TRAPEZOIDAL APPROXIMATION OF FUZZY NUMBERS

Transcription

1 BABES -BOLYAI UNIVERSITY, CLUJ-NAPOCA FACULTY OF MATHEMATICS AND COMPUTER SCIENCE TRAPEZOIDAL APPROXIMATION OF FUZZY NUMBERS Ph.D. Thesis Summary Scienti c Supervisor: Professor Ph.D. BLAGA PETRU Ph.D. Student BRÂNDAŞ ADRIANA CLUJ-NAPOCA

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3 Table of contents Key-words Introduction. Preliminaries. Fuzzy sets. Fuzzy numbers.3 Real numbers and intervals attached to fuzzy numbers.3. Ambiguity, value and expected interval.3. The entropy of fuzzy numbers.4 Approximation criteria. Approximation operators that preserve expectancy and level sets. Trapezoidal approximation which preserves the expected interval and the core. Trapezoidal approximation which preserves the expected interval and the support.3 Trapezoidal approximation which preserves expected value and core 3. Approximation operators which preserve ambiguity value and level sets 3. Trapezoidal approximation which preserves the ambiguity and the value 3. Trapezoidal approximation which preserves the ambiguity, the value and the core 3.3 Trapezoidal approximation which preserves the ambiguity, the value and the support 4. Weighted approximation 4. Weighted approximation which preserves the support 4. Weighted approximation which preserves the expected interval References 3

4 KEY-WORDS Fuzzy sets, fuzzy numbers, triangular fuzzy number, trapezoidal fuzzy number, ambiguity, value, expectancy interval, expectancy value, medium value, the entropy of fuzzy numbers, trapezoidal approximation, weighted approximation. INTRODUCTION Fuzzy Mathematics became known in the second half of the th century, after Lot A. Zadeh published the rst article on fuzzy Mathematics [8], 965. The article generated lots of research and implicitly lead to the founding of new branches in Mathematics. Fuzzy Mathematics may be regarded as a parallel to the classic Mathematics or as a natural continuation of the latter, but, in most cases, it is considered a new Mathematics, very useful in solving problems expressed 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 either ungradable or too complex to be adequately modeled through traditional methods. If Zadeh is the international promoter of the fuzzy Mathematics, the rst concepts of fuzzy Mathematics in Romania belong to G. Moisil. A detailed presentation of Zadeh s bibliography and works is given in [3]. The study of the fuzzy Mathematics is organized on several subdomains. Among these, we can mention: the fuzzy logic, the fuzzy arithmetic, the fuzzy control theory and the fuzzy nodulations. Compared to other research domains, 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 potential of the fuzzy numbers is accepted and illustrated in exercises presented in [44] or [54]. Results obtained by using fuzzy numbers are practically much better than those obtained by classic methods. The study is oriented on: fuzzy sets and fuzzy logic ([7]; [8]; [83]; [84]) fuzzy numbers ([], [45]; [54]) operations on fuzzy numbers ([]; [8], [9]; [4]; [54]) ordering of the fuzzy numbers [], defuzzi cation measures [8], approximation operators of the fuzzy numbers ([9]; [5]; [75]), distance between two fuzzy numbers ([74]; [39]), fuzzy equations [73]; linguistic variables [34] of fuzzy application []. Trapezoidal approximation operators was studied in many papers ([], [], [3], [9], [5], [37], [49], [48], [5], [5], [68], [78], [8], [8], [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 of operations, arriving at the central element of this paper: the fuzzy number. We present a classi cation of fuzzy numbers and elementary operations introduced over the fuzzy numbers set. Further on, in Real numbers and intervals attached to fuzzy numbers, we discuss a set of measures which, applied to fuzzy numbers, 4

5 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 a method required to simplify the processing of data that contain fuzzy numbers and can be seen from di erent perspectives. Since there are many approximation methods and many algorithms that do these approximations, we consider that it is not important to nd the best approximation operator, but to estimate the features and characteristics of operators which preserve fuzzy measures or level sets and nd interesting use for these operators. The most brutal form of approximation is defuzzi cation and it is a procedure that attaches a real number to a fuzzy number [43], []. This manner of approximation leads to great losses of important information and we can call this method, according to [5], type I approximation. Another version of approximation is the attaching of a real numbers interval to a fuzzy number ( e.g. in [4]), so a type II approximation. A more generous approximation of the fuzzy number is with a triangular fuzzy number, so a type III approximation (e.g. in []), and the approximation with a trapezoidal will be called type IV approximation ( e.g. in [8], [9], [5], [5], [5], [53]). Trapezoidal operators of fuzzy numbers approximation have been studied by many authors, because fuzzy numbers have lots of uses in the e cient adjustment of imprecise information. It is obvious that there is an in nite number of methods for the approximation of fuzzy numbers with trapezoidal numbers. One of the pioneers of trapezoidal approximation, Delgado, in [37], suggests that an approximation must preserve at least some of the initial parameters of the fuzzy number. Any method of trapezoidal approximation is easy to suggest, because there is a great variety of characteristics that can be preserved from the beginning and, under these conditions, we can reach a new approximation. In [5], authors propose a trapezoidal operator of approximation and a list of criteria that an operator should ful ll in order to be good. The characteristics an approximation operator must ful ll, according to [5], are the following:. cut invariance;. Translation invariance; 3. Scalar invariance; 4. Monotony; 5. Identity; 6. Nearest criterion; 7. Expected value invariance; 8. Continuity; 9. Compatibility with extension principle;. Order invariance;. Correlation invariance;. Uncertain invariance. In many works that introduce approximation operators, problems of minimum have been required, in order to identify these operators, because, for a 5

