Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers

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1 Advances in Fuzzy Mathematics ISSN X Volume, Number 3 (7, pp Research India Publications Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers Sunil N Yadav a and Madhav S Bapat b, * a Research Scholar, Department of Mathematics, Shivaji University, Kolhapur 46 4 M S, India b Department of Mathematics, Willingdon College, Sangli, 46 45, MS, India Abstract In this paper we discuss the problem of multiplication approximation of trapezoidal fuzzy numbers We introduce the sufficiently effective and simpler notation for trapezoidal fuzzy number and we call it as trapezoidal epsilondelta fuzzy number This outwits computational complexity of fuzzy numbers and introduces simplicity An approximation of multiplication of trapezoidal fuzzy numbers which preserves the core, value, and ambiguity is obtained Some properties of newly introduced multiplication approximation of trapezoidal epsilon-delta fuzzy numbers are studied This paper examines for the distance between the fuzzy number obtained by using new multiplication approximation with respect to distance metric and the fuzzy number obtained by extension principle based multiplication The distance is minimum than it is between the fuzzy number obtained by Dubois and Prade method and the fuzzy number obtained by extension principle based multiplication, with respect to distance metric Finally, example is given to illustrate the simplicity and computational efficiency of the proposed approximation AMS (: 94D Keywords: Trapezoidal epsilon-delta fuzzy number; Multiplication of trapezoidal fuzzy numbers; Trapezoidal approximation INTRODUCTION The notion of fuzzy number was introduced by Dubois and Prade [9, ] Trapezoidal approximation of fuzzy number was considered by many scholars [, 4, 6, 8, 4, 5,

2 76 Sunil N Yadav and Madhav S Bapat 6], etc which preserves certain criteria for approximation, semi-trapezoidal approximation was proposed by Nasibov and Peker [8] and improved by Ban [, 3]Many applications have proved that fuzzy set theory let us successfully model and transform imprecise information But the main barrier in the development of applications is the computational complexity Hence more attention is needed to ease the arithmetic computation with fuzzy numbers Therefore, some approximate methods are needed to simplify multiplication operation in particular Operations on fuzzy sets are extended by the extension principle introduced by Lotfi Zadeh Therefore, arithmetic operation on fuzzy numbers can be obtained by using extension principle; but for triangular or trapezoidal fuzzy numbers some operations, particularly multiplication the resultant fuzzy number need not be triangular or trapezoidal Hence further manipulation becomes more complicated The problem of finding the nearest parametric approximation of a fuzzy number with respect to the average Euclidean distance is solved by A Ban et al [,, 3] P Grzegorzewski et al [,, 3], etc developed methods for triangular or trapezoidal approximation of fuzzy numbers Hence multiplication of two triangular or trapezoidal fuzzy numbers which is not triangular or trapezoidal according to extension principle can be approximated by a triangular or trapezoidal fuzzy numbers In this paper, we obtain trapezoidal approximation for multiplication of epsilon-delta fuzzy numbers which preserves core, value and ambiguity Finally, we discuss some properties of newly introduced trapezoidal approximation In Section we introduce basic notions In Section 3 we discuss extension principle-based operations between trapezoidal epsilon-delta fuzzy numbers on and their characteristics In Section 4 we give new approximation of multiplication of fuzzy numbers by trapezoidal epsilon-delta fuzzy number on Some properties (core, value, ambiguity of newly developed approximation are studied PRELIMINARIES A fuzzy subset A of a set X is a function A: X [,] For (,], the crisp set x X A( x is defined as suppa x : A( x is called -cut, of a fuzzy set A, denoted by A The support suppa If Ax ( then A is called normal If each α-cut of A is convex then the fuzzy set A is called convex We assume X, the set of real numbers A fuzzy number A is a fuzzy subset of which is normal, convex and upper semi-continuous with bounded support If left and right curves are linear then the fuzzy number is called triangular or a trapezoidal fuzzy number The triangular fuzzy number is a particular type of a trapezoidal fuzzy number in which core is a singleton set The membership function of a trapezoidal fuzzy number A is of the form

