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Distance measures for Interval Type-2 fuzzy numbers
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Distance measures for Interval Type-2 fuzzy numbers Juan Carlos Figueroa-García a,∗,1 , Yurilev Chalco-Cano b , Heriberto Román-Florez b a
Universidad Distrital Francisco José de Caldas, Bogotá, Colombia
b
Instituto de Alta Investigación, Universidad de Tarapacá, Arica, Chile
article
info
Article history: Received 28 February 2014 Received in revised form 11 September 2014 Accepted 17 November 2014 Available online xxxx
abstract This paper shows some distance measures for comparing Interval Type-2 fuzzy numbers. In addition, some definitions about ordering of Interval Type-2 fuzzy numbers based on their centroids, are provided. Some numerical examples are given, and some interpretation issues are explained. © 2014 Published by Elsevier B.V.
Keywords: Minkowski distance Interval Type-2 fuzzy numbers α -cuts Fuzzy ordering
1. Introduction and motivation Fuzzy sets theory has been applied to many problems with successful results, mostly in cases where statistical information is absent or is incomplete. In these cases, there is a need for obtaining information from other sources which includes the opinion of the experts and their perception about the problem, so its representation through fuzzy sets seems as an appropriate way to handle this kind of information. In some decision making problems, there are multiple experts analyzing the problem who add uncertainty. Examples of these problems include the resolution of fuzzy differential equations (see Chalco-Cano & Román-Flores [2]), fuzzy inequalities (see Rufián-Lizana et al. [18]), Type-2 fuzzy linear programming (see Figueroa-García [5–7]), fuzzy PERT problems (see Chen & Chang [4]), etc. This way, the use of Interval Type-2 Fuzzy numbers (IT2FN) is an appropriate way to handle uncertainty coming from judgement of multiple experts, or linguistic uncertainty coming from the perception of concepts. One of the most frequent questions when using fuzzy numbers regards to define a measure for efficiently comparing two fuzzy numbers. Although some methods for comparing fuzzy sets have been reported, there is a need for extending their results to a Type-2 fuzzy sets environment. Its applicability in different fields such as decision making, fuzzy inequalities and other problems where linguistic uncertainty appears, is wide. This paper focuses on defining some distances to compare IT2FNs, and a Type-2 fuzzy binary relation. Some examples are provided and their results are discussed. The paper is divided into six sections. Section 1 introduces the problem. In Section 2, some basic definitions about IT2FNs are provided; in Section 3, some distance measures are presented. Section 4 presents the concept of fuzzy ordering of IT2FNs; Section 5 presents an application example; and finally in Section 6, the concluding remarks of the study are presented.
∗
Corresponding author. E-mail addresses: [email protected] (J.C. Figueroa-García), [email protected] (Y. Chalco-Cano), [email protected] (H. Román-Florez).
1 LAMIC research group member. http://dx.doi.org/10.1016/j.dam.2014.11.016 0166-218X/© 2014 Published by Elsevier B.V.
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˜ Fig. 1. Interval Type-2 fuzzy set A.
2. Basic definitions of IT2FNs Firstly, we establish some basic notations. A classical fuzzy set, namely Type-1 Fuzzy Set (T1FS), is denoted by capital letters e.g. A with a membership function µA (x) defined over x ∈ X , while an Interval Type-2 Fuzzy Set (IT2FS) is denoted by emphasized capital letters A˜ with a membership function µA˜ (x) defined over x ∈ X . µA measures the affinity degree of a value x ∈ X regarding the concept/word/label A, and µA˜ (x) measures the uncertainty degree of a value x ∈ X regarding A, so A˜ measures uncertainty around A. A Type-2 fuzzy set is an ordered pair A˜ = {((x, u), Jx , fx (u)) | x ∈ X ; u ∈ Jx ⊆ [0, 1]}, where A˜ represents uncertainty around the word A, Jx is the primary membership of x, u is its domain of uncertainty, and F2 (X) is the class of all Type-2 fuzzy sets (see Mendel [11,12]). Their mathematical definitions are: A˜ : X → F([0, 1]) A˜ =
x∈X
fx (u)/(x, u),
Jx ⊆ [0, 1]
u∈Jx
where µA˜ (x) is composed by an infinite amount of embedded Type-1 fuzzy sets namely Ae . Every element x has associated a set of primary memberships Jx weighted by a Secondary fuzzy set fx (u) where u is the domain of uncertainty of x, u ∈ Jx ⊆ [0, 1]. Now, an IT2FS is a simplification of a T2FS since its secondary membership function is assumed to be 1, which is shown as follows A˜ =
x∈X
1/(x, u) = u∈Jx
x∈X
1/u
x,
(1)
u∈Jx
where x, u are the primary and secondary variables, and fx (u)/u = 1 is the secondary membership function. ˜ namely FOU(A˜ ), Uncertainty about the word A is conveyed by the union of all of Jx into the Footprint Of Uncertainty of A, which is bounded by two functions: An Upper membership function UMF (A˜ ) = µ ¯ A˜ (x) ≡ A¯ and a Lower membership function
LMF (A˜ ) = µ ˜ (x) ≡ A. FOU(A˜ ) is shown in Fig. 1. A ˜ supp (A˜ ) is In Fig. 1, A˜ is an IT2FS, the universe of discourse for the primary variable x is the set x ∈ X , the support of A, α the interval x ∈ [ˇx, xˆ ] and µA˜ is a triangular membership function with parameters xˇ , xˆ , xˇ , xˆ and x¯ . µ ¯ A˜ (x) is the degree of ¯ αµ (x) is the degree of membership of a specific value x regarding its membership of a specific value x regarding its UMF, A; A˜ LMF, A, and u ∈ Jx = [αµ ˜ (x), α µA˜ (x)]. A
2.1. Decomposition of an IT2FS One of the most comprehensive ways to represent a T1FS A is through α -cuts. The α -cut of a A, namely αA, is defined as A = {x | µA (x) > α}. Thus, a fuzzy set A is the union of its α -cuts, α∈[0,1] α · αA, where ∪ denotes union (Klir & Yuan [9]).
α
Now, the extension of α -cut of A to the α -cut of A˜ (see Figueroa [5]), is shown next.
Definition 1 (Primary α -cut of an IT2FS). The primary α -cut of an Interval Type-2 fuzzy set αA˜ is the union of all x ∈ X whose primary memberships Jx are greater than α, Jx > α , this is: α˜
A= x∈X
u∈Jx >α
1/u
x;
Jx ⊆ [0, 1], α ∈ [0, 1].
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˜ Fig. 2. αA˜ of the set A.
˜ The symbol Now, αAe = {x | µAe (x) > α} is an α -cut done over an embedded set Ae ∈ A. α
denotes fuzzy union, so αA˜ is
α˜
the union of all Ae . A graphical representation of A is provided in Fig. 2. These lead to define the boundaries of each α -cut, as follows.
˜ Consider an Interval Type-2 fuzzy set with known UMF and LMF, and denote inf αA˜ := αA˜ L Definition 2 (Crisp Bounds of αA). α˜ α˜ U and sup A := A as the left and right interval-valued bounds of αA˜ := [inf αA˜ , sup αA˜ ]. Let the bounds of αA˜ be defined as the boundaries of the α -cuts of the shapes of its UMF and LMF, as follows: α˜ L
A :=
inf α µA˜ (x, u) ; inf αµ ˜ (x, u) = A x x
α˜ L(+)
A
, αA˜ L(−)
α α˜ R α A := sup µ ˜ (x, u) ; sup µA˜ (x, u) = αA˜ R(−) , αA˜ R(+) . A
(3)
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x
x
2.2. Type-2 fuzzy numbers In this paper, a Type-2 fuzzy number (T2FN) is considered as the extension of a Type-1 fuzzy number. This means that A˜ is an T2FS whose UMF and LMF are fuzzy numbers (e.g. normal and convex fuzzy subsets of R, Zadeh [24]). αA is closed interval for all α ∈ [0, 1], and its support supp (A) is defined over R. This also means that a fuzzy number is a normal and convex fuzzy set, which is shown as follows. Definition 3 (Type-2 Fuzzy Number). Let A˜ ∈ F2 (R). Then, A˜ is a Type-2 Fuzzy Number (T2FN) iff there exists a closed interval [a, b] ̸= 0 for each UMF (µ ¯ A˜ ) and LMF (µ ˜ ) such that A
1 l(x, u) r (x, u)
µA˜ (x, u) =
for x ∈ [a, b], for x ∈ [−∞, a], for x ∈ [b, ∞],
u ∈ Jx ⊆ [0, 1] u ∈ Jx ⊆ [0, 1] u ∈ Jx ⊆ [0, 1]
(5)
where l : (−∞, a) → F([0, 1]), u ∈ Jx ⊆ [0, 1] is monotonic non-decreasing, continuous from the right, and l(x, u) = 0 for x < ω1 , and r : (b, ∞) → F([0, 1]), u ∈ Jx ⊆ [0, 1] is monotonic non-increasing, continuous from the left, and r (x, u) = 0 for x > ω2 . Now, A˜ is an T2FN if both their UMF and LMF are fuzzy numbers, which is equivalent to the definition given by Mitchell [13]. The set A˜ shown in Fig. 1 is also an Interval Type-2 Fuzzy Number (IT2FN). 3. Distance-based Similarity between two IT2FNs Our approach is based on the idea of comparing the UMF and LMF of two IT2FNs to establish if they are equal or not. This way, we use the classic axioms of distance since αA˜ can be seen as a crisp set. Let R be the set of real numbers, and R+ be its non-negative orthant. X is the universal set; F2 (X ) is the class of all IT2FNs of X ; P (X ) is the class of all crisp sets of X , then the distance between A and B namely d(A, B) is called to be a metric (or simply distance) if d(A, B) ∈ R+ and satisfies the following three axioms: D1: d(A, B) = d(B, A), D2: d(A, A) = 0, D3: d(A, C ) 6 d(A, B) + d(B, C ).
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Now, in the p-dimensional Euclidean space Rp , Minkowski defined a family of Lm metric distances given by dm (A, B) =
p
1/m m
|xi − yi |
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i =1
where A, B are two points in Rp with coordinates xi and yi . In the space of the primary variable of an IT2FN, every α -cut returns a set of intervals whose supports are bounded over R (see Definition 2). This makes the definition of a distance between two IT2FNs using α -cuts easier, since we can decompose A˜ and B˜ into α -cuts to compute the distance between αA˜ and αB˜ as the distance between two intervals. Now, we use the following definition of distance between two intervals: Definition 4. Let A ∈ [a, a] and B ∈ [b, b] two interval sets defined over R+ , the L1 distance between A and B is defined as follows: d(A, B) = a − b + a − b .
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In general for crisp (or interval) sets, Kosko [10], and Nguyen & Kreinovich [15] used A ∩ B ⊆ A ∪ B as a useful measure to evaluate whether A = B or not, so A = B if and only if A ∩ B = A ∪ B. Hence, for crisp (or interval) finite sets, we can check if A is equal to B using the following ratio:
|A ∩ B| . |A ∪ B| This ratio is equal to 1 iff A = B and <1 if A ̸= B. As this ratio is smaller, more elements from A ∪ B are not part of the intersection A ∩ B. This can be extended to IT2FNs, since A˜ = B˜ iff µA˜ = µB˜ , as follows:
¯ A˜ (x, u) = µ ¯ B˜ (x, u) ˜A = B˜ ⇔ µ µA˜ (x, u) = µB˜ (x, u). ˜ and A˜ ̸= B˜ ; C (A˜ ) = C (B˜ ), so there is the possibility of having A˜ ̸= B˜ with equal or similar centroids. In other case A˜ ̸= B, Hence, the comparison of their centroids is a useful measure for decision making. Several similarity measures for IT2FSs have been proposed by Nguyen & Kreinovich [15], Zheng et al. [25], Xuecheng [23], and Hung & Yang [8]. Their results are focused to establish similarity between two IT2FSs based on the distance among memberships, accomplishing the basic axioms of a similarity measure. Our proposal compares intervals coming from αA˜ using an L1 Minkowski distance, as shown in next sections. 3.1. α -cut based distance of IT2FNs ˘ Ramík & Rimánek [16] defined that two sets A, B are equal iff αA = α B for every α ∈ [0, 1], αA := [inf αA, sup αA] and α α B := [inf B, sup B]. Given the concept of α -cut of an IT2FN (see Definition 2), we can intuitively think that A˜ = B˜ iff αA˜ = αB˜ for every α ∈ [0, 1], this is
α
A˜ = B˜ ⇔
|αA˜ L ∩ αB˜ L | |αA˜ R ∩ αB˜ R | = = 1, α L α L | A˜ ∪ B˜ | |αA˜ R ∪ αB˜ R |
∀ α ∈ [0, 1].