6 better approximation, authors used minimization of distance between the fuzzy number and its approximate. This problem was solved using Lagrange s multipliers method [5] or the Karush-Kuhn-Tuker theorem [9], [53]. Another manner of approach is given in [85], where trapezoidal approximation is studied from the angle of weighted distance. In [53], they discuss the weighted trapezoidal approximation of fuzzy numbers that preserve expected interval and new algorithms of trapezoidal approximation of fuzzy numbers are introduced, keeping a distance based on weighted bisymmetrical functions. In chapter II, we introduced and studied the trapezoidal approximation operator which preserves the expected interval and the fuzzy number core. In identifying this operator, the four conditions at the beginning for identifying the parameters of the approximation operator lead to a determined, one solution system. Although the shape of this operator is very simple and easy to obtain, we deal with a linear operator, a rare feature among trapezoidal approximation operators, therefore a valuable feature. Beside that, the operator is invariant in translations, it respects the criteria of identity, it preserves the order induced into the set of fuzzy numbers by the expected value, is invariant in relations based on expected value or expected interval, so it meets many of the criteria mentioned in [5]. However, the operator does not preserve the ambiguity of the fuzzy number, nor its value. Another studied characteristic is continuity, in connection to two metrics. In order to re ect the utility of this operator, we gave a calculation formula for fuzzy numbers, introduced by Bodjanova in [] and a set of other exercises and examples: solving the fuzzy equation 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 the fuzzy number [6]. If, in the case of the rst operator, all fuzzy numbers could be approximated to a trapezoidal number, in this operator s case, only a family of fuzzy numbers can be approximated with trapezoidal numbers, preserving the requested characteristics: preservation of the zero level set and the expected interval. The study of this operator is made from the perspective of important 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 multiplicity defect. We close the chapter with an approximation operator, called the trapezoidal operator that preserves expected value and core of the fuzzy number [8]. In the cases of the rst two operators, problems were reduced to a system of four equations and four unknowns, in the case of this operator from the condition of minimization of the distance between the fuzzy number and the trapezoidal one, the preservation of the expected value and of the core, the solving is done using a minimization theory, and we have chosen the Karush-Kuhn-Tucker theory. Beside generating this operator, we also studied their characteristics and generated examples. Chapter III introduces approximation operators that preserve value, ambiguity and a level set. The rst is the operator that preserves the ambiguity and 6

7 the value and it is the nearest trapezoidal approximation of a fuzzy number, it was introduced in [3]. Even though the operator is introduced by minimizing the distance between the fuzzy number and the trapezoidal that approximates it, we do not use minimization theories, but an extended trapezoidal. We study the properties of this operator, introduce calculation algorithms, demonstrate continuity and give examples. For the two operators that preserve value, ambiguity and a level set, introduced in [4], [9], we studied properties and presented a set of examples. Chapter IV studies approximation weighted operators. First, we present the weighted operator that preserves the fuzzy number support, introduced in [5]. The operator is generated by Lagrange s multipliers method, we present its particular cases, its properties and a set of examples. The weighted operator that preserves the expected interval is introduced in [3]. As a method of solving the 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,3,4, and a large part of paragraph.3.. Preliminaries. Fuzzy sets De nition Let X be a set. A fuzzy set is characterized by a function called membership function and de ned as the [8]: f X : X! [; ] associating each element of X to a real number in the range [:]:. Fuzzy numbers De nition A fuzzy number A is fuzzy subset of a real line R with the membership function A which is (see [4]) normal, fuzzy convex, upper semicontinous, supp A is bound, where the support of A, denoted by supp A; is the closure of the set fx X : A (x) > g : Any fuzzy number can be expressed (see [?]) in the form 8 g (x) ; when x [a; b) >< ; when x [b; c] A (x) = h (x) ; when x (c; d] >: ; otherwise, where a; b; c; d R such that a < b c < d; g is a real valued function that is increasing and upper semicontinuous and h is a real valued function that is decreasing and lower semicontinuous. Let us denote F (R) the set of fuzzy numbers. 7