3 Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers 77 ( x c if c x a, a c if a x b, Ax ( x d if b x d, b d, otherwise The above trapezoidal fuzzy number is denoted by A ( c, a, b d ( where For a fuzzy number A every α-cut is a closed interval Let A [ AL(, AU( ] A ( inf ( L x A x ( A ( sup ( U x A x (3 for and A [ AL(, AU(] The set of all fuzzy numbers is denoted by ( For a fuzzy number A we have the following characteristics The value Val( A, ambiguity Amb( A, width wa ( are introduced by (see [7], [8], [], [7]The expected interval EI ( A and the expected value EV ( A are given by (see [], [7] Val( A ( AL( AU( d (4 Amb( A ( AU( AL( d (5 w( A ( AU( AL( d (6 EI( A AL( d, AU( d (7 EV ( A AL ( d A ( d U (8 For a fuzzy number A and B with α-cut A [ AL(, AU( ] and B [ BL(, BU( ] respectively, a metric on the set of fuzzy numbers, which is an extension of the Euclidean distance, is defined by (see [], [3]

4 78 Sunil N Yadav and Madhav S Bapat d ( A, B AL ( BL ( AU ( BU ( d (9 In practice, fuzzy numbers with simple membership functions are preferred The most often used fuzzy numbers in many applications are triangular or trapezoidal fuzzy numbers If r is a real number then ε-fuzzy number r, for some,(ε is a symmetric fuzzy set whose support is the interval ( r, r The membership function of r, ε-fuzzy number is defined by(see [5] x( rε, if r ε x r, ε x ( r ε r ( x, if r x r ε, ε, otherwise The set of all epsilon fuzzy numbers is denoted by ( ( 3 TRAPEZOIDAL EPSILON-DELTA FUZZY NUMBERS AND ARITHMETIC OPERATIONS BETWEEN THEM 3 Trapezoidal epsilon-delta fuzzy number on Nonlinear membership functions of a fuzzy set causes problems with arithmetic computations, in particular with multiplication Therefore a suitable approximation is needed to replace the non linear membership function by linear function We introduce trapezoidal fuzzy number a, b, ( a, b ; a b for some,,(ε, called as trapezoidal ε-δ fuzzy number which is a notational simplification of a trapezoidal fuzzy number This simplification is computationally inexpensive under the arithmetic operations on fuzzy numbers Definition 3 If a, b are real numbers then trapezoidal ε-δ fuzzy number [ ab, ], for some,,(ε, is a fuzzy set [ ab, ], : [,] defined by x( a for a x a, for a x b, [ a, b], ( x x ( b for b x b, otherwise (

5 Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers 79 A trapezoidal fuzzy number A ( c, a, b, d in above notation is denoted by A [ a, b] ac, d b Also, [ a, b], ( a, a, b, b Thus, the fuzzy number A ( c, a, b, d and [ a, b], ( a, a, b, b are equivalent If a bwe obtain epsilon fuzzy number [5], the conditions ( a aand b( b imply the closed interval (rectangular trapezoidal epsilon-delta fuzzy number and if ( a a b ( b r we get a crisp number ie, a fuzzy number represents a precise value that can be identified with the proper real number r The set of all T trapezoidal epsilon-delta fuzzy numbers is denoted by, ( Pictorial representation of trapezoidal epsilon-delta fuzzy number is as shown in Fig [ a, b] ( x, a a b b Fig: Membership functions of the trapezoidal epsilon-delta fuzzy number [ ab, ], The level-cut of trapezoidal ε-δ fuzzy number [ ab, ], for some,,(ε, is ([ ab, ],, [,] is given by ([ a, b], = [ a ε( α, b ( α] The support of trapezoidal ε-δ fuzzy numbera, b,, a b isa (, b (, ab,,, and, Therefore A a( and A b ( ( ab, The length of, is denoted by, l (( a, b ( b a ( (, L l( a, b U and it is given by, which is independent of ( b a Note that for ε =, δ = the trapezoidal -fuzzy number[ ab, ] is the characteristic function of{( b a} If we put a binto ( then we get triangular representation of epsilon-delta fuzzy numbers as a, ( a, a, a which is effective notational simplification of a triangular fuzzy number A ( l, m, n Definition 3 Trapezoidal ε-δ fuzzy number [ ab, ], for some,,(ε, is said to be positive if a, and negative if b Note that a fuzzy number need not be either positive or negative

