Solving Fuzzy Equations Using Neural Nets with a New Learning Algorithm

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1 Joural of Advaces Compuer Research Quarerly ISSN: Sar Brach, Islamc Azad Uversy, Sar, I.R.Ira (Vol. 3, No. 4, November 212), Pages: Solvg Fuzzy Equaos Usg Neural Nes wh a New Learg Algorhm Ahmad Jafara 1, Safa Measoomy a 1, Raheleh Jafar 2 (1)Deparme of Mahemacs, Urma Brach, Islamc Azad Uversy, Urma, Ira (2)Deparme of Mahemacs, scece ad research Brach, Islamc Azad Uversy, Arak, Ira jafara5594@yahoo.com; measoomy@yahoo.com; jafar3339@yahoo.com Receved: 212/4/17; Acceped: 212/1/6 Absrac Arfcal eural eworks have he advaages such as learg, adapao, faul-olerace, parallelsm ad geeralzao. Ths paper maly eds o offer a ovel mehod for fdg a soluo of a fuzzy equao ha supposedly has a real soluo. For hs scope, we appled a archecure of fuzzy eural eworks such ha he correspodg coeco weghs are real umbers. The suggesed eural e ca adjus he weghs usg a learg algorhm ha based o he grade desce mehod. The proposed mehod s llusraed by several examples wh compuer smulaos. Keywords: Fuzzy equaos, Fuzzy feed-forward eural ework (FFNN), Cos fuco, Learg algorhm 1. Iroduco Fuzzy equaos are very useful for solvg may problems several appled felds lke mahemacal ecoomcs ad opmal corol heory, because may mahemacal formulaos of physcal pheomea coa hese kds of equaos. Therefore, varous approaches for solvg hese problems have bee proposed. Oe approach o drec soluo s usg fuzzy eural eworks (FNNs). I rece years he FNN model has bee rapdly developed ad pu o use a wde varey felds [9, 1, 21]. FNNs are smplfed models of he bologcal ervous sysem ad herefore have draw her movao from he kd of compug performed by a huma bra. FNNs are geeral capable of close approxmao of he predco model whou he eed of s explc (mahemacal) formulao coras o sascal approaches. Frs me, Buckley [8] appled a srucure of FNNs for solvg fuzzy equaos. Ishbuch e al. [13] defed a cos fuco for every par of fuzzy oupu vecor ad s correspodg fuzzy arge vecor ad also desged a learg algorhm of fuzzy eural eworks wh ragular ad rapezodal fuzzy weghs. Hayash e al. [12] fuzzfed he dela rule ad summarzed much of he work fuzzy eural eworks. Buckley ad Eslam [7] employed eural es o solve fuzzy problems wh boh real ad complex fuzzy umbers. Moreover, Lear ad olear fuzzy equaos have bee solved [1, 2, 3, 6]. Jafara e al. [16] proposed a umercal scheme o solve fuzzy lear volerra egral equaos sysem. The opc of umercal soluo of fuzzy 33

