Applied Mathematical Sciences, Vol. 9, 2015, no. 22, 1069 - 1076
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2015.411982
Convex Combination and its Application to
Fuzzy Sets and Interval-Valued Fuzzy Sets II
Omar Salazar and Jairo Soriano
Universidad Distrital Francisco Jose de Caldas, Bogota, Colombia
c 2014 Omar Salazar and Jairo Soriano. This is an open access article
Copyright distributed under the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Abstract
This paper showed a short characterization of embedded fuzzy sets
of an interval-valued fuzzy set. We proved that any embedded fuzzy set
can be expressed as a convex combination of three fuzzy sets always.
We established two basic results (existence and uniqueness) in the representation as a convex combination. We showed three examples from
the literature, and we showed they can be deduced by using the theory
presented herein.
Mathematics Subject Classification: 03B52, 03E72
Keywords: convex combination, fuzzy set, interval-valued fuzzy set, interval type-2 fuzzy set, embedded fuzzy set, membership function
1
Introduction
Type-2 fuzzy sets (T2FS) were introduced by Zadeh [24] as a generalization of ordinary fuzzy sets [23] (also called type-1 fuzzy sets). According
to Mendel et al. [14, 15, 16], a T2FS Ã, defined over a nonempty universal set X, has a three-dimensional membership function (MF) of the form
µÃ : X × [0, 1] 7→ [0, 1] : hx, ui 7→ µÃ (x, u). Each x ∈ X is put on the first
dimension, each primary membership u ∈ [0, 1] is put on the second dimension,
and each secondary membership µÃ (x, u) ∈ [0, 1] is put on the third dimension.
When secondary memberships are all equal to one, the resulting T2FS is
called interval type-2 fuzzy set (IT2FS) [15, 16]. An IT2FS has intervals as
1070
Omar Salazar and Jairo Soriano
level-sets [24, pp. 243], that is, level-sets have interval-valued membership functions (IVMF). Because secondary memberships provide no additional information, an IT2FS can be considered on two dimensions only, and it is completely
described by an IVMF [14, 16]. Gehrke et al. [2, 3, 4, 5, 6, 7] called such sets
interval-valued fuzzy sets (IVFS). Mendel et al. [16, footnote 3] claimed there
is equivalence between IT2FSs and IVFSs.
An IVFS has an IVMF whose lower and upper MFs are denoted µÃ and µÃ .
All those possible MFs between µÃ and µÃ are called embedded membership
functions (EMF), denoted µAe , which define embedded fuzzy sets (EFS). The
footprint of uncertainty (FOU) was defined as the set of all EMFs of an IVFS
[16, pp. 812]. The FOU is important because it represents the uncertainty in
selecting an unique EMF as representative membership degrees for all x ∈ X
[12, pp. 16].
Mendel et al. [10, 15, 16] used EMFs in a wavy-slice representation of the
FOU, and perhaps, the main purpose of EMFs was in formulating the centroid
of an IT2FS [14]. Melgarejo et al. [1, eq. (20)–(21)], [13, eq. (11)–(16)] used
EMFs to propose alternative algorithms in order to find the centroid of an
IT2FS. Nie and Tan [20, eq. (13)] used an EMF, obtained as the mean of µÃ
and µÃ , to propose an alternative type-reduction method. Greenfield et al. [8,
Th. 7], [9, Th. 1] proposed the collapsing method, which deduces the MF of a
representative embedded set whose defuzzified value closely approximates that
of the T2FS.
This paper focuses on a short characterization of EFSs by means of the
convex combination operation [23, pp. 345]. We used some results presented
in [21, 22]. We proved that any EMF can be expressed as convex combination
of three MFs, namely, µÃ , µÃ , and a third arbitrary µΛ . We proved that each
µΛ generates an EMF whenever µÃ and µÃ are fixed. We proved that the
EMFs shown by Melgarejo et al. [1, 13], Nie and Tan [20], and Greenfield
et al. [8, 9], can be deduced by using the theory presented herein.
This paper is organized as follows: Section 2 presents the main topic. This
section focuses on how an EMF is related to the convex combination of the
lower and upper MFs of an IVFS. Section 3 gives some examples. Finally,
conclusions are drawn in Section 4.
2
Convex Combination on IVFSs
Definition 2.1 (Interval-valued fuzzy set (IVFS)) [12] Given a nonempty universal set X, an interval-valued fuzzy set à is defined by a function
µÃ : X 7→ E([0, 1]), where E([0, 1]) = {[a, b] | a, b ∈ [0, 1], a ≤ b, [a, b] ⊆ [0, 1]}.
