Operations on Triangle Type-2 Fuzzy Sets

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Procedia

Engineering

Procedia Engineering 00 (2011) 000–000 www.elsevier.com/locate/procedia

Advanced in Control Engineering and Information Science

Operations on Triangle Type-2 Fuzzy Sets

Xue Ling, Yunjie Zhang a*

Department of Mathematics,Dalian Maritime University,Dalian 116026,China

Abstract

Triangle type-2 fuzzy sets, whose complexity is between general and interval type-2 fuzzy sets, can achieve balance between complexity-reduced and information-protected. However, operations on triangle type-2 fuzzy sets that exist in the literatures have some drawbacks in theory. In this paper, we reconstruct the framework of set-theoretic operations on triangle type-2 fuzzy sets by presenting polygon type-2 fuzzy sets, and give manageable and simplified formulas for operations on triangle type-2 fuzzy sets.

Keywords:Triangle type-2 fuzzy sets; polygon type-2 fuzzy sets; Set-theroetic operations

1. Introduction

Since Zadeh published his new classic paper in 1975 [1], type-2 fuzzy set (T2 FS) theory has gained more and more attention from researchers in a wide range of scientific areas, especially in the past few years [2, 3]. The operations on T2 FSs are convincing. However, it still be facing the computational bottleneck as the number of primary memberships in a T2 FS is very large. In order to solve this problem in a tractable yet approximate manner, Liang and Mendel introduce the interval type-2 fuzzy sets (IT2 FSs) [4]. Though the IT2 FS can remarkably reduce the complexity and have the simple and manageable operations [4], it loses much information to describe fuzzy phenomena at the same time. In order to achieve balance between complexity-reduced and information-protected as much as possible, Lv et al have brought forward the theory of triangle type-2 fuzzy sets (TT2 FSs) [5]. However, as a special case of

* Corresponding author. Tel.: 18741124765. E-mail address: nmgdylx@163.com.

Open access under CC BY-NC-ND license.

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general T2 FSs, the set-theoretic operations on TT2 FSs are inconsistent with those on T2 FSs. Consequently, the set-theoretic operations on TT2 FSs, which suggested by Lv, have some drawbacks.

In this paper we try to reconstruct the framework of set-theoretic operations on TT2 FSs. Newly operational formulas for union of TT2 FSs are derived by extension principle.

2. Operations on general and convex type-2 fuzzy sets

Set-theoretic operations of T2 FSs, which are derived by extension principle, are in reality operations for the secondary membership functions of each element in the universe. Consider two T2 FSs Ã and B �, in the universe X ] 1 , 0 [ , / ) ( ~₌

_[

_]

_⊆

∫ ∫

∈ ∈ u x X x u Ju_x fx u u x J A , ~=

_{∫ ∫}

[

( )/

]

, ⊆[0,1 ] ∈ ∈ w x X x u J_xwgx u u x J B

Using extension principle, we have

X x w u w g u f F x u x J u wJ_xw x x x J x B A∪~( )=

∫

∈ ( ) =

∫ ∫

∈ ∈ [ ( )∧ ( )]( ∨ ,) ∀ ∈ ~ _θ θ θ θ μ (1)

For the convex T2 FS, Tahayori and Antoni provided a simplified method of operations for union [6, 7]. Let Ã, B� be convex T2 FSs defined over universe X, and for a certain element x∈X

} ) ( ) ( | { arg , )) ( ) ( ( Supfx gx M H Jx fx gx M x J ∧ = = ∈ ∧ = ∈ θ θ θ θ θ θ } ) ( | { arg , )) ( ( Sup }, ) ( | { arg , )) ( ( Sup x B B x x B x J A x x A A x x J f =h V = ∈J f =h ∈ g =h V = ∈J g =h ∈ θ θ θ θ θ θ θ θ Then, we have

Theorem 1: H is between VA and VB, i.e., either VB ≤ H ≤ VA or VA ≤ H ≤ VB.