2 Solvg Fuzzy Equaos Usg Neural Nes A. Jafara; S. Measoomy a; R. Jafar polyomals by FNN vesgaed by Abbasbady e al. [4], coss of fdg soluo o polyomals lke a 1x a x = a where x R ad a,a1,..., a are fuzzy umbers. Jafara ad Jafar [15] appled fuzzy feed-back eural ework mehod for approxmao of he crsp soluo of dual fuzzy polyomals. We refer he reader o [2, 22] for more formao o fuzzy polyomals. The objecve of hs paper s prmarly o desg a ew model based o FNNs for approxmae soluo of fuzzy equaos. I hs work, a archecure of FFNN2 (fuzzy feed-forward eural ework wh fuzzy se pu sgals, fuzzy oupu sgal ad real umber weghs) equvale o fuzzy equao of he form A1 f1(x) A f (x) = A s bul, where A, A,..., are fuzzy umbers ad f (x) (for = 1,..., ) are real 1 A fucos. The proposed eural ework has wo layers ha he pu-oupu relao of each u s defed by he exeso prcple of Zadeh [23]. The coeffces of he fuzzy equao ad he rgh had fuzzy umber are cosdered as pu sgals ad arge oupu, respecvely. The oupu from he eural ework whch s also a fuzzy umber, s umercally compared wh he arge oupu. Nex a cos fuco s defed ha measures he dfferece bewee he fuzzy arge oupu ad correspodg acual fuzzy oupu. The he suggesed eural e usg a learg algorhm ha based o he grade desce mehod adjuss he crsp coeco weghs o ay desred degree of accuracy. The remader of he paper s orgazed accordg o he followg oule: I Seco 2, some basc defos are preseed. I seco 3, he fuzzy equao s brefly descrbed. I hs seco, we descrbe how o fd a real soluo of he fuzzy equao by usg FFNs. I seco 4, he applcably of he mehod s llusraed by several examples whch he exac soluo ad he compued resuls are compared wh each oher. Fally, cocluso s descrbed seco Prelmares I hs seco he mos basc used oaos fuzzy calculus are brefly roduced. We sared by defg he fuzzy umber. R such ha:. u s upper sem-couous,. u ( x) = ousde some erval [a, d],. There are real umbers b, c : a b c d, for whch: 1. u (x) s mooocally creasg o [a, b], 2. u (x) s mooocally decreasg o [c, d], 3. u (x) = 1, b x c. The se of all fuzzy umbers (as gve by defo 1) s deoed by E 1 [11, 19]. A alerave defo whch yelds he same E 1 s gve by Kaleva [17] ad Ma e al. [18]. 1 Defo 1. A fuzzy umber s a fuzzy I = [,1] Defo 2. A fuzzy umber v s a par ( vv), of fuco vr () ad vr (): r 1, whch sasfy he followg requremes:. vr () s a bouded mooocally creasg, lef couous fuco o (, 1] ad rgh couous a, 34

3 Joural of Advaces Compuer Research (Vol. 3, No. 4, November 212) vr () s a bouded mooocally decreasg, lef couous fuco o (, 1] ad rgh couous a,. vr () vr (): r 1. A popular fuzzy umber s he ragular fuzzy umber v= ( vm, vl, vu) where v m deoes he modal value ad he real values vl ad vu represe he lef ad rgh fuzzess, respecvely. The membershp fuco of a ragular fuzzy umber s defed as follows: x vm + 1, vm vl x vm, vl vm x µ v( x) = + 1, vm x vm+ vu, vu, oherwse. Is paramerc form s: vr () = vm+ vl( r 1), vr () = vm+ vu(1 r), r 1. Tragular fuzzy umbers are fuzzy umbers LR represeao where he referece fucos L ad R are lear. 2.1 Operaos o fuzzy umbers We brefly meo fuzzy umber operaos defed by he exeso prcple [23,24]. µ A+ B() z = max µ A( x) µ B( y) z= x+ y, { } { } µ ( z) = max µ ( x) µ ( y) z= xy, f( Ne) A B where A ad B are fuzzy umbers, µ (.) deoes he membershp fuco of each fuzzy umber, s he mmum operaor, ad f s a couous acvao fuco (such as f ( x) = x ) of oupu u of our fuzzy eural ework. The above operaos o fuzzy umbers are umercally performed o level ses (.e. -cus). For < 1, a level se of a fuzzy umber A s defed as:,1 ad [ A] ( ] [ A]. deoe [ A] by [ A] = { x µ A x x R} ( ),, = Sce level ses of fuzzy umbers become closed ervals, we [ ] [ ] [ ] A A, A =, l u where [ A] l ad [ A ] u are he lower ad he upper lms of he -level se [ A ], respecvely. From erval arhmec [5], he above operaos o fuzzy umbers are wre for he -level ses as follows: [ A] + [ B] = [ A] [ A] + [ B] [ B] = [ A] + [ B] [ A] + [ B],,,, l u l u l l u u (1) f ( Ne ) = f ( Ne, Ne ) = f ( Ne ), f ( Ne ), [ ] [ ] [ ] [ ] [ ] l u l u 35