8 Starting at the general de nition of fuzzy numbers in [54] are presented the main sets of fuzzy numbers: L R fuzzy numbers, trapezoidal fuzzy numbers, triangular fuzzy numbers, gaussian fuzzy numbers, quasi-gaussian fuzzy numbers, quasi-quadric fuzzy numbers, exponential fuzzy numbers, quasi-exponential fuzzy numbers and single-ton fuzzy numbers. De nition 3 A particular case of L and Prade in 98 [43] is 8 >< A(x) = Let us use the notation A(x) = (a; b; c; d) n : >: x a R fuzzy numbers introduced by Dubois b a n ; if x [a; b] ; if x [b; c] d x d c n ; if x [c; d] ; else. () For the fuzzy number given by () has following cut A = a + np (b a) ; d n p (d c) ; The algebraic structure for fuzzy mathematics was introduced by Mizumoto and Tanaka in [66] and [67], by Dubois and Prade in [43], [4] and [4], by Ma, Friderman and Kandel in [64]. The algebraic operations can be derived from the so-called extended principles [65] or using the cuts ([6], [76]). In this paper we use the last method..3 Real numbers and intervals attached to fuzzy numbers In applications, sometimes for practical reasons, a fuzzy number is assigned a real value or an interval of real numbers. This process captures relevant information of a fuzzy number and leads to simplify the representation and handling fuzzy numbers. Next, we present some entropy measures and parameters attached to fuzzy numbers and used in fuzzy numbers theory..3. Ambiguity, value and expected interval fuzzy number A is fuzzy subset of a real line R with the membership function A which is ( see [4]): normal (i.e. there exists an element x such that A (x ) = ), fuzzy convex (i.e. A (x + ( ) x ) min ( A (x ) ; A (x )) ; 8x ; x R; [; ]), upper semicontinuous and supp A is bounded, where supp A = cl fx X : A (x) > g is the support of A and cl is the closure operator. The -cut of A ; ]; ] ; of a fuzzy number A is a crisp set de ned as: A = fx X : A (x) g : 8

9 Every -cut of a fuzzy number A is a closed interval A = [A L () ; A U ()] ; where A L () = inf fx X : A (x) g A U () = sup fx X : A (x) g : The support of a fuzzy number A is denoted by (see [5]): supp(a) = [A L () ; A U ()] : The pair of functions (A L ; A U ) gives a parametric representation of a fuzzy number A. For two arbitrary fuzzy numbers A and B with cut sets [A L (); A U ()] and [B L (); B U ()] ; [; ] the metrics s D (A; B) = [A L () B L ()] d + [A U () B U ()] d; and d(a; B) = sup fmax fja L () B L ()j ; ja U () B U ()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) = V al(a) = (A U () A L ())d; () (A U () + A L ())d: (3) The expected interval of a fuzzy number A; denoted with EI(A) was introduced in ([4], [55]) as follows: EI(A) = [E (A); E (A)] = A L () d; A U () d The middle of the expected interval is named the expected value, EV (A) was given in [55], as follows: EV (A) = E (A) + E (A) :.3. 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 9

10 the entropy was de ned based on the distance. Kaufmann in [59] proposed a way of measuring the degree of fuzzi cation of a fuzzy set using fuzzy distance between the nearest non-fuzzy set and the fuzzy set. Another way of measuring the entropy was proposed by Yager [77], and this is to measure the distance between the fuzzy set and its complementary. Another approach is to use the entropy 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 in recent years. This can be claimed on the basis of many papers on the theme of entropy [8], [6], [6], [76]. Let h : [; ]! [; ] be a function of calculating the entropy, which satis es the following properties: it is increasing on the interval ; and descressing on the interval ; ;with h() = h() = ; h = and h(u) = h( u): Let H be Z H(A) = h(a(x))dx in [76] is called entropy function. Some entropy functions h are de ned in ([69] and [76]): u; if u ; h (u) = ( u); if u ; ; ; h (u) = 4u( u); X h 3 (u) = u ln u ( u) ln( u); and ln := ; h 3 was known as Shannon s function. We will denote with H i fuzzy entropy which respects the entropy function h i : Using h i functions is easy to obtain fuzzy entropy of multiplication for two trapezoidal fuzzy numbers. Theorem 4 (Theorem 4., [8]) If A; B are fuzzy numbers (i) H (A B) = (ii) H (A B) = (iii) H 3 (A B) = (a b a b + a 3 b 3 a 4 b 4 ) ; 3 (a b a b + a 3 b 3 a 4 b 4 ) ; (a b a b + a 3 b 3 a 4 b 4 ) : Theorem 5 (Theorem 5., [8]) If A; B are fuzzy numbers (i) H (A=B) = (a b 4 a b 3 ) (b 4 b 3 ) ln (b 3 + b 4 ) 4b 3 b 4 + (a 4b a 3 b ) (b b ) ln (b + b ) 4b b :

