1 − ai ]. The extremes of function fi (t) will be among the solution of fi (t) = 0 t ∈ (0, 1 − ai ) and the values 0 and 1 − ai , i.e, among (1 − ai )/2, 0 and 1 − ai . The maximum √ value 1 − ai 2 is obtained when t = (1 − ai )/2, whereas the minimum value 1 − ai is obtained in both 0 and 1 − ai . We conclude that n n 2 2 arccos 1 − ai 2 ≤ ds (A, B) ≤ arccos 1 − ai π i=1 π i=1
When t = 0 or t = 1 − ai , we have, respectively, μA (ui ) = 0 and μB (ui ) = 1 − ai + ai = 1, which implies that one set among A and B has to be crisp in order to reach the maximum value under the given difference in their membership degrees. Following a similar reasoning, it is easy to prove that the same conclusion is obtained in the case μB (ui ) = μA (ui ) − ai . If μB (ui ) = μA (ui ) − ai for some i, and μB (uj ) = μA (uj ) + aj for some j , then we could separate the elements into two different groups, each of them satisfying the inequalities, and therefore their summation obviously satisfying it too. The normalized inequality is obtained just by dividing the first one by n. Because spherical distances are quite different from the traditional distances, the semantics associated with them also differ. For the same relative difference in membership degrees, the spherical distance varies with the locations of its two relevant sets in the membership degree space, 2D for fuzzy sets and 3D for intuitionistic fuzzy sets. The spherical distance achieves its maximum when one of the fuzzy sets is an extreme crisp set. The following example illustrates this effect: Example 1. Consider our previous example on human behavior. We can classify our behavior as perfect, good, acceptable, poor, and worst, with corresponding fuzzy membership of 1, 0.75, 0.5, 0.25, and 0, respectively. Let A = {u, 0.75 : u ∈ U }, B = {u, 0.5 : u ∈ U }, and E = {u, 1 : u ∈ U } be three fuzzy subsets, and U = {u} is a universe of discourse with one element only. From Section 3, we have d1 (A, B) = l1 (A, B) = e1 (A, B) = q1 (A, B) = 0.25 d1 (A, E) = l1 (A, E) = e1 (A, E) = q1 (A, E) = 0.25 International Journal of Intelligent Systems
DOI 10.1002/int
YANG AND CHICLANA
412
Obviously, we have d1 (A, B) = d1 (A, E),
l1 (A, B) = l1 (A, E)
e1 (A, B) = e1 (A, E),
q1 (A, B) = q1 (A, E)
From Definition 3, we have dns (A, B) = ds (A, B) =
n √ 2 arccos 0.75 ∗ 0.5 + (1 − 0.75)(1 − 0.5) = 0.17 π i=1
dns (A, E) = ds (A, E) =
n √ 2 arccos 0.75 ∗ 1 = 0.55 π i=1
We note that the traditional linear distance of fuzzy sets does not differentiate the semantic difference of a crisp set from a fuzzy set. However, ds (A, E) and dns (A, E) are much greater than ds (A, B) and dns (A, B). This demonstrates that the crisp set E is much more different from A than B although their membership difference are the same. Hence, the proposed spherical distance does show the semantic difference between a crisp set and a fuzzy set. This is useful when this kind of semantic difference is significant in the consideration. Figure 3 shows four comparisons between the spherical distance and the Hamming distance for two fuzzy subsets A, B with a universe of discourse with one element U = {u}. The curves represent the spherical distance, and straight lines denote Hamming distances. Figure 3a displays how the distance changes with respect to μB (x) when μA (x) = 0; Figure 3b uses the value μA (x) = 1; in Figure 3c the value μA (x) = 0.2 is used; finally μA (x) = 0.5 is used in Figure 3d. Clearly, the spherical distance changes sharply for values close to the two lower and upper memberships values, but slightly for values close to the middle membership value. In the case of the Hamming distance, the same rate of change is always obtained. Figure 4 displays how the spherical distance and Hamming distance changes with respect to μB (x) for all possible values of μA (x). Figure 4a shows that the spherical distance forms a curve surface, whereas a plane surface produced by the Hamming distance is shown in Figure 4b. Their contours in the bottom clearly show their differences. The contours for spherical distances are ellipses coming from (0, 0) and (1, 1) with curvatures increasing sharply near (0, 1) and (1, 0). Compared with these ellipses, the contours of Hamming distance are a set of parallel lines. These figures prove our conclusions in Lemma 4: The spherical distances do not remain constant as Hamming distances do when both sets experiment a same change in their membership degrees. International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS
413
Figure 3. Comparison between spherical and Hamming distances for fuzzy sets.
