BABES -BOLYAI UNIVERSITY, CLUJ-NAPOCAFACULTY OF MATHEMATICS AND COMPUTER SCIENCE

TRAPEZOIDALAPPROXIMATIONOF FUZZY NUMBERS

Ph.D. Thesis Summary

Scienti�c Supervisor:Professor Ph.D. BLAGA PETRU

Ph.D. StudentBRÂNDAS ADRIANA

CLUJ-NAPOCA

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Table of contents

Key-wordsIntroduction1. Preliminaries

1.1 Fuzzy sets1.2 Fuzzy numbers1.3 Real numbers and intervals attached to fuzzy numbers

1.3.1 Ambiguity, value and expected interval1.3.2 The entropy of fuzzy numbers

1.4 Approximation criteria2. Approximation operators that preserve expectancy and level

sets2.1 Trapezoidal approximation which preserves the expected interval

and the core2.2 Trapezoidal approximation which preserves the expected interval

and the support2.3 Trapezoidal approximation which preserves expected value and core

3. Approximation operators which preserve ambiguity value andlevel sets

3.1 Trapezoidal approximation which preserves the ambiguity and thevalue

3.2 Trapezoidal approximation which preserves the ambiguity, the valueand the core

3.3 Trapezoidal approximation which preserves the ambiguity, the valueand the support4. Weighted approximation

4.1 Weighted approximation which preserves the support4.2 Weighted approximation which preserves the expected interval

References

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KEY-WORDSFuzzy sets, fuzzy numbers, triangular fuzzy number, trapezoidal fuzzy num-

ber, ambiguity, value, expectancy interval, expectancy value, medium value, theentropy of fuzzy numbers, trapezoidal approximation, weighted approximation.

INTRODUCTION

Fuzzy Mathematics became known in the second half of the 20 th century,after Lot�A. Zadeh published the �rst article on fuzzy Mathematics [82], 1965.The article generated lots of research and implicitly lead to the founding of newbranches in Mathematics. Fuzzy Mathematics may be regarded as a parallelto the classic Mathematics or as a natural continuation of the latter, but, inmost cases, it is considered a new Mathematics, very useful in solving problemsexpressed through a vague language.The promoter of the fuzzy sets theory, professor L. Zadeh, states that fuzzy

Mathematics is a useful instrument in moulding problems which are eitherungradable or too complex to be adequately modeled through traditional meth-ods. If Zadeh is the international promoter of the fuzzy Mathematics, the �rstconcepts of fuzzy Mathematics in Romania belong to G. Moisil. A detailedpresentation of Zadeh�s bibliography and works is given in [23].The study of the fuzzy Mathematics is organized on several subdomains.

Among these, we can mention: the fuzzy logic, the fuzzy arithmetic, the fuzzycontrol theory and the fuzzy nodulations. Compared to other research do-mains, the fuzzy arithmetic gained great interest only during the last years.Fuzzy numbers and sets are tools that are successfully used in scienti�c areas,such as: decision problems, social sciences, control theory, etc. The potentialof the fuzzy numbers is accepted and illustrated in exercises presented in [44]or [54]. Results obtained by using fuzzy numbers are practically much betterthan those obtained by classic methods. The study is oriented on: fuzzy setsand fuzzy logic ([70]; [82]; [83]; [84]) fuzzy numbers ([21], [45]; [54]) operationson fuzzy numbers ([12]; [18], [19]; [42]; [54]) ordering of the fuzzy numbers [22],defuzzi�cation measures [18], approximation operators of the fuzzy numbers([9]; [51]; [75]), distance between two fuzzy numbers ([74]; [39]), fuzzy equa-tions [73]; linguistic variables [34] of fuzzy application [20].Trapezoidal approximation operators was studied in many papers ([1], [2],

[3], [9], [15], [37], [49], [48], [51], [52], [68], [78], [80], [81], [85]).The paper has 4 chapters. In chapter I, �Preliminaries�, we introduced

elementary theoretical concepts that we are going to use during the whole paper.Starting from the concept of fuzzy numbers, we made a short presentation ofoperations, arriving at the central element of this paper: the fuzzy number. Wepresent a classi�cation of fuzzy numbers and elementary operations introducedover the fuzzy numbers set. Further on, in �Real numbers and intervals attachedto fuzzy numbers�, we discuss a set of measures which, applied to fuzzy numbers,

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defuzzify it, or turn it into an interval of real numbers ( e.g. expected interval)or into a real number (entropy, medium value, ambiguity or value).Starting with the second chapter, we present methods of trapezoidal and

weighted approximation of fuzzy numbers. Fuzzy numbers approximation is amethod required to simplify the processing of data that contain fuzzy numbersand can be seen from di¤erent perspectives. Since there are many approximationmethods and many algorithms that do these approximations, we consider thatit is not important to �nd the best approximation operator, but to estimatethe features and characteristics of operators which preserve fuzzy measures orlevel sets and �nd interesting use for these operators. The most brutal formof approximation is defuzzi�cation and it is a procedure that attaches a realnumber to a fuzzy number [43], [22]. This manner of approximation leads togreat losses of important information and we can call this method, according to[51], �type I approximation�. Another version of approximation is the attachingof a real numbers interval to a fuzzy number ( e.g. in [41]), so a � type IIapproximation�. A more generous approximation of the fuzzy number is with atriangular fuzzy number, so a �type III approximation�(e.g. in [11]), and theapproximation with a trapezoidal will be called �type IV approximation�( e.g.in [8], [9], [50], [51], [52], [53]).Trapezoidal operators of fuzzy numbers approximation have been studied by

many authors, because fuzzy numbers have lots of uses in the e¢ cient adjust-ment of imprecise information. It is obvious that there is an in�nite numberof methods for the approximation of fuzzy numbers with trapezoidal numbers.One of the pioneers of trapezoidal approximation, Delgado, in [37], suggeststhat an approximation must preserve at least some of the initial parameters ofthe fuzzy number. Any method of trapezoidal approximation is easy to suggest,because there is a great variety of characteristics that can be preserved fromthe beginning and, under these conditions, we can reach a new approximation.In [51], authors propose a trapezoidal operator of approximation and a list ofcriteria that an operator should ful�ll in order to be �good�.The characteristics an approximation operator must ful�ll, according to [51],

