An LU-fuzzy Calculator for the Basic Fuzzy Calculus"

Transcription

1 WP-EMS # 07/01 An LU-fuzzy Calculator for the Basc Fuzzy Calculus" Lucano Stefann (U. Urbno) Laerte Sorn (U. Urbno) WP-EMS # 007/01

2 An LU-fuzzy calculator for the basc fuzzy calculus Lucano Stefann, Laerte Sorn Faculty of Economcs Unversty of Urbno "Carlo Bo" Va A. Sa 4, 6109 Urbno, Italy Abstract The LU-model for fuzzy numbers has been ntroduced n [4] and appled to fuzzy calculus n [9]; n ths paper we buld an LU-fuzzy calculator, n order to explan the use of the LU-fuzzy representaton and to show the advantage of the parametrzaton. The calculator produces the basc fuzzy calculus: the arthmetc operatons (scalar multplcaton, addton, subtracton, multplcaton, dvson) and the fuzzy extenson of many unvarate functons (power wth nteger postve or negatve exponent, exponental, logarthm, general power functon wth numerc or fuzzy exponent, sn, arcsn, cos, arccos, tan, arctan, square root, Gaussan and standard Gaussan functons, hyperbolc snh, cosh, tanh and nverses, erf error functon and complementary erfc error functon, cumulatve standard normal dstrbuton). The use of the calculator s llustrated. 1 Introducton The arthmetc operatons on fuzzy numbers are usually approached ether by the use of the extenson prncple (n the doman of the membershp functon) or by the nterval arthmetcs (n the doman of the cuts): The exact analytcal fuzzy mathematcs dates back from the early eghtes and are outlned by Dubos and Prade (see [1]); the same authors have ntroduced the well known L-R model and the correspondng formulas for the fuzzy operatons (see []). Very recent lterature on fuzzy numbers s rch of contrbutons on the fuzzy arthmetc operatons and the use of smple formulas to approxmate them; an extensve survey and bblography on fuzzy ntervals s n [3]. We suggest n [4] the use of monotonc splnes to approxmate the fuzzy numbers, usng several nterpolaton forms (monotonc ratonal nterpolators and mxed cubc-exponental nterpolator) and we derve a procedure to control 1

3 the locatons of the nodes so that the error of the approxmaton s controlled by the possble nserton of addtonal nodes nto the pecewse nterpolaton. We see that, wth only a few nodes, our approxmatons of fuzzy calculus mantan accurate results. The parametrc LU representaton of the fuzzy numbers allows a set of possble shapes (types of membershp functons) that seems to be much wder than the well-known L-R framework. The paper s organzed as follows: secton contans a bref descrpton of the fuzzy calculus; n secton 3 we descrbe the LU-fuzzy model and n secton 4 we descrbe the detaled algorthms whch mplement the LU-fuzzy extenson prncple. Secton 5 contans the descrpton of the LU-fuzzy calculator. Basc fuzzy calculus We adopt the so called a cut settng for the de nton of a fuzzy number: De nton 1 A contnuous fuzzy number (or nterval) u s any par (u ; u ) of functons u : [0; 1]! R satsfyng the followng condtons: () u :! u R s a bounded monotonc ncreasng (non decreasng) contnuous functon [0; 1] ; () u :! u R s a bounded monotonc decreasng (non ncreasng) contnuous functon [0; 1] ; () u u [0; 1] : If u 1 < u 1 we have a fuzzy nterval and f u 1 = u 1 we have a fuzzy number. The notaton: u = u ; u denotes explctly the cuts of u. We wll also refer to u and u as the left (lower) and the rght (upper) branches on u, respectvely. If u = (u ; u ) and v = (v ; v ) are two gven fuzzy numbers, the arthmetc operatons are de ned as follows: De nton (Addton) u v = u v ; u v or, n terms of cuts, for [0; 1] : (u v) = u v ; u v De nton 3 (Scalar Multplcaton) For gven k R ku = (ku ; ku ) f k > 0 ku = (ku ; ku ) f k < 0 or, for [0; 1] : (ku) = mn ku ; ku ; max ku ; ku (1) ()

4 R u 0 u α u 1 u 1 u 0 α u α 1 α Fgure 1: The cut representaton of a general fuzzy nterval. In partcular, f k = 1, we obtan: u = u ; u wth the cuts : De nton 4 (Subtracton) ( u) = u ; u, [0; 1] u v = u ( v) = u v ; u v or, n terms of the cuts : (u v) = u v ; u v, [0; 1] (3) De nton 5 (Multplcaton) uv = (uv) ; (uv) where, [0; 1] : (uv) = mn fu v ; u v ; u v ; u v g (uv) = max fu v ; u v ; u v ; u v g (4) 3