11 (ii) H (A=B) = 4(a b 4 a b 3 ) (b 4 b 3 ) + 4(a 4b a 3 b ) (b b ) b4 + b 3 ln b 4 b 4 b 3 b 3 b + b ln b b b b : (iii) Moreover, b 3 > b 4 and b < b H 3 (A=B) = ab ab3 (b 4 b 3)b 4 k= ab4 ab3 (b 4 b 3)b 3 a4b a3b (b b )b a4b a3b (b b )b X k k b4 b 3 (k + ) + b 4 X ( ) k+ k k b4 b 3 (k + ) + b 3 k= X ( ) k+ k k b b (k + ) + b X k k b b (k + ) : b k= k= Proposition 6 If Then A T; B T : H (A B) = (a b a b a b a 3 b 3 + a 3 b 4 + a 4 b 3 ) H (A B) = 3 (a b a b a b a 3 b 3 + a 3 b 4 + a 4 b 3 ) H 3 (A B) = (a b a b a b a 3 b 3 + a 3 b 4 + a 4 b 3 ): Proposition 7 If A T; B T H i (A B) = b H i (A) + a H i (B) k [(b 4 b 3 ) (a a 3 ) + (a 4 a 3 ) (b b 3 )] where k = ; if i f; 3g ; if i = : 3.4 Approximation criteria Throughout this paper we propose to approximate the fuzzy numbers with trapezoidal fuzzy numbers. Thus, we introduce some operators de ned on the set of trapezoidal fuzzy numbers with trapezoidal fuzzy numbers in the set

12 values. Since the trapezoidal approximation can be studied from various angles, looking for operators who perform a large part of the property as outlined in the criteria list from [5]. These criteria are: Translation invariance for every A F (R); Linearity, T (A + z) = T (A) + z (4) T (A) = T (A) T (A + B) = T (A) + T (B) for every A; B F (R) and R ; invariance (T (A)) = A ; (6) for example: support invariance or core invariance. Identity T (A) = A; (7) for every A F T (R) ; Expected interval invariance for every A F (R); Expected value invariance (5) EI (T (A)) = EI (A) ; (8) EV (T (A)) = EV (A) ; (9) for every A F (R); Order invariance with respect to ordering de ned in [77] so A B, EV (A) > EV (B) () A B, T (A) T (B) () for every A; B F (R); Order invariance with respect to ordering M de ned in [58] as follows: 8 < M (A; B) = : so for every A; B F (R); E (A) E (B); ; if E (A) E (B) < ; E (A) E (B) (E (A) E (B)) ; if [E (A) E (B); E (A) E (B)] ; ; if E (A) E (B) > ; () M (T (A) ; T (B)) = M (A; B) ; (3)

13 Nonspeci city invariance de ned in [33], as follows: so for every A F (R); Correlation invariance w (A) = A (x)dx; (4) w (A) = w (T (A)) ; (5) (T (A) ; T (B)) = (A; B) (6) for every A; B F (R); where (A; B) denotes a correlation of two fuzzy numbers A and B, de ned in [57] as follows: (A; B) = E (A)E (B) + E (A)E (B) q(e (A)) + (E (A)) q(e (B)) + (E (B)) : (7) Monotony Nearness criterion where d is a metric and A; B F (R); Continuity if A B T (A) T (B) : (8) d (A; T (A)) d (A; B) ; (9) 8" > ; 9 > d (A; B) < ) d (T (A) ; T (B)) < "; () where d is a metric and A; B 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 [7], pp. 8-83) Let f; g ; g ; :::; g m : R n! R be convex and di erentiable functions. Then x solves the convex programming problem min f (x) subject to g i (x) b i ; i f; ; 3; :::; mg if and only if there exist i ; where i f; ; 3; :::; mg, such that mx (i) rf x + i rg i x = ; i= (ii) g i x b i ; (iii) i ; (iv) i b i g i x = : 3

14 Approximation operators that preserve expectancy and level sets In this chapter we introduce three operators that preserve approximation Expectancy and level set, as follows: trapezoidal approximation operator which preserves the expected interval and the core, the approximation operator which preserves and support and the expected interval in [6] and the trapezoidal approximation which preserves the expected value and the core, in paper [8]. For those operators we study properties like: translation invariance, continuity, preserving the expected value or expected interval, identity criterion.. Trapezoidal approximation which preserves the expected interval and the core. Trapezoidal approximation which preserves the expected interval and the support Let us denote F ES (R)= A F (R)j Theorem 9 If A F ES (R) T (A) = A L () ; [A U () A L ()] d A U () A L () A L () d A L () ; A U () d A U () ; A U () is the trapezoidal approximation which preserves the expected interval and the support. Remark If A = F ES (R) it doesn t exist a trapezoidal fuzzy number which preserves the expected interval and the support of the fuzzy number A. Example Let A be a fuzzy number A L () = + ; A U () = 3 ; [; ] the trapezoidal approximation which preserves the expected interval and the support is T (A) = ; 53 ; 73 ; 3 : Example Let us consider a fuzzy number A given by A L () = + p ; A U () = p ; [; ] : 4