are the following:1. ��cut invariance;2. Translation invariance;3. Scalar invariance;4. Monotony;5. Identity;6. Nearest criterion;7. Expected value invariance;8. Continuity;9. Compatibility with extension principle;10. Order invariance;11. Correlation invariance;12. Uncertain invariance.In many works that introduce approximation operators, problems of min-

imum have been required, in order to identify these operators, because, for a

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better approximation, authors used minimization of distance between the fuzzynumber and its approximate. This problem was solved using Lagrange�s multi-pliers method [51] or the Karush-Kuhn-Tuker theorem [9], [53]. Another mannerof approach is given in [85], where trapezoidal approximation is studied fromthe angle of weighted distance. In [53], they discuss the weighted trapezoidalapproximation of fuzzy numbers that preserve expected interval and new algo-rithms of trapezoidal approximation of fuzzy numbers are introduced, keepinga distance based on weighted bisymmetrical functions.In chapter II, we introduced and studied the trapezoidal approximation op-

erator which preserves the expected interval and the fuzzy number core. Inidentifying this operator, the four conditions at the beginning for identifyingthe parameters of the approximation operator lead to a determined, one solu-tion system. Although the shape of this operator is very simple and easy toobtain, we deal with a linear operator, a rare feature among trapezoidal ap-proximation operators, therefore a valuable feature. Beside that, the operatoris invariant in translations, it respects the criteria of identity, it preserves theorder induced into the set of fuzzy numbers by the expected value, is invariantin relations based on expected value or expected interval, so it meets many ofthe criteria mentioned in [51]. However, the operator does not preserve theambiguity of the fuzzy number, nor its value. Another studied characteristicis continuity, in connection to two metrics. In order to re�ect the utility ofthis operator, we gave a calculation formula for fuzzy numbers, introduced byBodjanova in [22] and a set of other exercises and examples: solving the fuzzyequation systems, estimation of the multiplicity order for trapezoidal numbers.The second paragraph of this chapter introduces the trapezoidal operator

of approximation, which preserves the expected interval and the support of thefuzzy number [26]. If, in the case of the �rst operator, all fuzzy numbers could beapproximated to a trapezoidal number, in this operator�s case, only a family offuzzy numbers can be approximated with trapezoidal numbers, preserving therequested characteristics: preservation of the zero level set and the expectedinterval. The study of this operator is made from the perspective of impor-tant characteristics, such as: translation invariance, linearity, identity criteria,preservation of measures based on the preservation of the expected interval,continuity in connection with di¤erent metrics, the study of the multiplicitydefect.We close the chapter with an approximation operator, called the trapezoidal

operator that preserves expected value and core of the fuzzy number [28]. Inthe cases of the �rst two operators, problems were reduced to a system of fourequations and four unknowns, in the case of this operator from the conditionof minimization of the distance between the fuzzy number and the trapezoidalone, the preservation of the expected value and of the core, the solving is doneusing a minimization theory, and we have chosen the Karush-Kuhn-Tucker the-ory. Beside generating this operator, we also studied their characteristics andgenerated examples.Chapter III introduces approximation operators that preserve value, ambi-

guity and a level set. The �rst is the operator that preserves the ambiguity and

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the value and it is the nearest trapezoidal approximation of a fuzzy number, itwas introduced in [13]. Even though the operator is introduced by minimizingthe distance between the fuzzy number and the trapezoidal that approximatesit, we do not use minimization theories, but an extended trapezoidal. We studythe properties of this operator, introduce calculation algorithms, demonstratecontinuity and give examples. For the two operators that preserve value, ambi-guity and a level set, introduced in [24], [29], we studied properties and presenteda set of examples.Chapter IV studies approximation weighted operators. First, we present the

weighted operator that preserves the fuzzy number support, introduced in [25].The operator is generated by Lagrange�s multipliers method, then we presentits particular cases, its properties and a set of examples. The weighted operatorthat preserves the expected interval is introduced in [30]. As a method of solvingthe minimization problem, we have chosen the Karush-Kuhn-Tucker theorem,presenting particular cases, properties and examples that activate this operator.Our original contribution may be found in chapters 2,3,4, and a large part

of paragraph 1.3.2.

1 Preliminaries

1.1 Fuzzy sets

De�nition 1 Let X be a set. A fuzzy set is characterized by a function calledmembership function and de�ned as the [82]:

fX : X ! [0; 1]

associating each element of X to a real number in the range [0:1]:

1.2 Fuzzy numbers

De�nition 2 A fuzzy number A is fuzzy subset of a real line R with the member-ship function �A which is (see [41]) normal, fuzzy convex, upper semicontinous,supp A is bound, where the support of A, denoted by supp A; is the closure ofthe set fx 2 X : �A(x) > 0g :

Any fuzzy number can be expressed (see [?]) in the form

�A (x) =

8>><>>:g (x) ; when x 2 [a; b)1; when x 2 [b; c]

h (x) ; when x 2 (c; d]0; otherwise,

where a; b; c; d 2 R such that a < b � c < d; g is a real valued function thatis increasing and upper semicontinuous and h is a real valued function that isdecreasing and lower semicontinuous.Let us denote F (R) the set of fuzzy numbers.

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Starting at the general de�nition of fuzzy numbers in [54] are presented themain sets of fuzzy numbers: L�R fuzzy numbers, trapezoidal fuzzy numbers, tri-angular fuzzy numbers, gaussian fuzzy numbers, quasi-gaussian fuzzy numbers,quasi-quadric fuzzy numbers, exponential fuzzy numbers, quasi-exponential fuzzynumbers and single-ton fuzzy numbers.

De�nition 3 A particular case of L � R fuzzy numbers introduced by Duboisand Prade in 1981 [43] is

A(x) =

8>>>><>>>>:

�x�ab�a

�n; if x 2 [a; b]

1; if x 2 [b; c]�d�xd�c

�n; if x 2 [c; d]0; else.

(1)

Let us use the notation A(x) = (a; b; c; d)n :

For the fuzzy number given by (1) has following ��cut

A� =�a+ n

p� (b� a) ; d� n

p� (d� c)

�;

The algebraic structure for fuzzy mathematics was introduced by Mizumotoand Tanaka in [66] and [67], by Dubois and Prade in [43], [41] and [40], by Ma,Friderman and Kandel in [64]. The algebraic operations can be derived fromthe so-called extended principles [65] or using the �� cuts ([60], [76]). In thispaper we use the last method.