5 De nton 6 (Dvson) If 0 = v 0 ; v 0 : where, [0; 1] : < : u v u v u v = u v u ; v n = mn u ; u v v n = max u ; u v v o ; u ; u v v ; u v ; u v We denote by F the set of the fuzzy numbers/ntervals. 3 Basc LU-fuzzy calculus o (5) The parametrc LU representaton of a fuzzy number s de ned on a decomposton of the nterval [0; 1] 0 = 0 < 1 < ::::: < 1 < < :::: < N = 1 for both the lower u () and the upper u () branches of the fuzzy numbers nvolved. In each of the N subntervals the values of the two functons and of ther rst dervatves I = [ 1 ; ], = 1; ; :::; N u ( 1 ) = u 0;, u ( 1 ) = u 0; u ( ) = u 1;, u ( ) = u 1; u 0 ( 1 ) = d 0;, u 0 ( 1 ) = d 0; u 0 ( ) = d 1;, u 0 ( ) = d 1; are assumed to be known; we are nterested n famles of monotonc functons that satsfy the above eght Hermte-type condtons for each subnterval I. In general, by the use of the followng transformaton of each subnterval I nto the standard [0; 1] nterval, t = 1 1 ; I ; (6) we can determne each pece ndependently and obtan the general left-contnuous LU-fuzzy numbers. Globally contnuous or more regular C (1) fuzzy numbers can be obtaned drectly from the data f the followng condtons are met for the values: u 1; = u 0;1, u 1; = u 0;1 for = 1; ; :::; N 1 4

6 and possbly for the slopes d 1; = d 0;1, d 1; = d 0;1 for = 1; ; :::; N 1: Let p (t) denotes the approxmaton of u on a generc subnterval I of the decomposton, wth the transformaton t = 1 1 ; so that each subnterval s re-mapped to the standard nterval [0; 1] by p (t) = u( 1 t( 1 )) (7) p 0 (t) = u 0 ( 1 t( 1 ))( 1 ): For smplcty of notaton, we omt the subscrpt and we refer to I = [0; 1].e. to the two-pont Hermte nterpolaton problem of determnng the monotonc functon p (t), t [0; 1], such that p(0) = u 0, p 0 (0) = d 0, p(1) = u 1, p 0 (1) = d 1 : () If u 0 u 1 (.e. the data are ncreasng) then d 0 0 and d 1 0 are requred and, f u 0 u 1 (.e. the data are decreasng) then d 0 0 and d 1 0 are requred. In partcular, u 0 = u 1, d 0 = d 1 = 0: We wll use the notaton (m; n)-ratonal to mean the rato P (t) Q(t) of an m- degree polynomal P (t) to an n-degree polynomal Q(t): 3.1 Quadratc/quadratc ratonal splne The Delbourgo and Gregory (,)-ratonal monotonc splne has the followng form: where p (t) = P (t) Q(t) f u 1 6= u 0 u 0 f u 1 = u 0 ; P (t) = (u 1 u 0 ) u 1 t (u 0 d 1 u 1 d 0 ) t(1 t) (u 1 u 0 ) u 0 (1 t) (9) Q(t) = (u 1 u 0 ) t (d 1 d 0 ) t(1 t) (u 1 u 0 ) (1 t) : Wthout any addtonal parameters, the functon above sats es the Hermte nterpolaton condtons at the ponts t = 0 and t = Cubc/lnear ratonal splne The Shrvastava and Joseph (3,1)-ratonal monotonc splne s gven by p (t) = P (t) Q(t) ;where P (t) = vu 0 (1 t) 3 wu 1 t 3 [(v w) u 0 vd 0 ] t (1 t) (10) [(v w) u 1 wd 1 ] t (1 t) Q(t) = v (w v) t 5