4 Solvg Fuzzy Equaos Usg Neural Nes A. Jafara; S. Measoomy a; R. Jafar [ ] [ ] [ ] [ ] [ ] k A = k A, A = k A, k A, f k, l u l u [ ] [ ] [ ] [ ] [ ] k A = k A, A = k A, k A, f k <. l u l u For arbrary u = ( u, u) ad v= (, vv) we defe addo ( u+ v) ad mulplcao by k as [11, 19]: ( u+ v)( r) = ur () + vr (), ( u+ v)( r) = ur () + vr (), 3. Fuzzy equao ( ku)( r) = kur. (),( kυ)( r) = kur.(), f k, ( ku)( r) = kur.(),( kυ)( r) = kur. (), f k<. We are eresed fdg a real soluo of he fuzzy equao (f exs) geeral form Af ( x) + + Af ( x) = A, (3) 1 1 where 1 A E ad (x) f (for = 1,..., ) are real fucos. For geg a approxmae soluo, a archecure of feed-forward eural ework equvale o Eq. (3) s bul. The ework s show Fgure 1. (2) Fgure 1. The proposed eural ework. 3.1 Ipu-oupu relao of each u Cosder a wo-layer FFNN2 wh pu euros ad oe oupu euro. I s clear ha he pu vecor, he arge oupu are ragular fuzzy umbers ad coeco weghs are crsp umbers. Whe a fuzzy pu vecor A = (A 1, A 2,...,A ) s preseed o he FFNN2, he he pu-oupu relao of each u ca be wre as follows (see Fgure 1): Ipu us: The pu euros make o chage her pus, so: O = A, = 1, 2,...,. (4) 36

5 Joural of Advaces Compuer Research (Vol. 3, No. 4, November 212) Oupu u: Y = f (Ne), Ne = ( wo. ), (5) = 1 where A s a ragular fuzzy umber ad w s a crsp coeco wegh. The relaos bewee he pu euros ad he oupu euro Eqs. (4)-(5) are defed by he exeso prcple [23] as Hayash e al. [12] ad Ishbuch e al. [14]. 3.2 Calculao of fuzzy oupu The fuzzy oupu from of euro he secod layer s umercally calculaed for crsp weghs ad level ses of fuzzy pus. The pu-oupu relaos of he fuzzy eural ework as show Fgure 1 ca be wre for he -level ses as follows: Ipu us: O = A, = 1,...,. (6) Oupu u: Le f be a oe-o-oe acvao fuco. Now we have: [ Y ] = f [ Ne ] ( ), [ Ne] = w [ O ] = 1 (. ). From Eqs. (6)-(7), we ca coclude ha he -level ses of he fuzzy oupu Y are calculaed from hose of he fuzzy pus ad crsp weghs. From Eqs. (1)-(2), he above relaos are rasformed o followg form: Ipu us: [ O ] = [ O ],[ O ] = [ A],[ A], = 1,,. l u l u Oupu u: [ Y] = [ Y] [ Y] = f [ Ne] f [ Ne] where, ( ), ( ), l u l u [ Ne] [ Ne],[ Ne] = = l u (. ) (. ), (. ) (. ),, l u u l M = w, C = w < ad M C = 1,,. w [ O] + w [ O] w [ O] + w [ O] M C M C where { } { } { } Lemma 1. Le fuzzy equao Af. ( ) 1 x = A = has a soluo for real crsp umber x, he D ϕ where D = = 1 doma ( f( x)). Proof. Le x R be a soluo of Eq. (3). Cosequely, we ca wre: Af ( x ) + + Af ( x ) = A, 1 1 (7) (8) 37