1.3 Real numbers and intervals attached to fuzzy numbers

In applications, sometimes for practical reasons, a fuzzy number is assigneda real value or an interval of real numbers. This process captures relevantinformation of a fuzzy number and leads to simplify the representation andhandling fuzzy numbers.Next, we present some entropy measures and parameters attached to fuzzy

numbers and used in fuzzy numbers theory.

1.3.1 Ambiguity, value and expected interval

fuzzy number A is fuzzy subset of a real line R with the membership function�A which is ( see [41]): normal (i.e. there exists an element x0 such that�A (x0) = 1), fuzzy convex (i.e. �A (�x1 + (1� �)x2) � min (�A (x1) ; �A (x2)) ;8x1; x2 2 R; � 2 [0; 1]), upper semicontinuous and supp A is bounded, wheresupp A = cl fx 2 X : �A(x) > 0g is the support of A and cl is the closureoperator.The �-cut of A ;� 2 ]0; 1] ; of a fuzzy number A is a crisp set de�ned as:

A� = fx 2 X : �A(x) � �g :

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Every �-cut of a fuzzy number A is a closed interval A� = [AL (�) ; AU (�)] ;where

AL (�) = inf fx 2 X : �A(x) � �gAU (�) = sup fx 2 X : �A(x) � �g :

The support of a fuzzy number A is denoted by (see [5]):

supp(A) = [AL (0) ; AU (0)] :

The pair of functions (AL; AU ) gives a parametric representation of a fuzzynumber A.For two arbitrary fuzzy numbers A and B with ��cut sets [AL(�); AU (�)]

and [BL(�); BU (�)] ; � 2 [0; 1] the metrics

D (A;B) =

sZ 1

0

[AL (�)�BL (�)]2 d�+Z 1

0

[AU (�)�BU (�)]2 d�;

and

d(A;B) = sup0���1

fmax fjAL (�)�BL (�)j ; jAU (�)�BU (�)jgg :

are presented in [48].The ambiguity and the value of a fuzzy number denoted with Amb(A) and

V al(A) was introduced in [37] as follows

Amb(A) =

Z 1

0

�(AU (�)�AL(�))d�; (2)

V al(A) =

Z 1

0

�(AU (�) +AL(�))d�: (3)

The expected interval of a fuzzy number A; denoted with EI(A) was intro-duced in ([40], [55]) as follows:

EI(A) = [E�(A); E�(A)] =

�Z 1

0

AL (�) d�;

Z 1

0

AU (�) d�

�The middle of the expected interval is named the expected value, EV (A) wasgiven in [55], as follows:

EV (A) =E�(A) + E

�(A)

2:

1.3.2 The entropy of fuzzy numbers

Until now the fuzzy entropy calculus has been studied from di¤erent angles,which led to several methods for calculating the entropy. In the �rst stage

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the entropy was de�ned based on the distance. Kaufmann in [59] proposed away of measuring the degree of fuzzi�cation of a fuzzy set using fuzzy distancebetween the nearest non-fuzzy set and the fuzzy set. Another way of measuringthe entropy was proposed by Yager [77], and this is to measure the distancebetween the fuzzy set and its complementary. Another approach is to use theentropy function to calculate the entropy.It should be noted that entropy is a measure that characterizes the elements

of fuzzy (sets or numbers) and interest for this concept is increasingly grown inrecent years. This can be claimed on the basis of many papers on the theme ofentropy [18], [61], [62], [76].Let h : [0; 1]! [0; 1] be a function of calculating the entropy, which satis�es

the following properties: it is increasing on the interval�0; 12

�and descressing

on the interval�12 ; 1�;with h(0) = h(1) = 0; h

�12

�= 1 and h(u) = h(1� u):

Let H be

H(A) =

ZX

h(A(x))dx

in [76] is called entropy function.Some entropy functions h are de�ned in ([69] and [76]):

h1(u) =

�2u; if u 2

�0; 12

�2(1� u); if u 2

�12 ; 1�;;

h2(u) = 4u(1� u);h3(u) = �u lnu� (1� u) ln(1� u); and 0 ln 0 := 0;

h3 was known as Shannon�s function.We will denote with Hi fuzzy entropy which respects the entropy function

hi:Using hi functions is easy to obtain fuzzy entropy of multiplication for two

trapezoidal fuzzy numbers.

Theorem 4 (Theorem 4.1, [18]) If A; B are fuzzy numbers then

(i) H1(A �B) = �12(a1b1 � a2b2 + a3b3 � a4b4) ;

(ii) H2(A �B) = �23(a1b1 � a2b2 + a3b3 � a4b4) ;

(iii) H3(A �B) = �12(a1b1 � a2b2 + a3b3 � a4b4) :

Theorem 5 (Theorem 5.1, [18]) If A; B are fuzzy numbers then(i)

H1(A=B) =2 (a2b4 � a1b3)(b4 � b3)2

ln(b3 + b4)

2

4b3b4

+2(a4b2 � a3b1)(b2 � b1)2

ln(b2 + b1)

2

4b2b1:

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(ii)

H2(A=B) =4(a2b4 � a1b3)(b4 � b3)2

�b4 + b3b4 � b3

lnb4b3� 2�

+4(a4b2 � a3b1)(b2 � b1)2

�2b2 + b1b2 � b1

lnb1b1� 2�:

(iii) Moreover, 2b3 > b4 and b2 < b1 then

H3(A=B) =a2b4�a1b3(b4�b3)b4

1Xk=1

k

(k + 1)2

�b4 � b3b4

�k+

+ a2b4�a1b3(b4�b3)b3

1Xk=1

(�1)k+1 k(k + 1)

2

�b4 � b3b3

�k+

+ a4b2�a3b1(b2�b1)b1

1Xk=1

(�1)k+1 k(k + 1)

2

�b2 � b1b1

�k+

+ a4b2�a3b1(b2�b1)b2

1Xk=1

k

(k + 1)2

�b2 � b1b2

�k:

Proposition 6 IfA 2 T; B 2 T :

Then

H1(A�B) =1

2(2a2b2 � a2b1 � a1b2 � 2a3b3 + a3b4 + a4b3)

H2(A�B) =2

3(2a2b2 � a2b1 � a1b2 � 2a3b3 + a3b4 + a4b3)