Theorem 2: Suppose VA ≤ H ≤ VB, for ∀θ ∈Jx

(1) when VA ≤ θ ≤ H, gx(θ ) ∧ fx(θ ) = gx(θ ); (2) when H ≤ θ ≤ VB, gx(θ ) ∨ fx(θ ) = gx(θ ).

Theorem 3: Under max t-norm and min t-norm, the secondary membership function for union Ã∪B �at element x is:

⎩ ⎨ ⎧ > ∧ ∧ ∨ ≤ ∧ = H h h g f H g f F B A x x x x x _θ _θ _θ θ θ θ θ ), ( )) ( ) ( ( ), ( ) ( ) ( (2)

3. Triangle type-2 fuzzy sets

In [5], Lv et al proposed the concept of triangle type-2 fuzzy sets. Suppose Ã is a T2 FS in the universe

X, and for arbitrary x∈X, secondary membership function at x is a linear function defined as follow:

⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ≤ ≤ − − < ≤ − − ≤ < < ≤ = 3 2 3 2 3 2 1 1 2 1 3 1 , ) ( , ) ( 1 0 , 0 ) ( or a u a a a a u h a u a a a a u h u a a u u f A A x (3)

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fx(a2)∈[0, 1]. Ã and its secondary membership function fx can be expressed as point parameter forms as:

Ã = {(x, (a1, 0), (a2, hA), (a3, 0)) | x∈X}, fx = (x, (a1, 0), (a2, hA), (a3, 0))

Set-theoretic operations of TT2 FSs have been defined in the literature [5]. We know that the operations of general T2 FSs are obtained by extension principle. The TT2 FS is a particular case of the general T2 FS, so the operational results of method proposed by Lv should be in conformity with those using extension principle. But fact is not true. The counterexample on union is shown in Fig.1.

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 A B union 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A B union (a) (b) (c)

Fig. 1.(a) TT2 FS Ã and B�: Ã = {(x, (0,0), (0.18,0.4), (0.5, 0)) | x∈X}, B� = {(x, (0.06, 0), (0.24, 1), (0.38, 0)) | x∈X}; (b) Operation result using extension principle; (c) Operation result using method proposed by Lv.

4. New framework of set-theoretic operations of triangle type-2 fuzzy sets

In this section, the framework of set-theoretic operations on TT2 FSs is re-established. First, we define the polygon type-2 fuzzy set that will play a fundamental role in what follows. Then we explore the secondary membership functions of union in detail, and provide reasonable and manageable operational formulas.

Definition 1: Let Ã be a T2 FS. For arbitrary x∈X, if secondary membership function of Ã at x is a linear function defined as follow:

⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ≤ < − − ≤ < − − + − ≤ < − − + − ≤ ≤ − − = − − − − − − − − − − − − otherwise , 0 , ) ( , ) ( , ) ( , ) ( ) ( 1 1 1 -1 2 2 1 2 1 1 2 2 1 3 2 2 3 2 3 3 2 2 3 2 1 1 2 1 2 n n n n n n n n n n n n n n n n x a u a a a a u h a u a a a a h a h u h h a u a a a a h a h u h h a u a a a a u h u f M M (4)

Then, Ã is called a polygon type-2 fuzzy set or an n-gon type-2 fuzzy set (n-gon T2 FS), where 0 ≤ a1 ≤ a2

≤ … ≤ an ≤ 1, n ≥ 3, hi = fx(ai)∈[0, 1], i = 2, …, n−1. Just as TT2 FSs are, the value of n is different to

different x, the point parameter forms of n-gon T2 FS Ã and its secondary membership function fx are:

Ã = {(x, (a1, 0), (a2, h2),…,(an−1, hn−1), (an, 0))| x∈X}, fx = (x, (a1, 0), (a2, h2),…,(an−1, hn−1),(an,0))

TT2 FSs are convex, so we can discuss operations using results proposed by Tahayori for simple. Theorem 4: Let Ã and B� be two TT2 FSs in the universe X, the union Ã∪B �is an n-gon T2 FS in which 3 ≤ n ≤ 6.