6 7 Sunil N Yadav and Madhav S Bapat Proposition 33 If [ ab, ], is a trapezoidal ε-δ fuzzy number ab, ;, ;,, then it has following characteristics [5] (i α-cut: ([a,b], = [ a ε( α, b ( α] (ii Support: ([, ], ab = (iii Core: ([ ab, ], = [ a, b] a, b,, ;, (iv Value: Val ( [ ab, ], = ( a b ( a (, b ( ; a, b,, ;, 6 (v Ambiguity: Amb([ a, b], = ( b a ( 6 (vi Width: w([ a, b], = ( b a ( (vii Expected Interval: EI ([ a, b] a, b, (viii Expected Value: EV ([ a, b] ( a b (, 4 Proof Let a, b are real numbers then trapezoidal ε-δ fuzzy number [ ab, ], for some,,(ε, is a fuzzy set [ ab, ], : [,] defined by Then x( a for a x a, for a x b, [ a, b], ( x x ( b for b x b,, otherwise, (i α-cut: a, b = x :[ a, b ]( x x( a x( b x a ( and x b ( a, b = [ a ε( α, b ( α], (ii Support: a, b = x :[ a, b ]( x, x( a x( b x a and x b a, b = ( a ε, b, (iii Core: a, b = x :[ a, b ]( x,

7 Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers 7 a b x( a x aand, = [ a, b], Val a, b ( A ( A ( d (iv Value x( b x b, L U (( a ( ( b ( d ( a b ( 6 (v Ambiguity: Amb([ a, b] ( A ( A ( d (vi Width: (( b ( ( a ( d = 6 U ( b a (, U L w a, b ( A ( A ( d (( b ( ( a ( d ( b a ( = (vii Expected Interval: EI ([ a, b], AL( d, AU( d ( a ( d, ( b ( d a, b (viii Expected Value: EV ([ a, b], AL ( d A ( d ( a ( d ( b ( d ( a b ( 4 L U

8 7 Sunil N Yadav and Madhav S Bapat 3 Arithmetic operations on trapezoidal epsilon-delta fuzzy numbers The operation of addition is defined in [9, ] We define addition of two trapezoidal epsilon-delta fuzzy numbers as following (a Addition Addition of ε-δ trapezoidal fuzzy numbers[ ab, ] and[ cd, ] is trapezoidal fuzzy number given by,,,, [ a c, b d] [ a, b] [ c, d], Proposition 34 If [ ab, ] and [ cd, ] are trapezoidal ε-δ fuzzy numbers, then we, have the following properties, (i α-cut: a c b d, a b, c d, [, ] [, ] [, ], (ii Support: ([ a c, b d] a, b c, d (iii Core:,,, ([ a c, b d] a, b c, d,,, (iv Value: Val ([ a c, b d],,, ( a b c d ( ( 4 (v Ambiguity: Amb [ a c, b d], (( b d ( a c ( (viwidth: 6 w [ a c, b d] ( b d ( a c ( (viiexpected Interval: EI ([ a c, b d],, [( a c (,( b d ( ] = (viiiexpected Value: EV [ a c, b d], ( a b c d (( ( 4 Proof The proof follows from Proposition (33 (b Scalar Multiplication For any scalar,, then [ a, b], [ a, b], and if, then [ a, b], [ b, a],

9 Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers 73 (c Negation Negation of trapezoidal ε-δ fuzzy number [ ab, ], is defined as ([ a, b], [ b, a], (d Subtraction Subtraction of trapezoidal ε-δ fuzzy number [ a, b] [ c, d] [ a d, b c],,, [ ab, ] and, [ cd, ] defined as These operations are consistent with the extension principle We can obtain the properties of scalar multiplication, negation and subtraction similar to addition [Proposition 34], (emultiplication Let[ ab, ] and[ cd, ] be - fuzzy numbers, then,, ([ a, b] a (, b (,([ c, d] c (, d (,, Therefore ([ a, b], ([ c, d], a (, b ( c (, d ( a c a d b c b d max{ a ( c (, a ( d (, b c b d [min{ ( (, ( (, ( (, ( ( }, ( (, ( ( }] b c b d a c a d b c b d Let x [min{ a ( c (, a ( d (, ( (, ( ( }, max{ ( (, ( (, For positive - fuzzy numbers [ ab, ] and[, ] ( (, ( ( }], cd x [ ac ( a c ( (, bd ( b d ( ( ] ac ( a c ( ( x bd ( b d ( ( Now ac ( a c( ( x, yields,