H3(A�B) =1

2(2a2b2 � a2b1 � a1b2 � 2a3b3 + a3b4 + a4b3):

Proposition 7 IfA 2 T; B 2 T

then

Hi(A�B) = b2Hi(A) + a2Hi(B)� k [(b4 � b3) (a2 � a3) + (a4 � a3) (b2 � b3)]

where

k =

�12 ; if i 2 f1; 3g23 ; if i = 2:

1.4 Approximation criteria

Throughout this paper we propose to approximate the fuzzy numbers withtrapezoidal fuzzy numbers. Thus, we introduce some operators de�ned on theset of trapezoidal fuzzy numbers with trapezoidal fuzzy numbers in the set

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values. Since the trapezoidal approximation can be studied from various angles,looking for operators who perform a large part of the property as outlined inthe criteria list from [51]. These criteria are:Translation invariance

T (A+ z) = T (A) + z (4)

for every A 2 F (R);Linearity,

T (�A) = �T (A)T (A+B) = T (A) + T (B)

(5)

for every A; B 2 F (R) and � 2 R�;�0 invariance

(T (A))�0 = A�0 ; (6)

for example: support invariance or core invariance.Identity

T (A) = A; (7)

for every A 2 FT (R) ;Expected interval invariance

EI (T (A)) = EI (A) ; (8)

for every A 2 F (R);Expected value invariance

EV (T (A)) = EV (A) ; (9)

for every A 2 F (R);Order invariance with respect to ordering � de�ned in [77]

A � B , EV (A) > EV (B) (10)

soA � B , T (A) � T (B) (11)

for every A; B 2 F (R);Order invariance with respect to ordering M de�ned in [58] as follows:

M (A;B) =

8<:0; if E�(A)� E�(B) < 0;

E�(A)�E�(B);E�(A)�E�(B)�(E�(A)�E�(B)) ; if 0 2 [E�(A)� E

�(B); E�(A)� E�(B)] ;1; if E�(A)� E�(B) > 0;

(12)so

M (T (A) ; T (B)) =M (A;B) ; (13)

for every A;B 2 F (R);

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Nonspeci�city invariance de�ned in [33], as follows:

w (A) =

Z 1

�1�A(x)dx; (14)

sow (A) = w (T (A)) ; (15)

for every A 2 F (R);Correlation invariance

� (T (A) ; T (B)) = � (A;B) (16)

for every A;B 2 F (R); where � (A;B) denotes a correlation of two fuzzy num-bers A and B, de�ned in [57] as follows:

� (A;B) =E�(A)E�(B) + E

�(A)E�(B)q(E�(A))

2+ (E�(A))

2q(E�(B))

2+ (E�(B))

2: (17)

Monotonyif A � B then T (A) � T (B) : (18)

Nearness criteriond (A; T (A)) � d (A;B) ; (19)

where d is a metric and A;B 2 F (R);Continuity

8" > 0;9 � > 0 d (A;B) < � ) d (T (A) ; T (B)) < "; (20)

where d is a metric and A;B 2 F (R);The below version of well-known Karush-Kuhn-Tucker theorem is useful in

the solving of the proposed problem.

Theorem 8 (see Rockafellar [71], pp. 281-283) Let f; g1; g2; :::; gm : Rn ! Rbe convex and di¤erentiable functions. Then

�x solves the convex programming

problemmin f (x)

subject to gi (x) � bi; i 2 f1; 2; 3; :::;mg if and only if there exist �i; wherei 2 f1; 2; 3; :::;mg, such that

(i) rf��x�+

mXi=1

�irgi��x�= 0;

(ii) gi

��x�� bi � 0;

(iii) �i � 0;(iv) �i

�bi � gi

��x��= 0:

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2 Approximation operators that preserve expectancyand level sets

In this chapter we introduce three operators that preserve approximation Ex-pectancy and level set, as follows: trapezoidal approximation operator whichpreserves the expected interval and the core, the approximation operator whichpreserves and support and the expected interval in [26] and the trapezoidalapproximation which preserves the expected value and the core, in paper [28].For those operators we study properties like: translation invariance, continuity,preserving the expected value or expected interval, identity criterion.

2.1 Trapezoidal approximation which preserves the ex-pected interval and the core

2.2 Trapezoidal approximation which preserves the ex-pected interval and the support

Let us denote

FES(R)=�A 2 F (R)j 2

Z 1

0

[AU (�)�AL (�)] d� � AU (0)�AL (0)�

Theorem 9 If A 2 FES(R) then

T (A) =

�AL (0) ; 2

Z 1

0

AL (�) d��AL (0) ; 2Z 1

0

AU (�) d��AU (0) ; AU (0)�

is the trapezoidal approximation which preserves the expected interval and thesupport.

Remark 10 If A =2 FES(R) then it doesn�t exist a trapezoidal fuzzy numberwhich preserves the expected interval and the support of the fuzzy number A.

Example 11 Let A be a fuzzy number

AL (�) = 1 + �2;

AU (�) = 3� �2; � 2 [0; 1]

then the trapezoidal approximation which preserves the expected interval and thesupport is

T (A) =

�1;5

3;7

3; 3

�:

Example 12 Let us consider a fuzzy number A given by

AL (�) = 1 +p�;

AU (�) = 45� 35p�; � 2 [0; 1] :

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Because

2

Z 1

0

[AU (�)�AL (�)] d� = 40 < 44 = AU (0)�AL (0)

we get A =2 FES(R) and not exists a trapezoidal fuzzy number which preservesthe expected interval and the support of A.

Remark 13 If A = (a; b; c; d)n ; is a fuzzy number, n 2 R�+ and

2n (b� c) � (n� 1) (a� d)

then

T (A) =

�a;2bn� an+ a

n+ 1;2cn� dn+ d

n+ 1; d

�is the trapezoidal fuzzy number which preserves the expected interval and thesupport of A:

Example 14 Let B = (5; 8; 12; 14) 13be a fuzzy number then the trapezoidal

approximation which preserves the expected interval and the support is:

T (B) =

�5;13

2; 13; 14

�:

The main properties of the trapezoidal approximation operator which pre-serves the expected interval and the support are introduced in the followingtheorem.