This concept leads to the following proposition:
˜ and αB˜ is the α -cut of B, ˜ α ∈ [0, 1], then A˜ = B˜ if only if Proposition 1. Let A˜ , B˜ ∈ F2 (R) be two IT2FNs. If αA˜ is the α -cut of A, α˜
A = αB˜ ∀ α ∈ [0, 1].
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Proof. A˜ = B˜ implies that αA˜ L = αB˜ L and αA˜ R = αB˜ R . We can show that if the equality of the intervals αA˜ L = αB˜ L and αA˜ R = αB˜ R hold ∀ α ∈ [0, 1], then by Definition 4, d(αA˜ L ,αB˜ L ) = 0, d(αA˜ R ,αB˜ R ) = 0 hold ∀ α ∈ [0, 1], so d(A˜ L , B˜ L ) = 0, d(A˜ R , B˜ R ) = 0 and the proof is complete. Based on the results proposed by Chaudhuri & RosenFeld [3], we propose the distance dα based on α -cuts and the L1 Minkowski distance (see Eq. (6)) to measure how different A˜ and B˜ are.
˜ ˜ Proposition 2. Let A˜ , B˜ ∈ F2 (R) be two IT2FNs. The distance (metric) dα between A and B given a set of n α -cuts, {α1 , α2 , . . . , αn } and Λ = ni=1 αi , is:
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dα (A˜ , B˜ ) ,
N 1
Λ
i=1
1
0
–
αi αiA˜ L(+) − αiB˜ L(+) + αiA˜ L(−) − αiB˜ L(−) + αiA˜ R(−) − αiB˜ R(−) + αiA˜ R(+) − αiB˜ R(+)
if α is continuous, then Λ = dα (A˜ , B˜ ) , 2
)
1 0
5
(9)
α dα = 1/2, so dα is defined as:
α αA˜ L(+) − αB˜ L(+) + αA˜ L(−) − αB˜ L(−) + αA˜ R(−) − αB˜ R(−) + αA˜ R(+) − αB˜ R(+) dα.
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Proof. Consider a single α -cut (αi ) and Eq. (9). It is easy to verify axioms D1 and D2 since it is based on the L1 Minkowski distance (see Eq. (6)). To proof D3 we take the extreme case that A˜ = B˜ which verifies dαi (A˜ , B˜ ) = 0 since αiA˜ L(+) = αiB˜ L(+) ,
= αiB˜ L(−) , αiA˜ R(−) = αiB˜ R(−) and αiA˜ R(+) = αiB˜ R(+) ; when B˜ = C˜ then dαi (B˜ , C˜ ) = 0 since αiB˜ L(+) = αiC˜ L(+) , αiB˜ L(−) = C , B = αiC˜ R(−) and αiB˜ R(+) = αiC˜ R(+) , so dαi (A˜ , C˜ ) = dαi (A˜ , B˜ ); when A˜ ̸= B˜ and B˜ ̸= C˜ then A˜ ̸= C˜ , dαi (A˜ , B˜ ) > 0 and dαi (B˜ , C˜ ) > 0, so dαi (A˜ , C˜ ) < dαi (A˜ , B˜ ) + dαi (B˜ , C˜ ); and finally if A˜ = C˜ then dαi (A˜ , C˜ ) = 0 since αiA˜ L(+) = αiC˜ L(+) , αi ˜ L(−) A = αiC˜ L(−) , αiA˜ R(−) = αiC˜ R(−) and αiA˜ R(+) = αiC˜ R(+) , so 0 6 dαi (A˜ , B˜ ) + dαi (B˜ , C˜ ) which is equivalent to 0 6 2dαi (B˜ , C˜ ), αi ˜ L(−)
A
αi ˜ L(−) αi˜ R(−)
and the proof is complete. 3.2. Centroid-based distance of IT2FNs The distance dα presented in Definition 2 can be restrictive in some applications (combinatorics, complex optimization problems, etc.) that require high computational efforts. To deal with, we propose the use of the distance between the centroids of two IT2FNs, whose concepts are shown next. ˜ its centroid C (A˜ ) is an interval bounded by two values min{C (Ae )} = Cl (A˜ ) and max{C (Ae )} = Cr (A˜ ). Every Given a set A, ˜ so there is an infinite amount of centroids enclosed into point enclosed into C (A˜ ) = [Cl (A˜ ), Cu (A˜ )] is a possible centroid of A, C (A˜ ) (see Wu & Mendel [20], and Melgarejo [1]). Two numbers A˜ and B˜ are centroid-equal if their centroids are equal, this is C (A˜ ) = C (B˜ ). Now, we need a measure to compare C (A˜ ), C (B˜ ) in order to say if they are either equal or not. To do so, we use the distance between two intervals X = [¯x, x] and Y = [¯y, y], d(X , Y ) = max{|¯x − y¯ |, |x − y|} proposed by Moore, Kearfott and Cloud [14]. Now, the distance between C (A˜ ) and C (B˜ ) is: d(C (A˜ ), C (B˜ )) = max{|Cl (A˜ ) − Cl (B˜ )|, |Cu (A˜ ) − Cu (B˜ )|}.
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This distance measures the max distance between the boundaries of C (A˜ ) and C (B˜ ), so the min distance is out of d(C (A˜ ), C (B˜ )). An alternative way to measure the distance between C (A˜ ) and C (B˜ ) is by adding the distance among all boundaries. This distance can be defined as follows:
˜ The L1 distance dc between C (A˜ ), C (B˜ ) is: Proposition 3. Let C (A˜ ), C (B˜ ) be the centroids of A˜ , B. dc (C (A˜ ), C (B˜ )) = |Cl (A˜ ) − Cl (B˜ )| + |Cu (A˜ ) − Cu (B˜ )|.
(12)
Proof. Axioms D1 and D2 are easy to proof by simple deduction. To proof D3 we will take the extreme case that C (A˜ ) = C (B˜ ) which immediately verifies d(C (A˜ ), C (B˜ )) = 0, and d(C (A˜ ), C (B˜ )) = d(C (A˜ ), C (C˜ )); when B˜ = C˜ , then d(C (B˜ ), C (C˜ )) = 0, so d(C (A˜ ), C (C˜ )) = d(C (A˜ ), C (B˜ )); when C (A˜ ) ̸= C (B˜ ) and C (B˜ ) ̸= C (C˜ ), then C (A˜ ) ̸= C (C˜ ), d(C (A˜ ), C (B˜ )) > 0 and d(C (B˜ ), C (C˜ )) > 0, so d(C (A˜ ), C (C˜ )) < d(C (A˜ ), C (B˜ )) + d(C (B˜ ), C (C˜ )); and finally if C (A˜ ) = C (C˜ ) then d(C (A˜ ), C (C˜ )) = 0, so 0 6 d(C (A˜ ), C (B˜ )) + d(C (B˜ ), C (C˜ )) which is equivalent to 0 6 2d(C (B˜ ), C (C˜ )), and the proof is complete. The distance dc preserves the information of both boundaries, since Eq. (12) adds both max and min distances. This means that the size of dc determines how far C (A˜ ) and C (B˜ ) are. Now we can define if C (A˜ ) = (B˜ ) using d(C (A˜ ), C (B˜ )) and dc (C (A˜ ), C (B˜ )), as follows.
˜ It is said that A˜ , B˜ are centroid-equal if: Definition 5. Let C (A˜ ), C (B˜ ) be the centroids of A˜ , B. C (A˜ ) = C (B˜ ) ⇔ d(C (A˜ ), C (B˜ )) = 0
(13)
C (A˜ ) = C (B˜ ) ⇔ dc (C (A˜ ), C (B˜ )) = 0.
(14)
Note that A˜ ̸= B˜ ; C (A˜ ) ̸= C (B˜ ), so conversely A˜ = B˜ ⇒ C (A˜ ) = C (B˜ ). This means that A˜ and B˜ can be different, so dα (A˜ , B˜ ) > 0 (see Definition 2) while having equal centroids, C (A˜ ) = C (B˜ ). 3.3. Subsethood of the centroid of two IT2FNs Another way to measure the relation between two IT2FNs is by using the concept of subsethood introduced by Kosko [10], and Nguyen & Kreinovich [15]. Given two crisp sets A and B, the set equality d= is defined as d= = |A ∩ B|/|A ∪ B| and the
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subsethood d⊆ is defined as d⊆ = |A ∩ B|/|A|. The set equality and subsethood of two centroids C (A˜ ), C (B˜ ) can be defined as follows: d= =
|C (A˜ ) ∩ C (B˜ )| , |C (A˜ ) ∪ C (B˜ )|
(15)
d⊆ =
|C (A˜ ) ∩ C (B˜ )| . |C (A˜ )|
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The case when C (A˜ ) = C (B˜ ) implies that d= = 0 and d⊆ = 0, so the use of d= and d⊆ makes sense when C (A˜ )∩ C (B˜ ) ̸= 0. Therefore, we can assert that if C (A˜ ) ∩ C (B˜ ) ̸= 0, then A˜ and B˜ have a degree of set equality based on their centroids. 4. Fuzzy max order of an IT2FN ˘ The binary relation . for classical fuzzy sets has been proposed and investigated by Ramík & Rimánek [16]. Let A and B be two fuzzy numbers. Then A . B if and only if sup αA 6 sup αB and inf αA 6 inf αB for each α ∈ [0, 1], where αA and αB are α -cuts of A and B respectively, and αA := [inf αA, sup αA] and αB := [inf αB, sup αB]. This binary relation satisfies the axioms of a partial order relation on F (R) and is called fuzzy max order. It is possible to extend the fuzzy max order to IT2FNs, as follows: Definition 6. Given two IT2FNs A˜ , B˜ ∈ F2 (R), the binary relation -, A˜ - B˜ holds iff αA˜ 6 αB˜ ∀ α ∈ [0, 1]. This is equivalent to the following result sup αA˜ 6 sup αB˜ ∀ α ∈ [0, 1], α˜
α˜
inf A 6 inf B ∀ α ∈ [0, 1],
(17) (18)
where αA˜ and αB˜ are α -cuts of A˜ and B˜ respectively. This way, A˜ - B˜ implies that every α -cut done over A˜ , B˜ can be seen as the crisp sets inf αA˜ := [αA˜ L(+) , αA˜ L(−) ], sup αA˜ := [ A , A ], inf αB˜ := [αB˜ L(+) , αB˜ L(−) ], sup αB˜ := [αB˜ R(−) , αB˜ R(+) ], αA˜ := [inf αA˜ , sup αA˜ ], and αB˜ := [inf αB˜ , sup αB˜ ]. This implies that αA˜ L(+) 6 αB˜ L(+) , αA˜ L(−) 6 αB˜ L(−) ,α A˜ R(−) 6 αB˜ R(−) and αA˜ R(+) 6 αB˜ R(+) hold ∀ α ∈ [0, 1]. Wang & Kerre [19] defined some reasonable properties to be fulfilled when ranking fuzzy sets, which were later extended by Wu & Mendel [21,22], and Mendel & Wu [12] to IT2FSs. We extend them to the binary relation A˜ - B˜ since the α -cuts of an IT2FN are intervals. This leads us to the following result: α˜ R(−) α˜ R(+)
˜ B, ˜ and C˜ be IT2FNs such as A˜ - B. ˜ The binary relation - satisfies the following ordering properties: Theorem 4. Let A, P1. P2. P3. P4.