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respectively

fx(u) = (x, (a1, 0), (a2, hA), (a3, 0)), gx(w) = (x, (b1, 0), (b2, hB), (b3, 0))

Since the set-theoretic operations on T2 FSs are essentially the operations of secondary membership functions of each element in the universe, the following discussion will mainly focus on the secondary membership Fx of union. It can be seen from analysis that the shape of Fx varies with position relations

between fx and gx, and the position relations amount to 16. Without loss of generality, we just talk about

cases (1): a1 < b1, a2 < b2, a3 < b3, hA < hB. In this case, there are still two subcases: a3 > a1 and a3 ≤ a1.

We just briefly discuss the subcase shown in Fig. 3(a) and the other is similar and omitted.

hA hB 0 a1 a2b1H Ja3 b2 K b31 hA hB 0 a1 a2b1H Ja3 b2 K b31 (a) (b)

Fig. 2. (a) Schematic illustration of Case; (b) The outline of Fx.

From Theorem 1, we have a2 ≤ H ≤ b2. Let J = arg{w∈[b1, b2] | gx(w) = hA}, K = arg{w∈[b2, b3] | gx(w) = hA}, then B A h b b h b J ( 2 1) 1 − + = , B A h b b h b K ( 2 3) 3 − + = , ⎩ ⎨ ⎧ ∉ < ∈ ≥ ] , [ , ] , [ , ) (w _hh _ww _JJ _KK g A A x

Hence there are six possible solutions:

1) When 0 ≤ θ < b1, since θ ≤ H and gx(θ ) = 0, from equation (2) of Theorem 3 and Theorem 2(1), we

have Fx(θ ) = fx(θ )∧gx(θ ) = 0.

2) When b1 ≤ θ ≤ H, since θ ≤ H, from equation (2) of Theorem 3 and Theorem 2(1), we have Fx(θ ) =

fx(θ )∧gx(θ ) = gx(θ ) = hB(θ −b1)/(b2−b1).

3) When H < θ < J, since θ > H and gx(θ ) < hA, from equation (2) of Theorem 3 and Theorem 2(2), we

have Fx(θ ) = (fx(θ )∨gx(θ )))∧(hA∧hB) = gx(θ )∧hA = gx(θ ) = hB(θ −b1)/(b2−b1).

4) When J ≤ θ ≤ K, since θ > H and gx(θ ) ≥ hA, from equation (2) of Theorem 3 and Theorem 2(2), we

have Fx(θ ) = (fx(θ )∨gx(θ )))∧(hA∧hB) = gx(θ )∧hA = hA.

5) When K < θ ≤ b3, since θ > H and gx(θ ) < hA, from equation (2) of Theorem 3 and Theorem 2(2), we

have Fx(θ ) = (fx(θ )∨gx(θ )))∧(hA∧hB) = gx(θ )∧hA = gx(θ ) = hB(θ −b3)/(b2−b3).

6) When b3 < θ ≤ 1, since θ > H and fx(θ ) = gx(θ ) = 0, from equation (2) of Theorem 3 and Theorem

2(2), we have Fx(θ ) = (fx(θ )∨gx(θ )))∧(hA∧hB) = 0∧hA = 0.

Therefore, in this case, the second membership function Fx of union Ã∪B �at x is expressed as

⎪ ⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎨ ⎧ ≤ < − − ≤ ≤ < ≤ − − = otherwise , 0 ) ( , , ) ( ) ( 3 3 2 3 1 1 2 1 b K b b b h J K h J b b b b h F B A B x θ θ θ θ θ θ ⎪ ⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎨ ⎧ ≤ < − − ≤ ≤ < ≤ − − = otherwise , 0 ) ( , , ) ( 3 3 3 1 1 1 b K b K b h J K h J b b J b h A A A θ θ θ θ θ

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certain element x is in the shape of an 4-gon over the interval [0, 1].