Theorem 15 The trapezoidal approximation operator preserving the expectedinterval and the support T : F (R) �! FT (R):(i) is invariant to translation, that is

T (A+ z) = T (A) + z

for every A 2 F (R) ;(ii) is a linear operator, that is

T (�A) = �T (A)

T (A+B) = T (A) + T (B)

for every A;B 2 F (R) and � 2 R�;(iii) ful�lls the identity criterion, that is

T (A) = A;

for every A 2 FT (R) ;(iv)is expected interval invariant, that is

EI (T (A)) = EI (A) ;

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for every A 2 F (R) ;(v) is order invariant with respect to the preference relation � de�ned by (see

[77])A � B , EV (A) > EV (B)

that isA � B , T (A) � T (B)

for every A; B 2 F (R) ;(vi) is order invariant with respect to the preference relation M de�ned by

(see [58])

M (A;B) =

8<:0; if E�(A)� E�(B) < 0

E�(A)�E�(B);E�(A)�E�(B)�(E�(A)�E�(B)) ; if 0 2 [E�(A)� E

�(B); E�(A)� E�(B)]1; if E�(A)� E�(B) > 0

that isM (T (A) ; T (B)) =M (A;B)

for every A;B 2 F (R) ;(vii) is uncertainty invariant with respect to the nonspeci�city measure de-

�ned by (see [?] )

w (A) =

Z 1

�1�A(x)dx

that isw (A) = w (T (A))

for every A 2 F (R) ;(viii) is correlation invariant, that is

� (T (A) ; T (B)) = � (A;B)

for every A;B 2 F (R) ; where � (A;B) denotes the correlation coe¢ cient be-tween A and B, de�ned as (see [57] )

� (A;B) =E�(A)E�(B) + E

�(A)E�(B)q(E�(A))

2+ (E�(A))

2q(E�(B))

2+ (E�(B))

2

2.3 Trapezoidal approximation which preserves expectedvalue and core

Theorem 16 If A is a fuzzy number then T (A) = (t1; t2; t3; t4) is the nearesttrapezoidal approximation of A; with respect the metric D; preserving the coreand the expected value.(i) IfZ 1

0

[(2� 6�)AU (�) + (6�� 10)AL (�)] d�+ 7AL (1) +AU (1) < 0 (21)

16

and Z 1

0

[AL (�) +AU (�)] d� � AL (1) +AU (1) ; (22)

then

t1 = t2 = AL (1)

t3 = AU (1)

t4 = 2

Z 1

0

AL (�) d�+ 2

Z 1

0

AU (�) d�� 2AL (1)�AU (1) :

(ii) IfZ 1

0

[(2� 6�)AL (�) + (6�� 10)AU (�)] d�+AL (1) + 7AU (1) > 0 (23)

and Z 1

0

[AL (�) +AU (�)] d� � AL (1) +AU (1) ; (24)

then

t1 = 2

Z 1

0

AL (�) d�+ 2

Z 1

0

AU (�) d��AL (1)� 2AU (1)

t2 = AL (1)

t3 = AU (1)

t4 = AU (1) :

(iii) IfZ 1

0

[(2� 6�)AU (�) + (6�� 10)AL (�)] d�+ 7AL (1) +AU (1) � 0 (25)

si Z 1

0

[(2� 6�)AL (�) + (6�� 10)AU (�)] d�+AL (1) + 7AU (1) � 0 (26)

then

t1 = �3

2

Z 1

0

� (AL (�)�AU (�)) d��3

4AL (1)+

1

2

Z 1

0

(5AL (�)�AU (�)) d��1

4AU (1)

t2 = AL (1)

t3 = AU (1)

t4 =3

2

Z 1

0

� (AL (�)�AU (�)) d��1

4AL (1)�

1

2

Z 1

0

(AL (�)� 5AU (�)) d��3

4AU (1) :

17

Example 17 Let A be a fuzzy number

A� =�1 + 99

p�; 200� 95

p��

then T (A) = (t1; t2; t3; t4) is

t1 = �32

Z 1

0

� (AL (�)�AU (�)) d��3

4AL (1) +

+1

2

Z 1

0

(5AL (�)�AU (�)) d��1

4AU (1)

=923

30

t2 = AL (1) = 100

t3 = AU (1) = 105

t4 = AU (1) = 105

Theorem 18 If A = (a; b; c; d)n is a fuzzy number then T (A) = (t1; t2; t3; t4)is the nearest trapezoidal fuzzy number preserving the core and the expected value(i) If

(n� 1) (d� c) + (17n+ 7) (b� a) < 0and

a� b� c+ d � 0;then

t1 = t2 = b

t3 = c

t4 =2a� 2b� c+ 2d+ cn

n+ 1:

(ii) If(b� a) (1� n)� (17n+ 7) (d� c) > 0

anda� b� c+ d � 0;

then

t1 =2a� b� 2c+ 2d+ bn

n+ 1t2 = b

t3 = c

t4 = c:

(iii) If(n� 1) (d� c) + (17n+ 7) (b� a) � 0

18

and(b� a) (1� n)� (17n+ 7) (d� c) � 0

then

t1 =8bn2 + 17an� 5bn+ cn� dn+ 7a� 3b� c+ d

4 (n+ 1) (2n+ 1)

t2 = b

t3 = c

t4 =8cn2 � an+ bn� 5cn+ 17dn+ a� b� 3c+ 7d

4 (n+ 1) (2n+ 1):

3 Approximation operators which preserve am-biguity and value

3.1 Trapezoidal approximation which preserves the ambi-guity and the value

Sometimes (see [80]) it is useful to denote a trapezoidal fuzzy number by

T = [l; u; x; y]

with l; u; x; y 2 R such that x; y � 0; x+ y � 2(u� l);

TL(�) = l + x(��1

2);

TU (�) = u� y(��1

2);

for every � 2 [0; 1]:It is immediate that

l =t1 + t22

; (27)

u =t3 + t42

; (28)

x = t2 � t1; (29)

y = t4 � t3 (30)

and T 2 FS(R) if and only if x = y. Also, by direct calculation we get

Amb(T ) =�6l + 6u� x� y

12; (31)

V al(T ) =6l + 6u+ x� y

12: (32)

19

The distance between T; T 0 2 FT (R); T = [l; u; x; y] and T 0 = [l0; u0; x0; y0]becomes ([79])

D2(T; T 0) = (l � l0)2 + (u� u0)2 + 1

12(x� x0)2 + 1

12(y � y0)2: (33)

