Trapezoidal approximations of fuzzy numbers—revisited

Fuzzy Sets and Systems 158 (2007) 757 – 768 www.elsevier.com/locate/fss

Trapezoidal approximations of fuzzy numbers—revisited Przemysław Grzegorzewski∗ , Edyta Mrówka Systems Research Institute, Polish Academy of Sciences, 1 ul. Newelska 6, 01-447 Warsaw, Poland Received 6 October 2006; received in revised form 15 November 2006; accepted 16 November 2006 Available online 14 December 2006

Abstract Fuzzy number approximation by trapezoidal fuzzy numbers which preserve expected interval is discussed. The previously proposed approximation operator is improved so as to always produce a well formed trapezoidal fuzzy number. © 2006 Elsevier B.V. All rights reserved. Keywords: Fuzzy numbers; Approximation of fuzzy numbers; Expected interval

1. Introduction In [11] we have discussed broadly the problem of the trapezoidal approximation of fuzzy numbers. We have formulated a list of desirable criteria which approximation operators should possess. We have also suggested a new approach to trapezoidal approximation of fuzzy numbers. The proposed operator, called the nearest trapezoidal approximation operator preserving expected interval, possesses many desired properties: it is simple and natural, it is continuous and invariant to translations and scale transformations. Moreover, it preserves ordering, correlation, uncertainty amount and the canonical representation of fuzzy number expressed by value-ambiguity indices suggested by Delgado et al. [4]. Unfortunately, we have not noticed that in some situations our operator may fail to lead a trapezoidal fuzzy number. Allahviranloo and Adabitabar Firozja [2] showed two examples illustrating such situations. Thus in the present paper we propose a corrected version of our trapezoidal approximation operator which fills the gap. It appears that this corrected operator reveals some new interesting properties. 2. Fuzzy numbers Let us consider a fuzzy number A, i.e. such fuzzy subset A of the real line R with membership function A : R → [0, 1] which is (see [5]): normal (i.e. there exist an element x0 such that A (x0 ) = 1), fuzzy convex (i.e. A (x1 + (1 − )x2 )A (x1 ) ∧ A (x2 ) ∀x1 , x2 ∈ R, ∀ ∈ [0, 1]), A is upper semicontinuous, supp A is bounded, where supp A = cl({x ∈ R : A (x) > 0}), and cl is the closure operator. A space of all fuzzy numbers will be denoted by F(R). Moreover, let A = {x ∈ R : A (x)} denote an -cut of a fuzzy number A. As it is known, every -cut of a fuzzy number is a closed interval, i.e. A = [AL (), AU ()], where AL () = inf{x ∈ R : A (x) } and AU () = sup{x ∈ R : A (x)}. ∗ Corresponding author. Tel.: +48 22 3735 78; fax: +48 22 3727 72.

E-mail addresses: [email protected] (P. Grzegorzewski), [email protected] (E. Mrówka). 0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2006.11.015

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P. Grzegorzewski, E. Mrówka / Fuzzy Sets and Systems 158 (2007) 757 – 768

Let us also recall that an expected interval EI(A) of a fuzzy number A is given by (see [6,12])  1   1 EI(A) = AL () d, AU () d . 0

(1)

0

In [11] we have discussed the problem of the trapezoidal approximation of fuzzy numbers. Roughly speaking we have shown how to substitute arbitrary fuzzy number by so called, trapezoidal fuzzy number, i.e. by a fuzzy number with linear sides and the membership function having a following form: ⎧ 0 if x < t1 , ⎪ ⎪ ⎪ ⎪ ⎪ x − t1 ⎪ ⎪ ⎪ if t1 x < t2 , ⎪ ⎪ ⎪ ⎨ t2 − t1 if t2 x t3 , A (x) = 1 (2) ⎪ ⎪ ⎪ t − x ⎪ 4 ⎪ ⎪ if t3 < x t4 , ⎪ ⎪ t4 − t3 ⎪ ⎪ ⎪ ⎩ 0 if t4 < x. Since the trapezoidal fuzzy number is completely characterized by four real numbers t1 t2 t3 t4 it is often denoted in brief as A(t1 , t2 , t3 , t4 ). A family of all trapezoidal fuzzy number will be denoted by FT (R) (of course, FT (R) ⊂ F(R)). By (1) and (2) the expected interval of the trapezoidal fuzzy number is given by   t1 + t2 t3 + t4 EI(B) = , . (3) 2 2 For two arbitrary fuzzy numbers A and B with -cuts [AL (), AU ()] and [BL (), BU ()], respectively, the quantity  1  1 2 d(A, B) = (AL () − BL ()) d + (AU () − BU ())2 d (4) 0