The set E([0, 1]) has a partial ordering relation ≤ that is defined in terms
of the full order of [0, 1], i.e., hE([0, 1]), ≤i is a partially ordered set. If
Convex combination and its application: part 2
1071
[a, b], [c, d] ∈ E([0, 1]) then [a, b] ≤ [c, d] if and only if a ≤ c and b ≤ d [19, 24].
Some elements in E([0, 1]) are incomparable, e.g., [0, 1], [0.25, 0.75] ∈ E([0, 1])
are incomparable. Mukaidono [19] and Gehrke et al. [2, 3, 5, 6, 7] related
the interval [a, b] ⊆ [0, 1] with the pair ha, bi of its endpoints. Therefore, an
IVFS can be defined by a function µÃ : X 7→ [0, 1][2] , where [0, 1][2] = {ha, bi |
a, b ∈ [0, 1], a ≤ b}. The function Φ : E([0, 1]) 7→ [0, 1][2] : [a, b] 7→ ha, bi, which
relates the interval [a, b] ∈ E([0, 1]) with the pair ha, bi ∈ [0, 1][2] , is a bijection. If Φ([a, b]) = ha, bi = hc, di = Φ([c, d]) then a = c and b = d, therefore,
[a, b] = {x | a ≤ x ≤ b} = {x | c ≤ x ≤ d} = [c, d] (Φ is injective); and if
ha, bi ∈ [0, 1][2] then the interval {x | a ≤ x ≤ b} = [a, b] ∈ E([0, 1]) is such
that Φ([a, b]) = ha, bi (Φ is surjective).
The function µÃ : X 7→ E([0, 1]) is called membership function and the
interval µÃ (x) = [µÃ (x), µÃ (x)] ∈ E([0, 1]) is called membership degree 1 of x in
Ã. MFs µÃ (x) and µÃ (x) are called lower and upper membership functions [16,
pp. 810], respectively, and they define ordinary fuzzy sets A and A which are
the lower and upper fuzzy sets of Ã. By definition [µÃ (x), µÃ (x)] ∈ E([0, 1]),
and this means µÃ (x), µÃ (x) ∈ [0, 1] and µÃ (x) ≤ µÃ (x) for all x ∈ X. In
other words µÃ , µÃ ∈ [0, 1]X and A ⊆ A. We are interested in fuzzy sets Ae
such that A ⊆ Ae ⊆ A (embedded fuzzy sets).
Definition 2.2 (Embedded fuzzy set (EFS)) Given an interval-valued
fuzzy set Ã, defined by µÃ : X 7→ E([0, 1]) : x 7→ [µÃ (x), µÃ (x)], an embedded
fuzzy set Ae of à is a fuzzy set such that its membership function µAe ∈ [0, 1]X
satisfies µÃ ≤ µAe ≤ µÃ .
Definition 2.3 (Footprint of uncertainty (FOU)) Given an intervalvalued fuzzy set Ã, defined by µÃ : X 7→ E([0, 1]) : x 7→ [µÃ (x), µÃ (x)], its
footprint of uncertainty is FOU(Ã) = {µAe | µAe ∈ [0, 1]X , µÃ ≤ µAe ≤ µÃ }.
It follows immediately that FOU(Ã) ⊆ [0, 1]X , therefore, FOU(Ã) inherits
the partial order from [0, 1]X , i.e., FOU(Ã) is a partially ordered subset of
[0, 1]X . MFs µÃ and µÃ are lower and upper bounds of FOU(Ã). In particular,
the equalities µÃ = inf FOU(Ã) and µÃ = sup FOU(Ã) hold. Additionally, if
µÃ = µÃ (à reduces to an ordinary fuzzy set) then FOU(Ã) = {µÃ } = {µÃ }
and FOU(Ã) has one element only (one EMF that coincides with the lower
and upper MFs of Ã). This trivial case establishes there is no uncertainty in
selecting a MF (because there is one only) that represents the membership
degrees for all x ∈ X.
1
Mendel et al. [10, 11, 14, 15, 16, 17, 18] call this interval primary membership degree.
1072
Omar Salazar and Jairo Soriano
Theorem 2.4 (of existence) Given an interval-valued fuzzy set Ã, for
each embedded fuzzy set Ae of à there exists at least one fuzzy set Λ such
that Ae = hA, A; Λi.