7 wth v; w > 0 and w v: If v = w we obtan the ordnary cubc splne. A choce of the tenson parameters v and w that guarantees the global monotoncty of p on [0; 1] s v w d 1 u 1 u 0 u 1 u 0 d 0 so that a smple choce may be, for example, v = 1 and u1 u 0 d 0 w = max d 1 (u 1 u 0 ) ; where s a nonnegatve small number, say [0; :1]. 3.3 Cubc/quadratc ratonal splne The (3,)-ratonal form s suggested by Gregory p (t) = P (t) Q(t) wth (11) P (t) = u 0 (1 t) 3 (wu 0 d 0 ) t (1 t) (wu 1 d 1 ) t (1 t) u 1 t 3 Q(t) = 1 t (1 t) (w 3) and a choce for the tenson parameter w 0 to have global monotoncty s w = d0d1 u 1 u 0 ; obtanng the ordnary cubc splne f w = Cubc/cubc ratonal splne The (3,3)-ratonal splne has been proposed by Sarfraz p (t) = P (t) Q(t) wth (1) P (t) = u 0 (1 t) 3 (vu 0 d 0 ) t (1 t) (wu 1 d 1 ) t (1 t) u 1 t 3 Q(t) = (1 t) 3 vt (1 t) wt (1 t) t 3 wth v = r d0d1 u 1 u 0 and w = s d0d1 u 1 u 0, r; s 1: If v = w then (1) becomes (11). 3.5 Mxed cubc-exponental nterpolaton The monotonc Hermte-type nterpolator s based on a mxed cubc-exponental splne; n ts smpler form, t s gven by d 0 d 1 p(t) = u 0 (u 1 u 0 )t (3 t) a d 0 d 0 a a (1 t)a d 1 a ta 6

8 where a = 1 w d0d1 u 1 u 0 0 to have monotoncty. We use w = d0d1 u 1 u 0 0 or, to work wth nteger exponents, w = nt( d0d1 u 1 u 0 ). The lnear case (.e. trangular fuzzy numbers) s obtaned by puttng d 0 = d 1 = u 1 u 0 and a = 3: t s easy to see that the model becomes p(t) = u 0 d 0 t: If the data are quadratc,.e. d 0 d 1 = (u 1 u 0 ); then a = 3 and the model becomes quadratc, p(t) = u 0 d 0 t (d 1 u 1 u 0 )t. 3.6 Parametrzaton and fuzzy operatons If the slopes d 0 and d 1 are not avalable, we can proceed by choosng them such that, for a gven postve nteger n; d 0 d 1 = n(u 1 u 0 ): In partcular, f we x an nteger n and a parameter [0; 1], we can select (provded that u 1 u 0 6= 0) d0 = n(1 )(u 1 u 0 ) d 1 = n(u 1 u 0 ) (13) so that d 0 d 1 = n(u 1 u 0 ) and a = n 1: If = 0 or = 1 the model gves two extreme shapes havng d 0 = 0 or d 1 = 0: Usng one of the prevous forms to represent the lower and the upper branches of the fuzzy number u = (u ; u ) we can wrte the general form of the representaton u = (u 0; ; d 0; ; u 1; ; d 1; ; u 0; ; d 0; ; u 1; ; d 1; ) =1;:::;N m u = [p (t ; u 0; ; d e 0; ; u 1; ; d e 1; ); p (t ; u 0; ; d e 0; ; u 1; ; d e 1; )] =1;;:::;N (14) where the functons p (t ; u 0; ; e d 0; ; u 1; ; e d 1; ) and p (t ; u 0; ; e d 0; ; u 1; ; e d 1; ) are obtaned by monotonc models, wth d e k; = d k; ( 1 ), = 0; 1 and t = 1 1 for [ 1 ; ]. For N 1 we have a total of N parameters u 0;1 u 1;1 u 0; u 1; ::: u 0;N u 1;N, d k; 0 de nng the ncreasng lower branch u and u 0;1 u 1;1 u 0; u 1; ::: u 0;N u 1;N, d k; 0 de nng the decreasng upper branch u (obvously, also u 1;N u 1;N s requred). A smpl caton of (14) can be obtaned by requrng contnuous or dfferentable branches; n the rst case, u 1; = u 0;1 and u 1; = u 0;1 for = 1; ; :::; N 1 whle, to have d erentablty, also the condtons d k; = d k;1, d k; = d k;1 are requred. For the two cases we then have 6N or 4N 4 parameters, respectvely. In the strcly monotonc case, the membershp functon (x) of the LUfuzzy number u, gven by the equatons (u ) = (u ) = for [0; 1]; s the followng. (x) = 0 f x = [u 0 ; u 0 ] (15) = f x = u or x = u for a gven [0; 1[ = 1 f x [u 1 ; u 1 ]: 7

