Approximations by LR-type fuzzy numbers

doi:10.1016/j.fss.2013.09.004

Fuzzy Sets and Systems

Volume 257, 16 December 2014, Pages 23-40

https://doi.org/10.1016/j.fss.2013.09.004 Get rights and content

Abstract

Recently, many scholars investigated interval, triangular, and trapezoidal approximations of fuzzy numbers. These publications can be grouped into two classes: Euclidean distance class and non-Euclidean distance class. Most approximations in Euclidean distance class can be calculated by formulas, but calculating approximations in the other class is more complicated. Furthermore, approximations in Euclidean distance class can be divided into two subclasses. One is to study approximations of fuzzy numbers without constraints, the other one is to study approximations preserving some attributes. In this paper, we use LR-type fuzzy numbers to approximate fuzzy numbers. The proposed approximations will generalize all recent approximations without constraints in Euclidean class. Also, an efficient formula is provided.

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 $u + v \in Ω$ and $r u \in Ω$ for all $u, v \in Ω$ and all $r \in R$ ,
(2)
Ω is convex iff $r u + (1 - r) v \in Ω$ for all $u, v \in Ω$ and all $r \in [0, 1]$ ,
(3)
Ω is Chebyshev iff for each $u \in H$ there exists a unique element $P_{Ω} (u) \in Ω$ such that $d (u, P_{Ω} (u)) ⩽ d (u, x), \forall x \in Ω,$ and then $P_{Ω} (u)$ 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 $\tilde{A}$ is a subset of the real line $R$ with membership function $μ_{\tilde{A}} : R \to [0, 1]$ such that:

(1)
$\tilde{A}$ is normal, i.e. there is an $x_{0} \in R$ with $μ_{\tilde{A}} (x_{0}) = 1$ ,
(2)
$\tilde{A}$ is fuzzy convex, i.e. $μ_{\tilde{A}} (λ x + (1 - λ) y) ⩾ \min {μ_{\tilde{A}} (x), μ_{\tilde{A}} (y)}$ for all $x, y \in R$ , $λ \in [0, 1]$ ,
(3)
$μ_{\tilde{A}}$ is upper semicontinuous, i.e. $μ_{\tilde{A}}^{- 1} ([α, 1])$ is closed for all $α \in [0, 1]$ ,
(4)
the support of $μ_{\tilde{A}}$ is bounded, i.e. the closure of ${x \in R | μ_{\tilde{A}} (x) > 0}$ is bounded,

see [16]. Let

L, R : [0, 1] \to [0, 1]

be two fixed functions which are both upper semicontinuous and decreasing such that

L (0) =

Weighted LR-approximations

Let $\tilde{A} \in F (R)$ . An LR-type fuzzy number $T_{L, R} (\tilde{A}) \in T_{L, R}$ is called weighted LR-approximation of $\tilde{A}$ with respect to $d_{λ} (\cdot, \cdot)$ if it satisfies $d (\tilde{A}, T_{L, R} (\tilde{A})) ⩽ d (\tilde{A}, \tilde{X}), \forall \tilde{X} \in T_{L, R} .$

Proposition 4.1

$T_{L, R}$ 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 $T_{L, R} (\tilde{A})$ exists and is unique. By taking the following shape functions the corresponding weighted LR-approximations can be applied:

(1)
By taking $L (x) = R (x) = 1 - x$ , $T_{L, R}$ is the weighted

Weighted symmetric f-approximations

A fuzzy number $\tilde{A} = [A_{L} (t), A_{U} (t)]$ is called symmetric if there exists a constant $c \in R$ such that $\frac{1}{2} (A_{L} (t) + A_{U} (t)) = c, \forall t \in [0, 1] .$ Suppose that the LR-type fuzzy number $\tilde{A} = [a - σ L^{- 1} (t), b + β R^{- 1} (t)]$ is symmetric. It implies that $β R^{- 1} (t) - σ L^{- 1} (t)$ is constant and denoted by c. Hence its t-cuts are of the form $\tilde{A} = [a - σ L^{- 1} (t), b + c + σ L^{- 1} (t)] .$ So, in this section we always assume that $L = R = f and λ_{L} = λ_{U} = λ .$ A fuzzy number of the form $\tilde{A} = [a - σ f^{- 1} (t), b^{'} + σ f^{- 1} (t)]$ is called symmetric f-type. Let $T_{f}^{s}$ 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 $\tilde{A} \in F (R)$ . From definition of weighted LR-approximation (LR-unimodal approximation, symmetric f-approximation, symmetric f-unimodal approximation) of $\tilde{A}$ we see that it is the nearest LR-type fuzzy number to $\tilde{A}$ . Hence, if $\tilde{A}$ 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 $L_{2}^{λ_{L}} [0, 1] \times L_{2}^{λ_{U}} [0, 1]$ and applying linear algebra and approximation theory. Note that, in our proof we do not use the last criterion of fuzzy numbers: $\tilde{A}$ has bounded support. So, if $\tilde{A} \in L_{2}^{λ_{L}} [0, 1] \times L_{2}^{λ_{U}} [0, 1]$ , we can still find its weighted LR-trapezoidal approximation by applying our proposed theorem. Furthermore, let $L, R : [0, \infty) \to [0, 1]$ be $L_{2} [0, + \infty)$ -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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Approximations by LR-type fuzzy numbers

Abstract

Introduction

Section snippets

Preliminaries: best approximations in Hilbert spaces

LR-type fuzzy numbers

Weighted LR-approximations

Weighted symmetric f-approximations

Properties

Conclusions

Acknowledgements

Appl. Math. Comput.

Int. J. Approx. Reason.

Appl. Math. Comput.

Comput. Math. Appl.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Comput. Math. Appl.

Int. J. Approx. Reason.

Inf. Sci.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.