Proof. By definition an EFS Ae of à satisfies A ⊆ Ae ⊆ A. The theorem
follows from [22, Th. 3.4] with A = A, B = A and C = Ae , and the fact that
A ∩ A = A and A ∪ A = A.
Theorem 2.5 (of uniqueness) Given an interval-valued fuzzy set Ã, for
each embedded fuzzy set Ae of à the membership degree µΛ (x) is unique in
the representation Ae = hA, A; Λi for each x ∈ S only, where S = {x ∈ X |
µÃ (x) < µÃ (x)}.
Proof. The theorem follows from [22, Th. 3.5] with A = A, B = A and
C = Ae , and the fact that A ∩ A = A and A ∪ A = A.
3
Some Examples
Example 3.1 Suppose that X = {0, 1, 2, 3, 4, 5, 6}. The lower µÃ (x) and
upper µÃ (x) MFs of an IVFS à are given in Table 1. The fuzzy set Λ generates
Ae in the form µAe (x) = µÃ (x) + µΛ (x)(µÃ (x) − µÃ (x)) for all x ∈ X, i.e.,
µAe (0) = µÃ (0) + µΛ (0)(µÃ (0) − µÃ (0)) = 0.1 + 0.5(0.3 − 0.1) = 0.2,
µAe (1) = µÃ (1) + µΛ (1)(µÃ (1) − µÃ (1)) = 0.3 + 0(0.4 − 0.3) = 0.3,
µAe (2) = µÃ (2) + µΛ (2)(µÃ (2) − µÃ (2)) = 0.5 + 0.25(0.7 − 0.5) = 0.55,
µAe (3) = µÃ (3) + µΛ (3)(µÃ (3) − µÃ (3)) = 0.5 + 0.6(1 − 0.5) = 0.8,
µAe (4) = µÃ (4) + µΛ (4)(µÃ (4) − µÃ (4)) = 0.4 + 0.5(1 − 0.4) = 0.7,
µAe (5) = µÃ (5) + µΛ (5)(µÃ (5) − µÃ (5)) = 0.2 + 0.5(0.8 − 0.2) = 0.5,
µAe (6) = µÃ (6) + µΛ (6)(µÃ (6) − µÃ (6)) = 0 + 1(0.9 − 0) = 0.9.
Note that S = {x ∈ X | µÃ (x) < µÃ (x)} = X. Then, µΛ (x) is unique for each
x ∈ X.
Table 1: Convex combination on interval-valued fuzzy sets (see text for more
details)
x
µÃ (x)
µÃ (x)
µAe (x)
µΛ (x)
0
0.1
0.3
0.2
0.5
1
0.3
0.4
0.3
0
2
0.5
0.7
0.55
0.25
3
0.5
1
0.8
0.6
4
0.4
1
0.7
0.5
5
0.2
0.8
0.5
0.5
6
0
0.9
0.9
1
1073
Convex combination and its application: part 2
Example 3.2 If Λ has a constant MF µΛ (x) = k ∈ [0, 1] for all x ∈ X then
µAe is µAe (x) = kµÃ (x) + (1 − k)µÃ (x) for all x ∈ X. In particular, if k = 0.5
we have
µAe (x) = 0.5µÃ (x) + (1 − 0.5)µÃ (x) = (µÃ (x) + µÃ (x))/2.
(1)
Equation (1) was used by Nie and Tan [20, eq. (13)] on their proposal of an
alternative type-reduction method.
Example 3.3 Suppose that X = R. Let xl , xr ∈ X be real numbers such
that Λl and Λr are defined by:
(
(
1, if x ≤ xl ,
0, if x ≤ xr ,
µΛl (x) =
and µΛr (x) =
0, if x > xl ,
1, if x > xr .
Let à be an arbitrary IVFS. The corresponding EFSs are:
(
µÃ (x),
µAel (x) = µΛl (x)µÃ (x) + (1 − µΛl (x))µÃ (x) =
µ (x),
( Ã
µÃ (x),
µAer (x) = µΛr (x)µÃ (x) + (1 − µΛr (x))µÃ (x) =
µÃ (x),
if x ≤ xl ,
if x > xl .
(2)
if x ≤ xr ,
if x > xr .
(3)
Equations (2)–(3) are important for computing the centroid of an IVFS [10,
14, 17, 18]. Melgarejo et al. [1, eq. (20)–(21)], [13, eq. (11)–(16)] used (2)–(3)
to propose alternative algorithms in order to find the centroid of an IT2FS.