An extended trapezoidal fuzzy number ([79]) is an order pair of polynomialfunctions of degree less than or equal to 1. An extended trapezoidal fuzzynumber may not be a fuzzy number, but the distance between two extendedtrapezoidal fuzzy numbers is similarly de�ned as in (??) or (33). In addition,we de�ne the value and the ambiguity of an extended trapezoidal fuzzy numberin the same way as in the case of a trapezoidal fuzzy number. We denote byFTe (R) the set of all extended trapezoidal fuzzy numbers, that is

FTe (R)=�T = [l; u; x; y] : TL (�) = l + x

��� 1

2

�;

TU (�) = u� y��� 1

2

�; � 2 [0; 1] ; l; u; x; y 2 R

;

where TL and TU have the same meaning as above.The extended trapezoidal approximation Te(A) = [le; ue; xe; ye] of a fuzzy

number A is the extended trapezoidal fuzzy number which minimizes the dis-tance d(A;X) where X 2 FTe (R). In the paper [5] the authors proved thatTe(A) is not always a fuzzy number. The extended trapezoidal approximationTe(A) = [le; ue; xe; ye] of a fuzzy number A is determined ([79]) by the followingequalities

le =

Z 1

0

AL(�)d�; (34)

ue =

Z 1

0

AU (�)d�; (35)

xe = 12

Z 1

0

(�� 12)AL(�)d�; (36)

ye = �12Z 1

0

(�� 12)AU (�)d�: (37)

The real numbers xe and ye are non-negative (see [79]) and from the de�nitionof a fuzzy number we have le � ue:In the paper [78] the author proved two very important distance properties

for the extended trapezoidal approximation operator, as follows.

Proposition 19 ([78], Proposition 4.2) Let A be a fuzzy number. Then

D2(A;B) = D2(A; Te(A)) +D2(Te(A); B) (38)

for any trapezoidal fuzzy number B:

Proposition 20 ([78], Proposition 4.4.) D(Te(A); Te(B)) � D(A;B) for allfuzzy numbers A;B.

20

Remark 21 Let A;B 2 F (R) and Te(A) = [le; ue; xe; ye]; Te(B) = [l0e; u0e; x0e; y0e]the extended trapezoidal approximations of A and B. Proposition 20 and (33)imply

(le � l0e)2 + (ue � u0e)2 � D2(A;B)

and(xe � x0e)2 + (ye � y0e)2 � 12D2(A;B):

Therefore T (A) = [lT ; uT ; xT ; yT ] if and only if (lT ; uT ; xT ; yT ) 2 R4 is asolution of the problem

min

�(l � le)2 + (u� ue)2 +

1

12(x� xe)2 +

1

12(y � ye)2

�(39)

under the conditions

x > 0; (40)

y > 0; (41)

x+ y 6 2u� 2l; (42)

�6l + 6u� x� y = �6le + 6ue � xe � ye; (43)

6l + 6u+ x� y = 6le + 6ue + xe � ye; (44)

where le; ue; xe; ye are given by (34)-(37). We immediately obtain that (39)-(44)is equivalent to

min�(x� xe)2 + (y � ye)2

�(45)

under the conditions

x > 0; (46)

y > 0; (47)

x+ y 6 3ue � 3le �1

2xe �

1

2ye: (48)

In addition,

l = �16(x� xe) + le (49)

andu =

1

6(y � ye) + ue: (50)

Let us consider the set

MA =

�(x; y) 2 R2 : x > 0; y > 0; x+ y 6 3ue � 3le �

1

2xe �

1

2ye

�;

dE the Euclidean metric on R2 and let us denote by PM (Z) the orthogonalprojection of Z 2 R2 on non-empty set M � R2, with respect to dE .

Theorem 22 (Theorem 5, [13])The problem (45)-(48) has an unique solution.

21

Theorem 23 (Theorem 7, [13])Let A 2 F (R) ; A� = [AL(�); AU (�)]; � 2 [0; 1],and T (A) = [lT ; uT ; xT ; yT ] the nearest trapezoidal fuzzy number to A whichpreserves ambiguity and value.(i) If Z 1

0

(3�� 1)AL(�)d��Z 1

0

(3�� 1)AU (�)d� � 0 (51)

then

xT = 6

Z 1

0

(2�� 1)AL(�)d�;

yT = �6Z 1

0

(2�� 1)AU (�)d�;

lT =

Z 1

0

AL(�)d�;

uT =

Z 1

0

AU (�)d�:

(ii) If Z 1

0

(3�� 1)AL(�)d�+Z 1

0

(�� 1)AU (�)d� > 0 (52)

then

xT = �6Z 1

0

�AL(�)d�+ 6

Z 1

0

�AU (�)d�

yT = 0

lT = 3

Z 1

0

�AL(�)d��Z 1

0

�AU (�)d�

uT = 2

Z 1

0

�AU (�)d�

(iii) If Z 1

0

(�� 1)AL(�)d�+Z 1

0

(3�� 1)AU (�)d� < 0 (53)

then

xT = 0

yT = �6Z 1

0

�AL(�)d�+ 6

Z 1

0

�AU (�)d�

lT = 2

Z 1

0

�AL(�)d�

uT = �Z 1

0

�AL(�)d�+ 3

Z 1

0

�AU (�)d�

22

(iv) If Z 1

0

(3�� 1)AL(�)d��Z 1

0

(3�� 1)AU (�)d� > 0 (54)Z 1

0

(3�� 1)AL(�)d�+Z 1

0

(�� 1)AU (�)d� � 0 (55)

and Z 1

0

(�� 1)AL(�)d�+Z 1

0

(3�� 1)AU (�)d� � 0 (56)

then

xT = 3

Z 1

0

(�� 1)AL(�)d�+ 3Z 1

0

(3�� 1)AU (�)d�

yT = �3Z 1

0

(3�� 1)AL(�)d�� 3Z 1

0

(�� 1)AU (�)d�

lT =1

2

Z 1

0

(3�+ 1)AL(�)d��1

2

Z 1

0

(3�� 1)AU (�)d�

uT = �1

2

Z 1

0

(3�� 1)AL(�)d�+1

2

Z 1

0

(3�+ 1)AU (�)d�:

Corollary 24 (i) If A 2 1, then

T (A) = (4I � 6L; 6L� 2I; 6U � 2S; 4S � 6U) :

(ii) If A 2 2, then

T (A) = (6L� 4U; 2U; 2U; 2U) :