10 74 Sunil N Yadav and Madhav S Bapat ( a c ( a c 4( ( ac x ( Taking into consideration [,] Now, therefore ( a c ( a c 4( ( ac x ( a c ( a c 4( ( ac x ac ( a c x ac Similarly for x bd ( b d ( (, yields ( b d ( b d 4 ( bd x (3 Taking into consideration [,] Now, implies ( b d ( b d 4 ( bd x ( b d ( b d 4 ( bd x From equations ( and (3 the multiplication of trapezoidal epsilon-delta fuzzy numbers [ ab, ] and[ cd, ] is defined by, ([ a, b], [ c, d], ( x, ( a c ( a c 4( ( ac x ( if ac ( a c x bd, ( b d ( b d 4 ( bd x ( if bd x bd ( b d, otherwise Multiplication given by (4 is extension principle based, which gives polygon curve (4

11 Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers 75 Multiplication of two trapezoidal ε-δ fuzzy numbers[ ab, ] and[ cd, ] by extension principle method need not be a trapezoidal fuzzy number Since multiplication is not trapezoidal fuzzy number we approximate it by trapezoidal fuzzy number,, Proposition 35 Multiplication of ε-δ fuzzy numbers[ ab, ] and[ cd, ] by extension principle method has the following properties (i -cut: [ a, b], [ c, d], [ a, b], [ c, d], [ ac ( a c ( ( (, bd ( b d ( ( ( ],, (iisupport: [ a, b], [ c, d], [ a, b], [ c, d], ( ac ( a c, bd ( b d (iii Core: [ a, b] [ c, d] [ a, b] [ c, d] ac, bd,,,, (ivvalue: Val [ a, b], [ c, d], ( ac bd ( b d ( a c ( (vambiguity: 6 Amb [ a, b] [ c, d] ( bd ac ( a c b d ( (viwidth:,, 6 w [ a, b] [ c, d] ( bd ac ( b d a c (,, 3 (viiexpected Interval: EI [ a, b], [ c, d], ac ( a c 3 (, bd ( b d 3 ( (viii Expected Value: EV [ a, b] [ c, d] ( ac bd (( b d ( a c (,, 4 6 Proof Let[ ab, ] and[ cd ] be trapezoidal ε-δ fuzzy numbers, by extension principle we have,,,,

12 76 Sunil N Yadav and Madhav S Bapat ([ a, b], [ c, d], ( x (iif ac ( a c x ac then,,,, ( a c ( a c 4( ( ac x ( if ac ( a c x ac, ( b d ( b d 4 ( bd x ( if bd x bd ( b d, otherwise ([ a, b] [ c, d] x : ([ a, b] [ c, d] ( x implies that ( a c ( a c 4( ( ac x [ a, b], [ c, d], ( [ a, b], [ c, d], ( ( a c ( a c 4( ( ac x ( a c a c ac x ( ( 4 ( 4 ( x ac ( a c( ( ( (5 and, if bd x bd ( b d, then ( b d ( b d 4 ( bd x [ a, b], [ c, d], ( ( ( b d ( b d 4 ( bd x ( b d b d bd x ( ( ( 4 ( 4 ( x bd ( b d( ( ( (6 Therefore, from (5 and (6 we obtain [ a, b], [ c, d],

13 Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers 77 [ ac ( a c ( ( (, bd ( b d ( ( ( ],, (ii Let x ac ( a c( ( (, bd ( b d ( ( ( ac ( a c ( ( ( x bd ( b d ( ( ( [ a, b], [ c, d], ( ac ( a c, bd ( b d (iii Core: ([ a, b], [ c, d], x : ([ a, b], [ c, d], ( x and ( a c ( a c 4( ( ac x ( if ac ( a c x ac, ( b d ( b d 4 ( rs x ( x [ ac, bd] if bd x bd ( b d ([ a, b] [ c, d] [ ac, bd],, Since, AL ac ( a c( ( and AU bd ( b d( ( We have (iv The value, Val( A ( AL( AU( d ( ac ( a c( ( Val([ a, b], [ c, d], d ( bd ( b d( ( ( ac bd ( b d ( a c ( 6

