Fuzzy Forecasting Applications on Supply Chains

Transcription

1 WSEAS TANSACTINS o SYSTEMS Haa Toza Fuzzy Forecag Applcao o Supply Cha HAKAN TZAN ZALP VAYVAY eparme of Idural Egeerg Turh Naval Academy 3494 Tuzla / Iabul TUKIYE hoza@dhoedur Abrac: - emad forecag; whch oud ba for deco mag proce, amog he ey acve ha drecly affec he upply cha performace A he demad paer vare from yem o yem, deermao of he approprae forecag model ha be f he demad paer a hard deco maageme of upply cha The whplah effec ca be expre a he varably of he demad formao bewee age of he upply cha ad he creae of h varably a he formao move upream hrough he cha The uage of proper demad forecag model ha adequae for he demad paer a mpora ep for moohg h uderable varably Th paper evaluae he effec of fuzzy lear regreo, fuzzy me ere ad fuzzy grey GM (, forecag model o upply cha performace quafyg he demad varably (e whplah hrough he age of a ear beer game upply cha mulao model expaded wh fuzzy parameer Key-Word: - Forecag, Supply cha, Fuzzy regreo, Fuzzy me ere, Grey fuzzy Iroduco Supply cha (Sc model are dyamc complex yem cludg may ophcaed acve uch a coa o me formao, chedulg, produco, drbuo, ervce ad deco mag procee whch are all requred for afyg he cuomer demad A mple defo for hee complex yem ca be expre a he ewor of orgazao ha are volved, hrough upream ad dowream lage, he dffere proce ad acve ha produce value he form of produc ad ervce he had of ulmae cuomer [] Alhough he formao flow Sc yem co of cumulave daa abou co parameer, produco acve, veory yem ad level, logc acve ad may oher relaed procee; behg of he defo expoe ha he performace of a ucceful Sc yem cocer moly wh accurae ad approprae demad formao a h val daa fluece all deco mag procee of Sc [] A well-ow pheomeo of Sc yem he varably of he demad formao bewee he age of he upply cha (e he whplah or bullwhp effec (WE Th varably creae a he demad daa move upream from he cuomer o he oher age of he SC yem egederg uderable exce veory level, defecve labor force, co creae, overload error produco acve ad ec Forreer [3, 4] wh a mple Sc mulao coed of realer, wholealer, drbuor ad facory level, dcovered he exece of WE a demad amplfcao Sc model He argued abou he poble caue ad uggeed ame dea o corol he WE He emphazed o he deco mag proce each phae of SC ad berayed ha h proce could be he ma reao he demad amplfcao hrough he cha from he realer o he facory level [] He alo deoed ha, me lag for clercal wor, purchag ad raporao, lead me ad pecfc facory capably could be oher lely reao of WE Le Forreer, Serma [5, 6] alo focued o he exece ad caue of WE He ued a expermeal Sc mulao ha mulae he beer drbuo a mple Sc co of four echelo; realer, wholealer, drbuor, facory whch he became a well-ow Sc mulao model; Beer rbuo Game, wdely ued for eachg he behavor, cocep ad rucure of Sc The model wa o mple bu depe o mplcy, uccefully howed he mpac of he deco proce echelo o he demad varably Ma objecve o gover each echelo achevg he dered veory ad ppele level mmzg he oal co cludg veory ad horage co Fg llurae he geeral yem rucure of a beer game [7] Serma appred ha accurae judgme made by he parcpa ad he adopo of hee judgme o he Sc yem are he ma reao of h pheomeo Lare eal [8]; ug a Serma baed beer game model, howed ha he rucure of a produco-drbuo cha produce broad varey of dyamc behavor I he udy, ISSN: Iue 5, Volume 7, May 8