0

is the distance between A and B (for more details we refer the reader to [7]). 3. Trapezoidal approximation In this section we propose an approximation operator T : F(R) → FT (R) which produces a trapezoidal fuzzy number that is the closest to given original fuzzy number among all trapezoidal fuzzy numbers having identical expected interval as the original one. Therefore, this operator will be called the nearest trapezoidal approximation operator preserving expected interval. Suppose A is a fuzzy number and [AL (), AU ()] is its -cut. Given A we will try to find a trapezoidal fuzzy number T (A) which is the nearest to A with respect to metric d (4). Let [TL (), TU ()] denote the -cut of T (A). Thus we have to minimize  1  1 2 d(A, T (A)) = (AL () − TL ()) d + (AU () − TU ())2 d (5) 0

0

with respect to TL () and TU (). However, since a trapezoidal fuzzy number is completely described by four real numbers that are borders of its support and core, our goal reduces to finding such real numbers t1 t2 t3 t4 that characterize T (A) = T (t1 , t2 , t3 , t4 ). It is easily seen that the -cut of T (A) is equal to [t1 + (t2 − t1 ), t4 − (t4 − t3 )]. Therefore (5) reduces to  1  1 d(A, T (A)) = [AL () − (t1 + (t2 − t1 ))]2 d + [AU () − (t4 − (t4 − t3 ))]2 d, (6) 0

and we will try to minimize (6) with respect to t1 , t2 , t3 , t4 .

0

P. Grzegorzewski, E. Mrówka / Fuzzy Sets and Systems 158 (2007) 757 – 768

759

However, we want to find a trapezoidal fuzzy number which is not only closest to given fuzzy number but which preserves the expected interval of that fuzzy number. We justify this additional requirement by the significant role of the expected interval in many situations and applications (see, e.g., [3,6–8,12–15]). Thus our problem is to find such real numbers t1 t2 t3 t4 that minimize (6) with respect to condition EI(T (A)) = EI(A).

(7)

By (1) and (3) we can rewrite (7) as follows:    1   1 t1 + t2 t3 + t4 , = AL () d, AU () d . 2 2 0 0

(8)

It is easily seen that in order to minimize d(A, T (A)) it suffices to minimize function f (t1 , t2 , t3 , t4 ) = d 2 (A, T (A)) with respect to following conditions:  1 t1 + t2 AL () d = 0, (9) − 2 0  1 t3 + t4 AU () d = 0. (10) − 2 0 Moreover, we have to assure that t2 t3 . Thus, to sum up, we wish to minimize function  1  1 2 f (t) = [AL () − (t1 + (t2 − t1 ))] d + [AU () − (t4 − (t4 − t3 ))]2 d 0

(11)

0

subject to

  h(t) = t1 + t2 − 2

1



0

T

1

AL () d, t3 + t4 − 2

AU () d

= 0T ,

(12)

0

g(t) = t2 − t3 0,

(13)

where t ∈ R4 . By the Karush–Kuhn–Tucker theorem, if t∗ is a local minimizer for the problem of minimizing f subject to h(t) = 0 and g(t) 0, then there exist the Lagrange multiplier vector  and the Karush–Kuhn–Tucker multiplier  such that Df (t∗ ) + T Dh(t∗ ) + Dg(t∗ ) = 0T ,

(14)

g(t∗ ) = 0,

(15)

 0.