15 Because [A U () A L ()] d = 4 < 44 = A U () A L () we get A = F ES (R) and not exists a trapezoidal fuzzy number which preserves the expected interval and the support of A. Remark 3 If A = (a; b; c; d) n ; is a fuzzy number, n R + and T (A) = n (b c) (n ) (a d) a; bn an + a ; n + cn dn + d ; d n + is the trapezoidal fuzzy number which preserves the expected interval and the support of A: Example 4 Let B = (5; 8; ; 4) be a fuzzy number the trapezoidal 3 approximation which preserves the expected interval and the support is: T (B) = 5; 3 ; 3; 4 : The main properties of the trapezoidal approximation operator which preserves the expected interval and the support are introduced in the following theorem. Theorem 5 The trapezoidal approximation operator preserving the expected interval and the support T : F (R)! F T (R): (i) is invariant to translation, that is for every A F (R) ; (ii) is a linear operator, that is T (A + z) = T (A) + z T (A) = T (A) T (A + B) = T (A) + T (B) for every A; B F (R) and R ; (iii) ful lls the identity criterion, that is T (A) = A; for every A F T (R) ; (iv)is expected interval invariant, that is EI (T (A)) = EI (A) ; 5

16 for every A F (R) ; (v) is order invariant with respect to the preference relation de ned by (see [77]) A B, EV (A) > EV (B) that is A B, T (A) T (B) for every A; B F (R) ; (vi) is order invariant with respect to the preference relation M de ned by (see [58]) 8 < M (A; B) = : that is E (A) E (B); ; if E (A) E (B) < E (A) E (B) (E (A) E (B)) ; if [E (A) E (B); E (A) E (B)] ; if E (A) E (B) > M (T (A) ; T (B)) = M (A; B) for every A; B F (R) ; (vii) is uncertainty invariant with respect to the nonspeci city measure de- ned by (see [?] ) that is w (A) = A (x)dx w (A) = w (T (A)) for every A F (R) ; (viii) is correlation invariant, that is (T (A) ; T (B)) = (A; B) for every A; B F (R) ; where (A; B) denotes the correlation coe cient between A and B, de ned as (see [57] ) (A; B) = E (A)E (B) + E (A)E (B) q(e (A)) + (E (A)) q (E (B)) + (E (B)).3 Trapezoidal approximation which preserves expected value and core Theorem 6 If A is a fuzzy number T (A) = (t ; t ; t 3 ; t 4 ) is the nearest trapezoidal approximation of A; with respect the metric D; preserving the core and the expected value. (i) If [( 6) A U () + (6 ) A L ()] d + 7A L () + A U () < () 6

17 and [A L () + A U ()] d A L () + A U () ; () (ii) If t = t = A L () t 3 = A U () t 4 = A L () d + A U () d A L () A U () : [( 6) A L () + (6 ) A U ()] d + A L () + 7A U () > (3) and [A L () + A U ()] d A L () + A U () ; (4) (iii) If t = t = A L () t 3 = A U () t 4 = A U () : A L () d + A U () d A L () A U () şi [( 6) A U () + (6 ) A L ()] d + 7A L () + A U () (5) t = 3 [( 6) A L () + (6 ) A U ()] d + A L () + 7A U () (6) (A L () A U ()) d 3 4 A L ()+ (5A L () A U ()) d 4 A U () t = A L () t 3 = A U () t 4 = 3 (A L () A U ()) d 4 A L () (A L () 5A U ()) d 3 4 A U () : 7

18 Example 7 Let A be a fuzzy number A = + 99 p ; 95 p T (A) = (t ; t ; t 3 ; t 4 ) is t = 3 + = 93 3 (A L () A U ()) d (5A L () A U ()) d 3 4 A L () + 4 A U () t = A L () = t 3 = A U () = 5 t 4 = A U () = 5 Theorem 8 If A = (a; b; c; d) n is a fuzzy number T (A) = (t ; t ; t 3 ; t 4 ) is the nearest trapezoidal fuzzy number preserving the core and the expected value (i) If (n ) (d c) + (7n + 7) (b a) < and a b c + d ; (ii) If and (iii) If t = t = b t 3 = c t 4 = a b c + d + cn : n + (b a) ( n) (7n + 7) (d c) > a b c + d ; t = a b c + d + bn n + t = b t 3 = c t 4 = c: (n ) (d c) + (7n + 7) (b a) 8

