Best proximation of fuzzy real numbers

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1 214 (214) 1-6 Available online at Volume 214, Year 214 Article ID jfsva-23, 6 Pages doi:1.5899/214/jfsva-23 Research Article Best proximation of fuzzy real numbers Z. Rohani 1, H. Mazaheri 1, S. J. Jesmani 2 (1) Faculty of Mathematics, Yazd University, Yazd, Iran (2) Education of Mathematics, Farss, Niriz, Iran Copyright 214 c Z. Rohani, H. Mazaheri and S. J. Jesmani. This is an open access article 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 Fuzzy appear as the most popular family of fuzzy sets useful both for theoretical considerations as well as diverse practical applications. In this note we show that the set of fuzzy numbers is a closed and convex set of L p, and therefore is a Chebyshev set. Keywords: Approximation of fuzzy numbers, Fuzzy numbers, Piecewise linear approximation, Proximinal, Best approximation, 1-knot. 1 Introduction The theory of fuzzy sets was introduced by L. Zadeh [18]. Then, many mathematicians have studied fuzzy normed spaces from several angles. However, complicated membership functions have many drawbacks in processing imprecise information modeled by fuzzy numbers including problems with calculations, computer implementation, etc. Moreover, handling too complex membership functions entails difficulties in interpretation of the results too. This is the reason that a suitable approximation of fuzzy numbers is so important. (See [1]-[17]) 2 Preliminaries and notations Fuzzy numbers are the most commonly used fuzzy subset of the real line. The membership function of a fuzzy numbers A is given: x a 1, l A (x) a 1 x a 2, µ A (x) = 1 a 2 x a 3, (2.1) r A (x) a 3 x a 4, x a 4, where a 1,a 2,a 3,a 4 R,l A [a 1,a 2 ] [,1] is a nondecreasing upper semicontinuous function such that Corresponding author. address: z.rohani85@yahoo.com

2 Page 2 of 6 and is a nonincreasing upper semicontinuous function l A (a 1 ) = o,l A (a 2 ) = 1, l A 1, r A : [a 3,a 4 ] [o,1] r A (a 3 ) = 1,r A (a 4 ) =. l A and r A are called the left and right sides of A, respectively. For any α (,1], the α cut of a fuzzy numbers A is a crisp set defined as The support or -cut, A, of a fuzzy numbers is defined as A α = {x R : A(x) α}. A = {x R : A(x) }. It is easily seen that for each α [,1] every α cut of a fuzzy numbers is a closed interval where A α = [A L (α),a U (α)], and A L (α) = inf{x R : A(x) α} A U (α) = sup{x R : A(x) α}. If the sides of the fuzzy numbers A are strictly monotone, then A L and A U are real inverse function of l A and r A on [,1], respectively. Two fuzzy numbers A and B are equal if A L (α) = B L (α) and A U (α) = B U (α) almost everywhere, α [,1]. The set of all fuzzy numbers will be denoted by F(R). In a family of fuzzy numbers we may define addition and scalar multiplication. Let A,B F(R), α [,1] and λ R. Then the sum of two fuzzy numbers A and B is a fuzzy numbers A + B with the α cut (A + B) α = A α + B α = [A L (α) + B L (α),a U (α) + B U (α)], while the scalar multiplication λ.a is defined by λa L (α),λa U (α),λ, (λ.a) α = λa α = λa U (α),λa L (α),λ. Let (X,d) be a metric space and G a closed subset of X. For x X, let d(x,g) = inf y G d(x,y). A point y G such that d(x,y ) = d(x,g) is called a point of best approximation to x. We shall denote by P G (x) the set of best approximation elements of x in G, i.e., P G (x) = {y G : d(x,y) = d(x,g)} The set G is said to be proximinal if each point of X has a best approximation in G and it is said to be Chebyshev if each point of X has a unique best approximation in G. Example 2.1. Let (B, ) be a Banach space. Then B is reflexive if and only if every nonempty closed convex subset K B is a proximinal set.

