TRAPEZOIDAL OPERATOR PRESERVING THE EXPECTED INTERVAL AND THE SUPPORT OF FUZZY NUMBERS

. The problem to ﬁnd the trapezoidal fuzzy number which preserves the expected interval and the support of a given fuzzy number is discussed. Properties of this new trapezoidal approximation operator are studied.


INTRODUCTION
In [12] the trapezoidal approximation of a fuzzy number is treated as a reasonable compromise between two opposite tendencies: to lose too much information and to introduce too sophisticated form of approximation from the point of view of computation. Many approximation methods of fuzzy numbers with trapezoidal fuzzy numbers were proposed in last years (see [2], [3], [4], [12], [13], [14], [18], [20]). In each case the authors attached to a fuzzy number a trapezoidal fuzzy number by preserving some parameters and/or minimizing the distance between them.
In this paper we propose a trapezoidal approximation operator which preserves the expected interval and the support of a given fuzzy number (in Section 3). We conclude that the approximation is computationally inexpensive and it is not possible for any fuzzy number. Following the list of criteria in [12], in Section 4 we examine important properties of this new trapezoidal approximation operator: translation invariance, linearity, identity, expected value invariance, order invariance with respect different preference relations, uncertainty invariance, correlation invariance, monotonicity and continuity.

PRELIMINARIES
A fuzzy number A is a fuzzy subset of the real line R with the membership function µ A which is (see [9]) normal, fuzzy convex, upper semicontinuous, supp A is bounded, where the support of A, denoted by supp A, is the closure of the set {x ∈ X : µ A (x) > 0} .
The α-cut, α ∈ (0, 1] of a fuzzy number A is a crisp set defined as: and cl is the closure operator. We denote by F (R) the set of fuzzy numbers.
is defined by (see [10], [15]) and the expected value by (see [15]) The core of a fuzzy number A is introduced by (see [1]): A trapezoidal fuzzy number T is characterized by four real numbers t 1 t 2 t 3 t 4 . It is denoted by T = (t 1 , t 2 , t 3 , t 4 ) and has the parametric 1] and the expected interval We denote by F T (R) the set of trapezoidal fuzzy numbers. Let A, B ∈ F (R) , The quantity ) 2 dα gives a distance between A and B (see, e.g., [11]). We consider the sum A + B and the scalar multiplication λ · A by (see [8]) respectively, for every α ∈ [0, 1]. In the case of the trapezoidal fuzzy numbers T = (t 1 , t 2 , t 3 , t 4 ) and S = (s 1 , s 2 , s 3 , s 4 ) we obtain Another kind of fuzzy number (see [5]) is defined by where n > 0, and denoted by A = (a, b, c, d) n . We have

MAIN RESULT AND EXAMPLES
For a fuzzy number A, A α = [A L (α) , A U (α)] , α ∈ [0, 1] , the problem is to find the trapezoidal fuzzy number, T (A) = (t 1 (A) , t 2 (A) , t 3 (A) , t 4 (A)) = (t 1 , t 2 , t 3 , t 4 ) , which preserves the expected interval and the support of A, that is Let us denote the trapezoidal fuzzy number which preserves the expected interval and the support of the fuzzy number A, is given by

Proof. Conditions (3.1) and (3.2) imply that
under restriction as T (A) is a trapezoidal fuzzy number, that is We obtain and The condition Example 2. Let us consider a fuzzy number A given by Because A ∈ F ES (R), the trapezoidal fuzzy number which preserves the expected interval and the support of A is then it doesn't exist a trapezoidal fuzzy number which preserves the expected interval and the support of the fuzzy number A, as the following example proves.
Example 3. Let us consider a fuzzy number A given by we get A / ∈ F ES (R) and not exists a trapezoidal fuzzy number which preserves the expected interval and the support of A.
is the trapezoidal fuzzy number which preserves the expected interval and the support of A.
Proof. For a fuzzy number A = (a, b, c, d) n we have according to Theorem 1 the conclusion is immediate.

PROPERTIES
In [12] Grzegorzewski and Mrowka proposed a number of criteria which the trapezoidal approximations should or just possess: α-cut invariance, translation invariance, identity, nearest criterion, expected value invariance, expected interval invariance, continuity, compatibility with the extension principle, order invariance.
Theorem 6. The trapezoidal operator preserving the expected interval and the support T : F ES (R) −→ F T (R) given in Theorem 1 has the following properties: (i) is invariant to translations, that is for every A, B ∈ F ES (R) and λ ∈ R * ; (iii) fulfills the identity criterion, that is F T (R) ⊂ F ES (R) and for every A ∈ F T (R) ; (iv) is order invariant with respect to the preference relation defined by (see [19]) (v) is order invariant with respect to the preference relation M defined by (see [17]) for every A, B ∈ F ES (R); (vi) is uncertainty invariant with respect to the nonspecificity measure defined by (see [6]) for every A, B ∈ F ES (R), where ρ (A, B) denotes the correlation coefficient between A and B, defined as (see [16])

Proof. (i) Let
According to Theorem 1 we obtain (ii) In the case λ > 0 and A ∈ F ES (R) we have Then ((2.5) is used here), In the case λ < 0 and A ∈ F ES (R), Then ((2.5) is used here), If A, B ∈ F ES (R) then Applying Theorem 1 and (2.4) we get, According to Theorem 1 the trapezoidal fuzzy number which preserves the expected interval and the support of A is Two important parameters, ambiguity and value, were introduced to capture the relevant information, to simplify the task of representing and handling fuzzy numbers. The ambiguity of a fuzzy number A (see [7]), denoted by Amb (A), is defined by and the value of a fuzzy number A (see [7]), denoted by V al (A), is defined by For a trapezoidal fuzzy number we have The trapezoidal operator T in Theorem 1 does not preserve the value and the ambiguity, that is there exists A ∈ F ES (R) such that V al(A) = V al(T (A)) and Amb(A) = Amb(T (A)), as the following example proves.
Unfortunately, the operator in Theorem 1 is not continuous with respect to metric D (see (2.1)) and not monotonic, as the following examples prove.

CONCLUSION
In this paper we have introduced the trapezoidal fuzzy number preserving the expected interval and the support of a fuzzy number. We have proved that this trapezoidal approximation fulfills properties like: translation invariance, linearity and identity, it does not preserve the value and the ambiguity of the fuzzy number and the criteria of continuity and monotony are not satisfied. The expected value invariance, order invariance, correlation invariance and uncertainty invariance are true because their definitions are based on the expected value.