9 In partcular, correspondng to the nodes of the decomposton, (u ) = (u ) = for = 0; 1; :::; N and, for the d erentable case (the general pecewse d erentable case s smlar) 0 (u ) = 1 ; 0 (u d ) = 1 d for = 0; 1; :::; N: In the applcatons to fuzzy calculus descrbed n [9] we consder only the dfferentable case, for whch we use the representaton: wth the data and the slopes u = (u ; d ; u ; d ) =0;1;:::;N (16) u 0 u 1 ::: u N u N u N 1 ::: u 0 (17) d 0; d 0: (1) By the Lower-Upper representatons we can de ne correspondng spaces of fuzzy numbers, on whch the standard operatons and metrcs can be ntroduced by the use of the standard fuzzy calculus. Denote by F N = uj u = (u 0; ; d 0; ; u 1; ; d 1; ; u 0; ; d 0; ; u 1; ; d 1; ) =1;:::;N or, n the d erentable case, b F N = uj u = (u ; d ; u ; d ) =0;1;:::;N the set of LU-fuzzy numbers. b F N s a 4(N 1)-dmensonal space. Gven two LU-fuzzy numbers u = (u ; d ; u ; d ) =0;1;:::;N and v = (v ; e ; v ; e ) =0;1;:::;N the arthmetc operators assocated to the LU representaton can be obtaned easly. The addton s de ned by: u v = (u v ; d e ; u v ; d e ) =0;1;:::;N : The scalar multplcaton s de ned as follows: f k 0 then f k < 0 then ku = (ku ; kd ; ku ; kd ) =0;1;:::;N ; ku = (ku ; kd ; ku ; kd ) =0;1;:::;N : In partcular, f k = 1, we have u = ( u ; d ; u ; d ) =0;1;:::;N

10 and the subtracton s de ned by u v = u ( v): For the fuzzy multplcaton we ntroduce an easy to mplement algorthm, based of the applcatons of the exact fuzzy multplcaton at the nodes of the subdvson; de ne and set the followng: (uv) = mnfu v ; u v ; u v ; u v g (19) (uv) = maxfu v ; u v ; u v ; u v g (0) uv = w ; f ; w ; f =0;1;:::;N To mplement the multplcaton we can proceed as follows: let (p ; q ) be the par assocated to the combnaton of superscrpts and gvng the mnmum (uv) n (19), and smlarly let (p ; q ) the par assocated to the combnaton of and gvng the maxmum (uv) n (0); then we obtan: w = u p v q and w = u p v q f = d p v q u p e q and f = d p v q u p where we use the product dervatve rule to obtan the new slopes. Analogous formulas can be deduced for the dvson: u=v = z ; g ; z ; g =0;1;:::;N e q (u=v) = mnfu =v ; u =v ; u =v ; u =v g and (1) (u=v) = maxfu =v ; u =v ; u =v ; u =v g. Let (r ; s ) be the par assocated to the combnaton of and gvng the mnmum n (u=v) and smlarly let (r ; s ) be the par assocated to the combnaton of and gvng the maxmum n (u=v) ; then t follows: z = u r =v s and z = u r =v s g = (d r v s u r e s )=(v s ) and g = (d r v s u r e s )=(v s ) : We note explctly that the scalar multplcaton s always reproduced exactly n all the models for all [0; 1] but, n general, ths s not true for the addton as the sum of ratonal or mxed functons s not a ratonal or a mxed functon of the same orders. 9

11 As ponted out by the results of the expermentaton reported n [4], the operatons above are exact at the nodes of the representaton and have very small global errors on [0; 1]: Further, t s easy to control the error by ntroducng addtonal nodes nto the representaton or by usng a su cently hgh number of nodes wth max f 1 g su cently small. To control the error of the approxmaton, we can proceed by ncreasng the number N 1 of ponts; a possble strategy s to double the number of ponts by usng N = K and by movng automatcally to N = K1 f a better precson s necessary. For all the computatons, as n [4], the llustrated algorthms are mmedate to mplement and also a standard spreadsheet can be used. The results n [4] of the parametrc operators have shown that both the ratonal (1) and the mxed (??) models perform very well, wth a percentage average error for a sngle multplcaton and dvson of the order of 0:1%: 4 Fuzzy extenson of unvarate functons In the general form the LU-representaton can be wrtten as: X = ; x ; x ; x ; x =0;1;:::;N where the nodes of the representaton are shown explctly (the symbol s used to denote the slopes). The fuzzy extenson of a sngle (real) varable (d erentable) functon f : R! R to a fuzzy argument u = [u ; u ] has cuts f (u) = [mn ff (x) j x u g ; max ff (x) j x u g] : () Note that f f s monotonc ncreasng we obtan f (u) = [f (u ) ; f (u )] whle, f f s monotonc decreasng, f (u) = [f (u ) ; f (u )] : For ths spec c case, we ntroduce a notaton smlar to the one used for multplcaton or dvson: let p ; p f ; g be de ned as follows p = p = f mn ff (u ) ; f (u )g = f (u ) f mn ff (u ) ; f (u )g = f (u ) f max ff (u ) ; f (u )g = f (u ) f max ff (u ) ; f (u )g = f (u ) We smplfy p p, = 0; 1; :::; N; n the ponts of the decomposton. So, we have f(u) = f u p and f(u) = f u p : If X s the LU-fuzzy number X = x ; D ; x ; D then ts mage f(x) s f(x) = =0;1;:::;N ; f(x p ); f 0 (x p )D p ; f(x p ); f 0 (x p )D p =0;1;:::N : The case of the fuzzy extenson of a nonmonotonc functon s handled n a smlar way. In ths case, the ponts where the mnmum and the maxmum 10