(16)

In our case, after some calculations, we get   1  1 2 1 ∗ Df (t ) = t1 + t2 + 2 AL () d − 2 AL () d, 3 3 0 0 2 1 t1 + t2 − 2 3 3 1 2 t3 + t4 − 2 3 3 2 1 t3 + t4 + 2 3 3



1

AL () d,

0



1

AU () d,

0



1 0



1

AU () d − 2

 AU () d ,

0

(17)

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P. Grzegorzewski, E. Mrówka / Fuzzy Sets and Systems 158 (2007) 757 – 768



1 1 0 0

Dh(t∗ ) =

0 0 1 1

 (18)

,



Dg(t∗ ) = 0, 1, −1, 0 .

(19)

Therefore, we can rewrite the Karush–Kuhn–Tucker conditions in a following way:  1  1 2 1 AL () d − 2 AL () d + 1 = 0, t1 + t2 + 2 3 3 0 0  1 2 1 AL () d + 1 +  = 0, t1 + t2 − 2 3 3 0  1 1 2 AU () d + 2 −  = 0, t3 + t4 − 2 3 3 0  1  1 2 1 AU () d − 2 AU () d + 2 = 0, t3 + t4 + 2 3 3 0 0  1 AL () d = 0, t1 + t2 − 2 

(20) (21) (22) (23) (24)

0 1

t3 + t4 − 2

AU () d = 0,

(25)

0

(t2 − t3 ) = 0,

(26)

 0.

(27)

To find points that satisfy the above conditions, we first try  > 0, which implies t2 − t3 = 0. Thus we are faced with a system of six linear equations  1  1 2 1 AL () d − 2 AL () d + 1 = 0, (28) t1 + t2 + 2 3 3 0 0  1 2 1 AL () d + 1 +  = 0, (29) t1 + t2 − 2 3 3 0  1 1 2 AU () d + 2 −  = 0, (30) t2 + t4 − 2 3 3 0  1  1 2 1 AU () d − 2 AU () d + 2 = 0, (31) t2 + t4 + 2 3 3 0 0  1 AL () d = 0, (32) t1 + t2 − 2 0



1

t2 + t4 − 2

AU () d = 0.

(33)

0

Solving the above system of equations we get  1  1  t1 = 3 AL () d − 3 AL () d − 3 0



t2 = t3 = 3 0

0 1

1



0



1

AL () d + 3 0

1

AU () d +

AU () d, 0



1

AU () d − 0



(34)

1

AL () d −

AU () d, 0

(35)

P. Grzegorzewski, E. Mrówka / Fuzzy Sets and Systems 158 (2007) 757 – 768



1

t4 = 3

 AU () d − 3

0

1 =

1 3



1

1



1

1

AL () d − 3

0

AL () d −

0

 

1 0

1

AU () d +

0

AL () d +

0



AU () d −

1 3



761

AL () d,

(36)

AU () d,

(37)

0 1

0

 1  1   1 1 1 1 AL () d + AL () d − AU () d + AU () d, 2 = − 3 0 3 0 0 0    1  1 2 1 2 1 AU () d − AL () d + AU () d. AL () d − 2 =2 3 0 3 0 0 0

(38) (39)

However, by the assumption that  > 0, we have a legitimate solution to the Karush–Kuhn–Tucker conditions if and only if    1  1 1 1 1 1 AL () d − AL () d > AU () d − AU () d (40) 3 0 3 0 0 0 or, in other words,    1 1 (AU () − AL ()) −  d > 0, 3 0 and then we get a solution t∗ = (t1 , t2 , t3 , t4 ), given by (34)–(36). In the second try we assume that  = 0. Thus we have to solve a following system of equations:  1  1 1 2 t1 + t2 + 2 AL () d − 2 AL () d + 1 = 0, 3 3 0 0  1 2 1 AL () d + 1 = 0, t1 + t2 − 2 3 3 0  1 1 2 AU () d + 2 = 0, t3 + t4 − 2 3 3 0  1  1 2 1 AU () d − 2 AU () d + 2 = 0, t3 + t4 + 2 3 3 0 0  1 AL () d = 0, t1 + t2 − 2 

(41)

(42) (43) (44) (45) (46)

0 1

t3 + t4 − 2

AU () d = 0.