˜ then A˜ = B. ˜ If A˜ - B˜ and B˜ - A, If A˜ - B˜ and B˜ - C˜ , then A˜ - C˜ . ˜ then A˜ % B. ˜ If A˜ ∩ B˜ = 0 and A˜ is on the right of B, The order of A˜ and B˜ is not affected by the other centroids under comparison.
Proof. See Appendix A. Basically, A˜ - B˜ if every boundary of every α -cut of A˜ , B˜ holds the binary relation 6 for every α -cut. This condition ensures that every value of A˜ is less or equal than every value of B˜ given any uncertainty degree α ∈ [0, 1]. This is a hard requirement in some applications since it is possible that not all α -cuts fulfill this condition. In cases where not all α -cuts meet the partial ˜ This way, we use the centroids of A˜ and B˜ to measure how different they order 6, then A˜ is not strictly less or equal than B. are (in average).
˜ It is said that A˜ is centroid less or equal than B, ˜ A˜ -c B, ˜ iff C (A˜ ) 6 C (B˜ ), Definition 7. Let C (A˜ ), C (B˜ ) be the centroid of A˜ and B. which implies that: Cu (A˜ ) 6 Cl (B˜ ).
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˜ i.e. Cl (A˜ ) 6 Cu (B˜ ) comes from partial ordering of intervals (see Moore, Kearfott & Cloud [14]). Note that The idea of A˜ -c B, the following two cases: Cu (A˜ ) 6 Cl (B˜ ) and Cl (B˜ ) 6 Cu (A˜ ) has been considered. There is the possibility of having that Cl (B˜ ) 6 Cl (A˜ ) and Cu (A˜ ) 6 Cu (B˜ ). This means that C (B˜ ) ⊆ C (A˜ ), so we cannot assert that C (A˜ ) -c C (B˜ ) or C (B˜ ) -c C (A˜ ), but we can say that they are similar (in some sense). At this point, if C (B˜ ) ⊆ C (A˜ ) (or conversely C (A˜ ) ⊆ C (B˜ )), then a proper way to compare A˜ and B˜ is by computing its degree of equality and subsethood as defined in Section 3.3. As the centroid of an IT2FN is a crisp set, we briefly show that at least the first four properties proposed by Wang & Kerre [19] are satisfied by d(C (A˜ ), C (B˜ )) and dc (C (A˜ ), C (B˜ )), from a crisp point of view:
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Table 1 Parameters and centroid of each IT2FN A˜ j . Set j
aˇ j
aj
aˆ j
aˇ j
aj
aˆ j
Cl (A˜ j )
Cu (A˜ j )
c (A˜ j )
A˜ 1 A˜ 2 A˜ 3 A˜ 4 A˜ 5
2 0 0 0 0
5 6 3 5 2
9 9 7 9 8
3 3 1 1 1
5 6 3 5 2
6 7 5 6 6
4.329 4.294 2.666 3.66 2.665
5.68 6 3.671 5 3.686
5.0045 5.147 3.1685 4.33 3.1755
˜ The binary relation -c satisfies the following Theorem 5. Let C (A˜ ), C (B˜ ), and C (C˜ ) be the centroids of three IT2FNs such as A˜ -c B. ordering properties: P1. P2. P3. P4.
If C (A˜ ) > C (B˜ ) and C (B˜ ) > C (A˜ ), then C (A˜ ) = C (B˜ ). If C (A˜ ) > C (B˜ ) and C (B˜ ) > C (C˜ ), then C (A˜ ) > C (C˜ ). If C (A˜ ) ∩ C (B˜ ) = 0 and C (A˜ ) is on the right of C (B˜ ), then C (A˜ ) > C (B˜ ). The order of C (A˜ ) and C (B˜ ) is not affected by the other centroids under comparison.
Proof. See Appendix B.
˜ This does Those properties ensures a logical ordering of centroids as a base for comparing the expected values of A˜ and B. not imply that C (A˜ ) = C (B˜ ) ⇒ A˜ = B˜ since there exists the possibility of having A˜ ̸= B˜ while C (A˜ ) = C (B˜ ). This leads us to propose the following corollary. Corollary 1. Let A˜ , B˜ ∈ F2 (R) be two IT2FNs. The binary relation A˜ - B˜ implies that C (A˜ ) 6 C (B˜ ), A˜ - B˜ H⇒
Cl (A˜ ) 6 Cl (B˜ ) Cu (A˜ ) 6 Cu (B˜ ).
This corollary is a natural extension of the concept of dα in the sense that two IT2FNs whose distance is large cannot have the ˜ then their centroids (which are computed using µ same centroid. If - is strictly true for A˜ and B, ¯ and µ) should be different, more precisely C (A˜ ) 6 C (B˜ ). It is also clear that if A˜ = B˜ then C (A˜ ) = C (B˜ ), and conversely C (A˜ ) = C (B˜ ) does not mean ˜ in other words: A˜ = B, A˜ = B˜ ⇒ C (A˜ ) = C (B˜ ).
˜ Also note that dα = 0 implies C (A˜ ) = C (B˜ ). Even when it is not a common Conversely we have C (A˜ ) = C (B˜ ) ; A˜ = B. case, there is a chance of having C (A˜ ) = C (B˜ ) while dα ̸= 0. This means that µA˜ ̸= µB˜ while having equal centroids. Remark 1. Román-Flores & Chalco-Cano [17] have discussed about H-continuity for defuzzification measures (centroid, in our case). Given the distance measure dα between two IT2FNs, then there exists a sequence of A˜ n IT2FNs that converge to a single fuzzy set A˜ τ if and only if dα (A˜ n , A˜ τ ) → 0 when limn→∞ . This also leads us to think that if their centroids C (A˜ ), C (B˜ ) are compatible in dα metric to A˜ , B˜ then it should be true that if dα (A˜ n , A˜ τ ) → 0 when limn→∞ then dα (C (A˜ n ), C (A˜ τ )) → 0 when limn→∞ . 5. Application example In this section, we compare five IT2FNs with triangular LMFs and UMF, defined by the following parameters: LMF = T (ˇa, a, aˆ ) and UMF = T (a¯ˇ , a¯ , a¯ˆ ). To do so, we apply the proposed distances dα , d, d= , and d⊆ . The idea is to see how different (by pairs) are all IT2FNs and their centroids (computed using the proposal of Figueroa [5]). Table 1 shows all parameters used in the comparison. Now, we compute dα , d and dc
0
1.46 0
dα =
0
dc =
0.36 0
3.83 5 0
0.66 1.83 3.17 0
4.83 6 1.17, 4.17 0
3.67 3.96 0
1.35 1.63 2.32 0
3.66 3.94 0.02, 2.31 0
0
d =
0.32 0
2.01 2.33 0
0.68 1 1.33 0
1.99 2.31 0.02, 1.31 0
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Fig. 3. Sets A˜ 2 and A˜ 5 .
the degree of equality d= and subsethood d⊆ between centroids are presented next:
0
0.979 0
d= =
0 0 0
0.332 0.302 0.005 0
0 0 0.984 , 0.011 0
1 0.792 d⊆ = 0 0.501 0
1 1 0 0.527 0
0 0 1 0.008 0.984
0.497 0.414 0.011 1 0.025
0 0 1 , 0.019 1
and finally, we compare the centroids of A˜ j using the center of centroid c (A˜ ) proposed by Wu & Mendel [22]. To do so, we compute c (A˜ ) = (Cl (A˜ ) + Cu (A˜ ))/2 to rank them as crisp numbers, which leads to the following rank: c (A˜ 3 ) 6 c (A˜ 5 ) 6 c (A˜ 4 ) 6 c (A˜ 1 ) 6 c (A˜ 2 ).
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5.1. Discussion of the results By using the above results, we can see differences among IT2FNs. The farthest fuzzy numbers are A˜ 2 and A˜ 5 since dα (A˜ 2 , A˜ 5 ) = 6, which is concordance to the distance between their centroids. For instance, d(C (A˜ 2 ), C (A˜ 5 )) = 2.314 and dc (C (A˜ 2 ), C (A˜ 5 )) = 3.943 which are the second largest distances, and of course d= (C (A˜ 2 ), C (A˜ 5 )) = 0, d⊆ (C (A˜ 2 ), C (A˜ 5 )) = 0, and d⊆ (C (A˜ 5 ), C (A˜ 2 )) = 0. This is a consequence of a large distance between two sets, since as dα increases, their centroids are more distant. An example of two different IT2FNs is shown in Fig. 3. The closest IT2FNs are A˜ 1 and A˜ 4 since dα (A˜ 1 , A˜ 4 ) = 0.666 which is a sign of its similarity. Their centroids have a relationship to dα since d(C (A˜ 1 ), C (A˜ 4 )) = 0.679 and dc (C (A˜ 1 ), C (A˜ 4 )) = 1.349 which are relatively small. Their degrees of equality and subsethood d= (C (A˜ 1 ), C (A˜ 4 )) = 0.332, d⊆ (C (A˜ 1 ), C (A˜ 4 )) = 0.497 and d⊆ (C (A˜ 4 ), C (A˜ 1 )) = 0.501 indicate that their centroids are similar. This comes from H-continuity concepts (see Remark 1), since small values of dα lead to similar centroids. Seen as centroids, the closest IT2FNs are A˜ 1 and A˜ 2 since their degrees of equality and subsethood d= (C (A˜ 1 ), C (A˜ 2 )) = 1, d⊆ (C (A˜ 1 ), C (A˜ 2 )) = 1 and d⊆ (C (A˜ 2 ), C (A˜ 1 )) = 0.792 are large, and their distances d(C (A˜ 1 ), C (A˜ 2 )) = 0.32 and dc (C (A˜ 1 ), C (A˜ 2 )) = 0.355 are smaller than other centroids. Also note that C (A˜ 3 ) ≈ C (A˜ 5 ) while A˜ 3 ̸= A˜ 5 , so dα = 1.167. This is a possible case (see Corollary 1) since different combinations of shapes of IT2FNs can lead to equal centroids, conversely case is not possible. To determine which IT2FN is less/higher or equal than others, Definition 6 provides information to define its fuzzy binary relation -. The obtained results are shown as follows:
−
- = 1 1 1
− 1 1 1
0 0
0 0 1
0
−
1
0 0
− , 0
−
− −
1
−
1
1
-c = 1
0 0
− −
− − −
0 0
− . −
−
In matrix -, ‘1’ means -, ‘0’ means %, and ‘–’ means that they are not fully comparable. This results show us some interesting information: A˜ 1 and A˜ 2 are greater than others. IT2FN A˜ 1 is also less or equal to all others except from A˜ 5 . Since there are some ‘–’, we cannot make a decision about the difference between numbers. To do so, we use -c , which shows if a centroid is less or equal than another. For instance, C (A˜ 3 ) 6 C (A˜ 1 ), so it is represented by ‘1’, likewise C (A˜ 1 ) > C (A˜ 3 ) is represented by ‘0’. The symbol ‘–’ means that the binary relation -c is not present in the comparison (which implies that d⊆ > 0).
