A new approach to similarity and inclusion measures between general type-2 fuzzy sets

Soft ComputDOI 10.1007/s00500-013-1101-z

METHODOLOGIES AND APPLICATION

A new approach to similarity and inclusion measuresbetween general type-2 fuzzy sets

Tao Zhao · Jian Xiao · YiXing Li · XueSong Deng

© Springer-Verlag Berlin Heidelberg 2013

Abstract Interval type-2 fuzzy similarity and inclusionmeasures have been widely studied. In this paper, theaxiomatic definitions of general type-2 fuzzy similarity andinclusion measures are given on the basis of interval type-2 fuzzy similarity and inclusion measures. To improve theshortcomings of the existing general type-2 fuzzy similarityand inclusion measures, we define two new general type-2fuzzy similarity measures and two new general type-2 fuzzyinclusion measures based on α-plane representation the-ory, respectively, and discuss their related properties. Unlikesome existing measures, one of the proposed similarity andinclusion measures are expressed as type-1 fuzzy sets, andtherefore these definitions are consistent with the highlyuncertain nature of general type-2 fuzzy sets. The theoreticalproof is also given to illustrate that the proposed measuresare natural extensions of the most popular type-1 fuzzy mea-sures. In the end, the performances of the proposed similarityand inclusion measures are examined.

Keywords General type-2 fuzzy sets · Similarity measure ·Inclusion measure · α-Plane theory

Communicated by H. Hagras.

T. Zhao (B)School of Transportation and Logistics, Southwest JiaotongUniversity, Chengdu 610031, Chinae-mail: [email protected]

J. Xiao · Y. Li · X. DengSchool of Electrical Engineering, Southwest Jiaotong University,Chengdu 610031, China

1 Introduction

Type-1 fuzzy sets (Zadeh 1965) extend the classical notionof set and express the vagueness of real life. After decadesof development, type-1 fuzzy sets have been successfullyapplied in the fields of automatic control, system identifica-tion, fault diagnosis, etc. To enhance the ability of systems fordealing with uncertainties and experience more freedom indesign than type-1 fuzzy sets, Zadeh (1975) proposed type-2 fuzzy sets. The membership grades of type-2 fuzzy setsare type-1 fuzzy sets on [0, 1]. In highly uncertain situa-tions, the performances of type-2 fuzzy sets are obviouslybeyond the type-1 fuzzy sets (Karnik et al. 1999). From thetime Zadeh presented type-2 fuzzy set theory, its applica-tions were limited due to the computational complexity. Onthe contrary, interval type-2 fuzzy set, as a simplified versionof type-2 fuzzy set, has been made great development. Thesecondary membership grade of interval type-2 fuzzy set is 1,which makes it relatively simple. Thus, most of researcherswere interested in the interval type-2 fuzzy systems (Liangand Mendel 2000). The general type-2 fuzzy sets can handlecomplex and changing systems, and therefore must be betterthan the interval type-2 fuzzy sets to deal with uncertainties(John and Coupland 2007).

So far, the investigations about general type-2 fuzzy setsare relatively less due to the computational complexity. Inorder to simplify the calculation for general type-2 fuzzy sets,researchers proposed various methods. The representation ofgeneral type-2 fuzzy sets is one of these methods. There arefive kinds of popular representation for general type-2 fuzzysets: vertical-slice representation, wave-slice representation,geometric representation, α-plane representation, and z-slicerepresentation. Vertical-slice representation of type-2 fuzzyset was firstly recognized by researchers, and so type-2 fuzzysystems were studied on the basis of Vertical-slice theory at

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the beginning (Mendel 2001). Subsequently, Mendel et al.(2006) proposed Wave-slice notation, and claimed that theset operations of type-2 fuzzy sets can be defined under thepremise of not using the extension principle. Next, Couplandand John (2004a,b, 2007) proposed computational geome-try representation for type-2 fuzzy sets, which is essentiallyapproximate calculation method based on some discretiza-tions to a certain extent, but also can reduce the calculationcomplexity. Recently, Liu (2008) proposed the α-plane rep-resentation of type-2 fuzzy sets where centroid computationof type-2 fuzzy sets by using the α-plane representation wasstudied, and it was claimed that the α-plane representationcan greatly reduce the computational workload. Almost atthe same time, the theory of z-slice method has also beenproposed in (Wagner and Hagras 2008, 2009, 2010), butMendel et al. (2009) proved that z-slice method and α-planemethod are essentially same. The α-plane method has beenextensively studied (Mendel 2010; Zhai and Mendel 2011),because it can take advantages of the interval type-2 fuzzysets theory to study the general type-2 fuzzy sets.

Similarity and inclusion measures between general type-2fuzzy sets are important from applications point of view. Incontrast to the investigations discussing similarity and inclu-sion measures between interval type-2 fuzzy sets (Wu andMendel 2007; Zeng and Li 2006; Zeng and Guo 2008; Zhanget al. 2009; Wu and Mendel 2008, 2009; Liu and Xiong 2002),the research about similarity and inclusion measures betweengeneral type-2 fuzzy sets is relatively scarce (Mitchell 2005;Yang and Lin 2009; Hwang et al. 2011; Rickard et al. 2009).Mitchell (2005) defined a similarity measure for general type-2 fuzzy sets, but it has the following drawbacks: (1) it doesnot satisfy reflexivity, i.e., the similarity measure between Aand A does not equal to 1; (2) the symmetry can not be satis-fied; (3) the similarity measure may change from experimentto experiment. In addition, similarity and inclusion mea-sures for general type-2 fuzzy sets based on the vertical slicetheory have also been defined in (Yang and Lin 2009; Hwanget al. 2011), but the primary membership of type-2 fuzzy setsin each primary variable must be same. That is to say, the foot-print of uncertainty must be same, which limits applicationscopes of these results. Rickard et al. (2009) used Zadeh’sextension principle to extend Kosko’s definition of the type-1fuzzy inclusion measure, and defined type-2 fuzzy inclusionmeasure. However, their inclusion measure does not satisfyreflexivity (Mendel and Wu 2010), i.e., the inclusion measurebetween A and A does not equal to 1.

In order to expand the theoretical researches of similar-ity and include measures for general type-2 fuzzy sets, andalso to improve the shortcomings of the existing measures, inthis paper, we define two new general type-2 fuzzy similaritymeasures and two new general type-2 fuzzy inclusion mea-sures based on α-plane representation theory, respectively.Different from some existing general type-2 fuzzy similar-

ity and inclusion measures, we put forward a new definitionfor general type-2 fuzzy similarity and inclusion measures,i.e., one of the proposed general type-2 fuzzy similarity andinclusion measures are expressed as type-1 fuzzy sets. Theabove definition method is consistent with the highly uncer-tain nature of general type-2 fuzzy set. However, it is not intu-itive when the general type-2 fuzzy similarity and inclusionmeasures are expressed as type-1 fuzzy sets. Thus, anothergeneral type-2 fuzzy similarity and inclusion measures aredefined as certain value in [0, 1]. The theoretical proof isalso given to illustrate that the proposed measures are nat-ural extensions of the most popular type-1 fuzzy measures.The rest of our work is organized as follows. In Sect. 2, thedefinitions and basic terminologies on general type-2 fuzzysets will be reviewed briefly, and the axiomatic definitions ofgeneral type-2 fuzzy similarity and inclusion measures willalso be given. In Sect. 3, the new general type-2 fuzzy similar-ity and inclusion measures will be defined, and their relatedproperties will also be derived. Furthermore, in Sect. 4 weuse numerical examples to examine the performances of theproposed similarity and inclusion measures. The last sectionconcludes this paper.