(47)

0

Solving the above equations we obtain  1  1 AL () d + 4 AL () d, t1 = −6 0



1

t2 = 6 



AL () d,

(49)

AU () d,

(50)

0

1

t3 = 6

1

AL () d − 2

0

(48)

0



1

AU () d − 2

0



t4 = −6 0

0 1



1

AU () d + 4

AU () d, 0

(51)

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P. Grzegorzewski, E. Mrówka / Fuzzy Sets and Systems 158 (2007) 757 – 768

1 = 0,

(52)

2 = 0.

(53)

One can notice that the solution t∗∗ = (t1 , t2 , t3 , t4 ), given by (48)–(51), coincides with the solution given in [11]. Now we have to verify that both points t∗ and t∗∗ satisfy the second-order sufficient conditions. For this we form a matrix L(t, , ) = D 2 f (t) + [D 2 h(t)] + D 2 g(t),

(54)

where [D 2 h(t)] = 1 D 2 h1 (t) + 2 D 2 h2 (t) and D 2 hi (t) is the Hessian of hi (t). In our case we get ⎤ ⎡2 1 0 0 3 3 ⎥ ⎢ ⎥ ⎢1 2 ⎢3 3 0 0⎥ ⎥ ⎢ L(t, , ) = ⎢ ⎥. ⎢0 0 2 1⎥ ⎢ 3 3 ⎥ ⎦ ⎣ 2 1 0 0 3 3

(55)

One check easily that for both points t∗ and t∗∗ we have yT L(t, , )y > 0 for all vectors y in the tangent space to the surface defined by active constraints, i.e. {y : Dh(t)y = 0, D 2 g(t)y = 0}. Therefore, we conclude that if condition    1 1 (AU () − AL ()) −  d > 0, (56) 3 0 holds then the nearest trapezoidal fuzzy number to fuzzy number A with respect to metric d preserving the expected interval is a trapezoidal fuzzy number T1 (A) = T1 (t1 , t2 , t3 , t4 ), with t1 , t2 , t3 , t4 given by  1  1  1  1 t1 = 3 AL () d − 3 AL () d − 3 AU () d + AU () d, (57) 0

0

 t2 = t3 = 3

1

0

 AL () d + 3

0



1

t4 = 3

1

0

 AU () d −

0



1

AU () d − 3

0

1

1

AL () d − 3 0

1

AL () d −

0



0



AU () d,

(58)

0



1

AU () d +

AL () d.

(59)

0

Otherwise, the nearest trapezoidal fuzzy number to fuzzy number A with respect to metric d preserving the expected interval is a trapezoidal fuzzy number T2 (A) = T2 (t1 , t2 , t3 , t4 ), with t1 , t2 , t3 , t4 given by  1  1 t1 = −6 AL () d + 4 AL () d, (60) 0



1

t2 = 6

0



0



AL () d,

(61)

AU () d,

(62)

0 1

t3 = 6

1

AL () d − 2 

1

AU () d − 2

0

0

 t4 = −6 0

1



1

AU () d + 4

AU () d. 0

(63)

P. Grzegorzewski, E. Mrówka / Fuzzy Sets and Systems 158 (2007) 757 – 768

763

Example 1. Let us consider a fuzzy number A, discussed in [2], with membership function ⎧ (x + 1)2 if − 1 x 0, ⎪ ⎪ ⎨ A (x) = (1 − x)2 if 0 x 1, ⎪ ⎪ ⎩ 0 otherwise. 1 √ √ Thus an -cut of that fuzzy number is A = [ −1, 1− ] for  ∈ [0, 1]. One can easily see that 0 AL () d = − 13 , 1 1 1 1 1 1 1 1 . Hence, 0 AL () d − 13 0 AL () d = 30 > , 0 AU () d = 13 and 0 AU () d = 10 AL () d = − 10 0  1 1 1 1 0 AU () d− 3 0 AU () d = − 30 which means that condition (40) holds, so the nearest trapezoidal approximation is given by (57)–(59). Finally we obtain a trapezoidal fuzzy number T1 (A) = T1 (− 23 , 0, 0, 23 ), i.e. a membership function T1 (A) of T1 (A) is ⎧3 x + 1 if − 23 x 0, ⎪ ⎪ ⎨2 T1 (A) (x) = 1 − 23 x if 0 < x  23 , ⎪ ⎪ ⎩ 0 otherwise. Example 2. Suppose a fuzzy number A has a following membership function  A (x) =

1 − x 2 if − 1 x 1, 0

otherwise.