12 values () are taken, can be nternal to the nterval u or concdent wth one of the extremal values. In the last case, the cut of the extenson s obtaned as descrbed above. Let f(x) = ; f ; f ; f ; f =0;1;:::N denote the fuzzy extenson n the general case and suppose that the functon f s d erentable. If the mnmum value, de nng the lower branch f (u), s taken at an nternal pont x ]u ; u [), then f = f(x ) and f = 0; f the maxmun value, de nng the upper branch f (u) pont x ]u ; u [), then, s taken at an nternal f = f(x ) and f = 0: The detals for the fuzzy extensons of the basc elementary functons and some specal functons of partcular nterest are gven n the rest of ths secton. In some computatons, we requre a bg constant to represent n nty (postve or negatve): we denote t as BIG and n our computatons we use BIG= Fuzzy extenson of X! X n (the nput are the fuzzy X and a postve nteger n ): Denote Y = X n wthn the LU-representaton framework Y = ; Y ; Y ; Y ; Y =0;1;:::;N case 1.n = 3; 5; ::: (odd) For each = 0; 1; :::; N Y = X n n 1 Y = n X X Y = X n Y = n X n 1 X 11

13 case. n = ; 4; ::: (even) For each = 0; 1; :::; N n Y = X Y = X n If x 0 then n 1 Y = n X X Y = n X n 1 X Y = X n else f x Y n = X 0 then Y = n X n 1 X Y n 1 = n X X Y = 0 Y = 0 ( Y = X n else f x then x Y = n X ( Y n = X else 4. Fuzzy extenson of X! X n = 1 X n n 1 X Y = n X n 1 X (the nput are the non-zero fuzzy X and a postve nteger n 1): Valdty test can be stated as follows: X s non-zero f ether X 0 > 0 or X 0 < 0; n other words t means that 0 = x 0 ; x 0 Denote Y = ; Y ; Y ; Y ; Y =0;1;:::;N wth case 1. n = 1; 3; 5; ::: (odd) For each = 0; 1; :::; N Y = X n Y = 1 (X ) n Y = n (X ) n1 X Y = 1 (X ) n Y = n (X ) n1 X 1

14 case. n = ; 4; ::: (even) For each = 0; 1; :::; N If x If x > 0 then < 0 then Y = 1 (X ) n Y = 1 (X ) n Y = n (X ) n1 X Y = n (X ) n1 X Y = 1 (X ) n Y = 1 (X ) n Y = n (X ) n1 X Y = n (X ) n1 X 4.3 Fuzzy extenson of X! exp (X) (the nput s the fuzzy X): Valdty test can be stated as follows: X s non-zero f ether X 0 X 0 < 0; n other words t means that 0 = x 0 ; x 0 : Let > 0 or Y = exp (X) For each = 0; 1; :::; N Y Y = exp X = exp X Y = exp X X Y = exp X X 4.4 Fuzzy extenson of X! ln (X) (the nput s the postve fuzzy X): Valdty test: X 0 > 0. Let For each = 0; 1; :::; N Y = ln (X) Y Y Y Y = ln X = ln X = X X = X X 4.5 Fuzzy extenson of (X; Y )! X Y (the nput are the postve fuzzy X and the fuzzy Y ): We use the de nton X Y = exp (Y ln ((X))) 13

15 and we compute by the sequence of operatons: Z = X Y () (natural logarthm) Z ln (X) () (standard multplcaton) Z Y Z () (exponental) Z exp (Z) 4.6 Fuzzy extenson of X! sn (X) on the nvertblty doman (the nput s the fuzzy X wth support n Valdty test: X 0 X 0 Let Y = sn (X) For each = 0; 1; :::; N Y Y = sn X = sn X Y = cos X X Y = cos X X ; ): 4.7 Fuzzy extenson of X! arc sn (X) (the nput s the fuzzy X wth support n [ 1; 1]): Valdty test: 1 X 0 X 0 1 Let Y = arc sn (X) For each = 0; 1; :::; N; (f necessary, use the constant BIG) Y = arc sn X Y = arc sn X ( BIG f X = 1 orx = 1 Y = 1 X p1 X f 1 < X < 1 ( BIG f X Y = 1 orx = 1 = p 1 X f 1 < X < 1 1 X 4. Fuzzy extenson of X! cos (X) on the nvertblty doman (the nput s the fuzzy X wth support n [0; ]): Valdty test: 0 X 0 X 0 Let Y = cos (X) 14