2 Background

In this section, we review the definitions and basic termi-nologies on general type-2 fuzzy sets, and give the axiomaticdefinitions of general type-2 fuzzy similarity and inclusionmeasures.

2.1 α-Plane representation theory for general type-2 fuzzysets

Definition 2.1 (Mendel 2001) A type-2 fuzzy set, denotedA, is expressed as:

A =∫

x∈X

u A(x)/x =∫

x∈X

⎡⎢⎢⎣

∫

u0∈J Ax

fx (u0)/u0

⎤⎥⎥⎦

/x J A

x ⊆ [0, 1].

where J Ax is the primary membership of x; fx (u0) is a sec-

ondary membership grade.

In the following sections, the class of all general type-2fuzzy sets of the universe x is denoted as F2(X).

Definition 2.2 (Mendel 2001) The footprint of uncertainty(F OU ) of a type-2 fuzzy set, A, is the union of all primarymemberships, i.e.,

FOU( A) = ∪x∈X

J Ax .

Definition 2.3 (Mendel 2001) The upper membership func-tion is associated with the upper bound of F OU ( A), and is

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denoted by u A(x). The lower membership function is asso-ciated with the lower bound of FOU(A), and is denoted byu A(x).

Definition 2.4 (Mendel et al. 2009) An α-plane for generaltype-2 fuzzy set A, which is denoted by Aα , is defined asfollow:

Aα =∫

x∈X

∫

u0∈J Ax

{(x, u0)| fx (u0) ≥ α}

=∫

x∈X

⎡⎢⎢⎣

∫

u0∈[S AL (x |α),S A

R (x |α)]

u0

⎤⎥⎥⎦

/x α ∈ [0, 1].

where [S AL (x |α), S A

R (x |α)] denote an α-cut of the secondarymembership function u A(x).

From the Definitions 2.2 and 2.4, we have F OU ( A) =A0. That is to say, the FOU of a type-2 fuzzy set is theα = 0 plane. Furthermore, α/ Aα can be regarded as a spe-cial interval type-2 fuzzy set whose secondary membershipgrade is equal to α.

Definition 2.5 (Mendel et al. 2009) The α-plane represen-tation (theorem) for type-2 fuzzy set A is

A = ∪α∈[0,1]

α/ Aα.

From the above definition, we can find that a general type-2 fuzzy set can be decomposed into some special intervaltype-2 fuzzy sets. As a result, the advantages of interval type-2 fuzzy sets can be used to study general type-2 fuzzy sets,and many complex definitions and representations can also besimplified by using α-plane approach. For example, the set-theoretic operations based α-plane representation are moreintuitionistic than those based on extension principle.

Theorem 2.1 (Mendel et al. 2009) Let ( A ∪ B)α and ( A ∩B)α be α-plane of A ∪ B and A ∩ B, respectively, we have

A ∪ B = ∪α∈[0,1]

α/( Aα ∪ Bα);A ∩ B = ∪

α∈[0,1]α/( Aα ∩ Bα).

where

Aα ∪ Bα =∫

x∈X

⎡⎢⎢⎣

∫

u0∈[S AL (x |α)∨SB

L (x |α),S AR (x |α)∨SB

R (x |α)]

u0

⎤⎥⎥⎦

/x;

Aα ∩ Bα =∫

x∈X

⎡⎢⎢⎣

∫

u0∈[S AL (x |α)∧SB

L (x |α),S AR (x |α)∧SB

R (x |α)]

u0

⎤⎥⎥⎦

/x

Obviously, S A∪BL (x |α)= S A

L (x |α)∨SBL (x |α), S A∪B

R (x |α) =S A

R (x |α)∨ SBR (x |α), S A∩B

L (x |α) = S AL (x |α)∧ SB

L (x |α) and

S A∩BR (x |α) = S A

R (x |α) ∧ SBR (x |α) hold.

Definition 2.6 (Hamrawi and Coupland 2010) Let A, B ∈F2(X), define A ⊆ B if S A

L (x |α) ≤ SBL (x |α) and S A

R (x |α) ≤SB

R (x |α) hold for any α ∈ [0, 1] and x ∈ X.It can be seen that the above containment concept of gen-

eral type-2 fuzzy sets satisfies the following properties: (1) ifA ⊆ B and B ⊆ A, then A = B; (2) if A ⊆ B and B ⊆ C,then A ⊆ C; (3) A ⊆ B ⇔ A∩ B = A and A∪ B = B. Thus,the above containment definition of general type-2 fuzzy setsis intuitive and justified. Certainly, the containment conceptof type-2 fuzzy sets is still to be properly defined by usingother approaches.

In Zhai and Mendel (2011), five uncertainty measures ofgeneral type-2 fuzzy sets were studied by using α-planerepresentation theory. Similar to the literature (Zhai andMendel 2011), in this paper we also make some assump-tions on the secondary membership functions. For any A =∫

x∈X [∫u0∈J Ax

fx (u0)/u0]/x, it must satisfy:

fx (u0)

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

gx (u0) u0 ∈ [S AL (x |0), S A

L (x |1)]1 u0 ∈ [S A

L (x |1), S AR (x |1)]

hx (u0) u0 ∈ [S AR (x |1), S A

R (x |0)]0 u0 ∈ (−∞, S A

L (x |0)) ∪ (S AR (x |0),∞)

,

where gx (u0) ∈ [0, 1] is monotonically non-decreasing, andhx (u0) ∈ [0, 1] is monotonically non-increasing. That is tosay, each secondary membership function of A must be anormal and convex type-1 fuzzy set.

2.2 Axiomatic definitions of general type-2 fuzzy similarityand inclusion measures

Similarity and inclusion measures have important applica-tions in the fields of clustering analysis, word computing,fault diagnosis, etc. In contrast to type-1 fuzzy similarity andinclusion measures, there are two explanations about gen-eral type-2 fuzzy similarity and inclusion measures. First,general type-2 fuzzy similarity and inclusion measures aretype-1 fuzzy sets on [0, 1]. Second, general type-2 fuzzysimilarity and inclusion measures are certain value in [0, 1].In Zhai and Mendel (2011), five uncertainty measures ofgeneral type-2 fuzzy sets were studied, and the ultimatemanifestation of these measures are type-1 fuzzy sets. Infact, type-2 fuzzy sets are mainly used to describe highlyuncertain phenomenon, so it is coincident with the natureof law when the uncertainty measures of type-2 fuzzy setsare expressed as type-1 fuzzy sets. Similarly, the general

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type-2 fuzzy similarity and inclusion measures should berepresented as type-1 fuzzy sets. However, if the generaltype-2 fuzzy similarity and inclusion measures are expressedas uncertain amounts, then they may be subject to certainrestrictions in practical applications because that they arenot intuitive. One solution is that these type-1 fuzzy sets aredefuzzified, so that one can get some certain values. Anothersolution is that we directly define these type-2 fuzzy simi-larity and inclusion measures as certain value in [0, 1]. Insummary, if the type-2 fuzzy similarity and inclusion mea-sures are expressed as uncertain amount, they are more in linewith the nature of type-2 fuzzy sets. If type-2 fuzzy similar-ity and inclusion measures are expressed as certain values,they are more intuitive and play an important role in practi-cal applications. In the following, the two situations will bediscussed.