6 Solvg Fuzzy Equaos Usg Neural Nes A. Jafara; S. Measoomy a; R. Jafar ad follows from he above relao ha f x ) exss. I s clear ha x doma ( 1 x ( f ( x)) ad cosequely x = doma ( f ( )) φ Corollary 1. The ecessary codo for exsece soluo of Eq. (3) s D φ. A archecure of FFNN2 soluo o Eq. (3) has bee gve Fgure 1. The modellg scheme s desged wh he smple ad versale fuzzy eural ework archecure. Now le A be he arge oupu correspodg o he fuzzy pu vecor A = (A 1,..., A ). We wa o defe a cos fuco for he -level ses of he fuzzy oupu Y ad he correspodg arge oupu A : e = e + e, (9) where e = l l u [ A] [ Y] 2 ( ) l l, 2 [ A] [ Y] 2 ( ) u u. eu = 2 I he cos fuco (9), e l ad e u ca be vewed as he squared errors for he lower lms ad he upper lms of he -level ses of he fuzzy oupu Y ad arge oupu A, respecvely. Now he cos fuco for he pu-oupu par {A; A } s obaed as: e= e. (1) 3.3 Learg algorhm of he FFNN2 Le a real quay x s alzed a radom value for varable x. The ma am of hs subdvso s o offer a algorhm for adjusg he parameer x ad coeco wegh w. For real parameer x adjusme rule ca be wre as follows: x ( + 1) = x () + x (), (11) e x () = η + γ x ( 1), x where s he umber of adjusmes, η s he learg rae ad γ s he momeum erm cosa. Thus our problem s o calculae he dervave e x (12) (12). The gve dervave ca be calculaed from he cos fuco e ad by usg he pu-oupu e relaos (6)-(7). The dervave ca be calculaed as follows: x e e l e = + u, x x x where (13) 38

7 Joural of Advaces Compuer Research (Vol. 3, No. 4, November 212) e x e x ad [ Y] [ Ne] [ Y] [ Ne] [ Ne] l e l l l = [ Y] x l l [ Ne] u e u u u = [ Y] x u u [ Ne ] [ Ne] l l w [ ] [ ] l [ Ne ] [ Ne] w u u j [ ] [ ] u,, = ( ) = ( A f ( x )) + ( A f ( x )), x w x u = 1 M C = ( ) = ( A f ( x )) + ( A f ( x )). x w x l = 1 M C Cosequely [ ] [ ] [ ] [ ] [ ] [ ] x () = η (( A Y ) A + ( A Y ) A f ( x ())) + l l l u u u M [ ] [ ] [ ] [ ] [ ] [ ] η (( A Y ) A + ( A Y ) A f ( x())) + γ x( 1), l l u u u l C = = < ad M C = { 1,, }. Afer adjusg he where M { w }, C { w } parameer x by usg Eqs. (11)-(12), he coeco weghs are updaed as follows: w ( + 1) = f( x ( + 1)), = 1,,. Le us assume ha pu-oupu par {A; A } where A = (A 1,...,A ) are gve as rag daa ad also m values of level ses (.e.,, 1 2, m ) are used for he learg of fuzzy eural ework. The he learg algorhm ca be summarzed as follows: Learg algorhm Sep 1: η >, γ > ad Emax > are chose. Also crsp quay x D s alzed a radom value. Sep 2: Le := where s he umber of eraos of he learg algorhm. The he rug error E s se o. Sep 3: Calculae he crsp coeco weghs as follows: w () = f ( x ()), = 1,,. Sep 4: Le := + 1. Repea Sep 5 for =,, 2,.. Sep 5: 1 m. Forward calculao: Calculae he -level se of he fuzzy oupu Y by preseg he level se of he fuzzy pu vecor A.. Back-propagao: Adjus he parameer x usg he cos fuco (9) for he -level ses of he fuzzy oupu Y ad he arge oupu A. The updae he weghs by usg Sep 3. Sep 6: Cumulave cycle error s compued by addg he prese error o E. Sep 7: The rag cycle s compleed. For E < Emax ermae he rag sesso. If E > Emax he E s se o ad we ae a ew rag cycle by gog back o Sep 4. 39