14 78 Sunil N Yadav and Madhav S Bapat (v The ambiguity, Amb( A ( A ( A ( d U L ( bd ( b d( ( Amb([ a, b], [ c, d], d ( ac ( a c( ( ( bd ac ( a c b d ( 6 (vi The width, w( A ( A ( A ( d U L ( bd ( b d( ( w([ a, b], [ c, d], d ( ac ( a c( ( ( bd ac ( b d a c ( 3 (vii The expected interval, EI( A AL( d, AU( d EI [ a, b] [ c, d] ( ( (, ac a c d,, bd ( b d( ( d (viii The expected value, EV ( A AL ( d AU ( d EV [ a, b] [ c, d] ac ( a c( ( d,, bd ( b d( ( d ( ac bd (( b d ( a c ( 4 6

15 Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers 79 IV TRAPEZOIDAL APPROXIMATION OF MULTIPLICATION OF EPSILON-DELTA FUZZY NUMBERS The new approximation presented in this section is based upon an analysis of the extension based product s significant functional characteristics and then conceptualization of a computationally simpler form with similar characteristics The aim is to develop an expression which better approximates the extension based product while maintaining computational efficiency and simplicity Examining the shape of the actual product we note that it is a polynomial curve whereas the new approximation is a straight line Approximating the polynomial curve with a linear curve is the main source of work The computational task is cumbersome and computationally demanding This is the rationale for the development and use of the new approximate Because of the presence of nonlinear term in the expression of multiplication of fuzzy numbers by extension principle it yields higher degree polygon curve From the point of view of computation we give linear trapezoidal approximation of fuzzy numbers which preserves the core, the value and the ambiguity In many works that introduced approximation operators, problem of minimum had been required, in order to identify these operators, because, for a better approximation, we used minimization of distance between the extension principle based fuzzy number and approximate We solve this problem by using distance metric The extension principle based multiplication of two positive trapezoidal ε-δ fuzzy numbers[ ab, ] and[ cd, ] is defined by,, [ a, b], [ c, d], ac ( a c( ( (, bd ( b d( ( (,, The term ( in the multiplication operation between trapezoidal fuzzy numbers can be made linear by using (, Definition 4 Trapezoidal approximation of multiplication of two positive trapezoidal ε-δ fuzzy numbers[ ab, ] and[ cd, ] is defined by,, [ a, b] [ c, d] [ ac, bd] a c t b d t,, (, ( Definition 4 Multiplication of two positive trapezoidal - fuzzy numbers [ ab, ] and [ cd, ] is defined by,, [ a, b] [ c, d] [ ac, bd] a c b d,,, Definition 43 Multiplication of two negative - fuzzy numbers[, ] ab, and

16 73 Sunil N Yadav and Madhav S Bapat [ cd, ] is defined by, [ a, b] [ c, d] [ bd, ac] b d c,, (, (a Definition 44 Multiplication of negative and a positive - fuzzy numbers [ a, b], a b; b and[ c, d],( c d, c is defined by,, [ a, b] [ c, d] [ ad, bc] a d b c,, (, ( In most of the applications we deal with positive trapezoidal - fuzzy numbers and hence we restrict the properties of multiplication to positive trapezoidal - fuzzy numbers only For t trapezoidal approximation formula for multiplication of positive trapezoidal epsilon-delta fuzzy numbers is given by definition (4 Some properties of this trapezoidal approximation are given below Proposition 4 Fort trapezoidal approximation of multiplication of positive trapezoidal ε-δ fuzzy numbers[ ab, ] and[ cd, ] has following characteristics, (i -cut: ac, bd ( a c,( b d = ac a c bd b d [ ( ( ( (, ( ( ( ( ] Support: ([ ac, bd] a c, b d, ( ac ( a c, bd ( b d 3 Core: ([ ac, bd], [ ac, bd] a c b d 4Value: Val([ ac, bd] ( a c, ( b d ( ac, bd ( b d a c ( 6 5 Ambiguity Amb([ ac, bd] ( a c, ( b d : ( bd ac ( a c b d ( 6 6Width: w([ ac, bd] ( a c, ( b d ( bd ac ( a c b d ( 4 7Expected Interval: EI ([ ac, bd] ( a c, ( b d [ ac ( a c (, bd ( b d ( ] 4 4