16 For each = 0; 1; :::; N Y = cos X Y = cos X Y = sn X X Y = sn X X 4.9 Fuzzy extenson of X! arccos (X) (the nput s the fuzzy X wth support n [ 1; 1]): Valdty test: 1 X 0 X 0 1 Let Y = arccos (X) For each = 0; 1; :::; N; Y = arccos X Y = arccos X ( BIG f X = 1 orx = 1 Y = p X f 1 < X 1 X < 1 ( BIG f X Y = 1 orx = 1 = p X f 1 < X 1 X < Fuzzy extenson of X! tan (X) (the nput s the fuzzy X wth support n ] ; [): Valdty test: - < X 0 X 0 < Let Y = tan (X) For each = 0; 1; :::; N = tan X = tan X Y = 1 tan X X Y = 1 tan X X Y Y 4.11 Fuzzy extenson of X! arctan (X) (the nput s the fuzzy X wth support n R): Let Y = arctan (X) 15

17 For each = 0; 1; :::; N; Y Y = arctan X = arctan X Y = X 1(X ) Y = X 1(X ) 4.1 Fuzzy extenson of X! p X (the nput s the non negatve fuzzy X): Valdty test: X 0 0 Let Y = p X For each = 0; 1; :::; N q Y = Y = X q X Y = X = px Y = p X X ( BIG f X = 0 X f X Y > 0 ( BIG f X = 0 = f X > 0 X Y 4.13 Fuzzy extenson of X! X (the nput are the postve fuzzy X and the real ): We use the de nton X = exp ( ln (X)) and we compute by the sequence of three steps: Y = X () (logarthm) Y ln (X) () (scalar multplcaton) Y Y () (exponental) Y exp (Y ) 4.14 Fuzzy extenson of X! e X (the nput s the fuzzy X): The computaton Y = exp X 16

18 can be done by followng three steps: (square power functon) () Y = X (opposte) () Y Y (exponental) () Y exp (Y ) 4.15 Fuzzy extenson of the Gauss standard functon X! p 1 e X (the nput s the fuzzy X): The computaton can be done by followng four steps: (square power functon) () Y = X 1 (negatve scalar multplcaton) () Y Y (exponental) () Y exp (Y ) 1 (scalar multplcaton) (v) Y p Y 4.16 Fuzzy extenson of the hyperbolc snusodal functon X! snh(x) (the nput s the fuzzy X): Let Y = snh (X) = ex e X For each = 0; 1; :::; N Y = e X e X = Y = e X e X = Y = X e X e X = Y = X e X e X = 4.17 Fuzzy extenson of the hyperbolc cosnusodal functon X! cosh(x) (the nput s the fuzzy X): Let Y = cosh (X) = ex e X 17

19 For each = 0; 1; :::; N : Y = cosh X f X Y 0 then = cosh X Y = X snh X Y = X snh X Y = cosh X else f X Y 0 then = cosh X Y = X snh X Y = X snh X Y = 1, Y = 0 Y then = cosh X else f abs(x ) abs(x ) Y = X snh X Y else = cosh X Y = X snh X 4.1 Fuzzy extenson of the hyperbolc tangentod functon X! tanh(x) (the nput s the fuzzy X): Let Y = tanh (X) tanh(x) = and snh (x) cosh (x) = ex e x e x e x For each = 0; 1; :::; N = tanh X = tanh X Y = X = cosh X Y = X = cosh X Y Y 4.19 Fuzzy extenson of the nverse hyperbolc snusodal functon X! snh 1 (X): (the nput s the fuzzy X): Note that snh 1 (x) = ln x p 1 x d dx snh 1 (x) = 1 p 1 x 1

20 For each = 0; 1; :::; N Y = snh 1 X Y = snh 1 X q Y = X = 1 X Y = X = q1 X 4.0 Fuzzy extenson of the nverse hyperbolc cosnusodal functon X! cosh 1 (X): (the nput s the fuzzy x; wth support 1 and the valdty test s x 0 1): Note that cosh 1 (x) = ln x p x 1, x 1 d dx cosh 1 (x) = 1 p x 1 For each = 0; 1; :::; N Y = cosh 1 X Y = cosh 1 X q Y = X = X 1 (f X = 1 then Y = BIG) q Y = X = X 1 (f X = 1 then Y = BIG) 4.1 Fuzzy extenson of the nverse hyperbolc tangentod functon X! tanh 1 (X): (the nput s the fuzzy x; wth support n ] 1; 1[ and the valdty test s 1 < x 0 x 0 < 1): Note that tanh 1 (x) = 1 1 x ln ; x ] 1; 1[ 1 x For each = 0; 1; :::; N d dx tan 1 (x) = 1 1 x Y = tan 1 X Y = tan 1 X Y = X = 1 X Y = X 1 = X 19