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In general C (A˜ 3 ) and C (A˜ 5 ) are less or equal than almost all others, and C (A˜ 4 ) has a degree of subsethood to all other centroids. This agrees with their distances dα since A˜ 3 and A˜ 5 are the farthest sets compared to all others, and it is concordant with the ranking given by their centers (see Eq. (20)). 6. Concluding remarks Our proposal is based on the comparison of the intervals provided by every α -cut made over A˜ which allows us to define an L1 Minkowski distance between two IT2FNs which meets with distance axioms. The proposed approach differs from most of available methods which define a fuzzy measure to compare the memberships of two IT2FNs, since it is based on α -cuts instead of the memberships of a particular value. Alternatively, we propose an L1 Minkowski distance for comparing the centroids of two IT2FNs, and a subsethood measure between two centroids. This allows us to compare the expected value of two IT2FNs using their centroids, which are popular defuzzification measures. The proposed approach is applied to some triangular IT2FNs to illustrate its meaning. The obtained results are satisfactory since we can clearly identify when two IT2FNs are different and how distant they are. Alternatively we can compare the centroids of two IT2FNs and measure the distance between them. The proposed measures are specially useful in cases where fuzzy theory (fuzzy logic systems, fuzzy sets, fuzzy optimization, fuzzy relational equations, etc.) is applied using α -cuts and the extension principle instead of mapping the primary variable (support) of an IT2FN. This way, our proposal has a wide potential use in applications such as resolution of fuzzy differential equations (see Chalco-Cano & Román-Flores [2]), resolution of fuzzy linear systems, linear programming with constraints in the form aij xj - b˜ i (see Figueroa, Kalenatic & López [7]), constrained decision making, ordering of words, etc. Acknowledgments Yurilev Chalco-Cano and Heriberto Román-Florez have been partially supported by Conicyt-Chile through Projects Fondecyt 1120665 and 1120674. Appendix A Theorem 4 states that the binary relation - satisfies the following ordering properties:
˜ then A˜ = B. ˜ It is enough to proof that if αA˜ 6 αB˜ and αB˜ 6 αA˜ holds for all α ∈ [0, 1], then P1. If A˜ - B˜ and B˜ - A, α˜ α˜ A = B. Considering Proposition 1 and Definition 6, we can decompose αA˜ into inf αA˜ := [αA˜ L(+) , αA˜ L(−) ] and sup αA˜ := ˜ [αA˜ R(−) , αA˜ R(+) ], and αB˜ into inf αB˜ := [αB˜ L(+) , αB˜ L(−) ] and sup αB˜ := [αB˜ R(−) , αB˜ R(+) ], so if inf αA˜ 6 inf αB˜ and inf αB˜ 6 inf αA, ˜ then αA˜ L(+) = αB˜ L(+) and αA˜ L(−) = αB˜ L(−) which is equivalent to say inf αA˜ = inf αB˜ (same result for sup αA˜ = sup αB). ˜ and sup αA˜ 6 sup αB˜ , sup αB˜ 6 sup αA˜ ⇒ sup αA˜ = Hence, if both inf αA˜ 6 inf αB˜ , inf αB˜ 6 inf αA˜ ⇒ inf αA˜ = inf αB, α˜ ˜ This immediately proofs that dα (A˜ , B˜ ) = 0. sup B hold ∀α ∈ [0, 1], then A˜ = B. ˜ and αB˜ > αC˜ hold ∀α ∈ [0, 1], then inf αA˜ > inf αB, ˜ P2. If A˜ % B˜ and B˜ % C˜ , then A˜ % C˜ . Definition 6 states that if αA˜ > αB, α˜ α˜ α˜ α˜ α˜ α˜ inf B > inf C , sup A > sup B, and inf B > inf C , which implies that αA˜ L(+) > αB˜ L(+) , αA˜ L(−) > αB˜ L(−) , αA˜ R(−) > αB˜ R(−) , α˜ R(+) ˜ L(+) > A > αB˜ R(+) , αB˜ L(+) > αC˜ L(+) , αB˜ L(−) > αC˜ L(−) , αB˜ R(−) > αC˜ R(−) , αB˜ R(+) > αC˜ R(+) . This in turn implies that αA α ˜ L(+) α˜ L(−) α ˜ L(−) α˜ R(−) α ˜ R(−) α˜ R(+) α ˜ R(+) ˜ ˜ C , A > C , A > C , and A > C also holds ∀α ∈ [0, 1], which lead to A % C . This immediately verifies that dα (A˜ , B˜ ) > 0, dα (A˜ , C˜ ) > 0, so dα (A˜ , C˜ ) > dα (A˜ , B˜ ). ˜ then A˜ % B. ˜ By definition (see Eq. (5)), any IT2FN is convex and normal, so P3. If A˜ ∩ B˜ = 0 and A˜ is on the right of B, A˜ ∩ B˜ = 0 implies that supp(A˜ ) ∩ supp(B˜ ) = 0, where supp(A˜ ) := supp(µ ¯ A˜ ). Now if A˜ is on the right of B˜ and their ˜ and Definition 6 holds ∀ α ∈ [0, 1]. This also verifies that dα (A˜ , B˜ ) > 0, supports have no common elements, then A˜ % B, d(C (A˜ ), C (B˜ )) > 0, and dc (C (A˜ ), C (B˜ )) > 0. P4. The order of A˜ and B˜ is not affected by the other centroids under comparison. It comes directly from the fact that we are not defining any other point and/or set as reference point for comparison, so dα (A˜ , B˜ ) is a function of A˜ and B˜ without the influence of any other point/set. This concludes the proof. Appendix B The binary relation -c as stated in Theorem 5 satisfies the following ordering properties: P1. If C (A˜ ) > C (B˜ ) and C (B˜ ) > C (A˜ ), then C (A˜ ) = C (B˜ ). If C (A˜ ) > C (B˜ ), then Cl (A˜ ) > Cl (B˜ ) and Cu (A˜ ) > Cu (B˜ ), so if C (B˜ ) > C (A˜ ), then Cl (B˜ ) > Cl (A˜ ) and Cu (B˜ ) > Cu (A˜ ). This means that Cl (A˜ ) = Cl (B˜ ) and Cu (A˜ ) = Cu (B˜ ) which implies that d(C (A˜ ), C (B˜ )) = 0 and dc (C (A˜ ), C (B˜ )) = 0.
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P2. If C (A˜ ) > C (B˜ ) and C (B˜ ) > C (C˜ ), then C (A˜ ) > C (C˜ ). If C (A˜ ) > C (B˜ ), then Cl (A˜ ) > Cu (B˜ ), and if C (B˜ ) > C (C˜ ), then Cl (B˜ ) > Cu (C˜ ), so we can see that Cl (A˜ ) > Cu (B˜ ) ⇒ Cl (A˜ ) > Cu (C˜ ), so C (A˜ ) > C (C˜ ), which implies that d(C (A˜ ), C (C˜ )) > d(C (A˜ ), C (B˜ )), and dc (C (A˜ ), C (C˜ )) > dc (C (A˜ ), C (B˜ )). P3. If C (A˜ ) ∩ C (B˜ ) = 0 and C (A˜ ) is on the right of C (B˜ ), then C (A˜ ) > C (B˜ ). If C (A˜ ) ∩ C (B˜ ) = 0 and C (A˜ ) is on the right of C (B˜ ), then ∃x (Cu (B˜ ) 6 x 6 Cl (A˜ )) and Cl (A˜ ) > Cu (B˜ ) which verifies C (A˜ ) > C (B˜ ), d(C (A˜ ), C (B˜ )) > 0, and dc (C (A˜ ), C (B˜ )) > 0. P4. The order of C (A˜ ) and C (B˜ ) is not affected by the other centroids under comparison. It comes directly from the fact that we are not defining any other centroid as reference point for comparison, so d(C (A˜ ), C (B˜ )) and dc (C (A˜ ), C (B˜ )) are functions of A˜ and B˜ without the influence of any other set. This concludes the proof. References [1] C. Celemin, M. Melgarejo, A proposal to speed up the computation of the centroid of an interval Type-2 fuzzy set, Adv. Fuzzy Syst. 2013 17 (2013) (Article ID 158969). [2] Y. Chalco-Cano, H. Román-Flores, Comparation between some approaches to solve fuzzy differential equations, Fuzzy Sets and Systems 160 (11) (2009) 1517–1527. [3] B. Chaudhuri, A. RosenFeld, A modified hausdorff distance between fuzzy sets, Inform. Sci. 118 (1999) 159–171. [4] S.-M. Chen, T.-H. Chang, Finding multiple possible critical paths using fuzzy PERT, IEEE Trans. Syst. Man Cybern. B 31 (6) (2001) 930–937. [5] J.C. Figueroa-García, An approximation method for type reduction of an interval Type-2 fuzzy set based on α -cuts, in: IEEE (Ed.), Proceedings of FEDCSIS, IEEE, 2012, pp. 1–6. [6] J.C. Figueroa-García, A general model for linear programming with interval type-2 fuzzy technological coefficients, in: 2012 Annual Meeting of the North American Fuzzy Information Processing Society, NAFIPS, IEEE, 2012, pp. 1–6. [7] J.C. Figueroa-García, D. Kalenatic, C.A. Lopez-Bello, Multi-period mixed production planning with uncertain demands: Fuzzy and interval fuzzy sets approach, Fuzzy Sets and Systems 206 (2012) 21–38. [8] W.-L. Hung, M.-S. Yang, Similarity measures between type-2 fuzzy sets, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 12 (6) (2004) 827–841. [9] G.J. Klir, T.A. Folger, Fuzzy Sets, Uncertainty and Information, Prentice Hall, 1992. [10] B. Kosko, Fuzziness vs. probability, Int. J. Gen. Syst. 17 (1) (1990) 211–240. [11] J.M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, Prentice Hall, 2001. [12] J.M. Mendel, D. Wu, Perceptual Computing: Aiding People in Making Subjective Judgments, John Wiley & Sons, 2010. [13] H.B. Mitchell, Ranking Type-2 fuzzy numbers, IEEE Trans. Fuzzy Syst. 14 (2) (2006) 287–294. [14] R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction to Interval Analysis, SIAM, 2009. [15] H.T. Nguyen, V. Kreinovich, Computing degrees of subsethood and similarity for interval-valued fuzzy sets: Fast algorithms, in: 9th International Conference on Intelligent Technologies InTec08, IEEE, 2008, pp. 47–55. ˘ [16] J. Ramík, J. Rimánek, Inequality relation between fuzzy numbers and its use in fuzzy optimization, Fuzzy Sets and Systems 16 (1985) 123–138. [17] H. Román-Flores, Y. Chalco-Cano, H-continuity of fuzzy measures and set defuzzification, Fuzzy Sets and Systems 157 (2006) 230–242. [18] A. Rufián-Lizana, Y. Chalco-Cano, R. Osuna-Gómez, G. Ruiz-Garzón, On invex fuzzy mappings and fuzzy variational-like inequalities, Fuzzy Sets and Systems 200 (2012) 84–98. [19] X. Wang, E.E. Kerre, Reasonable properties for the ordering of fuzzy quantities (I), Fuzzy Sets and Systems 118 (1) (2001) 375–387. [20] D. Wu, J.M. Mendel, Enhanced Karnik–Mendel algorithms, IEEE Trans. Fuzzy Syst. 17 (4) (2009) 923–934. [21] D. Wu, J.M. Mendel, A vector similarity measure for interval type-2 fuzzy sets and type-1 fuzzy sets, Inform. Sci. 178 (1) (2008) 381–402. [22] D. Wu, J.M. Mendel, A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets, Inform. Sci. 179 (1) (2009) 1169–1192. [23] L. Xuecheng, Entropy, distance measure and similarity measure of fuzzy sets and their relations, Fuzzy Sets and Systems 52 (1992) 305–318. [24] L. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Inform. Sci. 8 (1975) 199–249. [25] G. Zheng, J. Wang, W. Zhou, Y. Zhang, A similarity measure between interval type-2 fuzzy sets, in: IEEE International Conference on Mechatronics and Automation, IEEE, 2010, pp. 191–195.