3 Page 3 of 6 In order for a closed convex subset to be a Chebyshev set, we need stronger conditions on the Banach space. Let (X,d) be a metric space. For x X, r, we let B(x,r) = {y X : d(x,y) r},b[x,r] = {y X : d(x,y) r}. Definition 2.1. A convex metric space (X,d) is called strictly convex if, for x,y B[z,r] with d(x,y) = λ, then B[x,(1 t)λ] B[y,tλ] B[z,r] for all t (,1), and all z X, r. Lemma 2.1. Let (B, ) be a reflexive Banach space. Then B is strictly convex if and only if every nonempty closed convex subset K B is a Chebyshev set. Lemma 2.2. Every closed convex subset of a uniformly convex Banach space is a Chebyshev set. 3 Main section The problem of the nearest approximation of fuzzy numbers by piecewise linear 1-knot fuzzy numbers is discussed. By using 1-knot fuzzy numbers one may obtain approximations which are simple enough and flexible to reconstruct the input fuzzy concepts under study. They might be also perceived as a generalization of the trapezoidal approximations. Moreover, these approximations possess some desirable properties. Apart from theoretical considerations approximation algorithms that can be applied in practice are also given. In practical problems like solving fuzzy equations, data analysis or ranking fuzzy numbers, an adequate metric over the space of fuzzy numbers should be considered. The flexibility of the space of fuzzy numbers allows for the construction of many types of metric structures over this space. In the area of fuzzy numbers approximation the most suitable metric is an extension of (L P ) distance defined by 1 d p (A,B) = ( max((a L (α) B L (α)),(a U (α) B U (α))) p dα) 1 p. (3.2) Lemma 3.1. (L P, P ) is a uniformly convex space. for 1 p. Proof. It is trivial. Although family F(R) is quite rich and consists of fuzzy numbers with diverse membership functions, fuzzy numbers with simpler membership functions are often preferred in practice. The most commonly used subclass of F(R) is formed by so-called trapezoidal fuzzy numbers, i.e. fuzzy numbers with linear sides. Thus a membership function of a trapezoidal fuzzy numbers is given by µ T (x) = x t 1, x t 1 t 2 t 1 t 1 x t 2, 1 t 2 x t 3, t 4 x t 4 t 3 t 3 x t 4, x t 4, where t 1 t 2 t 3 t 4. Since the membership function of a trapezoidal fuzzy numbers T is completely defined by these four real numbers we denote it usually as T = T (t 1,t 2,t 3,t 4 ). It is easy to prove that T L (α) = t 1 + (t 2 t 1 )α, T U (α) = t 4 (t 4 t 3 )α. The set of all trapezoidal fuzzy numbers is denoted by F T (R).