8 Solvg Fuzzy Equaos Usg Neural Nes A. Jafara; S. Measoomy a; R. Jafar 4. Numercal examples To show he behavor ad properes of he proposed mehod, four examples have bee solved hs seco. For each example, he compued values of he approxmae soluo are calculaed ad he cos fuco s ploed over a umber of eraos. Example 4.1. Cosder he followg fuzzy equao: x 3 (1,2,3) e + (2,3,4)s( x) + (3,4,5) x = (1,2,3)., wh he exac soluo s x = ad x R. I hs example, we apply he proposed mehod o approxmae soluo of hs fuzzy equao. The rag paer s as follows: AA, = (1,2,3),(2,3,4),(3,4,5);(1,2,3). { } { } The FFNN2 s raed wh hree pu us ad sgle oupu euro. The rag sars wh x = 1, η =.1 ad γ =.1. Table 1 shows he approxmaed soluo over a umber of eraos ad Fgures 2 ad 3 show he accuracy of he calculaed soluo x ( ). Table 1. The approxmaed soluos wh error aalyss for Example 4.1 x ( ) e x ( ) e Fgure 2. The cos fuco for Example 4.1 over he umber of eraos. 4

9 Joural of Advaces Compuer Research (Vol. 3, No. 4, November 212) Fgure 3. Covergece of he approxmaed soluo for Example 4.1. Example 4.2. Le fuzzy equao 2 4 ( 1,,1) x + ( 2,,2) x + ( 3,,3)s( x 1) = ( 3,,3), wh he exac soluo x = 1. Smlarly, we assumed ha x =. 5, η =.1 ad γ =.1. Numercal resul ca be foud Table 2. Also Fgures 4 ad 5 show he accuracy of he soluo x ( ). Table 2. The approxmaed soluos wh error aalyss for Example 4.2 x ( ) e x ( ) e Fgure 4. The cos fuco for Example 4.2 over he umber of eraos. 41

10 Solvg Fuzzy Equaos Usg Neural Nes A. Jafara; S. Measoomy a; R. Jafar Fgure 5. Covergece of he approxmaed soluo for Example 4.2. Example 4.3. We cosder he fuzzy equao x π e( x+ 1) ( 2, 1,1) e + (1,2,3)a( ( x+ 1)) + (3,5,6)l = (2,6,1), 4 Wh he exac soluo x =. Learg s sared wh x =. 5, η =.2 ad γ =.2. Smlarly, Numercal resul ca be foud Table 3. Fgures 6 ad 7 show he accuracy of he approxmae soluo x ( ). Table 3. The approxmaed soluos wh error aalyss for Example 4.3 x ( ) e x ( ) e Fgure 6. The cos fuco for Example 4.3 over he umber of eraos. 42

11 Joural of Advaces Compuer Research (Vol. 3, No. 4, November 212) Fgure 7. Covergece of he approxmaed soluo for Example 4.3. Example 4.4. Cosder he fuzzy equao π 2 4 (2,4,6)s( ) 2cos ( 1) ( 2, 1,) (2,5,8), 2 x + x + x = wh he exac soluo x = 1. Usg a smlar maer whch has bee descrbed prevous examples, rag s sared wh x =1. 5, η =.1 ad γ =.1. Numercal resul ca be foud Table 4. Smlarly, Fgures 8 ad 9 show he accuracy of he soluo x (). Table 4. The approxmaed soluos wh error aalyss for Example 4.4 x ( ) e x ( ) e Fgure 8. The cos fuco for Example 4.4 over he umber of eraos. 43