2 WSEAS TANSACTINS o SYSTEMS Haa Toza dffere ype of behavor are ummarzed wh Lypuov expoe There were wo mpora aumpo he model; he parameer of he deco rule coa, (e, rema he ame hrough he ere oluo, deco each age coued uder he formao avalable each age Afer howg he effec of veory corol polce o dyamc ad co, Lare eal cocluded ha a ophcaed maageme formao yem ad reducg me lag are mpora ey for reduco demad amplfcao ad co Alo hey clamed ha erlocg commo parameer orderg polce mgh caue hgh co ad rcacy dyamc Fg The geeral yem rucure of a beer game Lee eal [9,,, ] declared caue of WE a: prce flucuao, raog game, order bachg, lead me ad demad foreca updag; whch alo he ma cope of h paper Bagaha eal [3], Grave [4], rezer eal [5], Che eal[6] ad L eal [7] alo uded WE from he perpecve of formao harg ad demad forecag / updag Accurae demad forecag oe of he major mmzao ool for WE bu, fdg he adequae model for he demad paer arl [] Though pa he ude abou forecag brg o lgh ha, uder relavely few daa formao ad ucerae (ju le he uao may Sc yem he fuzzy forecag model uch a fuzzy me ere (FT [8, 9,,,, 3], fuzzy lear regreo (F [4, 5, 6], fuzzy grey GM(, (FGG [7] ad oher appled forecag model [8, 9] performed uccefully, o much aeo pad o hee yemac for her uage Sc maageme Th paper focue o he effec of eleced fuzzy forecag model o he Sc performace va demad varably he yem The re of h paper orgazed a follow I eco, F, FT ad FGG forecag model are roduced Seco 3 deal wh he quafcao of WE for evaluao of Sc performace I eco 4 ad 5 propoed Sc mulao model dcued ad he effec of eleced forecag model o Sc performace are examed Fally eco 6 reearch fdg ad cocluo are preeed F, FT, FGG Forecag Model F Forecag Model Lear regreo; whch how he relao bewee repoe or depede varable y ad depede or explaaory varable x, ca be formulae coderg he relao of y o x a a lear fuco of parameer wh Y f ( x θx where θ he vecor of coeffce ad X he marx of depede varable [] The applcao of lear regreo model uable for he yem whch he daa e oberved are drbued accordg o a acal model (e uoberved error erm muually depede ad decally drbued Bu geerally, fg he demad paer of a real Sc o a pecfc acal drbuo o poble The F model roduced by Taaa e al [4, 5] whch devao reflec he vaguee of he yem rucure expreed by he fuzzy parameer of he regreo model (e poblc uable for he declared demad paer ad bacally ca be formulae a: ( c, ( c, x ( c, x ( c x Y, where c he ceral value ad he pread value, of he h fuzzy coeffce; A ( c,, uually preeed a a ragular fuzzy umber (TFN Ad h repreeao fac ha relaxe he crp lear regreo model Ug fuzzy ragular memberhp fuco for A, he mmum fuzze for Y ca be maaed wh he followg lear programmg (LP model whch mmze he oal pread for he fuzzy oupu parameer [, 5, 3] m Z M m ( h x c c x x ( h ( h where x, h ad,,3,,,,3,, Noe ha he cora of he LP model afe he followg x x y y ( ( ISSN: Iue 5, Volume 7, May 8

3 WSEAS TANSACTINS o SYSTEMS Haa Toza equao whch defe he degree of belogg of obervao (ha characerzed by h y o Y µ Y ( y h,,,3, (3 FT Forecag Model For dyamc yem le Sc, whe horcal demad daa ha wll ue o calculae he derable foreca value are lguc value ad (or are mall amou, fuzzy me ere model be f he apec [,, 3] Sog eal [8, 9, ]; fuzzfyg he erollme of he Uvery of Alabama, ued fuzzy me ere forecag problem ad propoed a fr-order mevara fuzzy me ere wh fr-order me-vara fuzzy me ere for he oluo of he forecag problem Laer Sog eal [] roduced a ew FT model ad berayed ha be reul are held by applyg eural ewor for defuzzfyg daa Wag [], L eal[] ad Hwag eal [3] alo uccefully ued FT forecag model I h paper Hwag FT forecag model eleced ad ued for FT demad forecag The model ca be ummarzed a follow: Fr, he varao bewee wo couou horcal daa o be calculaed ad mmum / maxmum creae (e m / max are o be deermed, Nex ep o defe he uvere dcoure; U, wh followg equao ug m ad max d U d, ] (4 [ m max where ad are pove value ha f for eparag U d o equal legh The fuzzy e o U d are o be defed (e defg fuzzy me ere ( F ( ad varao daa o be fuzzfed efg F ( a p p p F ( Z Z Zm (5 u u um where he memberhp p z are p z The fuzzy e A of U he ca be repreeed a; Z Z p A p p Zm (6 u u um Fuzzfcao of varao are deermed accordg o u ha hey f v Ad he fal ep clude compog he relao marx; (, whch govered by operao ad crero marxe (e w (, Z( ad defuzzfyg he calculaed varao [,3] whch wll be ued for emag he forhcomg value ug he relao of he chace value gahered from relao marx I h ep he wdow ba; w, have o be deermed whch how he umber of perod of varao ha wll be ued for forecag For perod, w (, Z( ad ( defed a follow [3]: w [ Z, Z, Z, ] Z F( (7 ( 3 F ( F ( 3 ( F( - w - w xz xz xz xz ( xz xz ( w w w where j m ad j j x Z j, w The he emaed varao wll be deermed wh he followg equaly Z m m (8 (9 ( Where r j Max( j ;,,, w ad j,,,m The foreca value for he perod calculaed by defuzzfcao of F( ad addg h value o he acual daa of he perod ad h operao coclude he FT forecag mehod m m [ r, r, ] F ( r m wm m m m wm xz m xz m xz m ISSN: Iue 5, Volume 7, May 8