In conclusion, the union Ã∪B�is an n-gon type-2 fuzzy set (3 ≤ n ≤ 6), whose point parameter expression is

C� = Ã∪B� = {(x, (c1, 0), (c2, h2), (c3, h3), (c4, h4), (c5, h5), (c6, 0)) | x∈X}

where 0 ≤ c1 ≤ … ≤ c6 ≤ 1, hi = fx(ci)∈[0, 1], 0 ≤ hi ≤ 1, i = 2, …, 5, and c1 = max{a1, b1}，c6 = max{a6, b6}, h3 = h4 = min{hA, hB}

} ) ( , ) ( max{ 3 2 1 1 1 2 3 1 3 B A h b b h b h a a h a c = + − + − , ₄ max{₃ 3( 2 3), ₃ 3( 2 3)} B A h b b h b h a a h a c = + − + − ) ( ) ( ) ( ) ( 1 2 1 2 1 2 1 1 2 1 2 h_ha _bb _bb _hh b_aa _aa c B A B A − − − − − − = , ) ( ) ( ) ( 1 2 1 2 1 1 2 _h _b h_bh a_h b_a _a h B A B A − − − − = ) ( ) ( ) ( ) ( 3 2 3 2 3 2 3 3 2 3 5 h_ha _bb _bb _hhb_aa _aa c B A B A − − − − − − = , ) ( ) ( ) ( 3 2 3 2 3 3 5 _h _b h_bh a_h b_a _a h B A B A − − − − =

It must be noticed that, for i = 2, 5, the point (ci, hi) is designate as inexistence when ci∉[cj−1, cj+1].

5. Conclusion

According to the discussion of convex T2 FS operations proposed by Tahayori, in this paper, we give the definition of polygon type-2 fuzzy set and reconstruct the framework of set-theoretic operations on TT2 FSs. Newly manageable and simplified formulas for union on TT2FSs are derived, and the discussion for intersection are similar to union, so it is unnecessary to talk about. Results suggested may provide a basis for type-reduction and establishment of triangle type-2 fuzzy logic systems.

References

[1] L.A. Zadeh, The Concept of a Linguistic Variable and Its Application to Approximate Reasoning, Information Sciences, 1975, 8(3): 199–249.

[2] J.M. Mendel, Advances in Type-2 Fuzzy Sets and Systems, Information Sciences, 2001, 177(1): 84–110. [3] O. Castillo, P. Melin, Type-2 Fuzzy Logic −− Theory and Applications, Heidelberg: Springer-Verlag Berlin, 2008.

[4] Q. Liang, J.M. Mendel, Interval Type-2 Fuzzy Logic Systems: Theory and Design, IEEE Transactions on Fuzzy Systems, 2000, 8(5): 535–550.

[5] Z. Lv, H. Jin, P. Yuan, The Theory of Triangle Type-2 Fuzzy Sets, Proceedings of the 2009 IEEE International Conference on Computer and Information Technology, Piscataway: IEEE Service Center, 2009: 57–62.

[6] H. Tahayori, G. D. Antoni, A Simple Method for Performing Type-2 Fuzzy Set Operations Based on Highest Degree of Intersection Hyperplane, Proceedings of the 2007 Annual Meeting of the North American Fuzzy Information Processing Society, Piscataway: IEEE Service Center, 2007: 404–409.

[7] H. Tahayori, A. Sadeghian, G. D. Antoni, Operations on Type-2 Fuzzy Sets Based on the Set of Pseudo-Highest Intersection Points of Convex Fuzzy Sets, Proceedings of the 2010 Annual Meeting of the North American Fuzzy Information Processing Society, Piscataway: IEEE Service Center, 2010: 1–6.