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Note
Distance measures for Interval Type-2 fuzzy numbers Juan Carlos Figueroa-García a,∗,1 , Yurilev Chalco-Cano b , Heriberto Román-Florez b a
Universidad Distrital Francisco José de Caldas, Bogotá, Colombia
b
Instituto de Alta Investigación, Universidad de Tarapacá, Arica, Chile
article
info
Article history: Received 28 February 2014 Received in revised form 11 September 2014 Accepted 17 November 2014 Available online xxxx
abstract This paper shows some distance measures for comparing Interval Type-2 fuzzy numbers. In addition, some definitions about ordering of Interval Type-2 fuzzy numbers based on their centroids, are provided. Some numerical examples are given, and some interpretation issues are explained. © 2014 Published by Elsevier B.V.
Keywords: Minkowski distance Interval Type-2 fuzzy numbers α -cuts Fuzzy ordering
1. Introduction and motivation Fuzzy sets theory has been applied to many problems with successful results, mostly in cases where statistical information is absent or is incomplete. In these cases, there is a need for obtaining information from other sources which includes the opinion of the experts and their perception about the problem, so its representation through fuzzy sets seems as an appropriate way to handle this kind of information. In some decision making problems, there are multiple experts analyzing the problem who add uncertainty. Examples of these problems include the resolution of fuzzy differential equations (see Chalco-Cano & Román-Flores [2]), fuzzy inequalities (see Rufián-Lizana et al. [18]), Type-2 fuzzy linear programming (see Figueroa-García [5–7]), fuzzy PERT problems (see Chen & Chang [4]), etc. This way, the use of Interval Type-2 Fuzzy numbers (IT2FN) is an appropriate way to handle uncertainty coming from judgement of multiple experts, or linguistic uncertainty coming from the perception of concepts. One of the most frequent questions when using fuzzy numbers regards to define a measure for efficiently comparing two fuzzy numbers. Although some methods for comparing fuzzy sets have been reported, there is a need for extending their results to a Type-2 fuzzy sets environment. Its applicability in different fields such as decision making, fuzzy inequalities and other problems where linguistic uncertainty appears, is wide. This paper focuses on defining some distances to compare IT2FNs, and a Type-2 fuzzy binary relation. Some examples are provided and their results are discussed. The paper is divided into six sections. Section 1 introduces the problem. In Section 2, some basic definitions about IT2FNs are provided; in Section 3, some distance measures are presented. Section 4 presents the concept of fuzzy ordering of IT2FNs; Section 5 presents an application example; and finally in Section 6, the concluding remarks of the study are presented.
∗
Corresponding author. E-mail addresses: [email protected] (J.C. Figueroa-García), [email protected] (Y. Chalco-Cano), [email protected] (H. Román-Florez).
1 LAMIC research group member. http://dx.doi.org/10.1016/j.dam.2014.11.016 0166-218X/© 2014 Published by Elsevier B.V.
2
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˜ Fig. 1. Interval Type-2 fuzzy set A.
2. Basic definitions of IT2FNs Firstly, we establish some basic notations. A classical fuzzy set, namely Type-1 Fuzzy Set (T1FS), is denoted by capital letters e.g. A with a membership function µA (x) defined over x ∈ X , while an Interval Type-2 Fuzzy Set (IT2FS) is denoted by emphasized capital letters A˜ with a membership function µA˜ (x) defined over x ∈ X . µA measures the affinity degree of a value x ∈ X regarding the concept/word/label A, and µA˜ (x) measures the uncertainty degree of a value x ∈ X regarding A, so A˜ measures uncertainty around A. A Type-2 fuzzy set is an ordered pair A˜ = {((x, u), Jx , fx (u)) | x ∈ X ; u ∈ Jx ⊆ [0, 1]}, where A˜ represents uncertainty around the word A, Jx is the primary membership of x, u is its domain of uncertainty, and F2 (X) is the class of all Type-2 fuzzy sets (see Mendel [11,12]). Their mathematical definitions are: A˜ : X → F([0, 1]) A˜ =
x∈X
fx (u)/(x, u),
Jx ⊆ [0, 1]
u∈Jx
where µA˜ (x) is composed by an infinite amount of embedded Type-1 fuzzy sets namely Ae . Every element x has associated a set of primary memberships Jx weighted by a Secondary fuzzy set fx (u) where u is the domain of uncertainty of x, u ∈ Jx ⊆ [0, 1]. Now, an IT2FS is a simplification of a T2FS since its secondary membership function is assumed to be 1, which is shown as follows A˜ =
x∈X
1/(x, u) = u∈Jx
x∈X
1/u
x,
(1)
u∈Jx
where x, u are the primary and secondary variables, and fx (u)/u = 1 is the secondary membership function. ˜ namely FOU(A˜ ), Uncertainty about the word A is conveyed by the union of all of Jx into the Footprint Of Uncertainty of A, which is bounded by two functions: An Upper membership function UMF (A˜ ) = µ ¯ A˜ (x) ≡ A¯ and a Lower membership function
LMF (A˜ ) = µ ˜ (x) ≡ A. FOU(A˜ ) is shown in Fig. 1. A ˜ supp (A˜ ) is In Fig. 1, A˜ is an IT2FS, the universe of discourse for the primary variable x is the set x ∈ X , the support of A, α the interval x ∈ [ˇx, xˆ ] and µA˜ is a triangular membership function with parameters xˇ , xˆ , xˇ , xˆ and x¯ . µ ¯ A˜ (x) is the degree of ¯ αµ (x) is the degree of membership of a specific value x regarding its membership of a specific value x regarding its UMF, A; A˜ LMF, A, and u ∈ Jx = [αµ ˜ (x), α µA˜ (x)]. A
2.1. Decomposition of an IT2FS One of the most comprehensive ways to represent a T1FS A is through α -cuts. The α -cut of a A, namely αA, is defined as A = {x | µA (x) > α}. Thus, a fuzzy set A is the union of its α -cuts, α∈[0,1] α · αA, where ∪ denotes union (Klir & Yuan [9]).
α
Now, the extension of α -cut of A to the α -cut of A˜ (see Figueroa [5]), is shown next.
Definition 1 (Primary α -cut of an IT2FS). The primary α -cut of an Interval Type-2 fuzzy set αA˜ is the union of all x ∈ X whose primary memberships Jx are greater than α, Jx > α , this is: α˜
A= x∈X
u∈Jx >α
1/u
x;
Jx ⊆ [0, 1], α ∈ [0, 1].
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˜ Fig. 2. αA˜ of the set A.
˜ The symbol Now, αAe = {x | µAe (x) > α} is an α -cut done over an embedded set Ae ∈ A. α
denotes fuzzy union, so αA˜ is
α˜
the union of all Ae . A graphical representation of A is provided in Fig. 2. These lead to define the boundaries of each α -cut, as follows.
˜ Consider an Interval Type-2 fuzzy set with known UMF and LMF, and denote inf αA˜ := αA˜ L Definition 2 (Crisp Bounds of αA). α˜ α˜ U and sup A := A as the left and right interval-valued bounds of αA˜ := [inf αA˜ , sup αA˜ ]. Let the bounds of αA˜ be defined as the boundaries of the α -cuts of the shapes of its UMF and LMF, as follows: α˜ L
A :=
inf α µA˜ (x, u) ; inf αµ ˜ (x, u) = A x x
α˜ L(+)
A
, αA˜ L(−)
α α˜ R α A := sup µ ˜ (x, u) ; sup µA˜ (x, u) = αA˜ R(−) , αA˜ R(+) . A
(3)
(4)
x
x
2.2. Type-2 fuzzy numbers In this paper, a Type-2 fuzzy number (T2FN) is considered as the extension of a Type-1 fuzzy number. This means that A˜ is an T2FS whose UMF and LMF are fuzzy numbers (e.g. normal and convex fuzzy subsets of R, Zadeh [24]). αA is closed interval for all α ∈ [0, 1], and its support supp (A) is defined over R. This also means that a fuzzy number is a normal and convex fuzzy set, which is shown as follows. Definition 3 (Type-2 Fuzzy Number). Let A˜ ∈ F2 (R). Then, A˜ is a Type-2 Fuzzy Number (T2FN) iff there exists a closed interval [a, b] ̸= 0 for each UMF (µ ¯ A˜ ) and LMF (µ ˜ ) such that A
1 l(x, u) r (x, u)
µA˜ (x, u) =
for x ∈ [a, b], for x ∈ [−∞, a], for x ∈ [b, ∞],
u ∈ Jx ⊆ [0, 1] u ∈ Jx ⊆ [0, 1] u ∈ Jx ⊆ [0, 1]
(5)
where l : (−∞, a) → F([0, 1]), u ∈ Jx ⊆ [0, 1] is monotonic non-decreasing, continuous from the right, and l(x, u) = 0 for x < ω1 , and r : (b, ∞) → F([0, 1]), u ∈ Jx ⊆ [0, 1] is monotonic non-increasing, continuous from the left, and r (x, u) = 0 for x > ω2 . Now, A˜ is an T2FN if both their UMF and LMF are fuzzy numbers, which is equivalent to the definition given by Mitchell [13]. The set A˜ shown in Fig. 1 is also an Interval Type-2 Fuzzy Number (IT2FN). 3. Distance-based Similarity between two IT2FNs Our approach is based on the idea of comparing the UMF and LMF of two IT2FNs to establish if they are equal or not. This way, we use the classic axioms of distance since αA˜ can be seen as a crisp set. Let R be the set of real numbers, and R+ be its non-negative orthant. X is the universal set; F2 (X ) is the class of all IT2FNs of X ; P (X ) is the class of all crisp sets of X , then the distance between A and B namely d(A, B) is called to be a metric (or simply distance) if d(A, B) ∈ R+ and satisfies the following three axioms: D1: d(A, B) = d(B, A), D2: d(A, A) = 0, D3: d(A, C ) 6 d(A, B) + d(B, C ).
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Now, in the p-dimensional Euclidean space Rp , Minkowski defined a family of Lm metric distances given by dm (A, B) =
p
1/m m
|xi − yi |
(6)
i =1
where A, B are two points in Rp with coordinates xi and yi . In the space of the primary variable of an IT2FN, every α -cut returns a set of intervals whose supports are bounded over R (see Definition 2). This makes the definition of a distance between two IT2FNs using α -cuts easier, since we can decompose A˜ and B˜ into α -cuts to compute the distance between αA˜ and αB˜ as the distance between two intervals. Now, we use the following definition of distance between two intervals: Definition 4. Let A ∈ [a, a] and B ∈ [b, b] two interval sets defined over R+ , the L1 distance between A and B is defined as follows: d(A, B) = a − b + a − b .
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In general for crisp (or interval) sets, Kosko [10], and Nguyen & Kreinovich [15] used A ∩ B ⊆ A ∪ B as a useful measure to evaluate whether A = B or not, so A = B if and only if A ∩ B = A ∪ B. Hence, for crisp (or interval) finite sets, we can check if A is equal to B using the following ratio:
|A ∩ B| . |A ∪ B| This ratio is equal to 1 iff A = B and <1 if A ̸= B. As this ratio is smaller, more elements from A ∪ B are not part of the intersection A ∩ B. This can be extended to IT2FNs, since A˜ = B˜ iff µA˜ = µB˜ , as follows:
¯ A˜ (x, u) = µ ¯ B˜ (x, u) ˜A = B˜ ⇔ µ µA˜ (x, u) = µB˜ (x, u). ˜ and A˜ ̸= B˜ ; C (A˜ ) = C (B˜ ), so there is the possibility of having A˜ ̸= B˜ with equal or similar centroids. In other case A˜ ̸= B, Hence, the comparison of their centroids is a useful measure for decision making. Several similarity measures for IT2FSs have been proposed by Nguyen & Kreinovich [15], Zheng et al. [25], Xuecheng [23], and Hung & Yang [8]. Their results are focused to establish similarity between two IT2FSs based on the distance among memberships, accomplishing the basic axioms of a similarity measure. Our proposal compares intervals coming from αA˜ using an L1 Minkowski distance, as shown in next sections. 3.1. α -cut based distance of IT2FNs ˘ Ramík & Rimánek [16] defined that two sets A, B are equal iff αA = α B for every α ∈ [0, 1], αA := [inf αA, sup αA] and α α B := [inf B, sup B]. Given the concept of α -cut of an IT2FN (see Definition 2), we can intuitively think that A˜ = B˜ iff αA˜ = αB˜ for every α ∈ [0, 1], this is
α
A˜ = B˜ ⇔
|αA˜ L ∩ αB˜ L | |αA˜ R ∩ αB˜ R | = = 1, α L α L | A˜ ∪ B˜ | |αA˜ R ∪ αB˜ R |
∀ α ∈ [0, 1].