4 WSEAS TANSACTINS o SYSTEMS Haa Toza FGG Forecag Model The grey yem heory roduced by eg [3, 33] ca mply be ummarzed a a mehodology ha cocer wh he yem comprg ucerae ad lac of uffce amou of formao; whch, he erm grey dcae he yem formao ha le bewee he clearly ad ceraly ow oe ad he uow par of he yem []The dcree me equece daa ued o expoe a regular dffereal equao wh he accumulaed geerag operao (AG; whch eablhe a regulary o o had o daa ere, ad vere accumulaed geerc operao (IAG I grey dffereal model GM (,m, erm ad m repree he order of ordary dffereal equao ad he umber of grey varable repecvely, defg he order of AG ad IAG Grey forecag whch oe of he mo mpora par of he grey yem heory, ca bacally be defe a he ue of pa or curre daa of a yem o develop a grey model for predcg he fuure red of he yem oupu (demad, me, ec A creae ad m alo creae he compuao me expoeally caug lely correce defec, mo wdely ue model grey yem heory GM (, whch ha ame mpora advaage hoe ca be ummarzed a he uage for ay d of daa drbuo cludg mall daa e ad le requreme for compuao [] The compuao yem rucure of he Grey GM (, model whch alo form a bae for FGG forecag model ca be ummarzed a follow [7] Le how o had daa colleced from he yem a; ( where repree he umber of daa The geeraed AG ere of ;, he ca be deoed a where ( (,,,,,, (,,,, Compog a dffereal equao for a equao (3 ad depcg dffereal equao a a dcree me ere wh oe u amplg erval he relao decrbed equao (4 ca be coruced a b (3 d d 3 3,,, (4 where a ad b deoe he uow developed coeffce ad he uow grey corol varable repecvely For equao (4 equal o ad for,, 3, equao (3 ca be redeg a marx form a equao (6 by eg he ecod par of he fr order grey model o ; average (5 (6 Afer applyg lea quare mehod, oluo of a ad b ca be obaed ad ug h wo parameer equao (3 ca alo be olved ad oupu foreca value ca be deermed wh he followg equao where he emaed cumulaed value of ad he foreca value for average( / ( 3 average( average( average( (7 (8 I addo o he fac defed for he uage of grey yem ad grey forecag model above; f he daa e colleced from he yem are lguc, FGG forecag model may perform uccefully The FGG forecag model roduced by Taur [7]; whch aume daa ere colleced from he yem are ymmercal ragular fuzzy umber (TFN, very mlar o he crp oe ad ca be explaed wh he followg equao The orgal daa ere colleced from he model : (9 where he umber daa colleced from he yem Ad ; he ew fuzzy daa equece geeraed wh AG, ca be how wh he followg equao a: a b a ( ( b / a e ( b /, a ( 3 ( 3 ( ISSN: Iue 5, Volume 7, May 8