Figure 4. Grid of spherical and Hamming distances for fuzzy sets.
6. CONCLUSIONS An important issue related with the representation of intuitionistic fuzzy sets is that of measuring distances. Most existing distances are based on linear plane International Journal of Intelligent Systems
DOI 10.1002/int
414
YANG AND CHICLANA
representation of intuitionistic fuzzy sets, and therefore are also linear in nature, in the sense of being based on the relative difference between membership degrees. In this paper, we have looked at the issue of 3D representation of intuitionistic fuzzy sets. We have introduced a new spherical representation, which allowed us to define a new distance function between intuitionistic fuzzy sets: the spherical distance. We have shown that the spherical distance is different from those existing distances in that it is nonlinear with respect to the change of the corresponding fuzzy membership degrees, and thus it seems more appropriate than usual linear distances for non linear contexts in 3D spaces.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
19. 20.
Chiaoa JY, Bordeauxa AR, Ambady N. Mental representations of social status. Cognition 2004;93:B49–B57. Viglioccoa G, Vinsona DP, Damianb MF, Levelt W. Semantic distance effects on object and action naming. Cognition 2002;85:B61–B69. Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets Syst 1986;20:87–96. Atanassov KT. More on intuitionistic fuzzy sets. Fuzzy Sets Syst 1989;33:37–46. Atanassov KT. Intuitionistic fuzzy sets. Heidelberg west, Germany: Physica-Verlag; 1999. Hung W, Yang M. Fuzzy entropy on intuitionistic fuzzy sets. Int J Intell Syst 2006;21(4):443– 451. Hung W, Yang M. On similarity measures between intuitionistic fuzzy sets. Int J Intell Syst 2008;23(3):364–383. Kacprzyk J. Multistage fuzzy control. Chichester, UK: Wiley; 1997. Szmidt E, Kacprzyk J. On measuring distances between intuitionistic fuzzy sets. Notes IFS 1997;3:1–13. Szmidt E, Kacprzyk J. Distance between intuitionistic fuzzy sets. Fuzzy Sets Syst 2000;114(3):505–518. Szmidt E, Kacprzyk J. Similarity of intuitionistic fuzzy sets and the jaccard coefficient. In: IPMU’2004 Proceedings, Perugia, Italy, 2004; pp 1405–1412. Szmidt E, Baldwin J. Entropy for intuitionistic fuzzy set theory and mass assignment theory. Notes on IFS 2004;10(3):15–286. Szmidt E, Kacprzyk J. A new concept of a similarity measure for intuitionistic fuzzy sets and its use in group decision making. In: Torra V, Narukawa Y, Miyamoto S, editors. Modelling decisions for artificial intelligence. LNAI 3558, Berlin: Springer; 2005. pp 272–282. Szmidt E, Baldwin J. Assigning the parameters for intuitionistic fuzzy sets. Notes on IFS 2005;11(6):1–12. Szmidt E, Kacprzyk J. A new similarity measure for intuitionistic fuzzy sets: straightforward approaches may not work. In: Proceedings of Int. IEEE Conf: Fuzzy Systems, 2007. pp 1–6. Sambuc R. Functions φ -floues. application i’aide au diagnostic en patholologie thyroidienne, Ph.D. thesis, University of Marseille, France, 1975. Turksen B. Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst 1986;20:191– 210. Cornelis C, Atanassov KT, Kerre EE. Intuitionistic fuzzy sets and interval-valued fuzzy sets: a critical comparison. In: Wagenknecht M, Hampel R, editors. In: Third EUSFLAT Proceedings, European Society for Fuzzy Logic and Technology, Zittau, Germany, 2003. pp 159–163. Bustince H, Burillo P. Vague sets are intuitionistic fuzzy sets. Fuzzy Sets Syst 1996;79:403– 405. Deschrijver G, Kerre E. On the relationship between some extensions of fuzzy set theory. Fuzzy Sets Syst 2003;133(2):227–235. International Journal of Intelligent Systems
DOI 10.1002/int
INTUITIONISTIC FUZZY SETS 21.