19 and (b a) ( n) (7n + 7) (d c) t = 8bn + 7an 5bn + cn dn + 7a 3b c + d 4 (n + ) (n + ) t = b t 3 = c t 4 = 8cn an + bn 5cn + 7dn + a b 3c + 7d : 4 (n + ) (n + ) 3 Approximation operators which preserve ambiguity and value 3. Trapezoidal approximation which preserves the ambiguity and the value Sometimes (see [8]) it is useful to denote a trapezoidal fuzzy number by T = [l; u; x; y] with l; u; x; y R such that x; y ; x + y (u l); for every [; ]: It is immediate that T L () = l + x( T U () = u y( ); ); l = t + t ; (7) u = t 3 + t 4 ; (8) x = t t ; (9) y = t 4 t 3 (3) and T F S (R) if and only if x = y. Also, by direct calculation we get 6l + 6u x y Amb(T ) = ; (3) 6l + 6u + x V al(t ) = y : (3) 9

20 The distance between T; T F T (R); T = [l; u; x; y] and T = [l ; u ; x ; y ] becomes ([79]) D (T; T ) = (l l ) + (u u ) + (x x ) + (y y ) : (33) An extended trapezoidal fuzzy number ([79]) is an order pair of polynomial functions of degree less than or equal to. An extended trapezoidal fuzzy number may not be a fuzzy number, but the distance between two extended trapezoidal 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 number in the same way as in the case of a trapezoidal fuzzy number. We denote by Fe T (R) the set of all extended trapezoidal fuzzy numbers, that is F T e (R)= T = [l; u; x; y] : T L () = l + x ; T U () = u y ; [; ] ; l; u; x; y R ; where T L and T U have the same meaning as above. The extended trapezoidal approximation T e (A) = [l e ; u e ; x e ; y e ] of a fuzzy number A is the extended trapezoidal fuzzy number which minimizes the distance d(a; X) where X Fe T (R). In the paper [5] the authors proved that T e (A) is not always a fuzzy number. The extended trapezoidal approximation T e (A) = [l e ; u e ; x e ; y e ] of a fuzzy number A is determined ([79]) by the following equalities l e = u e = x e = y e = A L ()d; (34) A U ()d; (35) ( ( )A L()d; (36) )A U ()d: (37) The real numbers x e and y e are non-negative (see [79]) and from the de nition of a fuzzy number we have l e u e : In the paper [78] the author proved two very important distance properties for the extended trapezoidal approximation operator, as follows. Proposition 9 ([78], Proposition 4.) Let A be a fuzzy number. Then for any trapezoidal fuzzy number B: D (A; B) = D (A; T e (A)) + D (T e (A); B) (38) Proposition ([78], Proposition 4.4.) D(T e (A); T e (B)) D(A; B) for all fuzzy numbers A; B.

21 Remark Let A; B F (R) and T e (A) = [l e ; u e ; x e ; y e ]; T e (B) = [l e; u e; x e; y e] the extended trapezoidal approximations of A and B. Proposition and (33) imply (l e l e) + (u e u e) D (A; B) and (x e x e) + (y e y e) D (A; B): Therefore T (A) = [l T ; u T ; x T ; y T ] if and only if (l T ; u T ; x T ; y T ) R 4 is a solution of the problem min (l l e ) + (u u e ) + (x x e) + (y y e) (39) under the conditions x > ; (4) y > ; (4) x + y 6 u l; (4) 6l + 6u x y = 6l e + 6u e x e y e ; (43) 6l + 6u + x y = 6l e + 6u e + x e y e ; (44) where l e ; u e ; x e ; y e are given by (34)-(37). We immediately obtain that (39)-(44) is equivalent to min (x x e ) + (y y e ) (45) under the conditions In addition, and x > ; (46) y > ; (47) x + y 6 3u e 3l e x e y e: (48) l = 6 (x x e) + l e (49) u = 6 (y y e) + u e : (5) Let us consider the set M A = (x; y) R : x > ; y > ; x + y 6 3u e 3l e x e y e ; d E the Euclidean metric on R and let us denote by P M (Z) the orthogonal projection of Z R on non-empty set M R, with respect to d E. Theorem (Theorem 5, [3])The problem (45)-(48) has an unique solution.

22 Theorem 3 (Theorem 7, [3])Let A F (R) ; A = [A L (); A U ()]; [; ], and T (A) = [l T ; u T ; x T ; y T ] the nearest trapezoidal fuzzy number to A which preserves ambiguity and value. (i) If (ii) If (iii) If (3 ) A L ()d x T = 6 y T = 6 l T = u T = (3 ) A U ()d (5) ( ) A L ()d; ( ) A U ()d; A L ()d; A U ()d: (3 ) A L ()d + x T = 6 y T = l T = 3 u T = A L ()d + 6 A L ()d A U ()d ( ) A L ()d + x T = y T = 6 l T = u T = ( ) A U ()d > (5) A L ()d + 6 A L ()d A L ()d + 3 A U ()d A U ()d (3 ) A U ()d < (53) A U ()d A U ()d

