\(^{\ast }\)Babeş-Bolyai University, Faculty of Mathematics and Computer Science, Kogălniceanu no. 1, 400084 Cluj-Napoca, Romania, e-mail: adrianabrandas@yahoo.com

1 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.

2 Preliminaries

A fuzzy number \(A\) is a fuzzy subset of the real line \(\mathbb {R} \) with the membership function \(\mu _{A}\) 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 \(\left\{ x\in X:\mu _{A}(x){\gt}0\right\} .\)

The \(\alpha \)-cut\(,\) \(\alpha \in (0,1]\) of a fuzzy number \(A\) is a crisp set defined as:

Every \(\alpha \)-cut, \(\alpha \in (0,1]\) of a fuzzy number \(A\) is a closed interval \(A_{\alpha }=\left[ A_{L}\left( \alpha \right) ,A_{U}\left( \alpha \right) \right] ,\) where

We denote

where the support of \(A\) is defined by supp\((A)=cl\left\{ x\in \mathbb {R}:\mu _{A}(x){\gt}0\right\} \) and \(cl\) is the closure operator. We denote by \(F\left( \mathbb {R}\right) \) the set of fuzzy numbers.

The expected interval \(EI(A)\) of \(A\in F\left( \mathbb {R}\right) ,A_{\alpha }=\left[ A_{L}\left( \alpha \right) ,A_{U}\left( \alpha \right) \right] \) 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 \(t_{1}\leqslant t_{2}\leqslant t_{3}\leqslant t_{4}.\) It is denoted by \(T=\left( t_{1},t_{2},t_{3},t_{4}\right) \) and has the parametric representation \(\left[ T_{L}\left( \alpha \right) ,T_{U}\left( \alpha \right) \right] ,\) where

and the expected interval

We denote by \(F^{T}\left( \mathbb {R}\right) \) the set of trapezoidal fuzzy numbers.

Let \(A,B\in F\left( \mathbb {R}\right) ,\)

The quantity

gives a distance between \(A\) and \(B\) (see, e.g., [ 11 ] ). We consider the sum \(A+B\) and the scalar multiplication \(\lambda \cdot A\) by (see [ 8 ] )

and

respectively, for every \(\alpha \in \lbrack 0,1]\). In the case of the trapezoidal fuzzy numbers \(T=\left( t_{1},t_{2},t_{3},t_{4}\right) \) and \(S=\left( s_{1},s_{2},s_{3},s_{4}\right) \) we obtain

respectively

Another kind of fuzzy number (see [ 5 ] ) is defined by

where \(n{\gt}0,\) and denoted by\(\ A=\left( a,b,c,d\right) _{n}\). We have

3 Main result and examples

For a fuzzy number \(A,\) \(A_{\alpha }=\left[ A_{L}\left( \alpha \right) ,A_{U}\left( \alpha \right) \right] ,\alpha \in \left[ 0,1\right] ,\) the problem is to find the trapezoidal fuzzy number, \(T(A)=\left( t_{1}\left( A\right) ,t_{2}\left( A\right) ,t_{3}\left( A\right) ,t_{4}\left( A\right) \right)=\left( t_{1},t_{2},t_{3},t_{4}\right) ,\) which preserves the expected interval and the support of \(A,\) that is

and

Let us denote

and

under restriction as \(T\left( A\right) \) is a trapezoidal fuzzy number, that is

We obtain

and

Because \(A_{L}\left( 0\right) \leq A_{L}\left( \alpha \right) \) and \(A_{U}\left( 0\right) \geq A_{U}\left( \alpha \right) \), for every \(\alpha \in \left[ 0,1\right] \), we have

The condition

becomes

so \(T\left( A\right) \) is a trapezoidal fuzzy number if \(A\in F\left( \mathbb {R}\right) \) satisfies the condition

that is

If \(A\notin F_{ES}(\mathbb {R)}\) 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.

therefore

Because \(A\in F_{ES}(\mathbb {R)}\) if and only if

according to Theorem 1 the conclusion is immediate.

4 Properties

In [ 12 ] Grzegorzewski and Mrowka proposed a number of criteria which the trapezoidal approximations should or just possess: \(\alpha \)-cut invariance, translation invariance, identity, nearest criterion, expected value invariance, expected interval invariance, continuity, compatibility with the extension principle, order invariance.

and

for every \(\alpha \in \left[ 0,1\right] ,\) that is

Because \(A\in F_{ES}(\mathbb {R)}\) we have

and

so

According to Theorem 1 we obtain

and

therefore

(ii) In the case \(\lambda {\gt}0\) and \(A\in F_{ES}(\mathbb {R)}\) we have

and

so

Then ((2.5) is used here),

In the case \(\lambda {\lt}0\) and \(A\in F_{ES}(\mathbb {R)},\)

and

therefore

so

Then ((2.5) is used here),

If \(A,\) \(B\in F_{ES}(\mathbb {R)}\) then

and

imply that

therefore ((??) is used here)

so

Applying Theorem 1 and (2.4) we get,

(iii) For \(A\in F^{T}\left( \mathbb {R} \right) ,A=\left( t_{1},t_{2},t_{3},t_{4}\right) \), we have

and \(A\in F_{ES}(\mathbb {R)}\) because

is equivalent to

that is

According to Theorem 1 the trapezoidal fuzzy number which preserves the expected interval and the support of \(A\) is

where

and

therefore

(iv), (v), (vi), (vii) For \(A,B\in F_{ES}(\mathbb {R)}\ \)we have

and

therefore

and

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 \(Amb\left( A\right) \), is defined by

and the value of a fuzzy number \(A\) (see [ 7 ] ), denoted by \(Val\left( A\right) \), is defined by

For a trapezoidal fuzzy number

we have

and

The trapezoidal operator \(T\) in Theorem 1 does not preserve the value and the ambiguity, that is there exists \(A\in F_{ES}(\mathbb {R)}\) such that

and

as the following example proves.

The continuity and monotony are between the criteria which a trapezoidal approximation operator should or just can possess ( [ 12 ] ). The continuity constraint means if two fuzzy numbers are close (in some sense) then their approximations should also be close. The criterion of monotony is verified for an trapezoidal approximation operator \(T\) if for any \(A,B\in F\left( \mathbb {R}\right) ,A\subseteq B\) implies \(T\left( A\right) \subseteq T\left( B\right) \). Here, for two fuzzy numbers \(A\) and \(B,A\subseteq B\) if and only if \(\mu _{A}\left( x\right) \leq \mu _{B}\left( x\right) \), for every \(x\in \mathbb {R}\). In fact, \(A\subseteq B\) if and only if \(A_{L}\left( \alpha \right) \geq B_{L}\left( \alpha \right) \) and \(A_{U}\left( \alpha \right) \leq B_{U}\left( \alpha \right) \), for every \(\alpha \in \left[ 0,1\right] \) and it is immediate that \(\left( t_{1},t_{2},t_{3},t_{4}\right) \subseteq \left( t_{1}^{\prime },t_{2}^{\prime },t_{3}^{\prime },t_{4}^{\prime }\right) \) if and only if \(t_{1}\geq t_{1}^{\prime },t_{2}\geq t_{2}^{\prime },t_{3}\leq t_{3}^{\prime }\) and \(t_{4}\leq t_{4}^{\prime }\). 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.

5 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.