21 4. Fuzzy extenson of the erf and erf c error functons wth x! erf(x) = p xz = Z p exp t dt = x = 1 erf c (x) erf c (x) = 1 Z x exp t dt = (ncreasng) exp t dt (decreasng) For the erf c functon we use the followng approxmaton, wth a fractonal error less then and where (Korner rule) z = abs (x) t = z < t exp z p (t) f x 0 erf c = : t exp z p (t) f x < 0 p (t) = a 0 t (a 1 t (a t (a 3 t (a 4 t (a 5 t (a 6 t (a 7 t (a ta 9 )))))))) and the coe cents assume the followng values a 0 = 1:65513 a 5 = 0:7607 a 1 = 1: a 6 = 1: a = 0: a 7 = 1:45157 a 3 = 0: a = 0:153 a 4 = 0:1606 a 9 = 0: Let For each = 0; 1; :::; N Y = erf (X) Y = erf X Y = erf X Y = X p exp X Y = X p exp X 0

22 Let now For each = 0; 1; :::; N Y Y Y = erf c (X) = erf c X = erf c X p exp X Y = X Y = X p exp X 4.3 Fuzzy extenson of the cumulatve standard normal dstrbuton functon (x) = p 1 xz t exp dt The functon (x) ; x R, can be approxmated by the followng procedure: 1 1 erf xp f x 0 (x) = 1 1 erf xp f x < 0 1 Let For each = 0; 1; :::; N Y = (X) wth X fuzzy Y Y = X = X Y = X p1 exp( X Y = x 1 ) p exp( X ) 5 Implementaton of the LU-fuzzy calculator To mplement the LU-fuzzy calculator, we have wrtten a wndows-based frame smlar to a standard hand-calculator. Fgure shows a panoramc vew of the calculator; from left to rght we can see the grds of the fuzzy numbers X, Y and Z. Z s the result of the operatons whle X and/or Y are the operands. For each element u fx; Y; Zgthe grd contans the values ; u ; u ; u and u respectvely. To start the calculatons, we have mplemented a set of prede ned types, ncludng trangular, trapezodal, exponental, gamma, ect. For a gven type, t s possble to de ne the number N of subntervals (N 1 ponts) n the unform decomposton. 1

23 General wndow of the LU-fuzzy calculator The calculatons are performed by clckng the button of the correspondng operaton. The left group of buttons nvolves the bnary operatons (see gure 3.) Bnary operatons and assgnments The second group of operators (see gure 4.) requre the assgment of ether X or Y to the temporary K (see gure 5) and operate on K tself puttng the result nto Z. Extenson of unvarate functons

24 Selecton of the argument for fuzzy extenson functons It s possble to save a gven (X, Y or Z) temporary result nto a stored lst (Put n Lst button), by assgnng a name to t; a saved fuzzy number can be reloaded ether n X or Y for further use (Get from Lst button). The Plot button (see gure 6.) opens a popup wndow wth the graph of the membershp functon of the correspondng fuzzy number. Buttons to Plot, Put nto Lst and Get from saved Lst Fgure 7. llustrates how to select a fuzzy number from a lst of prede ned types. Selecton of prede ned types of fuzzy nymbers To obtan the graphs or other representatons, one of the models descrbed n secton 3. can be selected ( gure.). Choosng the monotonc splne model 3

25 We llustrate an example to show how the calculator works. 1. Frst (see gure 9.) select a trapezodal fuzzy number and set to 5 the number of subntervals n the decomposton (the hgher N the hgher the precson n the calculatons); the maxmal value of N s 100; typcal values are, 5 or 10. Example for trapezodal fuzzy number If the selecton s loaded nto the X-area, the corrspondng grd appears as n gure 10. Assgne the trapezodal fuzzy number to X. A second fuzzy number s loaded nto Y and the button correspondng to the operaton Z=X/Y s actvated. The Z-grd s calculated by the rules of the LU-fuzzy calculus (see gure 11.). 4

26 An example of a fuzzy dvson: Z=X/Y 3. To see the graphcal representaton of X, Y and/or Z, clck the correspondng Plot button and the popup wndows appear ( gure 1.). Plottng the operands and the result of an operaton 4. Now, we save the result Z of the prevous operaton and we call Fuzzy_Z ts name ( gure 13.). 5