√ √ An -cut of that fuzzy number is A = [− 1 − , 1 − ] and it is easily seen that condition (56) is not fulfilled. Thus we apply a triangular approximation operator T2 (A) given by (60)–(63) and we get a trapezoidal fuzzy number 4 4 16 T2 (A) = T2 (− 16 15 , − 15 , 15 , 15 ), i.e. a membership function T2 (A) of T2 (A) is

T2 (A) (x) =

⎧4 5 ⎪ 3 + 4 x if − ⎪ ⎪ ⎪ ⎪ ⎨1 if − ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

4 3

− 45 x if

0

16 4 15 x < − 15 , 4 4 15 x  15 , 4 16 15 < x  15 ,

otherwise.

It is known that for any fuzzy number A there exist four numbers a1 , a2 , a3 , a4 ∈ R and two functions lA , rA : R → [0, 1], where lA is nondecreasing and rA is nonincreasing, such that we can describe a membership function A in a following manner ⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ lA (x) ⎨ A (x) = 1 ⎪ ⎪ ⎪ ⎪ rA (x) ⎪ ⎪ ⎪ ⎪ ⎩ 0

if x < a1 , if a1 x < a2 , if a2 x a3 ,

(64)

if a3 < x a4 , if a4 < x.

Functions lA and rA are called the left-hand side and the right-hand side of a fuzzy number A, respectively. It seems that it would be useful to express our trapezoidal approximation of a fuzzy number in terms of functions lA and rA and notation used in (64).

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P. Grzegorzewski, E. Mrówka / Fuzzy Sets and Systems 158 (2007) 757 – 768

Let A be a fuzzy number with continuous and strictly monotonic sides lA and rA . It can be shown that  a2  1 lA (x) dx, AL () d = a2 − 0

a1

 1 a2 2 1 l (x) dx, a2 − 2 2 a1 A 0  1  a4 AU () d = a3 + rA (x) dx,



1

AL () d =

0

 0

a3 1



1 1 AL () d = a3 + 2 2

a4

rA2 (x) dx.

a3

Hence, by formulae (57)–(59) and (56), the nearest trapezoidal fuzzy number to fuzzy number A with respect to metric d preserving the expected interval is a trapezoidal fuzzy number T1 (A) = T1 (t1 , t2 , t3 , t4 ), with t1 , t2 , t3 , t4 given as follows:  a2  a4   3 3 a2 2 1 3 a4 2 lA (x) dx + lA (x) dx + a3 + rA (x) dx − r (x) dx, (65) t1 = a2 − 3 2 2 a1 2 2 a3 A a1 a3 t2 = t3 =

1 a2 + 2

1 t4 = − a2 − 2





a2

lA (x) dx −

a1

a2

lA (x) dx +

a1

provided condition  a2  a2 + 2 lA (x) dx − 3 a1

a2 a1



3 2



3 2

a2

a1 a2

a1

1 2 lA (x) dx + a3 − 2

3 2 lA (x) dx + a3 + 3 2 

2 lA (x) dx

> a3 − 2

a4 a3





a4

rA (x) dx +

a3 a4

rA (x) dx −

a3

 rA (x) dx + 3

a4 a3

3 2



3 2

a4

a3



a4

a3

rA2 (x) dx

rA2 (x) dx,

rA2 (x) dx,

(66)

(67)

(68)

holds. Otherwise, the nearest trapezoidal fuzzy number to fuzzy number A with respect to metric d preserving the expected interval is a trapezoidal fuzzy number T2 (A) = T2 (t1 , t2 , t3 , t4 ), with t1 , t2 , t3 , t4 given by  a2  a2 2 lA (x) dx − 4 lA (x) dx, (69) t1 = a2 + 3 a1

 t2 = a2 − 3

a2

a1



a4

t3 = a3 + 3

a1

 2 lA (x) dx + 2

t4 = a3 − 3

a4



a3

 r (x) dx + 4 2

lA (x) dx,

(70)

rA (x) dx,

(71)

rA (x) dx.