17 Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers 73 8 Expected Value: EV ([ ac, bd] ( a c, ( b d ( ac bd [( b d ( a c ] ( 4 8 Note that ([ a, b] [ c, d] ( [ a, b] [ c, d] ([ a, b] ( [ c, d] [ a, b] ( [ c, d],,,,,,,, Proposition 4 Fort trapezoidal approximation of multiplication of positive trapezoidal ε-δ fuzzy numbers[ ab, ],,[ cd, ], and[ e, f] 3, is commutative and 3 associative Proof Let[ ab, ],[ cd, ] and [ e, f], be a ε-δ fuzzy numbers whose support lies in, Then we have, 3 3 [ a, b] [ c, d] [ ac, bd] [ ca, db], by,, a c, b d c a, d b symmetry of expression Next, if [ ab, ], [ cd, ], [ e, f], [ a, b] [ c, d],, 3, 3 ε-δ fuzzy numbers whose support lies in ([ ac, bd] ([ e, f ] Therefore, a c, b d 3, 3 [ c, d] [ a, b] and,,,, we have ([ a, b] [ c, d] [ e, f ] are the =,, 3, 3 [ ace, bdf ] ac e a c a c bd f b d b d ( 3 ( 3 (,( 3 ( 3 ( [ ace, bdf ] a c e ce c e b d f df d f And, (7 ( 3 3 ( ( 3 3, ( 3 3( ( 3 3 [ a, b] ([ c, d] [ e, f ] [ a, b] [ ce, df ] c e d f,, 3, = 3,, [ ace, bdf ] a c e ce c e b d f df d f ( 3 3 ( ( 3 3, ( 3 3( ( 3 3 = [ ace, bdf ] a ( c 3 e 3 ( ce ( c 3 e 3, b ( d 3 f 3 ( df ( d 3 f (8 3 From (7 and (8 we have ([ a, b] [ c, d] [ e, f ] [ a, b] ([ c, d] [ e, f ] =,, 3, 3,, 3, 3 Proposition 43 Fort trapezoidal approximation of multiplication of positive trapezoidal ε-δ fuzzy numbers is distributive over addition Proof: Let[ ab, ],,[ cd, ], and [ e, f] 3, be a ε-δ fuzzy numbers whose support lies 3

18 73 Sunil N Yadav and Madhav S Bapat in ; a, b, c, d, e, f ;, ;, ;,,3 Then we have i i i i i [ a, b] ([ c, d] [ e, f ] [ a, b] ([ c e, d f ] and,, 3, 3, 3, 3 = [ a ( c e, b ( d f ] a ( 3 ( c e ( 3, b ( 3 ( d f ( 3 (9 ([ a, b] [ c, d] [ a, b] [ e, f ] [ ac, bd] [ ae bf ],,, 3, 3 a c, b d a 3 e 3, b 3 f 3 [ ac ae, bd bf ] a c a e b d b f 3 3, 3 3 [ ac ae, bd bf ] ( a c e b d f From (9 and ( we have,, 3, 3 ( 3 ( ( 3, ( 3 ( ( 3 [ a, b] ([ c, d] [ e, f ] = [ a, b] [ c, d] [ a, b] [ e, f ],,, 3, 3 Proposition 44 Fort trapezoidal approximation of multiplication of positive trapezoidal ε-δ fuzzy numbers preserves core, value and ambiguity Proof The proof follows from proposition (35, 4 Note 45The above propositions also hold for negative ε-δ fuzzy numbers Note 46The multiplication of ε-δ fuzzy numbers which are neither positive nor negative can also be obtained Proposition 47 Let[ ab, ] and [ cd, ] be positive epsilon-delta fuzzy numbers,, such that a c; b d and if P [ a, b] [ c, d] ([ ac, bd], where t, be a trapezoidal t,, a c t(, b d t T approximation of a multiplication Then with respect to metric d on, ( defined by L L U U, where A and B are d( A, B ( A ( B ( d ( A ( B ( d fuzzy numbers with -cuts A [ AL(, AU( ] and B [ BL(, BU( ] T respectively Then Pt has the order Pe P P with respect to metric d on, ( Proof Let P t ( t, denote approximation for multiplication for positive trapezoidal epsilon-delta fuzzy numbers[ ab, ] and [ cd, ] ( a c, b d We have,,