4 Page 4 of 6 Definition 3.1. For any fixed α (,1) an α -piecewise linear 1-knot fuzzy numbers S is a fuzzy numbers with the following membership function x s 1, x s α 1 s 2 s 1 s 1 x s 2, α + (1 α ) x s 2 µ s (x) = where s = (s 1,...,s 6 ) such that s 1... s 6. s 3 s 2 s 2 x s 3, 1 s 3 x s 4, α + (1 α ) s 5 x s 5 s 4 s 4 x s 5, α s 6 x s 6 s 5 s 5 x s 6, x s 6, Alternatively, α -piecewise linear 1-knot fuzzy numbers may be defined using its α-cut representation, i.e. and { s1 + (s 2 s 1 ) α α S L (α) = α [,α ), s 2 + (s 3 s 2 ) α α 1 α α [α,1]. { s5 + (s 6 s 5 ) α α α S U (α) = α [,α ), s 4 + (s 5 s 4 ) 1 α 1 α α [α,1]. (3.3) (3.4) Let us denote the set of all such fuzzy numbers by F π(α ) (R). By setting F π(α ) (R) = F π(1) (R) := F T (R) we also include the cases α {,1}. Please note that the inclusion F T (R) F π(α ) (R) holds for any α [,1]. Indeed, if T = T (t 1,t 2,t 3,t 4 ) is a trapezoidal fuzzy numbers and α (,1), then we have T = S(α,s) where s = (s 1,...,s 6 ) and s 1 = t 1,s 2 = t 1 + (t 2 t 1 )α,s 3 = t 2,s 4 = t 3,s 5 = t 4 (t 4 t 3 )α,s 6 = t 4. Moreover, to simplify notation, let F π[a,b] (R) denote the set of all α- piecewise linear 1-knot fuzzy numbers, where α [a,b] for some a b 1, i.e. F π[a,b] (R) := α [a,b] F π(α) (R). Lemma 3.2. The set F π[,1] (R) is a closed subset of the space L P [,1] L P [,1]. Lemma 3.3. If α [,1] is arbitrarily chosen then the set F π(α ) (R) is a closed convex subset of the space L P [,1] L P [,1]. We omit the simple proof of the convexity property (one may easily obtain that F π(α ) (R) is a closed set of L P [,1] L P [,1]. Lemma 3.4. F π(α ) (R) is Chebyshev in L P [,1] L P [,1]. Proof. From Lemma 3.4, F π(α ) (R) is a closed convex subset of L P [,1] L P [,1]. From Lemma 2.1, F π(α ) (R) is Chebyshev. Therefore, for any element from L P [,1] L P [,1] there exists a unique best approximation. Lemma 3.5. If A is an arbitrary fuzzy number and α [;1]. Then there exists a unique fuzzy number S α (A) F π(α ) (R) satisfying Proof. This is trivial. A S α (A)) p = min A S p. S F π(α )

5 Page 5 of 6 4 Example Example 4.1. Consider A F(R) defined as A L (α) = { α <.5,.5 α.5, We put B L = A L A L (), B U =. Then B is a best approximation for A. A U (α) = 1 (4.5) 5 Conclusion Using 1-knot fuzzy numbers, one may obtain approximations completely characterized by six points on the real line. They are simple enough and flexible to preserve the main properties of large class of fuzzy quantities. We have shown that the approximation operator producing piecewise linear 1-knot fuzzy numbers closest to the original fuzzy numbers possess some desirable properties. Thus in all situations when a trapezoidal approximation is not sufficient we recommend the approximation by 1-knot piecewise linear fuzzy numbers. Although we have discussed both a case with a fixed knot and the problem of the optimal choice of the knot of the piecewise linear fuzzy numbers, some problems are, of course, still open. First of all, sometimes a piecewise linear approximation of fuzzy numbers with some additional constraints would be more adequate. Moreover, one may be interested in approximation of fuzzy numbers by piecewise linear fuzzy numbers having more than one knot. There are also some interesting problems which can be related to the 1-knot piecewise linear approximation. For example, approximation of a fuzzy numbers. Acknowledgements This research project is supported by intelligent robust research center of yazd university. The authors are highly grateful to the referees for their valuable comments and suggestions for improving the paper. References [1] T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math, 11 (3) (23) [2] S. Abbasbandy, M. Amirfakhrian, The nearest approximation of a fuzzy quantity in parametric form, Applied Mathematics and Computation, 172 (26) [3] S. Abbasbandy, B. Asady, The nearest trapezoidal fuzzy numbers to a fuzzy quantity, Applied Mathematics and Computation, 156 (24) [4] S. Abbasbandy, T. Hajjari, Weighted trapezoidal approximation preserving cores of a fuzzy numbers, Computers and Mathematics with Applications, 59 (21) [5] A. I. Ban, Approximation of fuzzy numberss by trapezoidal fuzzy numberss preserving the expected interval, Fuzzy Sets and Systems, 159 (28) [6] A. I. Ban, On the nearest parametric approximation of a fuzzy numbersrevisited, Fuzzy Sets and Systems, 16 (29)

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