(72)

a1

r 2 (x) dx − 2

a3



a2

a4 a3 a4

a3

To sum up, one can notice that approximation operator T2 is identical with that proposed in [11] (actually, the solution proposed in [11] has been reduced to T2 ). However, it appears that in some situations, i.e. for fuzzy numbers satisfying condition (56), we receive another formula for the trapezoidal approximation operator–operator T1 . In the last case the trapezoidal approximation actually leads to triangular fuzzy numbers. In the next section we will consider whether our corrected approximation operator fulfills all requirements desired for such operators and discussed in [11]. Moreover, we will try to find a suitable interpretation for the condition (56).

P. Grzegorzewski, E. Mrówka / Fuzzy Sets and Systems 158 (2007) 757 – 768

765

4. Properties Since one can easily propose many approximation methods a natural question arises: How to construct a “good’’ approximation operator? We have discussed this problem in our previous paper [11] and we have proposed there a list of criteria that the approximation operator should or just can possess. Then we have proved that the nearest trapezoidal approximation operator preserving expected interval—denoted in this paper by T2 —is invariant to translations and scale invariant, is monotonic and fulfills identity criterion, preserves the expected interval and fulfills the nearness criterion with respect to metric (4) in subfamily of all trapezoidal fuzzy numbers with fixed expected interval, is continuous and compatible with the extension principle, is order invariant with respect to some preference fuzzy relations and is correlation invariant. Moreover, it preserves the width, value and ambiguity of fuzzy number (for details we refer the reader to [11]). Now a natural question arises whether operator T1 also preserves all the above mentioned properties. Calculations similar to those performed in [11] show that the approximation operator T1 fulfills many desired requirements, like: invariance to translations and scale, nearness, continuity, compatibility with the extension principle, expected interval invariance, order invariance, correlation invariance and width invariance. However, the identity criterion, by the definition, does not apply to T1 and T1 does not preserve the ambiguity of fuzzy number. Thus let us consider briefly these two cases. The identity criterion states that the approximation of a trapezoidal fuzzy number is equivalent to that number, i.e. if A ∈ FT (R) then T (A) = A.

(73)

As it was shown, we apply operator T1 provided condition (56) holds. However, one may check easily that no triangular fuzzy number satisfies (56).Actually, by (2) the -cut of a triangular fuzzy number A is A = [t1 +(t2 −t1 ), t4 −(t4 −t3 )] and hence    1 1 1 −  d = (t2 − t3 ) 0 (AU () − AL ()) 3 6 0 which contradicts (56). It means that the identity criterion simply does not apply to operator T1 . To simplify the task of representing and handling fuzzy numbers Delgado et al. [4] suggested two parameters— value and ambiguity—which represent some basic features of fuzzy numbers and hence they were called a canonical representation of fuzzy numbers. First parameter may be seen as a point that corresponds to the typical value of the magnitude that the fuzzy number A represents, while the ambiguity characterizes the vagueness of fuzzy number A (for more details we refer the reader to [4]). Using, so-called, regular reducing function the value and ambiguity of a fuzzy number A is defined as follows:  1 Val(A) = (AL () + AU ()) d, (74) 0



Amb(A) =

1

(AU () − AL ()) d.

(75)

0

We have shown in [11] that our approximation operator T2 preserves both indices, i.e. ∀A ∈ F(R) Val(A) = Val(T2 (A)),

(76)

Amb(A) = Amb(T2 (A)).

(77)

One can easily see that operator T1 also preserves value of a fuzzy number, i.e. Val(A) = Val(T1 (A)).

(78)

However, we get Amb(A) < Amb(T1 (A)).

(79)

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P. Grzegorzewski, E. Mrówka / Fuzzy Sets and Systems 158 (2007) 757 – 768

Indeed, ambiguity of a trapezoidal fuzzy number A is given by Amb(A) =

t3 − t 2 (t4 − t3 ) + (t2 − t1 ) + . 2 6

Thus for T1 (A) described by (57)–(59) we get  1 1 1 Amb(T1 (A)) = (t4 − t1 ) = (AU () − AL ()) d. 6 3 0

(80)

(81)