19 Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers 733 ( Pe ac ( a c ( (, bd ( b d ( ( and ( ( P ac ( a c ( t(, bd ( b d ( t( ( t Then d( P P ( t ( ( t ( d (3 for t, we have e t 5 d( P P ( ( (4 e 3 d( P P ( ( (5 e From (3 and (4 we have d( P P d( P P Therefore we have e e P P P e We note that triangular fuzzy numbers are a special cases of trapezoidal fuzzy numbers in which the mode is a point instead of a flat line Consequently, the results for triangular fuzzy numbers can be obtained from trapezoidal fuzzy numbers by using a point value for the mode set for the left and right curves Example Let ab, 3,5 and, 4,7,,7 cd be two positive trapezoidal ε-, 3, δ fuzzy numbers Then characteristics of,7 3, 3,5 4,7 by extension principle, Dubois Prade approximation, trapezoidal approximation fort, is depicted in the following Table 4 Characteristics Table 4: presents multiplication Extension Principle ( 3,5 4,7 Polygonal,7 3,,7 3, 3,5 4,7 Dubois-Prade Approximation for t = Trapezoidal Approximation for t = ½,35 3,59,35 3/,66 function Core [,35] [,35] [,35] Value Ambiguity Width

20 734 Sunil N Yadav and Madhav S Bapat 65,696 [55, 645] Expected Interval 65,68 Expected Value Distance Table 4 shows new multiplication approximation preserves the characteristics core, value, and ambiguity It also demonstrates the minimum of distance between the fuzzy numbers obtained by extension based multiplication and new approximation V CONCLUSION In the present paper we have considered the problem of multiplication approximation of trapezoidal fuzzy numbers by introducing simple notation for representation We have discussed some properties of the approximation The proposed multiplication approximation satisfies basic algebraic properties of multiplication Output given by the new multiplication approximation is close to output given by extension principle with respect to average Euclidean distance Therefore, the approximation is useful in many applications Because of the nonlinear membership functions of a multiplication operation between fuzzy numbers, it is open for fuzzy numbers In order to outwit this disadvantage, future work needs to be done to develop the new approximation of multiplication of fuzzy numbers which significantly avoid the loss of the information and computationally swift and simple The work performed here is restricted to the multiplication operation, for division and inverse a similar approach may be used to develop more accurate approximations REFERENCES [] S Abbasbandy, B Asady, The nearest trapezoidal fuzzy number to a fuzzy quantity, Appl Math Comput 56 ( [] A I Ban, Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval, Fuzzy Sets and Syst 59 ( [3] A I Ban, On the nearest parametric approximations of fuzzy numbers--- Revisited, Fuzzy Sets and Syst66 ( [4] A I Ban, A Brandas, L Coroianu, C Negrutiu, O Nica, Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the value and ambiguity, Comput Math Appl 6 ( [5] M S Bapat, S N Yadav, P N Kamble, Triangular approximations of fuzzy numbers, International Journal of Statistika and Mathematika, ISSN: E-ISSN:49 865, Volume 7, Issue 3, ( [6] L Coroianu, M Gagolewski, P Grzegorzewski, Nearest piecewise linear approximation of fuzzy numbers, Fuzzy Sets and Systems 33 (3 6 5

21 Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers 735 [7] S Chanas, On the interval approximation of a fuzzy number, Fuzzy Sets and Systems ( [8] M Delgado, M A Vila, W Voxman, On a canonical representation of a fuzzy numbers, Fuzzy Sets and Systems 93 ( [9] D Dubois, H Prade, Operations on fuzzy numbers, Int J Syst Sci 9 ( [] D Dubois, H Prade, The mean value of a fuzzy number, Fuzzy Sets Syst 4 ( [] D Dubois, H Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York (98 [] P Grzegorzewski, Metrics and order in space of a fuzzy numbers, Fuzzy Sets and Systems 97 ( [3] P Grzegorzewski, Nearest interval approximation of a fuzzy number, Fuzzy Sets and Systems 3 ( 3 33 [4] P Grzegorzewski, Natural trapezoidal approximations of a fuzzy numbers, Fuzzy Sets and Systems 5 (4 9 9 [5] PGrzegorzewski, E Mrowka, Trapezoidal approximations of fuzzy numbers, Fuzzy Sets and Systems 53 ( [6] PGrzegorzewski, E Mrowka, Trapezoidal approximations of fuzzy numbers-- revisited, Fuzzy Sets and Systems 58 ( [7] S Heilpern, The expected value of a fuzzy number, Fuzzy Sets and Systems 47 ( [8] E N Nasibov, S Peker On the nearest parametric approximation of a fuzzy number Fuzzy Sets and Systems 59 (

22 736 Sunil N Yadav and Madhav S Bapat