Similar to the axiomatic definitions of interval type-2fuzzy similarity and inclusion measures (Mendel and Wu2010), we give the axiomatic definitions of general type-2fuzzy similarity and inclusion measures in next part. Here wedefine type-2 fuzzy empty set ∅ = ∫

x∈X [∫u0∈[0,0] 1/u0]/x .

Obviously, S∅L(x |α) = S∅

R(x |α) = 0 holds for any x ∈ Xand α ∈ [0, 1].

Definition 2.7 A real function I : F2(X)× F2(X) → [0, 1]is called an inclusion measure of general type-2 fuzzy sets,if I satisfies the following axioms:

(I1) ∀ A ∈ F2(X), I ( A, ∅) = 0;(I2) ∀ A, B ∈ F2(X), A ⊆ B ⇔ I ( A, B) = 1;(I3) ∀ A, B, C ∈ F2(X), A ⊆ B ⊆ C ⇒ I (C, A) ≤

I (B, A), I (C, A) ≤ I (C, B).

Remark 1 If I (•, •) is a type-1 fuzzy set, then “0” in (I1) isexpressed as “1/0”, “1” in (I2) is expressed as “1/1”, and“≤” in (I3) represents inclusion relation of type-1 fuzzysets.

Definition 2.8 A real function S : F2(X)× F2(X) → [0, 1]is called a similarity measure of general type-2 fuzzy sets, ifS satisfies the following axioms:

(S1) ∀ A, B ∈ F2(X), S( A, B) = S(B, A);(S2) ∀ A ∈ F2(X), S( A, A) = 1;(S3) ∀ A, B, C ∈ F2(X), A ⊆ B ⊆ C ⇒ S( A, C) ≤

S( A, B), S( A, C) ≤ S(B, C).

Remark 2 If S(•, •) is a type-1 fuzzy set, then “1” in (S2)is expressed as “1/1”, and “≤” in (S3) represents inclusionrelation of type-1 fuzzy sets.

3 General type-2 fuzzy similarity and inclusionmeasures

3.1 Definitions of general type-2 fuzzy inclusion measures

The most popular inclusion measure for type-1 fuzzy setswas proposed by Kosko (1990), which is

IK (A, B) =∫

x∈X min(u A(x), u B(x))dx∫x∈X u A(x)dx

.

where A and B are type-1 fuzzy sets on X . The notation∫is an integral, for discrete universes of discourse X,

∫is

replaced by the summation∑

.In the following, based on the Definition 2.7 and the fore-

going analysis, we define new general type-2 fuzzy inclu-sion measures. Furthermore, the theoretical proof will alsobe given to illustrate that the proposed inclusion measuresare natural extensions of the type-1 fuzzy inclusion measureIK (•, •). The new general type-2 fuzzy inclusion measure isdefined as:

I1( A, B) = ∪α∈[0,1]

{α/[IL( A, B, a), IR( A, B, a)]}.

where

IL( A, B, a)=min

{∫x∈X min(S A

L (x |α), SBL (x |α))dx∫

x∈X S AL (x |α)dx

,

∫x∈X min(S A

R (x |α), SBR (x |α))dx∫

x∈X S AR (x |α)dx

};

IR( A, B, a)=max

{∫x∈X min(S A



,

∫x∈X min(S A



}.

Theorem 3.1 I1(•, •) is an inclusion measure on F2(X).

Proof (I1) ∀ A ∈ F2(X), α ∈ [0, 1] and x ∈ X , we have

S AL (x |α) ∧ S∅

L(x |α) = 0 and S AR (x |α) ∧ S∅

R(x |α) = 0.That is to say, for any α ∈ [0, 1], we have IL( A, B, a) =IR( A, B, a) = 0. Thus, I1( A, ∅) = 1/0.

(I2) if A ⊆ B, then S AL (x |α) ≤ SB

L (x |α) and S AR (x |α) ≤

SBR (x |α) hold for any α ∈ [0, 1] and x ∈ X . Thus,

IL( A, B, a) = 1 and IR( A, B, a) = 1 hold for any α ∈[0, 1]. We obtain I1( A, B) = 1/1. On the other hand, ifI1( A, B) = 1/1, then IL( A, B, a) = 1 and IR( A, B, a) = 1hold for any α ∈ [0, 1]. Thus, for any α ∈ [0, 1], we have

∫x∈X min(S A



=∫

x∈X min(S AR (x |α), SB

R (x |α))dx∫x∈X S A

R (x |α)dx= 1.

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Thus,∫

x∈X [S AL (x |α) − min(S A

L (x |α), SBL (x |α))]dx = ∫

x∈X

[S AR (x |α) − min(S A

R (x |α), SBR (x |α))]dx = 0.

Since S AL (x |α)−min(S A

L (x |α), SBL (x |α)) ≥ 0 and S A

R (x |α)−min(S A

R (x |α), SBR (x |α)) ≥ 0 hold for any α ∈ [0, 1]

and x ∈ X , we have S AL (x |α) = min(S A

L (x |α), SBL (x |α))

and S AR (x |α) = min(S A

R (x |α), SBR (x |α)). That is to say,

S AL (x |α) ≤ SB

L (x |α) and S AR (x |α) ≤ SB

R (x |α) hold for anyα ∈ [0, 1] and x ∈ X . Therefore, A ⊆ B.

(I3) if A ⊆ B ⊆ C , then S AL (x |α) ≤ SB

L (x |α) ≤ SCL (x |α)

and S AR (x |α) ≤ SB

R (x |α) ≤ SCR (x |α) hold for any α ∈

[0, 1] and x ∈ X . We have IL(C, A, a) ≤ IL(B, A, a)

and IR(C, A, a) ≤ IR(B, A, a) for any α ∈ [0, 1]. Thus,I1(C, A) ≤ I1(B, A). Similarly,I1(C, A) ≤ I1(C, B). Thewhole proof is completed. ��Theorem 3.2 If A and B are degenerated to the type-1 fuzzysets on X , then the general type-2 fuzzy inclusion measureI1(•, •)becomes the type-1 fuzzy inclusion measure IK (•, •).

Proof Since A and B are degenerated to the type-1 fuzzy seton X , we have

A =∫

x∈U

u A(x)/x =∫

x∈X

⎡⎢⎢⎣

∫

u0∈J Ax

1/u0

⎤⎥⎥⎦

/x, J A

x ∈ [0, 1],

B =∫

x∈U

u B(x)/x =∫

x∈X

⎡⎢⎢⎣

∫

u0∈J Bx

1/u0

⎤⎥⎥⎦

/x, J B

x ∈ [0, 1].

That is to say, the primary memberships J Ax and J B

x of x only

can take a sole value. Thus, we obtain S AL (x |α) = S A

R (x |α) =S A

L (x |1) and SBL (x |α) = SB

R (x |α) = SBL (x |1) for any x ∈ X

and α ∈ [0, 1]. If we set S AL (x |1) = u A(x) and SB

L (x |1) =u B(x), then u A(x) ∈ [0, 1] and u B(x) ∈ [0, 1] are functionsrelated to x ∈ X . Thus,

∫x∈X min(S A



=∫


,

∫x∈X min(S A



=∫


.

That is to say, for any α ∈ [0, 1], we have

IL( A, B, a) = IR( A, B, a) =∫


.