23 (iv) If (3 ) A L ()d (3 ) A L ()d + (3 ) A U ()d > (54) ( ) A U ()d (55) and ( ) A L ()d + x T = 3 y T = 3 l T = u T = ( ) A L ()d + 3 (3 ) A U ()d (56) (3 ) A L ()d 3 (3 + ) A L ()d Corollary 4 (i) If A, (ii) If A, (iii)if A 3, (iv) If A 4, (3 ) A U ()d (3 ) A L ()d + ( ) A U ()d (3 ) A U ()d (3 + ) A U ()d: T (A) = (4I 6L; 6L I; 6U S; 4S 6U) : T (A) = (6L 4U; U; U; U) : T (A) = (L; L; L; 6U 4L) : T (A) = (I 6U + S; 3L I + 3U S; Example 5 Let A and B be fuzzy numbers 3L I + 3U S; I 6L + S) : A L () = + p ; p A U () = 4 ; B L () = + p ; B U () = 35 3 p : 3

24 We obtain that A and B 4 9 T (A) = 5 ; 3 5 ; 34 5 ; T (B) = ; ; ; : 5 5 Since we obtain that and so (A + B) L () = + p (A + B) U () = 39 3 p ; T (A) + T (B) = T (A + B) = thus the operator T is not additive ; 6 5 ; 64 5 ; 495 ; ; 6 5 ; 73 5 ; 457 ; 5 T (A) + T (B) 6= T (A + B) ; Theorem 6 If A; B i ; i f; ; 3; 4g, T (A) + T (B) = T (A + B) : 3. Trapezoidal approximation which preserves the ambiguity, the value and the core Theorem 7 If A = [A L () ; A U ()] ; [; ] is a fuzzy number one has T (A) = (t ; t ; t 3 ; t 4 ) ; where t = 6 t = A L () t 3 = A U () t 4 = 6 A L () d A L () A U () d A U () is the trapezoidal fuzzy number preserving the core, the ambiguity and the value. Example 8 Let A be a fuzzy number, with the parametric representation A L () = + p A U () = 3 7 p : 4

25 The trapezoidal fuzzy number preserving the core, the ambiguity and the value is 7 T (A) = 5 ; ; 3; 96 5 Example 9 Let A be a fuzzy number, with the parametric representation A L () = + 7 p p A U () = 3 : The trapezoidal fuzzy number preserving the core, the ambiguity and the value is T (A) = ; 8; 9; 5 5 Theorem 3 If A = (a; b; c; d) n is a fuzzy number the trapezoidal fuzzy number preserving the core, the ambiguity and the value is bn + 3a b cn + 3d c T (A) = ; b; c; : n + n + Theorem 3 The trapezoidal fuzzy number preserving the core, the ambiguity and the value T : F (R)!F T (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 ambiguity, the value and the support 4 Weighted approximation 4. Weighted approximation which preserves the support The processing of fuzzy numbers is sometimes di cult, therefore several methods have been introduced for approximating the fuzzy numbers with trapezoidal fuzzy numbers. Each of these methods, either trapezoidal approximation [5], [9], [5], [5], [5], [49], [8] or weighted approximation [?], [?] bring some bene ts and important properties. The Karush-Kuhn-Tucker theorem is used in this paper to compute the nearest weighted trapezoidal fuzzy number with respect to weighted distance, such that the support of given fuzzy number is preserved. Important properties such translation invariance, scale invariance, identity, nearest criterion and continuity of the weighted trapezoidal approximation are studied. 5

26 De nition 3 For two arbitrary fuzzy numbers A and B with and [B L (); B U ()] ; [; ] the quantity e(a; B) = 4 3 f () g (A ; B )d5 cut sets [A L (); A U ()] (57) is the weighted distance between A and B; where g (A ; B ) = [A L () B L ()] + [A U () B U ()] ; and f is a non-negative function, increasing on [; ] such that f() = (58) named the weighting function. For a fuzzy number A; A = [A L () ; A U ()] ; [; ] ; the problem is to nd a trapezoidal fuzzy number T (A) = (t ; t ; t 3 ; t 4 ) ; which is the nearest to A with respect to metric e (see 57) and preserves the support of the fuzzy number A. The fuzzy number has to ful lls the following conditions: min e(a; T (A)) t = A L () t 4 = A U () : (59) The problem is to determine four real numbers t ; t ; t 3 ; t 4 ; which are solutions of system (59), where t t t 3 t 4 ; so we have to minimize the function D; where D (t ; t ; t 3 ; t 4 ) = f () [A L () (t + (t t ) )] d + + f () [A U () (t 4 + (t 3 t 4 ) )] d: We reformulate the problem by (59) as follows min h (t ; t 3 ) 6