Example 3.4 Suppose that X = R. This universal set has been discretized
into N points. The collapsing method proposed by Greenfield et al. [8, 9] uses
the Representative Embedded Set (RES) [8, Th. 7] whose membership degrees
approximates to µAe (xi ) ≈ µÃ (xi ) + ri for all i = 1, . . . , N , where
ri =
n−1
X
j=0
w̄ji bij ,
w̄ji
wji
= Pn−1
j=0
wji
,
wji
1
=
,
||A|| + Ri−1 + bij
Ri−1 =
i−1
X
rk ,
k=0
P
with R0 = 0, and where ||A|| = N
i=1 µÃ (xi ) is the scalar cardinality of A.
The difference µÃ (xi ) − µÃ (xi ) has been discretized into n points at distances
bi0 (= 0), bi1 ,. . . , bin−1 (= µÃ (xi ) − µÃ (xi )) from µÃ (xi ). This embedded set can
be written as a convex combination by using the following identity:
!
n−1
n−1
i
X
X
b
j
ri =
w̄ji bij =
w̄ji bin−1 = µΛ (xi )(µÃ (xi ) − µÃ (xi )),
i
b
j=0
j=0 n−1
Pn−1 i i
i
where bin−1 = µÃ (xi ) − µÃ (xi ) and µΛ (xi ) =
j=0 (bj /bn−1 )w̄j . Then, the
RES approximates to µAe (xi ) ≈ µÃ (xi ) + µΛ (xi )(µÃ (xi ) − µÃ (xi )) for all i =
1, . . . , N .
1074
4
Omar Salazar and Jairo Soriano
Conclusion
This paper proved that any embedded fuzzy set Ae can be expressed as convex
combination of three fuzzy sets, whose membership functions are: the lower
µÃ and upper µÃ membership functions, and a third arbitrary membership
function µΛ . This representation is possible for all x ∈ X, but µΛ (x) is unique
for each x ∈ S only, where S = {x ∈ X | µÃ (x) < µÃ (x)}. Our theory was
presented with no assumption about the nature of X. Three notable examples
from the literature were presented, where we showed they can be deduced by
using the theory presented herein.
References
[1] Hector Bernal, Karina Duran, and Miguel Melgarejo. A comparative
study between two algorithms for computing the generalized centroid
of an interval type-2 fuzzy set. In Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ 2008), pages 954–959, 2008.
http://dx.doi.org/10.1109/fuzzy.2008.4630484
[2] Mai Gehrke, Carol Walker, and Elbert Walker. Some comments on interval valued fuzzy sets. International Journal of Intelligent Systems,
11(10):751–759, October 1996. http://dx.doi.org/10.1002/(sici)1098111x(199610)11:10¡751::aid-int3¿3.0.co;2-y
[3] Mai Gehrke, Carol Walker, and Elbert Walker.
A mathematical setting for fuzzy logic.
International Journal of Uncertainty,
Fuzziness and Knowledge-Based Systems, 5(3):223–238, June 1997.
http://dx.doi.org/10.1142/s021848859700021x
[4] Mai Gehrke, Carol Walker, and Elbert Walker. Some comments on fuzzy
normal forms. In Proceedings of the ninth IEEE International Conference
on Fuzzy Systems, volume 2, pages 593–598, San Antonio, Texas, May
2000. http://dx.doi.org/10.1109/fuzzy.2000.839060
[5] Mai Gehrke, Carol Walker, and Elbert Walker. Normal forms and
truth tables for interval-valued fuzzy logic.
In Joint 9th IFSA
World Congress and 20th NAFIPS International Conference, volume 5,
pages 1327–1331, Vancouver, British Columbia, Canada, July 2001.
http://dx.doi.org/10.1109/nafips.2001.943740
[6] Mai Gehrke, Carol Walker, and Elbert Walker. Some basic theory of interval-valued fuzzy sets.