5 WSEAS TANSACTINS o SYSTEMS Haa Toza where ;,,,, a ymmercal TFN wh ceral ad pread value,,,, repecvely Ad he fuzzy GM(, model deoed a; ( where a he developg coeffce ad STFN b deoe he fuzzy grey pu wh he ceral value b ad he pread value b ad he memberhp fuco for b coruced a follow α b µ b( α b b - b α b b oherwe ( By eg he amplg erval oe u a crp model / d ca be rewre a or ;,3,, (3 / d d,,,, (4 where a fuzzy umber wh ceral ad pread value; ad, c,3,, Le he average of ad, Q ( whch a STFN wh he ceral value Q ad he pread value p be he ecod par of equao ( a ; where / ( Q p / d d a b / d d aq b, (5 wh he ceral value Q / ad he pread value (6 / (7 A he pread deerme fuzze; value of uow varable a, b ad b ca be obaed from he oluo of he followg LP model wh he objecve fuco ha mmze he pread value of STFN b (8 Afer olvg he LP problem; mlar o he crp grey GM(, model, Taur uggeed ha emaed fuzzy low umber wh lower boud, ad upper boud upr, low upr ;,, (,, could be obaed Fally he fuzzy foreca value for perod ;, low, upr,, could be deermed a follow; M z b ( b aq ( h ( b ap ( h ( b aq ( h ( b ap ( h ( for, g low, up (9 3 The Meaurg Sc Performace wh Quafed WE Th udy evaluae he Sc performace by quafyg he demad varably (e WE varou age of he propoed Sc mulao defg WE a he rao of demad varace of wo coeque age [6] The maller he WE he beer he Sc performace wll be Che e al[6]; aumg he cuomer demad perod o he realer ( a radom varable; defed wh he followg equao h ;, g, g, g (3 where µ ad ρ deoe a o egavy coa ad correlao parameer ( p < repecvely (a ρ dcae he relaohp bewee demad ρ beray he depede decally drbued (dd demad The varace of emerged a a, b, b µ ρ ε (3 Aumg he veory yem he realer order-up-o polcy wh fxed lead me ad he forecag echque ued mple movg average, he order placed by he realer a perod ; q ( whch a mpora varable owg o he fac ha WE quafed a Var (q relave o Var ( deoed a: δ ρ ISSN: Iue 5, Volume 7, May 8

6 WSEAS TANSACTINS o SYSTEMS Haa Toza (3 A q a perod ca be egave (equao (3 a mpora aumpo made a he exce veory reured whou co Che eal model Ug he emae of lead me demad ad adard devao of he foreca he perod, Equao (3 ca be rewre a follow; where ad L (33 L σ repree he emae of mea lead me ad he emae of adard devao of L perod foreca error repecvely ad e he oe perod foreca error Ug varace ad covarace Var q ca be deermed wh he followg equao ( x Cov( q q z σ q y y L L L L z( σ e σ e ad q / have he followg relao: (34 q L L p ( ρ p p (35 where p he umber of obervao Mau eal [34] propoed aoher approach for quafyg WE ug Lyapuov expoe (LPE Auhor aed ha LPE; whch may ue for quafcao of he rregulare of o-lear yem dyamc, may alo be ue for quafyg BWE for Cd ad ocd f LPE eed a a facor for expadg a error erm of a yem The relao bewee q ad ca be aed a ; (( L / p ( L / p( L / p, L e p σ L e ( L / p z( L / p xcov( ( p q ( j / j ( L j / x ( ρ j L, σ e (36 where deoe he umber of obervao for he calculao of foreca value of demad average ad L he lead me a age 4 The Smulao Model I h udy a ear beer drbuo game model ued; whch exeded from he Pa [35] model wh predeermed co em (holdg, eup / produco, veory rerco, produco rerco ad delay fuco The model propoed here, mply a wo aged Sc yem co of a realer ad a facory ue o commo ue ad ucceful prmed fuzzy fuco; MaLab he adjudcaed mulao ool emad formao each perod ca be eher crp or fuzzy depedg o he forecag model ha wll be aalyzed; mlar o our prevou model [] The model evaluae he Sc performace by compug he rao of demad varace of coeque age; e Var / Var, where deoe he demad S S from age S o upream age S (e c, r, f where c, r, f repree cuomer, realer, facory repecvely The co, delay ad facory produco capacy parameer are varable ad her value deped o he aalyzer The geerc deco rule each me perod ca be ummarzed wh he followg equao [, 6] Upream rder Quay Foreca Value Correco of Iveory Correco for Supply Le] [ S (37 The orderg deco yem of he beer game model lluraed Fg [6, 35] Smple expoeal moohg (EXS model ued a a crp forecag echque for comparo The formulao of EXS a follow; F F ( ( α α (38 where F he foreca value for perod, obervao of demad perod, F he calculaed foreca value of he prevou perod ad α he moohg coa; < α ISSN: Iue 5, Volume 7, May 8