This concept leads to the following proposition:
˜ and αB˜ is the α -cut of B, ˜ α ∈ [0, 1], then A˜ = B˜ if only if Proposition 1. Let A˜ , B˜ ∈ F2 (R) be two IT2FNs. If αA˜ is the α -cut of A, α˜
A = αB˜ ∀ α ∈ [0, 1].
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Proof. A˜ = B˜ implies that αA˜ L = αB˜ L and αA˜ R = αB˜ R . We can show that if the equality of the intervals αA˜ L = αB˜ L and αA˜ R = αB˜ R hold ∀ α ∈ [0, 1], then by Definition 4, d(αA˜ L ,αB˜ L ) = 0, d(αA˜ R ,αB˜ R ) = 0 hold ∀ α ∈ [0, 1], so d(A˜ L , B˜ L ) = 0, d(A˜ R , B˜ R ) = 0 and the proof is complete. Based on the results proposed by Chaudhuri & RosenFeld [3], we propose the distance dα based on α -cuts and the L1 Minkowski distance (see Eq. (6)) to measure how different A˜ and B˜ are.
˜ ˜ Proposition 2. Let A˜ , B˜ ∈ F2 (R) be two IT2FNs. The distance (metric) dα between A and B given a set of n α -cuts, {α1 , α2 , . . . , αn } and Λ = ni=1 αi , is:
J.C. Figueroa-García et al. / Discrete Applied Mathematics (
dα (A˜ , B˜ ) ,
N 1
Λ
i=1
1
0
–
αi αiA˜ L(+) − αiB˜ L(+) + αiA˜ L(−) − αiB˜ L(−) + αiA˜ R(−) − αiB˜ R(−) + αiA˜ R(+) − αiB˜ R(+)
if α is continuous, then Λ = dα (A˜ , B˜ ) , 2
)
1 0
5
(9)
α dα = 1/2, so dα is defined as:
α αA˜ L(+) − αB˜ L(+) + αA˜ L(−) − αB˜ L(−) + αA˜ R(−) − αB˜ R(−) + αA˜ R(+) − αB˜ R(+) dα.
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Proof. Consider a single α -cut (αi ) and Eq. (9). It is easy to verify axioms D1 and D2 since it is based on the L1 Minkowski distance (see Eq. (6)). To proof D3 we take the extreme case that A˜ = B˜ which verifies dαi (A˜ , B˜ ) = 0 since αiA˜ L(+) = αiB˜ L(+) ,
= αiB˜ L(−) , αiA˜ R(−) = αiB˜ R(−) and αiA˜ R(+) = αiB˜ R(+) ; when B˜ = C˜ then dαi (B˜ , C˜ ) = 0 since αiB˜ L(+) = αiC˜ L(+) , αiB˜ L(−) = C , B = αiC˜ R(−) and αiB˜ R(+) = αiC˜ R(+) , so dαi (A˜ , C˜ ) = dαi (A˜ , B˜ ); when A˜ ̸= B˜ and B˜ ̸= C˜ then A˜ ̸= C˜ , dαi (A˜ , B˜ ) > 0 and dαi (B˜ , C˜ ) > 0, so dαi (A˜ , C˜ ) < dαi (A˜ , B˜ ) + dαi (B˜ , C˜ ); and finally if A˜ = C˜ then dαi (A˜ , C˜ ) = 0 since αiA˜ L(+) = αiC˜ L(+) , αi ˜ L(−) A = αiC˜ L(−) , αiA˜ R(−) = αiC˜ R(−) and αiA˜ R(+) = αiC˜ R(+) , so 0 6 dαi (A˜ , B˜ ) + dαi (B˜ , C˜ ) which is equivalent to 0 6 2dαi (B˜ , C˜ ), αi ˜ L(−)
A
αi ˜ L(−) αi˜ R(−)
and the proof is complete. 3.2. Centroid-based distance of IT2FNs The distance dα presented in Definition 2 can be restrictive in some applications (combinatorics, complex optimization problems, etc.) that require high computational efforts. To deal with, we propose the use of the distance between the centroids of two IT2FNs, whose concepts are shown next. ˜ its centroid C (A˜ ) is an interval bounded by two values min{C (Ae )} = Cl (A˜ ) and max{C (Ae )} = Cr (A˜ ). Every Given a set A, ˜ so there is an infinite amount of centroids enclosed into point enclosed into C (A˜ ) = [Cl (A˜ ), Cu (A˜ )] is a possible centroid of A, C (A˜ ) (see Wu & Mendel [20], and Melgarejo [1]). Two numbers A˜ and B˜ are centroid-equal if their centroids are equal, this is C (A˜ ) = C (B˜ ). Now, we need a measure to compare C (A˜ ), C (B˜ ) in order to say if they are either equal or not. To do so, we use the distance between two intervals X = [¯x, x] and Y = [¯y, y], d(X , Y ) = max{|¯x − y¯ |, |x − y|} proposed by Moore, Kearfott and Cloud [14]. Now, the distance between C (A˜ ) and C (B˜ ) is: d(C (A˜ ), C (B˜ )) = max{|Cl (A˜ ) − Cl (B˜ )|, |Cu (A˜ ) − Cu (B˜ )|}.
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This distance measures the max distance between the boundaries of C (A˜ ) and C (B˜ ), so the min distance is out of d(C (A˜ ), C (B˜ )). An alternative way to measure the distance between C (A˜ ) and C (B˜ ) is by adding the distance among all boundaries. This distance can be defined as follows:
˜ The L1 distance dc between C (A˜ ), C (B˜ ) is: Proposition 3. Let C (A˜ ), C (B˜ ) be the centroids of A˜ , B. dc (C (A˜ ), C (B˜ )) = |Cl (A˜ ) − Cl (B˜ )| + |Cu (A˜ ) − Cu (B˜ )|.
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Proof. Axioms D1 and D2 are easy to proof by simple deduction. To proof D3 we will take the extreme case that C (A˜ ) = C (B˜ ) which immediately verifies d(C (A˜ ), C (B˜ )) = 0, and d(C (A˜ ), C (B˜ )) = d(C (A˜ ), C (C˜ )); when B˜ = C˜ , then d(C (B˜ ), C (C˜ )) = 0, so d(C (A˜ ), C (C˜ )) = d(C (A˜ ), C (B˜ )); when C (A˜ ) ̸= C (B˜ ) and C (B˜ ) ̸= C (C˜ ), then C (A˜ ) ̸= C (C˜ ), d(C (A˜ ), C (B˜ )) > 0 and d(C (B˜ ), C (C˜ )) > 0, so d(C (A˜ ), C (C˜ )) < d(C (A˜ ), C (B˜ )) + d(C (B˜ ), C (C˜ )); and finally if C (A˜ ) = C (C˜ ) then d(C (A˜ ), C (C˜ )) = 0, so 0 6 d(C (A˜ ), C (B˜ )) + d(C (B˜ ), C (C˜ )) which is equivalent to 0 6 2d(C (B˜ ), C (C˜ )), and the proof is complete. The distance dc preserves the information of both boundaries, since Eq. (12) adds both max and min distances. This means that the size of dc determines how far C (A˜ ) and C (B˜ ) are. Now we can define if C (A˜ ) = (B˜ ) using d(C (A˜ ), C (B˜ )) and dc (C (A˜ ), C (B˜ )), as follows.
˜ It is said that A˜ , B˜ are centroid-equal if: Definition 5. Let C (A˜ ), C (B˜ ) be the centroids of A˜ , B. C (A˜ ) = C (B˜ ) ⇔ d(C (A˜ ), C (B˜ )) = 0
(13)
C (A˜ ) = C (B˜ ) ⇔ dc (C (A˜ ), C (B˜ )) = 0.
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Note that A˜ ̸= B˜ ; C (A˜ ) ̸= C (B˜ ), so conversely A˜ = B˜ ⇒ C (A˜ ) = C (B˜ ). This means that A˜ and B˜ can be different, so dα (A˜ , B˜ ) > 0 (see Definition 2) while having equal centroids, C (A˜ ) = C (B˜ ). 3.3. Subsethood of the centroid of two IT2FNs Another way to measure the relation between two IT2FNs is by using the concept of subsethood introduced by Kosko [10], and Nguyen & Kreinovich [15]. Given two crisp sets A and B, the set equality d= is defined as d= = |A ∩ B|/|A ∪ B| and the
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subsethood d⊆ is defined as d⊆ = |A ∩ B|/|A|. The set equality and subsethood of two centroids C (A˜ ), C (B˜ ) can be defined as follows: d= =
|C (A˜ ) ∩ C (B˜ )| , |C (A˜ ) ∪ C (B˜ )|
(15)
d⊆ =
|C (A˜ ) ∩ C (B˜ )| . |C (A˜ )|
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The case when C (A˜ ) = C (B˜ ) implies that d= = 0 and d⊆ = 0, so the use of d= and d⊆ makes sense when C (A˜ )∩ C (B˜ ) ̸= 0. Therefore, we can assert that if C (A˜ ) ∩ C (B˜ ) ̸= 0, then A˜ and B˜ have a degree of set equality based on their centroids. 4. Fuzzy max order of an IT2FN ˘ The binary relation . for classical fuzzy sets has been proposed and investigated by Ramík & Rimánek [16]. Let A and B be two fuzzy numbers. Then A . B if and only if sup αA 6 sup αB and inf αA 6 inf αB for each α ∈ [0, 1], where αA and αB are α -cuts of A and B respectively, and αA := [inf αA, sup αA] and αB := [inf αB, sup αB]. This binary relation satisfies the axioms of a partial order relation on F (R) and is called fuzzy max order. It is possible to extend the fuzzy max order to IT2FNs, as follows: Definition 6. Given two IT2FNs A˜ , B˜ ∈ F2 (R), the binary relation -, A˜ - B˜ holds iff αA˜ 6 αB˜ ∀ α ∈ [0, 1]. This is equivalent to the following result sup αA˜ 6 sup αB˜ ∀ α ∈ [0, 1], α˜
α˜
inf A 6 inf B ∀ α ∈ [0, 1],
(17) (18)
where αA˜ and αB˜ are α -cuts of A˜ and B˜ respectively. This way, A˜ - B˜ implies that every α -cut done over A˜ , B˜ can be seen as the crisp sets inf αA˜ := [αA˜ L(+) , αA˜ L(−) ], sup αA˜ := [ A , A ], inf αB˜ := [αB˜ L(+) , αB˜ L(−) ], sup αB˜ := [αB˜ R(−) , αB˜ R(+) ], αA˜ := [inf αA˜ , sup αA˜ ], and αB˜ := [inf αB˜ , sup αB˜ ]. This implies that αA˜ L(+) 6 αB˜ L(+) , αA˜ L(−) 6 αB˜ L(−) ,α A˜ R(−) 6 αB˜ R(−) and αA˜ R(+) 6 αB˜ R(+) hold ∀ α ∈ [0, 1]. Wang & Kerre [19] defined some reasonable properties to be fulfilled when ranking fuzzy sets, which were later extended by Wu & Mendel [21,22], and Mendel & Wu [12] to IT2FSs. We extend them to the binary relation A˜ - B˜ since the α -cuts of an IT2FN are intervals. This leads us to the following result: α˜ R(−) α˜ R(+)
˜ B, ˜ and C˜ be IT2FNs such as A˜ - B. ˜ The binary relation - satisfies the following ordering properties: Theorem 4. Let A, P1. P2. P3. P4.