12 Solvg Fuzzy Equaos Usg Neural Nes A. Jafara; S. Measoomy a; R. Jafar 5. Coclusos Fgure 9. Covergece of he approxmaed soluo for Example 4.4. I hs paper, a ew archecure of feed-forward eural eworks has bee proposed o approxmae soluo of a fuzzy polyomal. Preseed FFNN2 hs sudy s a umercal mehod for calculag ukow coeffces he gve equao. I s clear ha o ge he bes approxmag soluo of he equao, umber of eraos mus be chose large eough. Wh he avalably of hs mehodology, ow wll be possble o vesgae he approxmae soluo of oher kds of fuzzy equaos. The aalyzed examples llusrae he ably ad relably of he prese mehod. The obaed soluos, comparso wh exac soluos adm a remarkable accuracy. 6. Refereces [1] S. Abbasbady ad M. Alav, A mehod for solvg fuzzy lear sysems, Ira. J. Fuzzy Sys. 2 (25) [2] S. Abbasbady ad B. Asady, Newo's mehod for solvg fuzzy olear equaos, Appl. Mah. Compu. 159 (24) [3] S. Abbasbady ad R. Ezza, Newo's mehod for solvg a sysem of fuzzy olear equaos, Appl. Mah. Compu. 175 (26) [4] S. Abbasbady ad M. Oad, Numercal soluo of fuzzy polyomals by fuzzy eural ework, Appl. Mah. Compu. 181 (26) [5] G. Alefeld ad J. Herzberger, Iroduco o Ierval Compuaos, Academc Press, New York, [6] B. Asady, S. Abbasbady ad M. Alav, Fuzzy geeral lear sysems, Appl.Mah. Compu. 169 (25) [7] J.J. Buckley ad E. Eslam, Neural e soluos o fuzzy problems: The quadrac quao, Fuzzy Ses Sys. 86 (1997) [8] J.J. Buckley ad Y. Qu, Solvg lear ad quadrac fuzzy equaos, Fuzzy Ses Sys. 35 (199) [9] A. Daa, V. Talukdar, A. Koar ad L.C. Ja, A eural ework based approach for proe srucural class predco, J. Iell. Fuzzy. Sys. 2 (29) [1] G. Fogga, T.T. Ha Pham, G.Warkozek ad F.Wurz, Opmzao eergy maageme buldgs wh eural eworks, I. J. Appl. Elecrom.3 (29) [11] R. Goeschel ad W. Voxma, Elemeary calculus, Fuzzy Ses Sys. 18(1986) [12] Y. Hayash, J.J. Buckley ad E. Czogala, Fuzzy eural ework wh fuzzy sgals ad weghs, I. J. Iell. Sys. 8 (1993)

13 Joural of Advaces Compuer Research (Vol. 3, No. 4, November 212) [13] H. Ishbuch, K.Kwo ad H.Taaka, A learg of fuzzy eural eworks wh ragular fuzzy weghs, Fuzzy Ses Sys. 71 (1995) [14] H. Ishbuch, H. Okada ad H. Taaka, Fuzzy eural eworks wh fuzzy weghs ad fuzzy bases, : Proc. ICNN 93 (Sa Fracsco), 4 (1993) [15] A. Jafara ad R. Jafar, Approxmae soluos of dual fuzzy polyomals by feed-back eural eworks, joural of Sof Compuao ad Applcaos, (212). do:1.5899/212/jsca-5 [16] A. Jafara, S. Measoomy Na ad S. Tava, A umercal scheme o solve fuzzy lear volerra egral equaos sysem, Joural of Appled Mahemacs, (212). do:1.1155/212/ [17] O. Kaleva, Fuzyy d ereal equaos, Fuzzy Ses Sys. 24 (1987) [18] M. Ma, M. Fredma, A. Kadel, A ew fuzzy arhmec, Fuzzy Ses Sys. 18 (1999) [19] H.T. Nguye, A oe o he exeso prcple for fuzzy ses, J. Mah. Aal. Appl. 64 (1978) [2] S.K. Oh, W. Pedrycz ad S.B. Roh, Geecally opmzed fuzzy polyomal eural eworks wh fuzzy se-based polyomal euros, Iform. Sc. 176 (26) [21] A.R. Tahavvor ad M. Yaghoub, Aalyss of aural coveco from a colum of cold horzoal cylders usg arfcal eural ework, Appl. Mah. Model. 36 (212) [22] C.C. Wag ad C.F. Tsa, Fuzzy processg usg polyomal bdrecoal heero assocave ework, Iform. Sc. 125 (2) [23] L.A. Zadeh, The cocep of a lgusc varable ad s applcao o approxmae reasog: Pars 1-3, Iform. Sc. 8 (1975) [24] L.A. Zadeh, Toward a geeralzed heory of uceray (GTU) a oule, Iform. Sc. 172 (25)

14 Solvg Fuzzy Equaos Usg Neural Nes A. Jafara; S. Measoomy a; R. Jafar 46