27 Savng of an ntermedate result for further use 5. Now we load the saved Fuzzy_Z nto the X-area, by gettng t from the lst of saved elements ( gures 14. and 15.). Loadng of a saved ntermedate fuzzy number nto X area Result of loadng a saved fuzzy number nto the X area 6

28 6. If we try to apply the fuzzy extenson of the log functon to a fuzzy number X, we rst select the assgment K=X and then we clck on the Z=log(K) button and on the Plot buttons of X and Z ( gure 16.). Calculatng and plottng X and Z=log(X) 6 Example of Use: Black-Sholes opton prcng In ths nal secton we llustrate the use of the calculator for the fuzz caton of Black-Sholes formula n the valuaton of a european put or call opton. Standard Black Scholes (B-S) formula for the European put/call optons (wthout dvdends) S t : (current) stock prce at tme t [0; T ] FUZZY T : tme to maturty CRISP K : exercse prce (strke prce) FUZZY r : nterest rate (contnuously compounded), FUZZY : (standard devaton) volatlty, FUZZY then FUZZY prce C t of (European) call opton at tme t s gven by: C t = S t (D 1 (S t ; K; r; ; T t)) Ke r(t t) (D (S t ; K; r; ; T t)) FUZZY prce P t of put opton at tme t (wth the same expry date T and strke prce K) s gven by: P t = S t [ (D 1 (S t ; K; r; ; T t)) 1] Ke r(t t) [ (D (S t ; K; r; ; T t)) 1]: 7

29 where (x) : cumulatve (ncreasng) standard normal functon and D 1 = ln S K rt p p t; t D = ln S K rt p p t: t Here, we apply exact extenson prncple for the calculaton of D 1 and D by extendng the two dmensonal functons (x; y)! x y y where x = ln S K rt and y = p t: The probablty densty functon for the opton value C t s gven by (lognormal type): D (C t ) = a e 1 b where a = b = r(t t) e and (T t) C t e r(t t) S t C ln te r(t t) k S r (T t) p : (T t) Also the fuzzy versons of the GREEKS can be easly calculated. The results obtaned by the LU-fuzzy decomposton wth N = 10 are exact up to decmal places. The data are from H.-C. WU, Computers and Operatons Research, 004. K = 30, T = 0:5, t = 0 and the fuzzy numbers are all trangular and symmetrc: S 0 = h3; 33; 34, r = h0:04; 0:05; 0:05 and mod ed to r LU = (0:04; 0:1; 0:1; 0:00; 0:05; 0:005; 0:05; 0:01) = h0:0; 0:1; 0:1 and mod ed to LU = (0:0; 0:1; 0:1; 0:005; 0:1; 0:01; 0:1; 0; 1)

30 Example Plot X = D 1, Plot Y = (D 1 ) Mod ed r and to LU: Plot X = r LU, Plot Y = LU 9

31 Plot X = C 0, Plot Y = ModfedC 0 References [1] D. Dubos, H. Prade, Fuzzy Sets and Systems: Theory and Applcatons, Academc Press, New York, (190). [] D. Dubos, H. Prade, Possblty Theory. An approach to Computerzed Processng of Uncertanty, Plenum Press, New York, (19). [3] D. Dubos, E. Kerre, R.Mesar, H. Prade, Fuzzy Interval Analyss, n: Fundamentals of Fuzzy Sets, D. Dubos, H. Prade, Rds. Kluwer, Boston, The Handbooks of Fuzzy Sets Seres, (000) [4] M.L. Guerra, L. Stefann, Approxmate Fuzzy Arthmetc Operatons Usng Monotonc Interpolatons, Fuzzy Sets and Systems,150/1, 5-33, 005. [5] M. Hanss, The transformaton method for the smulaton and analyss of systems wth uncertan parameters, Fuzzy Sets and Systems 130 (00) [6] A. Kandel, Fuzzy mathematcal technques wth applcatons, Addson Wesley, Readng, Mass, 000. [7] A. Kaufman M.M.Gupta, Introducton to fuzzy arthmetc, theory and applcatons, Van Nostrand Renhold Company, New York, 195. [] G. J. Klr, Fuzzy arthmetc wth requste constrants, Fuzzy Sets and Systems 91 (1997) [9] L. Stefann, L.Sorn, M.L.Guerra, Parametrc Representatons of Fuzzy Numbers and Applcatons, Workng Paper Seres EMS, n.95, Unversty of Urbno,

32 [10] L. Stefann, L.Sorn, M.L.Guerra, Smulatons of Fuzzy Dynamcal Systems usng the LU Representaton of Fuzzy Numbers, Chaos, Soltons and Fractals, forthcomng, 005. [11] L.A. Zadeh, Fuzzy Sets, Informaton and Control (1965)