˜ then A˜ = B. ˜ If A˜ - B˜ and B˜ - A, If A˜ - B˜ and B˜ - C˜ , then A˜ - C˜ . ˜ then A˜ % B. ˜ If A˜ ∩ B˜ = 0 and A˜ is on the right of B, The order of A˜ and B˜ is not affected by the other centroids under comparison.
Proof. See Appendix A. Basically, A˜ - B˜ if every boundary of every α -cut of A˜ , B˜ holds the binary relation 6 for every α -cut. This condition ensures that every value of A˜ is less or equal than every value of B˜ given any uncertainty degree α ∈ [0, 1]. This is a hard requirement in some applications since it is possible that not all α -cuts fulfill this condition. In cases where not all α -cuts meet the partial ˜ This way, we use the centroids of A˜ and B˜ to measure how different they order 6, then A˜ is not strictly less or equal than B. are (in average).
˜ It is said that A˜ is centroid less or equal than B, ˜ A˜ -c B, ˜ iff C (A˜ ) 6 C (B˜ ), Definition 7. Let C (A˜ ), C (B˜ ) be the centroid of A˜ and B. which implies that: Cu (A˜ ) 6 Cl (B˜ ).
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˜ i.e. Cl (A˜ ) 6 Cu (B˜ ) comes from partial ordering of intervals (see Moore, Kearfott & Cloud [14]). Note that The idea of A˜ -c B, the following two cases: Cu (A˜ ) 6 Cl (B˜ ) and Cl (B˜ ) 6 Cu (A˜ ) has been considered. There is the possibility of having that Cl (B˜ ) 6 Cl (A˜ ) and Cu (A˜ ) 6 Cu (B˜ ). This means that C (B˜ ) ⊆ C (A˜ ), so we cannot assert that C (A˜ ) -c C (B˜ ) or C (B˜ ) -c C (A˜ ), but we can say that they are similar (in some sense). At this point, if C (B˜ ) ⊆ C (A˜ ) (or conversely C (A˜ ) ⊆ C (B˜ )), then a proper way to compare A˜ and B˜ is by computing its degree of equality and subsethood as defined in Section 3.3. As the centroid of an IT2FN is a crisp set, we briefly show that at least the first four properties proposed by Wang & Kerre [19] are satisfied by d(C (A˜ ), C (B˜ )) and dc (C (A˜ ), C (B˜ )), from a crisp point of view:
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Table 1 Parameters and centroid of each IT2FN A˜ j . Set j
aˇ j
aj
aˆ j
aˇ j
aj
aˆ j
Cl (A˜ j )
Cu (A˜ j )
c (A˜ j )
A˜ 1 A˜ 2 A˜ 3 A˜ 4 A˜ 5
2 0 0 0 0
5 6 3 5 2
9 9 7 9 8
3 3 1 1 1
5 6 3 5 2
6 7 5 6 6
4.329 4.294 2.666 3.66 2.665
5.68 6 3.671 5 3.686
5.0045 5.147 3.1685 4.33 3.1755
˜ The binary relation -c satisfies the following Theorem 5. Let C (A˜ ), C (B˜ ), and C (C˜ ) be the centroids of three IT2FNs such as A˜ -c B. ordering properties: P1. P2. P3. P4.
If C (A˜ ) > C (B˜ ) and C (B˜ ) > C (A˜ ), then C (A˜ ) = C (B˜ ). If C (A˜ ) > C (B˜ ) and C (B˜ ) > C (C˜ ), then C (A˜ ) > C (C˜ ). If C (A˜ ) ∩ C (B˜ ) = 0 and C (A˜ ) is on the right of C (B˜ ), then C (A˜ ) > C (B˜ ). The order of C (A˜ ) and C (B˜ ) is not affected by the other centroids under comparison.
Proof. See Appendix B.
˜ This does Those properties ensures a logical ordering of centroids as a base for comparing the expected values of A˜ and B. not imply that C (A˜ ) = C (B˜ ) ⇒ A˜ = B˜ since there exists the possibility of having A˜ ̸= B˜ while C (A˜ ) = C (B˜ ). This leads us to propose the following corollary. Corollary 1. Let A˜ , B˜ ∈ F2 (R) be two IT2FNs. The binary relation A˜ - B˜ implies that C (A˜ ) 6 C (B˜ ), A˜ - B˜ H⇒
Cl (A˜ ) 6 Cl (B˜ ) Cu (A˜ ) 6 Cu (B˜ ).
This corollary is a natural extension of the concept of dα in the sense that two IT2FNs whose distance is large cannot have the ˜ then their centroids (which are computed using µ same centroid. If - is strictly true for A˜ and B, ¯ and µ) should be different, more precisely C (A˜ ) 6 C (B˜ ). It is also clear that if A˜ = B˜ then C (A˜ ) = C (B˜ ), and conversely C (A˜ ) = C (B˜ ) does not mean ˜ in other words: A˜ = B, A˜ = B˜ ⇒ C (A˜ ) = C (B˜ ).
˜ Also note that dα = 0 implies C (A˜ ) = C (B˜ ). Even when it is not a common Conversely we have C (A˜ ) = C (B˜ ) ; A˜ = B. case, there is a chance of having C (A˜ ) = C (B˜ ) while dα ̸= 0. This means that µA˜ ̸= µB˜ while having equal centroids. Remark 1. Román-Flores & Chalco-Cano [17] have discussed about H-continuity for defuzzification measures (centroid, in our case). Given the distance measure dα between two IT2FNs, then there exists a sequence of A˜ n IT2FNs that converge to a single fuzzy set A˜ τ if and only if dα (A˜ n , A˜ τ ) → 0 when limn→∞ . This also leads us to think that if their centroids C (A˜ ), C (B˜ ) are compatible in dα metric to A˜ , B˜ then it should be true that if dα (A˜ n , A˜ τ ) → 0 when limn→∞ then dα (C (A˜ n ), C (A˜ τ )) → 0 when limn→∞ . 5. Application example In this section, we compare five IT2FNs with triangular LMFs and UMF, defined by the following parameters: LMF = T (ˇa, a, aˆ ) and UMF = T (a¯ˇ , a¯ , a¯ˆ ). To do so, we apply the proposed distances dα , d, d= , and d⊆ . The idea is to see how different (by pairs) are all IT2FNs and their centroids (computed using the proposal of Figueroa [5]). Table 1 shows all parameters used in the comparison. Now, we compute dα , d and dc
0
1.46 0
dα =
0
dc =
0.36 0
3.83 5 0
0.66 1.83 3.17 0
4.83 6 1.17, 4.17 0
3.67 3.96 0
1.35 1.63 2.32 0
3.66 3.94 0.02, 2.31 0
0
d =
0.32 0
2.01 2.33 0
0.68 1 1.33 0
1.99 2.31 0.02, 1.31 0
8
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Fig. 3. Sets A˜ 2 and A˜ 5 .
the degree of equality d= and subsethood d⊆ between centroids are presented next:
0
0.979 0
d= =
0 0 0
0.332 0.302 0.005 0
0 0 0.984 , 0.011 0
1 0.792 d⊆ = 0 0.501 0
1 1 0 0.527 0
0 0 1 0.008 0.984
0.497 0.414 0.011 1 0.025
0 0 1 , 0.019 1
and finally, we compare the centroids of A˜ j using the center of centroid c (A˜ ) proposed by Wu & Mendel [22]. To do so, we compute c (A˜ ) = (Cl (A˜ ) + Cu (A˜ ))/2 to rank them as crisp numbers, which leads to the following rank: c (A˜ 3 ) 6 c (A˜ 5 ) 6 c (A˜ 4 ) 6 c (A˜ 1 ) 6 c (A˜ 2 ).
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5.1. Discussion of the results By using the above results, we can see differences among IT2FNs. The farthest fuzzy numbers are A˜ 2 and A˜ 5 since dα (A˜ 2 , A˜ 5 ) = 6, which is concordance to the distance between their centroids. For instance, d(C (A˜ 2 ), C (A˜ 5 )) = 2.314 and dc (C (A˜ 2 ), C (A˜ 5 )) = 3.943 which are the second largest distances, and of course d= (C (A˜ 2 ), C (A˜ 5 )) = 0, d⊆ (C (A˜ 2 ), C (A˜ 5 )) = 0, and d⊆ (C (A˜ 5 ), C (A˜ 2 )) = 0. This is a consequence of a large distance between two sets, since as dα increases, their centroids are more distant. An example of two different IT2FNs is shown in Fig. 3. The closest IT2FNs are A˜ 1 and A˜ 4 since dα (A˜ 1 , A˜ 4 ) = 0.666 which is a sign of its similarity. Their centroids have a relationship to dα since d(C (A˜ 1 ), C (A˜ 4 )) = 0.679 and dc (C (A˜ 1 ), C (A˜ 4 )) = 1.349 which are relatively small. Their degrees of equality and subsethood d= (C (A˜ 1 ), C (A˜ 4 )) = 0.332, d⊆ (C (A˜ 1 ), C (A˜ 4 )) = 0.497 and d⊆ (C (A˜ 4 ), C (A˜ 1 )) = 0.501 indicate that their centroids are similar. This comes from H-continuity concepts (see Remark 1), since small values of dα lead to similar centroids. Seen as centroids, the closest IT2FNs are A˜ 1 and A˜ 2 since their degrees of equality and subsethood d= (C (A˜ 1 ), C (A˜ 2 )) = 1, d⊆ (C (A˜ 1 ), C (A˜ 2 )) = 1 and d⊆ (C (A˜ 2 ), C (A˜ 1 )) = 0.792 are large, and their distances d(C (A˜ 1 ), C (A˜ 2 )) = 0.32 and dc (C (A˜ 1 ), C (A˜ 2 )) = 0.355 are smaller than other centroids. Also note that C (A˜ 3 ) ≈ C (A˜ 5 ) while A˜ 3 ̸= A˜ 5 , so dα = 1.167. This is a possible case (see Corollary 1) since different combinations of shapes of IT2FNs can lead to equal centroids, conversely case is not possible. To determine which IT2FN is less/higher or equal than others, Definition 6 provides information to define its fuzzy binary relation -. The obtained results are shown as follows:
−
- = 1 1 1
− 1 1 1
0 0
0 0 1
0
−
1
0 0
− , 0
−
− −
1
−
1
1
-c = 1
0 0
− −
− − −
0 0
− . −
−
In matrix -, ‘1’ means -, ‘0’ means %, and ‘–’ means that they are not fully comparable. This results show us some interesting information: A˜ 1 and A˜ 2 are greater than others. IT2FN A˜ 1 is also less or equal to all others except from A˜ 5 . Since there are some ‘–’, we cannot make a decision about the difference between numbers. To do so, we use -c , which shows if a centroid is less or equal than another. For instance, C (A˜ 3 ) 6 C (A˜ 1 ), so it is represented by ‘1’, likewise C (A˜ 1 ) > C (A˜ 3 ) is represented by ‘0’. The symbol ‘–’ means that the binary relation -c is not present in the comparison (which implies that d⊆ > 0).