Hence, I1( A, B) =∫

x∈X min(u A(x),u B (x))dx∫x∈X u A(x)dx

. The whole proof

is completed. ��The general steps for computing the general type-2 fuzzy

inclusion measure I1( A, B) can be summarized as:Step1: decide on the numbers of α-planes. Call that num-

ber � + 1. Regardless of � + 1, α = 0 and α = 1 mustalways be used.

Step2: for each α, compute α/[IL( A, B, a), IR( A, B, a)].Step3: compute I1( A, B).Noted that the inclusion measure I1(•, •) is a type-1 fuzzy

set, so we can obtain the centroid Centroid(I1(•, •)) ofI1(•, •) by using centroid defuzzification method. Based onCentroid(I1(•, •)), we can be very intuitive to see the degreeto which a type-2 fuzzy set is included in another type-2fuzzy set. As the foregoing analysis, another program is thatwe directly define general type-2 fuzzy inclusion measure ascertain value in [0, 1]. If the α is broken into �+1 values, thegeneral type-2 fuzzy inclusion measure I2(•, •) as a certainvalue, is defined as follow:

I2( A, B) = 1

� + 1

∑α=0, 1

�, 2�

,··· , �−1�

,1

∫x∈X min(S A

L (x |α), SBL (x |α))dx + ∫



L (x |α)dx + ∫x∈X S A

R (x |α)dx.

Theorem 3.3 I2(•, •) is an inclusion measure on F2(X).

Proof (I1) ∀ A ∈ F2(X), α ∈ [0, 1] and x ∈ X , we have

S AL (x |α) ∧ S∅

L(x |α) = 0 and S AR (x |α) ∧ S∅

R(x |α) = 0. Thatis to say, for any α ∈ [0, 1], we have

∫x∈X min(S A

L (x |α), S∅L (x |α))dx + ∫

x∈X min(S AR (x |α), S∅

R(x |α))dx∫x∈X S A


R (x |α)dx= 0.

Thus, I2( A, ∅) = 0.(I2) if A ⊆ B, then S A

L (x |α) ≤ SBL (x |α) and S A

R (x |α) ≤SB

R (x |α) hold for any α ∈ [0, 1] and x ∈ X . Thus, for anyα ∈ [0, 1], we have

∫x∈X min(S A

L (x |α), SBL (x |α))dx + ∫




R (x |α)dx

=∫

x∈X S AL (x |α)dx + ∫

x∈X S AR (x |α)dx∫

x∈X S AL (x |α)dx + ∫


= 1.

We obtain I2( A, B) = 1.On the other hand, if I2( A, B) = 1, then

� + 1 =∑

α=0, 1�

, 2�

,..., �−1�

,1

∫x∈X min(S A

L (x |α), SBL (x |α))dx + ∫




R (x |α)dx.

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Since∫

x∈X min(S AL (x |α),SB

L (x |α))dx+∫x∈X min(S A

R (x |α),SBR (x |α))dx∫

x∈X S AL (x |α)dx+∫


≤1, we have

∫x∈X [S A

L (x |α) − min(S AL (x |α), SB

L (x |α))]dx

+ ∫x∈X [S A

R (x |α) − min(S AR (x |α), SB

R (x |α))]dx = 0.

Furthermore, S AL (x |α)−min(S A

L (x |α), SBL (x |α)) ≥ 0 and

S AR (x |α) − min(S A

R (x |α), SBR (x |α)) ≥ 0 hold for any α ∈

[0, 1] and x ∈ X . Thus, S AL (x |α) ≤ SB

L (x |α) and S AR (x |α) ≤

SBR (x |α) hold for any α ∈ [0, 1] and x ∈ X . That is to say,

A ⊆ B.(I3) if A ⊆ B ⊆ C , then S A

L (x |α) ≤ SBL (x |α) ≤ SC

L (x |α)


R (x |α) ≤ SCR (x |α) hold for any α ∈ [0, 1]

and x ∈ X . That is to say, for anyα ∈ [0, 1], we have

I2(C, A) = 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1

∫x∈X min(SC

L (x |α), S AL (x |α))dx + ∫

x∈X min(SCR (x |α), S A

R (x |α))dx∫x∈X SC

L (x |α)dx + ∫x∈X SC

R (x |α)dx

= 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1

∫x∈X S A


R (x |α)dx∫x∈X SC


R (x |α)dx

≤ 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1

∫x∈X S A


R (x |α)dx∫x∈X SB

L (x |α)dx + ∫x∈X SB

R (x |α)dx

= 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1

∫x∈X min(SB

L (x |α), S AL (x |α))dx + ∫

x∈X min(SBR (x |α), S A

R (x |α))dx∫x∈X SB


R (x |α)dx

= I2(B, A).

Similarly, I2(C, A) ≤ I2(C, B). The whole proof is com-pleted. ��Theorem 3.4 If A and B are degenerated to the type-1 fuzzysets on X , then the general type-2 fuzzy inclusion measureI2(•, •)becomes the type-1 fuzzy inclusion measure IK (•, •).

Proof If A and B are degenerated to the type-1 fuzzy set onX , from the discussion of Theorem 3.2, we have S A

L (x |1) =u A(x) and SB

L (x |1) = u B(x), where u A(x) ∈ [0, 1] andu B(x) ∈ [0, 1] are functions related to x ∈ X . Thus, for anyα ∈ [0, 1], we have

∫x∈X min(S A

L (x |α), SBL (x |α))dx + ∫




R (x |α)dx

=∫

x∈X min(u A(x), u B (x))dx + ∫x∈X min(u A(x), u B (x))dx∫

x∈X u A(x)dx + ∫x∈X u A(x)dx

=∫

x∈X min(u A(x), u B (x))dx∫x∈X u A(x)dx

.

That is to say, I2( A, B) =∫

x∈X min(u A(x),u B (x))dx∫x∈X u A(x)dx

. The whole

proof is completed. ��

Remark 3 Rickard et al. (2009) used Zadeh’s extension prin-ciple to define a general type-2 fuzzy inclusion measure. Itshould be noted that the measure defined there is also rep-resented as type-1 fuzzy set. Rickard’s inclusion measurehas important application values. However, it does not sat-isfy reflexivity. From Theorem 3.1, we find that the proposedmeasure I1(•, •) can satisfy reflexivity. Furthermore, it willalso be seen that the proposed inclusion measures I1(•, •)

and I2(•, •) would obtain more desirable properties thanRickard’s inclusion measure from Sect. 3.3. The inclusionmeasures of general type-2 fuzzy sets based on the verticalslice theory have also been defined in (Yang and Lin 2009;Hwang et al. 2011), but it is required that the FOU of twogeneral type-2 fuzzy sets must be same. From the definitionsof I1(•, •) and I2(•, •), one can know that the proposed mea-sures have not such limitations.

3.2 Definitions of general type-2 fuzzy similarity measures

The most popular similarity measure for type-1 fuzzy setswas proposed by Jaccard (Wu and Mendel 2009), which is

SJ (A, B) =∫

x∈X min(u A(x), u B(x))dx∫x∈X max(u A(x), u B(x))dx

.

where A and B are type-1 fuzzy sets on X . The notation∫ is an integral, for discrete universes of discourse X , ∫ isreplaced by the summation

∑.