415
Dubois D, Ostasiewicz W, Prade H. Fuzzy set: history and basic notions. In: Dubois D, Prade H, editors. Fundamentals of fuzzy sets. Boston, MA: Kluwer; 2000. pp 21–124. Dubois D, Prade H. Fuzzy sets and systems: theory and applications. New York: Academic Press; 1980. Wang G, He Y. Intuitionistic fuzzy sets and l-fuzzy sets. Fuzzy Sets Syst 2000;110:271–274. Xu Z, Yager RR. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 2006;35(4):417–433. Abramowitz M, Stegum IS, editors. Handbook of mathematical functions with formulas, graphs and mathematical tables. New York: Dover; 1972. Szmidt E, Baldwin J. New similarity measures for intuitionistic fuzzy set theory and mass assignment theory. Notes on IFS 2003;9(3):60–76.
22. 23. 24. 25. 26.
APPENDIX: PROOF FOR LEMMA 2 According to Definition 3, we have n 2 ds (A, B) = arccos μA (ui )μB (ui ) + νA (ui )νB (ui ) + τA (ui )τB (ui ) π i=1
Because A and B satisfy μB (ui ) = μA (ui ) + ai νB (ui ) = νA (ui ) + bi then τB (ui ) = 1 − μA (ui ) − νA (ui ) − ai − bi and n 2 ds (A, B) = arccos μA (ui )(μA (ui ) + ai ) + νA (ui )(νA (ui ) + bi ) π i=1
√ + (1 − μA (ui ) − νA (ui ))(1 − μA (ui ) − νA (ui ) − ai − bi ) Let f (u, v) =
u(u + ai ) + v(v + bi ) + (1 − u − v)(1 − u − v − ai − bi ).
Denoting a = |ai | and b = |bi |, then we distinguish 4 possible cases: Case 1: ai ≥ 0 and bi ≥ 0. In this case, f (u, v) =
u(u + a) + v(v + b) + (1 − u − v)(1 − u − v − a − b). International Journal of Intelligent Systems
DOI 10.1002/int
416
YANG AND CHICLANA
Let f (u, v)|u = 0, we have u=
a(1 − a − b − v) a(v − 1) and v = . b 2a + b
Let f (u, v)|v = 0, then v=
b(u − 1) b(1 − a − b − u) and v = . a 2b + a
u=
b(u − 1) a(v − 1) < 0 and v = < 0, b a
Because
the valid solutions are u=
b(1 − a − b − u) a(1 − a − b − v) and v = . 2a + b 2b + a
Solving these equations, we have u=
b(1 − a − b) a(1 − a − b) and v = , 2(a + b) 2(a + b)
and therefore f0 = f
a(1 − a − b) b(1 − a − b) , 2(a + b) 2(a + b)
=
1 − (a + b)2 .
Obviously, the third square root in f (u, v) must be defined so 0 ≤ u ≤ 1 − a − b and 0 ≤ v ≤ 1 − a − b. The boundary points are reached when u and v get their minimum or maximum values. Denoting t = τA (ui ), the following needs to be verified u + v + t = 1. If u = v = 1 − a − b then t = 2a + 2b − 1, and (1 − u − v)(1 − u − v − a − b) = (2a + 2b − 1)(a + b − 1) ≤ 0, which implies that the third square root in f (u, v) is not defined. Therefore, we have three boundary points for A u = 0, v = 1 − a − b, t = a + b u = 1 − a − b, v = 0, t = a + b International Journal of Intelligent Systems
DOI 10.1002/int
417
INTUITIONISTIC FUZZY SETS
u = 0, v = 0, t = 1, and therefore f1 = f (0, 1 − a − b) = f2 = f (1 − a − b, 0) =
(1 − a)(1 − a − b)
(1 − b)(1 − a − b)
f3 = f (0, 0) =
√ 1 − a − b.
If ai ≥ 0 and bi ≥ 0, then a + b ≤ 1, and we have f1 ≤ f3 ≤ f0 f2 ≤ f3 ≤ f0 . The relationship between f1 and f2 depends on the relationship between a and b. Let c = max{a, b} and e = min{a, b}, then we have
(1 − c)(1 − a − b) ≤ f (u, v) ≤
1 − e2 .
Case 2: ai ≤ 0 and bi ≤ 0. In this case, following a similar reasoning to the above one, the same conclusion is obtained. Case 3: ai ≤ 0 and bi ≥ 0. In this case f (u, v) =
u(u − a) + v(v + b) + (1 − u − v)(1 − u − v + a − b).