Then combining (81) with (56) and substituting definition (75) we obtain  1  1 1 Amb(T1 (A)) = (AU () − AL ()) d > (AU () − AL ()) d = Amb(A), 3 0 0 which proves that (79) holds. Let us now combine the discussion on ambiguity and the interpretation of condition (56). As we have noticed above, we can rewrite (56) as follows:  1  1 1 (AU () − AL ()) d, (82) (AU () − AL ()) d < 3 0 0 so by definition (75), we get the equivalent form of condition (56)  1 1 Amb(A) < (AU () − AL ()) d. 3 0

(83)

Let B denote any triangular fuzzy number with support [t1 , t4 ], i.e. with -cuts B = [t1 + (b2 − t1 ), t4 − (t4 − b2 )], where t1 and t4 are given by (57)–(59). By (80) we get immediately Amb(B) = 16 (t4 − t1 ), and for our triangular fuzzy number B we obtain simultaneously  1 1 1 (BU () − BL ()) d = (t4 − t1 ). 3 0 6

(84)

(85)

Therefore, keeping in mind (81) we get an equivalent form of our condition (56) Amb(A) < Amb(B),

(86)

where B is any triangular fuzzy number with the same support as our triangular approximation of A. Inequality (86) has a nice interpretation. Namely, we approximate a fuzzy number A by the trapezoidal approximation operator T1 provided ambiguity of this fuzzy number is smaller than the ambiguity of any triangular fuzzy number with the same support as the support of our possible triangular approximation of the original fuzzy number A. It means that we apply operator T1 to fuzzy numbers less vague than given triangular fuzzy number. If a fuzzy number under study is more vague than given triangular fuzzy number then we have to apply the approximation operator T2 . But we can propose also another interpretation of our condition (56). One of the most natural nonspecificity measures is, so-called, the width of a fuzzy number (see [3]), defined as  ∞ A (x) dx. (87) w(A) = −∞

Provided fuzzy number A has continuous and strictly monotonic sides we obtain an equivalent formula for the width, i.e.  1 w(A) = (AU () − AL ()) d. (88) 0

P. Grzegorzewski, E. Mrówka / Fuzzy Sets and Systems 158 (2007) 757 – 768

767

Thus going back to (83) we obtain another version of condition (56): Amb(A) < 13 w(A),

(89)

which states that we approximate a fuzzy number A by the trapezoidal approximation operator T1 provided ambiguity of this fuzzy number is smaller than one-third of the width of that fuzzy number A. For more vague fuzzy numbers we should apply the approximation operator T2 . Finally, we may also obtain an interpretation of condition (56) based on the centroid point of a fuzzy number. Let y(A) denote the y-coordinate of the centroid point of a fuzzy number A. In [16] the authors showed that 1 (AU () − AL ()) d . (90) y(A) = 0 1 (A () − A ()) d U L 0 By (75) and (87) we may express y(A) in a form y(A) =

Amb(A) . w(A)

(91)

Therefore, we get immediately that our condition (56) is equivalent to the following one: y(A) < 13 .

(92)

It means that we approximate a fuzzy number A by the trapezoidal approximation operator T1 if the y-coordinate of the centroid point of A is lower than one-third. Otherwise, we apply operator T2 . Recently Yeh [17] has noticed that the approximation operator T2 preserves y-coordinate of the centroid point of a fuzzy number A, i.e. y(A) = y(T2 (A)).

(93)

Since the output of operator T1 is always a triangular fuzzy number and the y-coordinate y of the centroid point for any triangular fuzzy number is 13 , thus contrary to T2 the trapezoidal approximation operator T1 does not preserve y, i.e. y(A) < y(T1 (A)) = 13 .

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5. Conclusion In the present contribution we have improved our paper [11] devoted to trapezoidal approximation of fuzzy numbers. We have shown a corrected expression for the approximation operator, called the nearest trapezoidal approximation operator preserving expected interval. Our operator possesses many desired properties proved in the previous paper [11]. Even after corrections suggested in the present paper almost all these properties hold because they are strongly based on the expected interval invariance criteria. Some differences in properties are related to ambiguity invariance and y-coordinate of the centroid point invariance. It was shown that these very parameters appear the crucial ones for the appropriate choice of the form of the approximation operator. References [2] [3] [4] [5] [6] [7] [8] [11] [12] [13]

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