In the following, based on the Definition 2.8 and the fore-going analysis, we define new general type-2 fuzzy simi-larity measures. Furthermore, the theoretical proof will alsobe given to illustrate that the proposed similarity measuresare natural extensions of the type-1 fuzzy similarity measureSJ (•, •). The new general type-2 fuzzy similarity measureis defined as:

S1( A, B) = ∪α∈[0,1]

{α/[SL( A, B, a), SR( A, B, a)]}

where

SL( A, B, a) = min

{ ∫x∈X min(S A


x∈X max(S AL (x |α), SB

L (x |α))dx,

∫x∈X min(S A


x∈X max(S AR (x |α), SB

R (x |α))dx

};

SR( A, B, a) = max

{ ∫x∈X min(S A



L (x |α))dx,

∫x∈X min(S A



R (x |α))dx

}.

Theorem 3.5 S1(•, •) is a similarity measure on F2(X).

Proof (S1) and (S2) are trivial.

123


(S3) if A ⊆ B ⊆ C , then S AL (x |α) ≤ SB

L (x |α) ≤ SCL (x |α)



and x ∈ X . Thus, for any α ∈ [0, 1], we have

SL( A, C, a) = min

{ ∫x∈X min(S A

L (x |α), SCL (x |α))dx∫

x∈X max(S AL (x |α), SC

L (x |α))dx,

∫x∈X min(S A

R (x |α), SCR (x |α))dx∫

x∈X max(S AR (x |α), SC

R (x |α))dx

}

= min

{∫x∈X S A

L (x |α)dx∫x∈X SC

L (x |α)dx,

∫x∈X S A


R (x |α)dx

}

≤ min

{∫x∈X S A

L (x |α)dx∫x∈X SB

L (x |α)dx,

∫x∈X S A

R (x |α)dx∫x∈X SB

R (x |α)dx

}

= min

{ ∫x∈X min(S A



L (x |α))dx,

∫x∈X min(S A



R (x |α))dx

}.

That is to say, SL( A, C, a) ≤ SL( A, B, a). Similarly, weobtain SR( A, C, a) ≤ SR( A, B, a).

Therefore, S1( A, C) ≤ S1( A, B). Similarly, S1( A, C) ≤S1(B, C). The whole proof is completed. ��Theorem 3.6 If A and B are degenerated to the type-1fuzzy sets on X , then the general type-2 fuzzy similaritymeasure S1(•, •) becomes the type-1 fuzzy similarity mea-sureSJ (•, •).




SL( A, B, a) = SR( A, B, a) =∫

x∈X min(u A(x), u B(x))dx∫x∈X max(u A(x), u B(x))dx

.

That is to say, S1( A, B) =∫

x∈X min(u A(x),u B (x))dx∫x∈X max(u A(x),u B (x))dx

. The whole

proof is completed. ��Noted that the similarity measure S1(•, •) is a type-1 fuzzy

set, so we can obtain the centroid Centroid(S1(•, •)) ofS1(•, •) by using centroid defuzzification method. If the α isbroken into �+1 values, the general type-2 fuzzy similaritymeasure S2(•, •) as a certain value, is defined as follow:

S2( A, B) = 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1

∫x∈X min(S A

L (x |α), SBL (x |α))dx + ∫


R (x |α))dx∫x∈X max(S A

L (x |α), SBL (x |α))dx + ∫


R (x |α))dx.

Theorem 3.7 S2(•, •) is a similarity measure on F2(X).

Proof (S1) and (S2) are trivial.(S3) if A ⊆ B ⊆ C , then S A

L (x |α) ≤ SBL (x |α) ≤ SC

L (x |α)



and x ∈ X . Thus, for any α ∈ [0, 1], we have

S2( A, C) = 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1

∫x∈X min(S A

L (x |α), SCL (x |α))dx + ∫

x∈X min(S AR (x |α), SC


L (x |α), SCL (x |α))dx + ∫

x∈X max(S AR (x |α), SC

R (x |α))dx

= 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1

∫x∈X S A




R (x |α)dx

≤ 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1

∫x∈X

S AL (x |α)dx + ∫


∫x∈X SB


R (x |α)dx

= 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1

∫x∈X min(S A

L (x |α), SBL (x |α))dx + ∫



L (x |α), SBL (x |α))dx + ∫


R (x |α))dx

= S2( A, B).

Similarly, S2( A, C) ≤ S2(B, C). The whole proof is com-pleted. ��

Theorem 3.8 If A and B are degenerated to the type-1fuzzy sets on X , then the general type-2 fuzzy similarity mea-sure S2(•, •) becomes the type-1 fuzzy similarity measureSJ (•, •).




S2( A, B) = 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1

∫x∈X min(S A

L (x |α), SBL (x |α))dx + ∫



L (x |α), SBL (x |α))dx + ∫


R (x |α))dx

= 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1∫x∈X min(u A(x), u B (x))dx + ∫

x∈X min(u A(x), u B (x))dx∫x∈X max(u A(x), u B (x))dx + ∫

x∈X max(u A(x), u B (x))dx

=∫

x∈X min(u A(x), u B (x))dx∫x∈X max(u A(x), u B (x))dx

.

The whole proof is completed. ��

123

T. Zhao et al.

Remark 4 Observe that if A and B become interval type-2fuzzy sets, then S2(•, •) reduces to the interval type-2 fuzzysimilarity measure defined in Wu and Mendel (2009).

Remark 5 Mitchell (2005) defined a general type-2 fuzzysimilarity measure sets by using wave-slice representa-tion theory, but Mitchell’s similarity measure does not sat-isfy reflexivity and symmetry. However, from Theorem 3.5and Theorem 3.7, it is clear that the proposed similaritymeasures S1(•, •) and S2(•, •) have improved the short-comings. Furthermore, the proposed measures have alsono limitations to the FOU of two general type-2 fuzzysets.

3.3 Properties of general type-2 fuzzy similarityand inclusion measures

In this subsection, some properties of general type-2 fuzzysimilarity and inclusion measures will be derived.

Theorem 3.9 For any A, B, C ∈ F2(X), if B ⊆ C , thenthe following statements hold:

(1) I1( A, B) ≤ I1( A, C);(2) I2( A, B) ≤ I2( A, C).

Proof (1) Since B ⊆ C , we have SBL (x |α) ≤ SC

L (x |α) and

SBR (x |α) ≤ SC

R (x |α) for any α ∈ [0, 1] and x ∈ X . Thus, forany α ∈ [0, 1], we can obtain

IL( A, B, a) = min

{∫x∈X min(S A



,

∫x∈X min(S A



}

≤ min

{∫x∈X min(S A



,

∫x∈X min(S A



}

= IL( A, C, a).

IR( A, B, a) = max

{∫x∈X min(S A



,

∫x∈X min(S A



}

≤ max

{∫x∈X min(S A



,

∫x∈X min(S A



}

= IR( A, C, a).

Therefore, I1( A, B) ≤ I1( A, C).(2) From B ⊆ C , we have

I2( A, B) = 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1

∫x∈X min(S A

L (x |α), SBL (x |α))dx + ∫




R (x |α)dx

≤ 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1

∫x∈X min(S A

L (x |α), SCL (x |α))dx + ∫

x∈X min(S AR (x |α), SC



R (x |α)dx

= I2( A, C).


Theorem 3.10 For any A, B ∈ F2(X), the following state-ments hold:

(1) S1( A ∪ B, A ∩ B) = S1( A, B);(2) S2( A ∪ B, A ∩ B) = S2( A, B).