27 where + +t 3 h (t ; t 3 ) = +t f () A L () d t f () d + t A L () f () [A L () ( )] d +t 3 subject to + f () A U () d t 3 f () d f () A U () ( ) d + f () A L () d+ f () ( ) d f () A L () A L () ( f () A U () d f () A U () A U () ( A L () t t t 3 t 3 A U () : f () [A U () ( ) d ) d )] d: (6) (6) Theorem 33 Let A; A = [A L () ; A U ()] be a fuzzy number and T (A) = (t ; t ; t 3 ; t 4 ) the nearest (with respect to metric e) trapezoidal fuzzy number to fuzzy number A preserving the support of A: (i) If R f () [A L () + A U ()] d A L () R ( ) f () d A U () R ( + ) f () d > (6) (ii) If t = A L () (63) t = A U () (64) t 3 = A U () (65) t 4 = A U () : (66) R f () [A L () + A U ()] d+ +A L () R ( + ) f () d + A U () R ( ) f () d > (67) t = A L () (68) 7

28 (iii) If t = A L () (69) t 3 = A L () (7) t 4 = A U () : (7) f () [A L () A U ()] d [A L () A U ()] f () ( ) d (7) t = A L () (73) R t = L () d A L () R f () ( ) d R f () d (74) R t 3 = f () A U () d A U () R f () ( ) d R f () d (75) and t 4 = A U () : (76) (iv) If R f () [A L () A U ()] d [A L () A U ()] R f () ( ) d > (77) R f () [A L () + A U ()] d+ +A L () R ( + ) f () d + A U () R ( ) f () d (78) R f () [A L () + A U ()] d A L () R ( ) f () d A U () R ( + ) f () d ; (79) t = A L () t = R t 3 = L () + A U ()] d R f () d [A L () + A U ()] R f () ( ) d R f () d t 4 = A U () : Theorem 34 For a fuzzy number A = (a; b; c; d) n the nearest trapezoidal fuzzy number preserving the support of A, with f () = ; is T (A) = (t ; t ; t 3 ; t 4 ) : (i) If a d an + 4bn + 4cn 7dn > 8

29 and (ii) If (iii) If (iv) If t = a t = d t 3 = d t 4 = d: a d + 7an 4bn 4cn + dn > t = a t = a t 3 = a t 4 = d: a d an + 4bn 4cn + dn t = a t = a an + 4bn 3n + t 3 = d dn + 4cn 3n + t 4 = d: a d an + 4bn 4cn + dn > a d an + 4bn + 4cn 7dn a d + 7an 4bn 4cn + dn t = a t = t 3 = t 4 = d: a + d an + 4bn + 4cn dn (3n + ) Example 35 The case (i) of Theorem 34 is applicable to the fuzzy number and A = ( ; ; ; ) T (A) = ( ; ; ; ) is the nearest trapezoidal fuzzy number preserving the support of A: 9

30 Example 36 The case (ii) of Theorem 34 is applicable to the fuzzy number A = (5; ; ; 3) and T (A) = (5; 5; 5; 3) 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 and A = (; ; ; 3) T (A) = ; 8 7 ; 3 7 ; 3 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 and A = ( 3; ; 9; ) T (A) = 3; 6 7 ; 6 7 ; is the nearest trapezoidal fuzzy number preserving the support of A: 4. Weighted approximation which preserves the expected interval Let L ; U : [; ]! R be non-negative functions L () d > U () d > named weighting functions and the weighted distance s d (A; B) = L () [A L () B L ()] d + U () [A U () B U ()] d; where A; B are fuzzy numbers with A = [A L () ; A U ()] şi B = [B L () ; B U ()] ; [; ] : 3

31 Theorem 39 Let A; A = [A L () ; A U ()] ; [; ] be a fuzzy number and T (A) = (l; u; ; ) the nearest weighted trapezoidal approximation operator preserving the expected interval. (i) If e (! L ) + e (! U ) u e + l e (8) (ii) If l = l e e! L u = u e + e! U = e = e : e (! L ) + e (! U ) u e + l e > (8) c e (! U ) d e (! U ) (! L ) + d (u e l e ) (! L ) (8) d e (! L ) c e (! L ) (! U ) + c (u e l e ) (! U ) (83) l = l e c e (! U ) d e (! U ) (! L ) + d (u e l e ) (! L ) d (! L ) + c (! U ) u = u e + de (! L ) c e (! L ) (! U ) + c (u e l e ) (! U ) d (! L ) + c (! U ) = ce (! U ) d e (! U ) (! L ) + d (u e l e ) (! L ) d (! L ) + c (! U ) = de (! L ) c e (! L ) (! U ) + c (u e l e ) (! U ) d (! L ) + c (! U ) : (iii) If! L! U c e (! U ) d e (! U ) (! L ) + d (u e l e ) (! L ) < (84) l = l e u = u e + ue l e! U = = ue l e :! U! U 3

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