Approximations by LR-type fuzzy numbers
Introduction
In practice, fuzzy intervals are often used to represent uncertain or incomplete information. An interesting problem is to approximate general fuzzy numbers by specific fuzzy numbers, so as to simplify calculations. These approximations can be grouped into two classes: the Euclidean distance class and the non-Euclidean distance class. The Euclidean distance class includes the interval approximation (proposed by Grzegorzewski [18]), symmetric triangular approximation (proposed by Ma et al. [26]), trapezoidal approximation (proposed by Abbasbandy and Asady [1] and improved by Yeh [31]), weighted triangular approximation (proposed by Zeng and Li in 2007 [36] and improved by Yeh [31], [34]), weighted trapezoidal approximation [34], semi-trapezoidal approximation (proposed by Nasibov and Peker [27] and improved by Ban [6], [7]), weighted semi-trapezoidal approximation (proposed by Yeh [35]), and π functional approximation (proposed by Hassine et al. [24]). The non-Euclidean distance class includes the interval approximation under the Hamming distance (proposed by Chanas [12]), symmetric and non-symmetrical trapezoidal approximations under the Euclidean distance between the respective 1/2-levels (proposed by Delgado et al. [13]), trapezoidal approximation under the source distance (proposed by Abbasbandy and Amirfakhrian [2]), and polynomial approximation under the source distance (proposed by Abbasbandy and Amirfakhrian [3]). In addition, during the last years approximation of a fuzzy number preserving some attributes had been studied. For example, trapezoidal approximation preserving the expected interval was proposed by Grzegorzewski and Mrówka [19], [20], [21], [23] and improved by Ban [5] and Yeh [30], [32] independently, trapezoidal approximation preserving cores of a fuzzy number was proposed by Grzegorzewski and Stefanini [22] and further studied by Abbasbandy and Hajjari [4], trapezoidal approximation preserving the value and ambiguity was proposed by Ban et al. [8], trapezoidal approximation preserving ambiguity was proposed by Ban and Coroianu [10], symmetric trapezoidal approximation preserving the x-centroid was proposed by Wang and Li [29], and triangular approximation preserving the x-centroid was proposed by Li et al. [25]. Interval approximations, triangular approximations, and trapezoidal approximations are belong to linear-shape approximations. Besides, semi-trapezoidal approximations (or parametric approximations), π functional approximation [24], and polynomial approximations [3] are belong to nonlinear-shape approximations. The famous LR-type fuzzy numbers were first introduced by Dubois and Prade. In this paper, we use LR-type fuzzy numbers to approximate any fuzzy number without preserving any attribute with respect to weighted Euclidean distance, which will generalize all recent approximations without constraints in Euclidean class. In Section 2 several preliminaries about best approximations in Hilbert spaces are presented. In Section 3 basic definition and notation about LR-type fuzzy numbers are introduced. In Sections 4 and 5, we compute the proposed approximations: weighted LR-approximations and weighted symmetric f-approximations. Their corresponding formulas are presented, too. In Section 6 we study properties of the two approximations.
Section snippets
Preliminaries: best approximations in Hilbert spaces
A complete inner product space is often called a Hilbert space. Let Ω be a subset of a Hilbert space H, then we say that
- (1)
Ω is a subspace iff and for all and all ,
- (2)
Ω is convex iff for all and all ,
- (3)
Ω is Chebyshev iff for each there exists a unique element such that and then is called the best approximation of u in Ω.
Fact 2.1 (See [14, pp. 23–24].) Every closed convex subset (closed subspace, finite dimensional subspace) is
LR-type fuzzy numbers
A fuzzy number is a subset of the real line with membership function such that:
- (1)
is normal, i.e. there is an with ,
- (2)
is fuzzy convex, i.e. for all , ,
- (3)
is upper semicontinuous, i.e. is closed for all ,
- (4)
the support of is bounded, i.e. the closure of is bounded,
Weighted LR-approximations
Let . An LR-type fuzzy number is called weighted LR-approximation of with respect to if it satisfies
Proposition 4.1 is closed and convex.
The proof of Proposition 4.1 is similar to [35, Proposition 4.1].
Now, by applying Fact 2.1, Proposition 4.1 implies that exists and is unique. By taking the following shape functions the corresponding weighted LR-approximations can be applied:
- (1)
By taking , is the weighted
Weighted symmetric f-approximations
A fuzzy number is called symmetric if there exists a constant such that Suppose that the LR-type fuzzy number is symmetric. It implies that is constant and denoted by c. Hence its t-cuts are of the form So, in this section we always assume that A fuzzy number of the form is called symmetric f-type. Let denote the set of all symmetric f
Properties
Grzegorzewski and Mrówka [19] proposed many properties of trapezoidal approximations. Here we propose properties of weighted LR-approximations and weighted symmetric f-approximations.
Let . From definition of weighted LR-approximation (LR-unimodal approximation, symmetric f-approximation, symmetric f-unimodal approximation) of we see that it is the nearest LR-type fuzzy number to . Hence, if is of LR-type (LR-unimodal, symmetric f-type, symmetric f-unimodal) then its weighted
Conclusions
In the present paper, we determinate weighted LR-approximations via embedding the set of all fuzzy numbers into the Hilbert space and applying linear algebra and approximation theory. Note that, in our proof we do not use the last criterion of fuzzy numbers: has bounded support. So, if , we can still find its weighted LR-trapezoidal approximation by applying our proposed theorem. Furthermore, let be -integrable, upper
Acknowledgements
The author is very grateful to the anonymous referees for their detailed comments and valuable suggestions. This research has been supported by the National Science Council of Taiwan (NSC 101-2115-M-024-002).
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