Proof (1) For any α ∈ [0, 1], we have

SL ( A ∪ B, A ∩ B, a) = min

⎧⎨⎩

∫x∈X min(S A∪B

L (x |α), S A∩BL (x |α))dx∫

x∈X max(S A∪BL (x |α), S A∩B

L (x |α))dx,


R (x |α), S A∩BR (x |α))dx∫

x∈X max(S A∪BR (x |α), S A∩B

R (x |α))dx

⎫⎬⎭

= min

⎧⎨⎩

∫x∈X S A∩B

L (x |α)dx∫x∈X S A∪B

L (x |α)dx,

∫x∈X S A∩B

R (x |α)dx∫x∈X S A∪B

R (x |α)dx

⎫⎬⎭

= min

⎧⎨⎩

∫x∈X min(S A



L (x |α))dx,

∫x∈X min(S A



R (x |α))dx

⎫⎬⎭

= SL ( A, B, a).

Similarly, SR( A∪ B, A∩ B, a) = SR( A, B, a). Thus, S1( A∪B, A ∩ B) = S1( A, B).

(2) S2( A ∪ B, A ∩ B)

123


= 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1∫x∈X min(S A∪B

L (x |α), S A∩BL (x |α))dx + ∫

x∈X min(S A∪BR (x |α), S A∩B

R (x |α))dx∫x ∈ Xmax(S A∪B

L (x |α), S A∩BL (x |α))dx + ∫

x∈X max(S A∪BR (x |α), S A∩B

R (x |α))dx

= 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1∫x∈X S A∩B

L (x |α)dx + ∫x∈X S A∩B


L (x |α)dx + ∫x∈X S A∪B

R (x |α)dx

= 1

� + 1

∑α=0, 1

�, 2�

,..., �−1�

,1∫x∈X min(S A

L (x |α), SBL (x |α))dx + ∫



L (x |α), SBL (x |α))dx + ∫


R (x |α))dx

= S2( A, B).


Theorem 3.11 For any A, B ∈ F2(X), the following state-ments hold:

(1) I1( A ∪ B, A ∩ B) = S1( A, B);(2) I2( A ∪ B, A ∩ B) = S2( A, B);(3) S1( A, A ∩ B) = I1( A, B);(4) S2( A, A ∩ B) = I2( A, B).

Proof (1) For any α ∈ [0, 1], we have

IL( A ∪ B, A ∩ B, a)

= min

{∫x∈X min(S A∪B


x∈X S A∪BL (x |α)dx

,



x∈X S A∪BR (x |α)dx

}

= min

{∫x∈X S A∩B

L (x |α)dx∫x∈X S A∪B

L (x |α)dx,

∫x∈X S A∩B


R (x |α)dx

}

= min

{ ∫x∈X min(S A



L (x |α))dx,

∫x∈X min(S A



R (x |α))dx

}

= SL( A, B, a).

Similarly, IR( A ∪ B, A ∩ B, a) = SR( A, B, a). Thus,I1( A ∪ B, A ∩ B) = S1( A, B).

(2) I2( A ∪ B, A ∩ B)

= 1

� + 1

∑α=0, 1

�, 2

�,..., �−1

�,1


L (x |α), S A∩BL (x |α))dx +∫

x∈X min(S A∪BR (x |α), S A∩B

R (x |α))dx∫x∈X S A∪B

L (x |α)dx+∫x∈X S A∪B

R (x |α)dx

= 1

� + 1

∑α=0, 1

�, 2

�,..., �−1

�,1

∫x∈X S A∩B



L (x |α)dx + ∫x∈X S A∪B

R (x |α)dx

= 1

� + 1

∑α=0, 1

�, 2

�,..., �−1

�,1

∫x∈X min(S A

L (x |α), SBL (x |α))dx + ∫



L (x |α), SBL (x |α))dx + ∫


R (x |α))dx

= S2( A, B).

(3) For any α ∈ [0, 1], we have

SL( A, A ∩ B, a)

= min

{ ∫x∈X min(S A


x∈X max(S AL (x |α), S A∩B

L (x |α))dx,

∫x∈X min(S A


x∈X max(S AR (x |α), S A∩B

R (x |α))dx

}

= min

{∫x∈X S A∩B

L (x |α)dx∫x∈X S A

L (x |α)dx,

∫x∈X S A∩B

R (x |α)dx∫x∈X S A

R (x |α)dx

}

123

T. Zhao et al.

= min

{∫x∈X min(S A



,

∫x∈X min(S A



}

= IL( A, B, a).

Similarly, we have SR( A, A ∩ B, a) = IR( A, B, a). Thus,S1( A, A ∩ B) = I1( A, B).

(4) S2( A, A ∩ B)

= 1

� + 1

∑α=0, 1

�, 2

�,..., �−1

�,1

∫x∈X min(S A

L (x |α), S A∩BL (x |α))dx+∫

x∈X min(S AR (x |α), S A∩B


L (x |α), S A∩BL (x |α))dx+∫

x∈X max(S AR (x |α), S A∩B

R (x |α))dx

= 1

� + 1

∑α=0, 1

�, 2

�,..., �−1

�,1

∫x∈X S A∩B


R (x |α)dx∫x∈X S A


R (x |α)dx

= 1

� + 1

∑α=0, 1

�, 2

�,..., �−1

�,1

∫x∈X min(S A

L (x |α), SBL (x |α))dx + ∫




R (x |α)dx

= I2( A, B).

The whole proof is completed. ��Remark 6 From the Theorems 3.1 to 3.11, it can be seenthat the proposed measures improve the shortcomings of theexisting measures since that the proposed measures satisfymore desirable properties. Furthermore, the proposed mea-sures are natural extensions of the most popular type-1 fuzzymeasures. As a result, the proposed measures keep the char-acteristics of the most popular type-1 fuzzy measures. Unlikesome existing measures, the similarity measure S1(•, •) andinclusion measure I1(•, •) are expressed as type-1 fuzzy sets,and therefore the definitions are consistent with the highlyuncertain nature of general type-2 fuzzy sets.

4 Examples

In this section, we by several examples examine the perfor-mances of the proposed general type-2 fuzzy similarity andinclusion measures. By doing so, we hope to solve the fol-lowing issues:

(1) If I1(•, •) and S1(•, •) are type-1 fuzzy sets, do theshapes of these type-1 fuzzy sets have inevitable link withthe shapes of secondary membership functions of the generaltype-2 fuzzy sets?

Remark 7 As is well known, the secondary membershipfunctions of general type-2 fuzzy sets are type-1 fuzzy sets.

Furthermore, it is noted that the proposed measures I1(•, •)

and S1(•, •) are also expressed as type-1 fuzzy sets. Hence,one needs to investigate whether the shapes of I1(•, •) andS1(•, •) have inevitable link with the shapes of secondarymembership functions of the general type-2 fuzzy sets. Thesemay be guidance for the further applications of I1(•, •) andS1(•, •), for example, perceptual computing and clustering.

(2) Do Centroid(I1(•, •)) and Centroid(S1(•, •)) con-verge to real values as � increases?