7 WSEAS TANSACTINS o SYSTEMS Haa Toza B f INV B S (4 INV ele INV INV ( IS S (4 INV S SC x FW, FW xl (43 (44 SA B M MA, B (45 B B, (, I, (46 M M ( S I, (47 MA MA, ( S, IS, (48 Fg The orderg deco yem of he beer game The cuomer order receved from realer perod ae a a bae for forecag he forhcomg demad, ad each me he order receved, forecag fuco updae rucure accordg o he ew demad formao Afer emag he forhcomg demad; mulao model, ug he deco rule (37 ad oher parameer (e co, lead me, avalably, demad paer ec, mae a orderg deco o upper echelo of Sc Ad he rao of varably bewee he cuomer order, realer order ad maufacurg deco of facory how he performace of Sc yem baed o he eleced forecag model The value of adjume parameer ued for correco of veory ad upply le he ame a Serma ad Pa model The formulao of yem rucure each age a follow; FW α ( INV INV αβ ( S SA B I S (39 B (4 where, he order deco, FW he foreca value deermed from he eleced forecag model, INV ad INV are veory ad dered veory, SA ad S are he upply le ad dered upply le, B he order baclogged, I he comg order, S ad IS are ougog ad comg hpme, SC he afey coa, L he oal delay age a perod α ad β repree he adjume parameer for veory ad upply le repecvely 5 Experme The followg fgure he ex page llurae radomly geeraed c (he ame for all mulao ru, ad calculaed r ad f value derved from he mulao ug eleced forecag mehod for a me horzo of 3 perod Ad o reflec he repoe of Sc performace o he eleced forecag model, he calculaed adard devao value are gve Table- (varace value ca alo oba ug Table- I he mulao, he produco capacy of facory ae a u per perod ad o had veory each echelo a me zero e o aga u The e of value for 3 perod;, are gve below c e {6697; 6769; 766; 5654; 3654; 4735; 96 ; ; 53389; 38779; 36958; 83868; 49454; ; 43398; 4; 4984; 46548; 38547; 7585; 5888; 75688; 4454; 7657; 6653; 584; ; 4874; 983; 76784} e ISSN: Iue 5, Volume 7, May 8

8 WSEAS TANSACTINS o SYSTEMS Haa Toza Fg3 The demad repoe o EXS forecag model (e WE Fg3 emad repoe o F forecag model Fg emad repoe o F forecag model Fg emad repoe o F forecag model Forecag Model EXS F FT FGG Sd c Sd r Sd f Table- Sadard devao value ISSN: Iue 5, Volume 7, May 8