(iii)If A 2 3, then

T (A) = (2L; 2L; 2L; 6U � 4L) :

(iv) If A 2 4, then

T (A) = (2I � 6U + 2S; 3L� I + 3U � S;3L� I + 3U � S; 2I � 6L+ 2S) :

Example 25 Let A and B be fuzzy numbers

AL(�) = 1 +p�;

AU (�) = 4�p�;

BL(�) = 1 +p�;

BU (�) = 35� 31p�:

23

We obtain that A 2 1 and B 2 4

T (A) =

�19

15;31

15;34

15;76

15

�T (B) =

�29

15; 2; 2;

419

15

�:

Since

(A+B)L (�) = 2 + 2p�

(A+B)U (�) = 39� 32p�;

we obtain that

T (A) + T (B) =

�48

15;61

15;64

15;495

15

�;

and

T (A+B) =

�38

15;62

15;73

15;457

15

�;

soT (A) + T (B) 6= T (A+B) ;

thus the operator T is not additive.

Theorem 26 If A;B 2 i; i 2 f1; 2; 3; 4g, then

T (A) + T (B) = T (A+B) :

3.2 Trapezoidal approximation which preserves the ambi-guity, the value and the core

Theorem 27 If A� = [AL (�) ; AU (�)] ; � 2 [0; 1] is a fuzzy number then onehas T (A) = (t1; t2; t3; t4) ; where

t1 = 6

Z 1

0

�AL (�) d�� 2AL (1)

t2 = AL (1)

t3 = AU (1)

t4 = 6

Z 1

0

�AU (�) d�� 2AU (1)

is the trapezoidal fuzzy number preserving the core, the ambiguity and the value.

Example 28 Let A be a fuzzy number, with the parametric representation

AL (�) = 1 +p�

AU (�) = 30� 27p�:

24

The trapezoidal fuzzy number preserving the core, the ambiguity and the valueis

T (A) =

�7

5; 2; 3;

96

5

�Example 29 Let A be a fuzzy number, with the parametric representation

AL (�) = 1 + 27p�

AU (�) = 30�p�:

The trapezoidal fuzzy number preserving the core, the ambiguity and the valueis

T (A) =

�59

5; 28; 29;

148

5

�Theorem 30 If A = (a; b; c; d)n is a fuzzy number then the trapezoidal fuzzynumber preserving the core, the ambiguity and the value is

T (A) =

�2bn+ 3a� b2n+ 1

; b; c;2cn+ 3d� c2n+ 1

�:

Theorem 31 The trapezoidal fuzzy number preserving the core, the ambiguityand the value T : F (R)!FT (R) ful�lls the following properties:(i) is invariant to translations;(ii) is linear;(iii) ful�lls the identity criterion;

3.3 Trapezoidal approximation which preserves the ambi-guity, the value and the support

4 Weighted approximation

4.1 Weighted approximation which preserves the support

The processing of fuzzy numbers is sometimes di¢ cult, therefore severalmethods have been introduced for approximating the fuzzy numbers with trape-zoidal fuzzy numbers. Each of these methods, either trapezoidal approximation[5], [9], [51], [52], [50], [49], [80] or weighted approximation [?], [?] bring somebene�ts and important properties.The Karush-Kuhn-Tucker theorem is used in this paper to compute the near-

est weighted trapezoidal fuzzy number with respect to weighted distance, suchthat the support of given fuzzy number is preserved. Important properties suchtranslation invariance, scale invariance, identity, nearest criterion and continuityof the weighted trapezoidal approximation are studied.

25

De�nition 32 For two arbitrary fuzzy numbers A and B with ��cut sets [AL(�); AU (�)]and [BL(�); BU (�)] ; � 2 [0; 1] the quantity

e(A;B) =

24 1Z0

f (�) g2(A�; B�)d�

3512

(57)

is the weighted distance between A and B; where

g2(A�; B�) = [AL(�)�BL(�)]2 + [AU (�)�BU (�)]2 ;

and f is a non-negative function, increasing on [0; 1] such that

f(0) = 0 (58)

named the weighting function. For a fuzzy number A; A� = [AL (�) ; AU (�)] ; � 2[0; 1] ; the problem is to �nd a trapezoidal fuzzy number T (A) = (t1; t2; t3; t4) ;which is the nearest to A with respect to metric e (see 57) and preserves thesupport of the fuzzy number A.

The fuzzy number has to ful�lls the following conditions:

min e(A; T (A))t1 = AL (0)t4 = AU (0) :

(59)

The problem is to determine four real numbers t1; t2; t3; t4; which are solutionsof system (59), where t1 � t2 � t3 � t4; so we have to minimize the function D;where

D (t1; t2; t3; t4) =

1Z0

f (�) [AL (�)� (t1 + (t2 � t1)�)]2 d�+

+

1Z0

f (�) [AU (�)� (t4 + (t3 � t4)�)]2 d�:

We reformulate the problem by (59) as follows

minh (t2; t3)

26

where

h (t2; t3) =

1Z0

f (�)A2L (�) d�� 2t21Z0

�f (�)AL (�) d�+

+t22

1Z0

�2f (�) d�+ 2t2AL (0)

1Z0

�f (�) (1� �) d�

+

1Z0

f (�) [AL (0) (1� �)]2 d�� 21Z0

f (�)AL (�)AL (0) (1� �) d�

+

1Z0

f (�)A2U (�) d�� 2t31Z0

�f (�)AU (�) d�

+t23

1Z0

�2f (�) d�� 21Z0

f (�)AU (�)AU (0) (1� �) d�

+2t3

1Z0

�f (�)AU (0) (1� �) d�+1Z0

f (�) [AU (0) (1� �)]2 d�:

(60)

subject toAL (0) � t2t2 � t3

t3 � AU (0) :(61)

Theorem 33 Let A; A� = [AL (�) ; AU (�)] be a fuzzy number and T (A) =(t1; t2; t3; t4) the nearest (with respect to metric e) trapezoidal fuzzy number tofuzzy number A preserving the support of A:(i) If R 1

0�f (�) [AL (�) +AU (�)] d��

�AL (0)R 10� (1� �) f (�) d��AU (0)

R 10� (1 + �) f (�) d� > 0

(62)

thent1 = AL (0) (63)

t2 = AU (0) (64)

t3 = AU (0) (65)

t4 = AU (0) : (66)