Remark 8 Since I1(•, •) and S1(•, •) are expressed as type-1 fuzzy sets, they may be subject to certain restrictionsin practical applications since that they are not intuitive.Hence, we can obtain the centroid Centroid(I1(•, •)) ofI1(•, •) by using centroid defuzzification method. Basedon Centroid(I1(•, •)), one can be very intuitive to see thedegree to which a general type-2 fuzzy set is included inanother general type-2 fuzzy set. However, it should be notedthat Centroid(I1(•, •)) is relative to the numbers of α-planes, i.e., �. In real applications, we should decide on howmany α-planes will be used. Thus, it is necessary to judgethe convergence of Centroid(I1(•, •)).

(3) Do I2(•, •) and S2(•, •) converge to real values as �

increases?

Remark 9 Since I2(•, •) and S2(•, •) are also relative to thenumbers of α-planes, i.e., �. Of course, it is desirable toconsider the convergences of I2(•, •) and S2(•, •).

(4) The performance comparisons about I1(•, •) andI2(•, •), S1(•, •) and S2(•, •).

Remark 10 By comparisons, we hope to see the differencesabout I1(•, •) and I2(•, •), and S1(•, •) and S2(•, •), so thatwe can get some guidance in real applications.

Because Gaussian membership function, triangular mem-bership function and trapezoid membership function are thenormal situation in practical applications of type-2 fuzzysystems, various cases should be tested based on differentchoices made for these membership functions. In this sec-tion, we present nine cases.

Case 1: Gaussian function with randomly generatedtriangular vertical slice.

Let A, B be two general type-2 fuzzy sets on X = [0, 10].Their upper membership functions and lower membershipfunctions are Gaussian functions:

u A(x) = e− (x−6)28 ; u A(x) = e− (x−6)2

2 ; u B(x) = e− (x−4)218 ;

u B(x) = e− (x−4)22 .

where FOU of A are shown in Fig. 1a, and FOU of B areshown in Fig. 1b. For any x , the secondary membership

123


0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

x

u

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

x

u

(a) (b)

Fig. 1 a The FOU of A for case 1, case 2 and case 3. b The FOU of B for case 1, case 2 and case 3

function u A(x) is triangular membership function, whoseapex is determined by the following formula:

Apex = u A(x) + rand ∗ (u A(x) − u A(x)).

where rand ∈ [0, 1] is randomly generated value. For any x ,the secondary membership function u B(x) is also triangularmembership function, whose apex computation formula issame with the A.

The membership functions of I1(•, •) and S1(•, •) for� =10,000 is depicted in Fig. 4a. Centroid(I1(•, •)) andCentroid(S1(•, •)) for � ranging from 1 to 5,000 are shownin Fig. 5a. I2(•, •) and S2(•, •) for � ranging from 1 to 5,000are shown in Fig. 6a.

Case 2: Gaussian function with randomly generatedtrapezoid vertical slice.

Let A, B be two general type-2 fuzzy sets on X = [0, 10].Their FOU are same with Case 1, shown in Fig. 1a, b. Forany x , the secondary membership function u A(x) is trapezoidmembership function, whose top left and right endpoint aredetermined by the following formulas:

Lapex = u A(x) + 0.8 ∗ rand ∗ (u A(x) − u A(x));Rapex = u A(x) + rand ∗ (u A(x) − u A(x)).

where rand ∈ [0, 1] is randomly generated value. For anyx , the secondary membership function u B(x) is also trape-zoid membership function, whose top endpoint computationformulas are same with the A.

The membership functions of I1(•, •) and S1(•, •) for� =10,000 is depicted in Fig. 4b. Centroid(I1(•, •)) andCentroid(S1(•, •)) for � ranging from 1 to 5,000 are shownin Fig. 5b. I2(•, •) and S2(•, •) for � ranging from 1 to 5,000are shown in Fig. 6b.

Case 3: Gaussian function with randomly generatedGaussian vertical slice.

Let A, B be two general type-2 fuzzy sets on X = [0, 10].Their FOU are same with Case 1, shown in Fig. 1a, b. Forany x , the secondary membership function u A(x) is Gaussianmembership function, whose mean is determined by μ =

(u A(x) + u A(x))/2 and standard deviation σ is determinedby the following formula:

σ = 10 ∗ rand.

where rand ∈ [0, 1] is randomly generated value. For any x ,the secondary membership function u B(x) is also Gaussianmembership function, whose mean and standard deviationcomputation formulas are same with the A.

The membership functions of I1(•, •) and S1(•, •) for� =10,000 is depicted in Fig. 4c. Centroid(I1(•, •)) andCentroid(S1(•, •)) for � ranging from 1 to 5,000 are shownin Fig. 5c. I2(•, •) and S2(•, •) for � ranging from 1 to 5,000are shown in Fig. 6c.

Case 4: Triangular function with randomly generatedtriangular vertical slice.

Let A, B be two general type-2 fuzzy sets on X = [0, 10].Their upper membership functions and lower membershipfunctions are triangular functions: u A(x) = trimf(x, [−2, 2,

20]); u A(x) = 0.8 ∗ trimf(x[−1, 3, 15]); u B(x) = tr im f (x,

[−20, 5, 50]); u B(x) = 0.8 ∗ tr im f (x, [−10, 8, 35]). Wheretrimf denotes triangular function, the first parameter andthird parameter of [] denote bottom left and right endpoint,respectively, and the second parameter of [] denote apex.Where FOU of A are shown in Fig. 2a, and FOU of B areshown in Fig. 2b. For any x , the secondary membership func-tions u A(x) and u B(x) are triangular membership functions,whose apex computation formulas are same with Case 1.

The membership functions of I1(•, •) and S1(•, •) for� =10,000 is depicted in Fig. 4d. Centroid(I1(•, •)) andCentroid(S1(•, •)) for � ranging from 1 to 5,000 are shownin Fig. 5d. I2(•, •) and S2(•, •) for � ranging from 1 to 5,000are shown in Fig. 6d.

Case 5: Triangular function with randomly generatedtrapezoid vertical slice.

Let A, B be two general type-2 fuzzy sets on X = [0, 10].Their FOU are same with Case 4, shown in Fig. 2a, b. For anyx , the secondary membership functions u A(x) and u B(x) are

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trapezoid membership functions, whose top endpoint com-putation formulas are same with Case 2.

The membership functions of I1(•, •) and S1(•, •) for� =10,000 is depicted in Fig. 4e. Centroid(I1(•, •)) andCentroid(S1(•, •)) for � ranging from 1 to 5,000 are shownin Fig. 5e. I2(•, •) and S2(•, •) for � ranging from 1 to 5,000are shown in Fig. 6e.

Case 6: Triangular function with randomly generatedGaussian vertical slice.

Let A, B be two general type-2 fuzzy sets on X = [0, 10].Their FOU are same with Case 4, shown in Fig. 2a, b. For anyx , the secondary membership functions u A(x) and u B(x) areGaussian membership functions, whose mean and standarddeviation computation formulas are same with Case 3.

The membership functions of I1(•, •) and S1(•, •) for� =10,000 is depicted in Fig. 4f. Centroid(I1(•, •)) andCentroid(S1(•, •)) for � ranging from 1 to 5,000 are shownin Fig. 5f. I2(•, •) and S2(•, •) for � ranging from 1 to 5,000are shown in Fig. 6f.

Case 7: Trapezoid function with randomly generatedtriangular vertical slice.