9 WSEAS TANSACTINS o SYSTEMS Haa Toza 6 eearch Fdg ad Cocluo I h udy he effec of eleced fuzzy forecag model; F, FT ad FGG, o Sc performace are aalyzed ug compued demad varably a a rao of varace of coeque age (ewe For comparg he obaed reul of he mulao ug fuzzy forecag, EXS forecag model choe a a bae crp forecag model A mple umercal example made ug radom geeraed demad daa The reul expoed ha he fuzzy forecag model ued he udy; excep FGG, qucly capured he demad paer ad coderably creaed he performace of propoed Sc yem decreag demad varably ad hrough he cha Furher reearche ca be made ug fuzzy lead me a o more adap he model o complex real-world Sc yem eferece: [] Chropher M, Logc ad Supply Cha Maageme, Pma Publhg, Lodo, UK, 994, pp3 [] Toza H, Vayvay, Aalyzg emad Varably Through SC Ug Fuzzy egreo ad Grey GM(, Forecag Model, Iformao Scece 7, World Scefc, 7, pp [3] Forreer J, Idural yamc: A major breahrough for deco maer, Harvard Bue evew, 36, 958, pp [4] Forreer J, Idural yamc, MIT Pre, Cambrdge, MA, 96 [5] Serma J, Modelg Maageral Behavor: Mpercepo of Feedbac a yamc eco Mag Experme, Maageme Scece, 35, 989, pp3-399 [6] Serma J, eermc Chao a Expermeal Ecoomc Syem, Joural of Ecoomc Behavor ad rgazao,, 989, pp-8 [7] hp://beergameu-luaca/bghm, 8 [8] Lare E, Morecrof J, Thome JS, Complex Behavor a Produco-rbuo model, Europea Joural of perao eearch, 9, 999, pp 6-4 [9] Lee H L, Padmaabha V, Whag S, The Bullwhp Effec Supply Cha, MIT Sloa Maageme ev, 38, 997a, pp 93- [] Lee H L, Padmaabha V, Whag S, Iformao oro a Supply Cha: The bullwhp effec, Maageme Scece, 43, 997b, pp [] Lee H L, Algg Supply Cha Sraege wh Produc Ucerae, Calfora Maageme evew, 44,, pp 5--9 [] Lee H L, Padmaabha V, Whag S, Iformao oro a Supply Cha: The bullwhp effec, Maageme Scece, 5, 4, pp [3] Bagaha M, Cohe M, The Sablzg Effec of Iveory Supply Cha, perao eearch, 46, 998, pp [4] Grave S C, A Sgle Iveory Model foe a Noaoary emad Proce, Maufacurg &Servce perao Maageme,, 999, pp5-6 [5] rezer Z, ya J, Smch-Lev, Quafyg he Bullwhp Effec: The mpac of forecag, lead me ad formao, Maageme Scece, 46 (3,, pp [6] Che F, rezer Z, ya J, Smch-Lev,The Impac of Expoeal Smoohg Foreca o he Bullwhp Effec, Naval eearch Logc, 47,, pp [7] L G, Wag S, Ya H, Yu G, Iformao Traporao Supply Cha, Compuer ad perao eearch, 3, 5, [8] Sog Q, Chom B S, Forecag Erollme wh Fuzzy Tme Sere, Fuzzy Se ad Syem, 54, 993, pp -9 [9] Sog Q, Chom B S, Fuzzy Tme Sere ad I Model, Fuzzy Se ad Syem, 54, 993, pp [] Sog Q, Chom B S, Forecag Erollme wh Fuzzy Tme Sere Par II, Fuzzy Se ad Syem, 6, 994, pp -8 [] Wag CH, Predcg Tourm emad Ug Fuzzy Tme Sere ad Hybrd Grey Theory, Tourm Maageme, 5, 4, pp [] L S, Cheg Y, A Hdde Marov Model-baed Forecag Model for Fuzzy Tme Sere, WSEAS Traaco o Syem, 5, 6, pp [3] Kahrama C, Fuzzy Applcao Idural Egeerg,, h ed, Sprger, 6, pp4 55 [4] Taaa H, Uejma S, Aa K, Fuzzy Lear egreo Model, IEEE Tra Syem, Ma ad Cybere,, 98, pp [5] Taaa H, Waada J, Poblc Lear Syem ad Ther Applcao o he Lear egreo Model, Fuzzy Se ad Syem, 7, 988, pp75 89 [6] Wag, HF, Taur H, Igh of a Fuzzy egreo Model, Fuzzy Se ad Syem,,, pp ISSN: Iue 5, Volume 7, May 8

10 WSEAS TANSACTINS o SYSTEMS Haa Toza [7] Taur C, Fuzzy Grey GM(, Model Uder Fuzzy Syem, Ieraoal Joural of Compuer Mahemac, 8 No, 5, pp4-49 [8] Ela Ch N, Teoura G J, Kavaza S, Coax GC, A Mderm Eergy Forecag Mehod Ug Fuzzy Logc, WSEAS Traaco o Syem, 3, 4, pp 8-36 [9] George A, Ucec C, Forecag he Wd Eergy Produco Ug a Neuro-fuzzy Model, WSEAS Traaco o Evrome ad evelopme,, 6, pp [3] o T J, Fuzzy Logc wh Egeerg Applcao, h ed, Joh Wley ad So, 5, pp [3] Hwag J, Che SM, Lee CH, Hadlg Forecag Problem Ug Fuzzy Tme Sere, Fuzzy S ad Syem,, 998, pp7-8 [3] eg,,lj, Grey Predco ad eco ( Chee, Huazog Iue f Techology Pre, Wuha, Cha, 989 [33] eg,,lj, The Joural of Grey Syem,, 989, pp-4 [34] Mau A, Madad A, The Bullwhp Effec ad Lyapuov Expoe, Appled Mahemac ad Compuao, 89, 7, pp 35-4, [35] Pa SK, Aaly of he Caue of Bullwhp Effec Supply Cha: A Smulao Approach, George Wahgo Uvery, 3 ISSN: Iue 5, Volume 7, May 8