(ii) If

�R 10�f (�) [AL (�) +AU (�)] d�+

+AL (0)R 10� (1 + �) f (�) d�+AU (0)

R 10� (1� �) f (�) d� > 0

(67)

thent1 = AL (0) (68)

27

t2 = AL (0) (69)

t3 = AL (0) (70)

t4 = AU (0) : (71)

(iii) IfZ 1

0

�f (�) [AL (�)�AU (�)] d� � [AL (0)�AU (0)]Z 1

0

�f (�) (1� �) d� (72)

thent1 = AL (0) (73)

t2 =

R 10�f (�)AL (�) d��AL (0)

R 10�f (�) (1� �) d�R 1

0�2f (�) d�

(74)

t3 =

R 10�f (�)AU (�) d��AU (0)

R 10�f (�) (1� �) d�R 1

0�2f (�) d�

(75)

t4 = AU (0) : (76)

(iv) If R 10�f (�) [AL (�)�AU (�)] d��

� [AL (0)�AU (0)]R 10f (�)� (1� �) d� > 0

(77)

�R 10�f (�) [AL (�) +AU (�)] d�+

+AL (0)R 10� (1 + �) f (�) d�+AU (0)

R 10� (1� �) f (�) d� � 0

(78)

and R 10�f (�) [AL (�) +AU (�)] d��

�AL (0)R 10� (1� �) f (�) d��AU (0)

R 10� (1 + �) f (�) d� � 0;

(79)

then

t1 = AL (0)

t2 = t3 =

R 10�f (�) [AL (�) +AU (�)] d�

2R 10�2f (�) d�

�[AL (0) +AU (0)]

R 10�f (�) (1� �) d�

2R 10�2f (�) d�

t4 = AU (0) :

Theorem 34 For a fuzzy number A = (a; b; c; d)n the nearest trapezoidal fuzzynumber preserving the support of A, with f (�) = �; is T (A) = (t1; t2; t3; t4) :(i) If

a� d� an+ 4bn+ 4cn� 7dn > 0

28

then

t1 = a

t2 = d

t3 = d

t4 = d:

(ii) Ifa� d+ 7an� 4bn� 4cn+ dn > 0

then

t1 = a

t2 = a

t3 = a

t4 = d:

(iii) Ifa� d� an+ 4bn� 4cn+ dn � 0

then

t1 = a

t2 =a� an+ 4bn3n+ 1

t3 =d� dn+ 4cn3n+ 1

t4 = d:

(iv) Ifa� d� an+ 4bn� 4cn+ dn > 0a� d� an+ 4bn+ 4cn� 7dn � 0

anda� d+ 7an� 4bn� 4cn+ dn � 0

then

t1 = a

t2 = t3 =a+ d� an+ 4bn+ 4cn� dn

2 (3n+ 1)

t4 = d:

Example 35 The case (i) of Theorem 34 is applicable to the fuzzy number

A = (�100; 2; 10; 12)2and

T (A) = (�100; 12; 12; 12)is the nearest trapezoidal fuzzy number preserving the support of A:

29

Example 36 The case (ii) of Theorem 34 is applicable to the fuzzy number

A = (5; 10; 20; 310)2

andT (A) = (5; 5; 5; 310)

is the nearest trapezoidal fuzzy number preserving the support of A:

Example 37 The case (iii) of Theorem 34 is applicable to the fuzzy number

A = (0; 10; 20; 30)2

and

T (A) =

�0;80

7;130

7; 30

�is the nearest trapezoidal fuzzy number preserving the support of A:

Example 38 The case (iv) of Theorem 34 is applicable to the fuzzy number

A = (�30;�10;�9;�2)2

and

T (A) =

��30; �60

7;�607;�2

�is the nearest trapezoidal fuzzy number preserving the support of A:

4.2 Weighted approximation which preserves the expectedinterval

Let �L; �U : [0; 1]! R be non-negative functionsZ 1

0

�L (�) d� > 0Z 1

0

�U (�) d� > 0

named weighting functions and the weighted distance

d� (A;B) =

sZ 1

0

�L (�) [AL (�)�BL (�)]2 d�+Z 1

0

�U (�) [AU (�)�BU (�)]2 d�;

whereA; B are fuzzy numbers withA� = [AL (�) ; AU (�)] siB� = [BL (�) ; BU (�)] ;� 2 [0; 1] :

30

Theorem 39 Let A; A� = [AL (�) ; AU (�)] ; � 2 [0; 1] be a fuzzy number andT (A) = (l; u; �; �) the nearest weighted trapezoidal approximation operator pre-serving the expected interval.(i) If

�e (1� !L) + �e (1� !U )� ue + le � 0 (80)

then

l = le � �e�!L �

1

2

�u = ue + �e

�!U �

1

2

�� = �e

� = �e:

(ii) If�e (1� !L) + �e (1� !U )� ue + le > 0 (81)

c�e (1� !U )2 � d�e (1� !U ) (1� !L) + d (ue � le) (1� !L) � 0 (82)

d�e (1� !L)2 � c�e (1� !L) (1� !U ) + c (ue � le) (1� !U ) � 0 (83)

then

l = le � c�e (1� !U )2 � d�e (1� !U ) (1� !L) + d (ue � le) (1� !L)

d (1� !L)2 + c (1� !U )2�!L �

1

2

�u = ue +

d�e (1� !L)2 � c�e (1� !L) (1� !U ) + c (ue � le) (1� !U )d (1� !L)2 + c (1� !U )2

�!U �

1

2

�� =

c�e (1� !U )2 � d�e (1� !U ) (1� !L) + d (ue � le) (1� !L)d (1� !L)2 + c (1� !U )2

� =d�e (1� !L)2 � c�e (1� !L) (1� !U ) + c (ue � le) (1� !U )

d (1� !L)2 + c (1� !U )2:

(iii) If

c�e (1� !U )2 � d�e (1� !U ) (1� !L) + d (ue � le) (1� !L) < 0 (84)

then

l = le

u = ue +ue � le1� !U

�!U �

1

2

�� = 0

� =ue � le1� !U

:

31

(iv) If

d�e (1� !L)2 � c�e (1� !L) (1� !U ) + c (ue � le) (1� !U ) < 0 (85)

then

l = le � ue � le1� !L

�!L �

1

2

�u = ue

� =ue � le!L � 1

� = 0:

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