Let A, B be two general type-2 fuzzy sets on X = [0, 10].Their upper membership functions and lower member-ship functions are trapezoid functions: u A(x) = trapmf(x,

[−8, 2, 5, 20]);

u A(x) = 0.7 ∗ trapmf(x, [−5, 3, 8, 15]); u B(x) =trapmf(x, [−18, 3, 6, 25]); u B(x) = 0.6 ∗ trapmf(x, [−5,

6, 7, 19]). Where trapmf denotes trapezoid function, the firstparameter and four parameter of [] denote bottom left andright endpoint, respectively, and the second parameter andthird parameter of [] denote top left and right endpoint,respectively. Where FOU of A are shown in Fig. 3a, and FOUof B are shown in Fig. 3b. For any x , the secondary member-ship functions u A(x) and u B(x) are triangular membershipfunctions, whose apex computation formulas are same withCase 1.

The membership functions of I1(•, •) and S1(•, •) for� =10,000 is depicted in Fig. 4g. Centroid(I1(•, •)) andCentroid(S1(•, •)) for � ranging from 1 to 5,000 are shownin Fig. 5g. I2(•, •) and S2(•, •) for � ranging from 1 to 5,000are shown in Fig. 6g.

Case 8: Trapezoid function with randomly generatedtrapezoid vertical slice.

Let A, B be two general type-2 fuzzy sets on X = [0, 10].Their FOU are same with Case 7, shown in Fig. 3a, b. For anyx , the secondary membership functions u A(x) and u B(x) aretrapezoid membership functions, whose top endpoint com-putation formulas are same with Case 2.

The membership functions of I1(•, •) and S1(•, •) for� =10,000 is depicted in Fig. 4h. Centroid(I1(•, •)) and

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Fig. 4 a I1(•, •) and S1(•, •) for Case 1. b I1(•, •) and S1(•, •) forCase 2. c I1(•, •) and S1(•, •) for Case 3. d I1(•, •) and S1(•, •) forCase 4. (e) I1(•, •) and S1(•, •) for Case 5. f I1(•, •) and S1(•, •) forCase 6. g I1(•, •) and S1(•, •) for Case 7. h I1(•, •) and S1(•, •) forCase 8. i I1(•, •) and S1(•, •) for Case 9

Centroid(S1(•, •)) for � ranging from 1 to 5,000 are shownin Fig. 5h. I2(•, •) and S2(•, •) for � ranging from 1 to 5,000are shown in Fig. 6h.

Case 9: Trapezoid function with randomly generatedGaussian vertical slice.

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Fig. 5 a Centroid(I1(•, •)) and Centroid(S1(•, •)) for Case 1.b Centroid(I1(•, •)) and Centroid(S1(•, •)) for Case 2.c Centroid(I1(•, •)) and Centroid(S1(•, •)) for Case 3.d Centroid(I1(•, •)) and Centroid(S1(•, •)) for Case 4.e Centroid(I1(•, •)) and Centroid(S1(•, •)) for Case 5.f Centroid(I1(•, •)) and Centroid(S1(•, •)) for Case 6.g Centroid(I1(•, •)) and Centroid(S1(•, •)) for Case 7.h Centroid(I1(•, •)) and Centroid(S1(•, •)) for Case 8.i Centroid(I1(•, •)) and Centroid(S1(•, •)) for Case 9

Let A, B be two general type-2 fuzzy sets on X = [0, 10].Their FOU are same with Case 7, shown in Fig. 3a, b. For anyx , the secondary membership functions u A(x) and u B(x) are

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Fig. 6 a I2(•, •) and S2(•, •) for Case 1. b I2(•, •) and S2(•, •) forCase 2. c I2(•, •) and S2(•, •) for Case 3. d I2(•, •) and S2(•, •) forCase 4. e I2(•, •) and S2(•, •) for Case 5. f I2(•, •) and S2(•, •) forCase 6. g I2(•, •) and S2(•, •) for Case 7. h I2(•, •) and S2(•, •) forCase 8. i I2(•, •) and S2(•, •) for Case 9

Gaussian membership functions, whose mean and standarddeviation computation formulas are same with Case 3.

The membership functions of I1(•, •) and S1(•, •) for� =10,000 is depicted in Fig. 4i. Centroid(I1(•, •)) andCentroid(S1(•, •)) for � ranging from 1 to 5,000 are shownin Fig. 5i. I2(•, •) and S2(•, •) for � ranging from 1 to 5,000are shown in Fig. 6i.

Observe from Fig. 4a–i that, regardless of the natures ofthe secondary membership functions of A and B, the shapesof membership functions of S1(•, •) and I1(•, •) are alwayssimilar to the trapezoid when � =10,000. But noted thatwe only describe the membership functions of S1(•, •) andI1(•, •) when � =10,000. This is because that � is greater,S1(•, •) and I1(•, •) are more close to the true values. Infact, this can be derived from the definitions of I1(•, •) andS1(•, •). It is noted that we only discuss the cases that thesecondary membership functions of general type-2 fuzzy setsare Gaussian membership function, triangular membershipfunction or trapezoid membership function. Since Gaussianmembership function, triangular membership function andtrapezoid membership function are normal and convex, wehave [IL( A, B, 1), IR( A, B, 1)] �= ∅ from the definition ofI1(•, •). That is to say, the 1-cut set of I1(•, •) is a closedinterval. Furthermore, from the convexity of secondary mem-bership functions, we can see that I1(•, •) is convex. Hence,the shape of I1(•, •) is the trapezoid. It is similar to the analy-sis of S1(•, •).

From Fig. 5a–i, we find that Centroid(I1(•, •)) andCentroid(S1(•, •)) are always volatile. Even if � isincreased to 5,000, most of the situations for Centroid(I1(•, •)) and Centroid(S1(•, •)) are still not stabilized.Especially, for the cases three, seven, eight and nine,we almost do not see a trend of convergence. In fact,Centroid(I1(•, •)) and Centroid(S1(•, •)) are obtainedby using centroid defuzzification method to I1(•, •) andS1(•, •). It is noted that the shapes of I1(•, •) and S1(•, •)

are easily influenced by the value of �. Thus, Centroid(I1(•, •)) and Centroid(S1(•, •)) are always volatile as thevalue of �.

From Fig. 6a–i, we can obtain that I2(•, •) and S2(•, •)

as � increases, will soon stabilize and converge to a certainvalue. The above conclusions show that I2(•, •) and S2(•, •)

have more practical values. However, I1(•, •) and S1(•, •)

are coincident with the nature of and the semantic interpre-tation of general type-2 fuzzy sets, and therefore may havemore further research topics.

5 Conclusions

In this paper, the new general type-2 fuzzy similarity andinclusion measures have been defined by using α-plane the-ory. The proposed measures improve the shortcomings ofthe existing measures. Furthermore, the proposed measuresare natural extensions of the most popular type-1 fuzzy mea-sures. As a result, the proposed measures keep the character-istics of the most popular type-1 fuzzy measures. Since theproposed measures are relative to the numbers of α-planes,we should decide on how many α-planes will be used in realapplications. However, it is difficult to decide on the numbers

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of α-planes in theory. Of course, the larger the numbers of α-planes are, the more stable proposed measures are, whereasif the numbers of α-planes are too large, the computing timewould increase. Hence, it is of high interest to find efficientalgorithm to handle the problem, and further research willfocus on this issue.

Acknowledgments The authors would like to thank reviewers fortheir valuable comments and suggestions. This work was supported bythe National Natural Science Foundation of China(51177137,61134001)and the Fundamental Research Funds for the Central Universi-ties(SWJTU11CX034).

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A new approach to similarity and inclusion measures between general type-2 fuzzy sets