Let f (u, v)|u = 0, then u=
a(1 − v) a(1 + a − b − v) and v = . b 2a − b
Let f (u, v)|v = 0, then v=
b(1 − u) −b(1 + a − b − u) and v = . a a − 2b
For u = a(1 − v)/b and v = b(1 − u)/a, we get a = b, hence u + v = 1, and then both A and B are fuzzy sets. According to Lemma 4, we know that √ 1 − a ≤ f (u, v) ≤ 1 − a 2 . Note that the following also holds: (1 − a)(1 − a − b) ≤ (1 − a). International Journal of Intelligent Systems
DOI 10.1002/int
418
YANG AND CHICLANA
For u = [a(1 + a − b − v)]/(2a − b) and v = [−b(1 + a − b − u)]/(a − 2b), we have u=
a(1 + a − b) b(1 + a − b) and v = − . 2(a − b) 2(a − b)
If a > b then u ≥ 0 and v ≤ 0, and therefore the only possibility here is v = 0, i.e b = 0 in which case it is 1+a f0 = f , 0 = 1 − a2. 2 If a < b then u ≤ 0 and v ≥ 0, and therefore a = 0, in which case it is 1−b f1 = f 0, = 1 − b2 . 2 For u = a(1 − v)/b and v = [−b(1 + a − b − u)]/(a − 2b), we have u=
1−b a(1 + b) and v = 2b 2
and we obtain f
a(1 + b) 1 − b , 2b 2
=
1 − b2 = f1 .
Also, because u + v ≤ 1 we have that (a − b) · (1 + b) ≤ 0 and, therefore, it is a ≤ b. For u = [a(1 + a − b − v)]/(2a − b) and v = b(1 − u)/a, we have u=
b(1 − a) 1+a and v = 2 2a
and f
1 + a b(1 − a) , 2 2a
=
1 − a 2 = f0 .
In this case it is a ≥ b. The ranges for u and v are a ≤ u ≤ 1 and 0 ≤ v ≤ 1 − max{a, b} therefore, we have the following boundary points for A u = a, v = 1 − a, t = 0 International Journal of Intelligent Systems
DOI 10.1002/int
419
INTUITIONISTIC FUZZY SETS
u = a, v = 1 − b, t = b − a u = 1, v = 0, t = 0 u = a, v = 0, t = 1 − a Thus f2 = f (a, 1 − a) =
(1 − a)(1 − a + b)
f3 = f (a, 1 − b) =
√ 1−b
f4 = f (1, 0) =
√ 1−a
f5 = f (a, 0) =
(1 − a)(1 − b)
The following inequalities hold f2 ≤ f4 ≤ f0 f5 ≤ f4 ≤ f0 f5 ≤ f3 ≤ f1 Denoting fmin =
(1 − c)(1 − a − b) and fmax =
1 − e2
with e = min{a, b} and c = max{a, b}, then we have
1 − e2 ≥ f (u, v) ≥
(1 − c)(1 − a − b).
Case 4: ai ≥ 0 and bi ≤ 0. In this case, following a similar reasoning to the above one, the same conclusion is obtained. In the four cases, we conclude that n n 2 2 2 arccos 1 − ei ≤ ds (A, B) ≤ arccos (1 − ci )(1 − |ai | − |bi |) π i=1 π i=1 n n 2 2 2 arccos 1 − ei ≤ ds (A, B) ≤ arccos (1 − ci )(1 − |ai | − |bi |) nπ i=1 nπ i=1
where ci = max{|ai |, |bi |} and ei = min{|ai |, |bi |}. International Journal of Intelligent Systems
DOI 10.1002/int
420
YANG AND CHICLANA
According to the boundary points discussed in the above four cases, if ai bi ≥ 0 holds for each ui , we have μA (ui ) = 0, νA (ui ) = 1 − a − b, τA (ui ) = a + b or μA (ui ) = 1 − a − b, νA (ui ) = 0, τA (ui ) = a + b hence, set B has to satisfy μB (ui ) = a, νB (ui ) = 1 − a, τB (ui ) = 0 or μB (ui ) = 1 − b, νB (ui ) = b, τB (ui ) = 0 Obviously, B is a√fuzzy set. A similar conclusion can be drawn if ai bi < 0 holds √ for each ui and (1 − c)(1√ − c + e) ≤ (1 − a)(1 √ − b). However, if ai < 0 and bi > 0 hold for each ui and (1 − a)(1 − a + b) > (1 − a)(1 − b), we have μA (ui ) = a, νA (ui ) = 0, τA (ui ) = 1 − a hence, the set B satisfies μB (ui ) = 0, νB (ui ) = b, τB (ui ) = 1 − b Obviously, A is an intuitionistic fuzzy set with available information supporting only membership, whereas B is an intuitionistic fuzzy set with available information supporting only nonmembership.
International Journal of Intelligent Systems
DOI 10.1002/int