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In general C (A˜ 3 ) and C (A˜ 5 ) are less or equal than almost all others, and C (A˜ 4 ) has a degree of subsethood to all other centroids. This agrees with their distances dα since A˜ 3 and A˜ 5 are the farthest sets compared to all others, and it is concordant with the ranking given by their centers (see Eq. (20)). 6. Concluding remarks Our proposal is based on the comparison of the intervals provided by every α -cut made over A˜ which allows us to define an L1 Minkowski distance between two IT2FNs which meets with distance axioms. The proposed approach differs from most of available methods which define a fuzzy measure to compare the memberships of two IT2FNs, since it is based on α -cuts instead of the memberships of a particular value. Alternatively, we propose an L1 Minkowski distance for comparing the centroids of two IT2FNs, and a subsethood measure between two centroids. This allows us to compare the expected value of two IT2FNs using their centroids, which are popular defuzzification measures. The proposed approach is applied to some triangular IT2FNs to illustrate its meaning. The obtained results are satisfactory since we can clearly identify when two IT2FNs are different and how distant they are. Alternatively we can compare the centroids of two IT2FNs and measure the distance between them. The proposed measures are specially useful in cases where fuzzy theory (fuzzy logic systems, fuzzy sets, fuzzy optimization, fuzzy relational equations, etc.) is applied using α -cuts and the extension principle instead of mapping the primary variable (support) of an IT2FN. This way, our proposal has a wide potential use in applications such as resolution of fuzzy differential equations (see Chalco-Cano & Román-Flores [2]), resolution of fuzzy linear systems, linear programming with constraints in the form aij xj - b˜ i (see Figueroa, Kalenatic & López [7]), constrained decision making, ordering of words, etc. Acknowledgments Yurilev Chalco-Cano and Heriberto Román-Florez have been partially supported by Conicyt-Chile through Projects Fondecyt 1120665 and 1120674. Appendix A Theorem 4 states that the binary relation - satisfies the following ordering properties:
˜ then A˜ = B. ˜ It is enough to proof that if αA˜ 6 αB˜ and αB˜ 6 αA˜ holds for all α ∈ [0, 1], then P1. If A˜ - B˜ and B˜ - A, α˜ α˜ A = B. Considering Proposition 1 and Definition 6, we can decompose αA˜ into inf αA˜ := [αA˜ L(+) , αA˜ L(−) ] and sup αA˜ := ˜ [αA˜ R(−) , αA˜ R(+) ], and αB˜ into inf αB˜ := [αB˜ L(+) , αB˜ L(−) ] and sup αB˜ := [αB˜ R(−) , αB˜ R(+) ], so if inf αA˜ 6 inf αB˜ and inf αB˜ 6 inf αA, ˜ then αA˜ L(+) = αB˜ L(+) and αA˜ L(−) = αB˜ L(−) which is equivalent to say inf αA˜ = inf αB˜ (same result for sup αA˜ = sup αB). ˜ and sup αA˜ 6 sup αB˜ , sup αB˜ 6 sup αA˜ ⇒ sup αA˜ = Hence, if both inf αA˜ 6 inf αB˜ , inf αB˜ 6 inf αA˜ ⇒ inf αA˜ = inf αB, α˜ ˜ This immediately proofs that dα (A˜ , B˜ ) = 0. sup B hold ∀α ∈ [0, 1], then A˜ = B. ˜ and αB˜ > αC˜ hold ∀α ∈ [0, 1], then inf αA˜ > inf αB, ˜ P2. If A˜ % B˜ and B˜ % C˜ , then A˜ % C˜ . Definition 6 states that if αA˜ > αB, α˜ α˜ α˜ α˜ α˜ α˜ inf B > inf C , sup A > sup B, and inf B > inf C , which implies that αA˜ L(+) > αB˜ L(+) , αA˜ L(−) > αB˜ L(−) , αA˜ R(−) > αB˜ R(−) , α˜ R(+) ˜ L(+) > A > αB˜ R(+) , αB˜ L(+) > αC˜ L(+) , αB˜ L(−) > αC˜ L(−) , αB˜ R(−) > αC˜ R(−) , αB˜ R(+) > αC˜ R(+) . This in turn implies that αA α ˜ L(+) α˜ L(−) α ˜ L(−) α˜ R(−) α ˜ R(−) α˜ R(+) α ˜ R(+) ˜ ˜ C , A > C , A > C , and A > C also holds ∀α ∈ [0, 1], which lead to A % C . This immediately verifies that dα (A˜ , B˜ ) > 0, dα (A˜ , C˜ ) > 0, so dα (A˜ , C˜ ) > dα (A˜ , B˜ ). ˜ then A˜ % B. ˜ By definition (see Eq. (5)), any IT2FN is convex and normal, so P3. If A˜ ∩ B˜ = 0 and A˜ is on the right of B, A˜ ∩ B˜ = 0 implies that supp(A˜ ) ∩ supp(B˜ ) = 0, where supp(A˜ ) := supp(µ ¯ A˜ ). Now if A˜ is on the right of B˜ and their ˜ and Definition 6 holds ∀ α ∈ [0, 1]. This also verifies that dα (A˜ , B˜ ) > 0, supports have no common elements, then A˜ % B, d(C (A˜ ), C (B˜ )) > 0, and dc (C (A˜ ), C (B˜ )) > 0. P4. The order of A˜ and B˜ is not affected by the other centroids under comparison. It comes directly from the fact that we are not defining any other point and/or set as reference point for comparison, so dα (A˜ , B˜ ) is a function of A˜ and B˜ without the influence of any other point/set. This concludes the proof. Appendix B The binary relation -c as stated in Theorem 5 satisfies the following ordering properties: P1. If C (A˜ ) > C (B˜ ) and C (B˜ ) > C (A˜ ), then C (A˜ ) = C (B˜ ). If C (A˜ ) > C (B˜ ), then Cl (A˜ ) > Cl (B˜ ) and Cu (A˜ ) > Cu (B˜ ), so if C (B˜ ) > C (A˜ ), then Cl (B˜ ) > Cl (A˜ ) and Cu (B˜ ) > Cu (A˜ ). This means that Cl (A˜ ) = Cl (B˜ ) and Cu (A˜ ) = Cu (B˜ ) which implies that d(C (A˜ ), C (B˜ )) = 0 and dc (C (A˜ ), C (B˜ )) = 0.
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P2. If C (A˜ ) > C (B˜ ) and C (B˜ ) > C (C˜ ), then C (A˜ ) > C (C˜ ). If C (A˜ ) > C (B˜ ), then Cl (A˜ ) > Cu (B˜ ), and if C (B˜ ) > C (C˜ ), then Cl (B˜ ) > Cu (C˜ ), so we can see that Cl (A˜ ) > Cu (B˜ ) ⇒ Cl (A˜ ) > Cu (C˜ ), so C (A˜ ) > C (C˜ ), which implies that d(C (A˜ ), C (C˜ )) > d(C (A˜ ), C (B˜ )), and dc (C (A˜ ), C (C˜ )) > dc (C (A˜ ), C (B˜ )). P3. If C (A˜ ) ∩ C (B˜ ) = 0 and C (A˜ ) is on the right of C (B˜ ), then C (A˜ ) > C (B˜ ). If C (A˜ ) ∩ C (B˜ ) = 0 and C (A˜ ) is on the right of C (B˜ ), then ∃x (Cu (B˜ ) 6 x 6 Cl (A˜ )) and Cl (A˜ ) > Cu (B˜ ) which verifies C (A˜ ) > C (B˜ ), d(C (A˜ ), C (B˜ )) > 0, and dc (C (A˜ ), C (B˜ )) > 0. P4. The order of C (A˜ ) and C (B˜ ) is not affected by the other centroids under comparison. It comes directly from the fact that we are not defining any other centroid as reference point for comparison, so d(C (A˜ ), C (B˜ )) and dc (C (A˜ ), C (B˜ )) are functions of A˜ and B˜ without the influence of any other set. This concludes the proof. References [1] C. Celemin, M. Melgarejo, A proposal to speed up the computation of the centroid of an interval Type-2 fuzzy set, Adv. Fuzzy Syst. 2013 17 (2013) (Article ID 158969). [2] Y. Chalco-Cano, H. Román-Flores, Comparation between some approaches to solve fuzzy differential equations, Fuzzy Sets and Systems 160 (11) (2009) 1517–1527. [3] B. Chaudhuri, A. RosenFeld, A modified hausdorff distance between fuzzy sets, Inform. Sci. 118 (1999) 159–171. [4] S.-M. Chen, T.-H. Chang, Finding multiple possible critical paths using fuzzy PERT, IEEE Trans. Syst. Man Cybern. B 31 (6) (2001) 930–937. [5] J.C. Figueroa-García, An approximation method for type reduction of an interval Type-2 fuzzy set based on α -cuts, in: IEEE (Ed.), Proceedings of FEDCSIS, IEEE, 2012, pp. 1–6. [6] J.C. Figueroa-García, A general model for linear programming with interval type-2 fuzzy technological coefficients, in: 2012 Annual Meeting of the North American Fuzzy Information Processing Society, NAFIPS, IEEE, 2012, pp. 1–6. [7] J.C. Figueroa-García, D. Kalenatic, C.A. Lopez-Bello, Multi-period mixed production planning with uncertain demands: Fuzzy and interval fuzzy sets approach, Fuzzy Sets and Systems 206 (2012) 21–38. [8] W.-L. Hung, M.-S. Yang, Similarity measures between type-2 fuzzy sets, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 12 (6) (2004) 827–841. [9] G.J. Klir, T.A. Folger, Fuzzy Sets, Uncertainty and Information, Prentice Hall, 1992. [10] B. Kosko, Fuzziness vs. probability, Int. J. Gen. Syst. 17 (1) (1990) 211–240. [11] J.M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, Prentice Hall, 2001. [12] J.M. Mendel, D. Wu, Perceptual Computing: Aiding People in Making Subjective Judgments, John Wiley & Sons, 2010. [13] H.B. Mitchell, Ranking Type-2 fuzzy numbers, IEEE Trans. Fuzzy Syst. 14 (2) (2006) 287–294. [14] R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction to Interval Analysis, SIAM, 2009. [15] H.T. Nguyen, V. Kreinovich, Computing degrees of subsethood and similarity for interval-valued fuzzy sets: Fast algorithms, in: 9th International Conference on Intelligent Technologies InTec08, IEEE, 2008, pp. 47–55. ˘ [16] J. Ramík, J. Rimánek, Inequality relation between fuzzy numbers and its use in fuzzy optimization, Fuzzy Sets and Systems 16 (1985) 123–138. [17] H. Román-Flores, Y. Chalco-Cano, H-continuity of fuzzy measures and set defuzzification, Fuzzy Sets and Systems 157 (2006) 230–242. [18] A. Rufián-Lizana, Y. Chalco-Cano, R. Osuna-Gómez, G. Ruiz-Garzón, On invex fuzzy mappings and fuzzy variational-like inequalities, Fuzzy Sets and Systems 200 (2012) 84–98. [19] X. Wang, E.E. Kerre, Reasonable properties for the ordering of fuzzy quantities (I), Fuzzy Sets and Systems 118 (1) (2001) 375–387. [20] D. Wu, J.M. Mendel, Enhanced Karnik–Mendel algorithms, IEEE Trans. Fuzzy Syst. 17 (4) (2009) 923–934. [21] D. Wu, J.M. Mendel, A vector similarity measure for interval type-2 fuzzy sets and type-1 fuzzy sets, Inform. Sci. 178 (1) (2008) 381–402. [22] D. Wu, J.M. Mendel, A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets, Inform. Sci. 179 (1) (2009) 1169–1192. [23] L. Xuecheng, Entropy, distance measure and similarity measure of fuzzy sets and their relations, Fuzzy Sets and Systems 52 (1992) 305–318. [24] L. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Inform. Sci. 8 (1975) 199–249. [25] G. Zheng, J. Wang, W. Zhou, Y. Zhang, A similarity measure between interval type-2 fuzzy sets, in: IEEE International Conference on Mechatronics and Automation, IEEE, 2010, pp. 191–195.