In Joint 9th IFSA World
Congress and 20th NAFIPS International Conference, volume 5,
Convex combination and its application: part 2
1075
pages 1332–1336, Vancouver, British Columbia, Canada, July 2001.
http://dx.doi.org/10.1109/nafips.2001.943741
[7] Mai Gehrke, Carol Walker, and Elbert Walker. Normal forms and truth
tables for fuzzy logics. Fuzzy Sets and Systems, 138(1):25–51, August
2003. http://dx.doi.org/10.1016/s0165-0114(02)00566-3
[8] Sarah Greenfield, Francisco Chiclana, Simon Coupland, and Robert
John. The collapsing method of defuzzification for discretized interval type-2 fuzzy sets. Information Sciences, 179(13):2055–2069, 2009.
http://dx.doi.org/10.1016/j.ins.2008.07.011
[9] Sarah Greenfield, Francisco Chiclana, and Robert John. The collapsing
method: Does the direction of collapse affect accuracy? In Proceedings of
the Joint 2009 International Fuzzy Systems Association World Congress
and 2009 European Society of Fuzzy Logic and Technology Conference
(IFSA-EUSFLAT 2009), pages 980–985, Lisbon, Portugal, July 2009.
[10] Nilesh N. Karnik and Jerry M. Mendel. Centroid of a type-2 fuzzy set. Information Sciences, 132:195–220, 2001. http://dx.doi.org/10.1016/s00200255(01)00069-x
[11] Nilesh N. Karnik and Jerry M. Mendel.
Operations on type2 fuzzy sets.
Fuzzy sets and systems, 122:327–348, 2001.
http://dx.doi.org/10.1016/s0165-0114(00)00079-8
[12] George J. Klir and Bo Yuan. Fuzzy sets and fuzzy logic: theory and
applications. Prentice Hall PTR, New Jersey, 1995.
[13] Miguel Melgarejo. A fast recursive method to compute the generalized centroid of an interval type-2 fuzzy set. In Annual Meeting of the North American Fuzzy Information Processing Society
NAFIPS 2007, pages 190–194, San Diego, California, USA, June 2007.
http://dx.doi.org/10.1109/nafips.2007.383835
[14] Jerry M. Mendel. On centroid calculations for type-2 fuzzy sets. Appl.
Comput. Math., 10(1):88–96, 2011.
[15] Jerry M. Mendel and Robert I. John.
Type-2 fuzzy sets made
simple. IEEE Transactions on Fuzzy Systems, 10(2):117–127, 2002.
http://dx.doi.org/10.1109/91.995115
[16] Jerry M. Mendel, Robert I. John, and Feilong Liu.
Interval type-2 fuzzy logic systems made simple.
IEEE Transactions on Fuzzy Systems,
4(6):808–821,
December 2006.
http://dx.doi.org/10.1109/tfuzz.2006.879986
1076
Omar Salazar and Jairo Soriano
[17] Jerry M. Mendel and Feilong Liu. Super-exponential convergence of the
Karnik-Mendel algorithms for computing the centroid of an interval type2 fuzzy set. IEEE Transactions on Fuzzy Systems, 15(2):309–320, April
2007. http://dx.doi.org/10.1109/tfuzz.2006.882463
[18] Jerry M. Mendel and Hongwei Wu.
New results about the
centroid of an interval type-2 fuzzy set, including the centroid
of a fuzzy granule.
Information Sciences, 177:360–377, 2007.
http://dx.doi.org/10.1016/j.ins.2006.03.003
[19] Masao Mukaidono. Algebraic structures of interval truth values in fuzzy
logic. In Proceedings of the Sixth IEEE International Conference On
Fuzzy Systems, volume 3, pages 699–705, Barcelona, Spain, July 1997.
http://dx.doi.org/10.1109/fuzzy.1997.622797
[20] Maowen Nie and Woei Wan Tan. Towards an efficient type-reduction
method for interval type-2 fuzzy logic systems. In Proceedings of the
IEEE International Conference on Fuzzy Systems (FUZZ 2008), pages
1425–1432, 2008. http://dx.doi.org/10.1109/fuzzy.2008.4630559
[21] Omar Salazar and Jairo Soriano. Generating embedded type-1 fuzzy sets
by means of convex combination. In Proceedings of the 2013 IFSA World
Congress NAFIPS Annual Meeting, pages 51–56, Edmonton, Canada,
June 2013. http://dx.doi.org/10.1109/ifsa-nafips.2013.6608374
[22] Omar Salazar and Jairo Soriano. Convex combination and its application to fuzzy sets and interval-valued fuzzy sets I. Applied Mathematical
Sciences, 2015.
[23] Lofti A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965.
http://dx.doi.org/10.1016/s0019-9958(65)90241-x
[24] Lofti A. Zadeh. The concept of a linguistic variable and its application
to approximate reasoning-I. Information Sciences, 8(3):199–249, 1975.
http://dx.doi.org/10.1016/0020-0255(75)90036-5
Received: December 4, 2014; Published: February 4, 2015