NEURAL DATA ENVELOPMENT ANALYSIS: A SIMULATION

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1 Itratoal Joural f Idustral grg v NURAL ATA NVLPMNT ANALYSIS: A SIMULATIN Luz Bod Nto Marcos Prra stllta Ls la Goçalvs Goms João Carlos Corra Batsta Soars d Mllo 3 Fabao S. lvra. d g. ltrôca Tlcomucaçõs Uvrsdad do stado do Ro d Jaro Rua São Fracsco Xavr 54 Bl. A Sala 536 Maracaã 55-9 Ro d Jaro RJ Brazl. mal: {L. Bod Nto lbod@ur.br; F.S. lvra fabao@magl.com.br} Programa d g. d Produção Uvrsdad Fdral do Ro d Jaro Ctro d Tcologa F-5 Ilha do Fudão Ro d Jaro RJ Brazl. mal: {M.P.. Ls ls@.ufr.br;.g. Goms ggoms@.ufr.br} 3. d g. d Produção Uvrsdad Fdral Flums Rua Passo da Pátra 56 São omgos 4-4 Ntró RJ Brazl mal: {gmacsm@vm.uff.br} Ths ar vstgats th crato of ffccy masurmts structurs of dcso-mag uts MUs calld Nuro- A by usg hgh-sd otmzato moduls calld Nuro-LP srd th "hlosohy" of a ucovtoal artfcal ural tor ANN ad umrcal mthods. I addto th lar rogrammg roblm LPP s trasformd to a otmzato roblm thout costrats by usg a sudo-cost fucto hr's addd a trm of alty causg hgh cost all tm that o of th costrats gos volatd. Th roblm s covrtd to a dffrtal quatos systm. A ucovtoal ANN mlmts a umrcal soluto basd gradt mthod. Sgfcac: Th covrgc sd ca b trmly hgh f th roosd moduls as tgratd a ch VLSI Vry Larg Scal Itgrato or CMS - Comlmtary Mtal-d Smcoductor ad coctd to a fr slot a comutr. Kyords: A Mathmatcal Programmg Comutatoal Itllgc VLSI CMS. Rcvd ; Acctd. INTRUCTIN Th ata vlomt Aalyss A s a mathmatcal tchqu that has th obctv of aalyzg th dcso-mag uts MUs rformac. It also allos th valuato of th rlatv oratoal ffccy of orgazatos MUs cotmlatg ach MU rlatvly to all th othrs that comos th vstgatd MUs grou Chars Coor ad Sford 996. Th A tchqu comars th MU ffccs by thr ablts trasformg uts oututs masurg th rachd outut rlato trms of th rovso suld by th ut. I th d of th aalyss th A tchqu s abl to tll hch uts ar rlatvly ffct ad hch uts ar rlatvly ffct Agulo 998. Th A tchqu volvs th us of Lar Programmg LP to solv a grou of tr-rlatd of th lar rogrammg roblms LPPs as may as th MU umbrs fally obctfyg dtrmg th rlatv ffccy of ach MU stllta. tmzato moduls calld Nuro-LP ll b usd th ural modl roosd Nuro- A srd by th artfcal ural tor hlosohy Bod. Th A modls ca b ortd to uts or oututs ad ths ortato must b rvously chos by th aalyst as startg ot th A aalyss. Th ortato to uts dcats that at to rduc th uts g th oututs uaffctd. I th othr had th ortato to oututs dcats that at to cras th oututs thout affctg th uts Coll 998. Th most mortat modls ar th follog: CCR ad BCC A modls. CCR Modl rstd by Chars Coor ad Rhod 978 that bulds a o aramtrcal surfac lar by arts ovr th data ad dtrms th vstgatd MUs tchcal ffccy ovr ths surfac. It as cocvd as a ut ortd modl ad t ors th costat rtur of scal CRS hch mas that ach varato th uts roducs a roortoal varato th oututs. Th roblm cossts dtrmg th u ad v ght valus to mamz th lar combato of th oututs dvdd by th lar combato of th uts stllta 998. Th rocss must b ratd to ach of th MUs ad by ths rocsss dtrm th rlatv valu of ach MU ffccy. But quato f u ad v ar th otmum soluto vctors u ad v ll b otmum soluto vctors too ad cosqutly th roblm ll

2 Itratoal Joural f Idustral grg v rst ft solutos. To solv ths roblm Chars ad Coor 96 troducd a lar trasformato that allos trasformg lar fractoal roblms to LPPs cratg th modl calld Multlrs quato. Ma h subct to : u v hr : h r - r r - MU ffccy total amout of s - total amout of - total amout of Y X amout of amout of s u Y s v X u Y v ght to ut v X uts oututs MU u ght to outut... outut to MU ut to MU Ma h subct to : r s v X u Y u v s r u Y v X... It s ossbl drv th dual modls to multlr rmal. So th dual ll rst a smallr amout of costrats s+r < + bcaus th A modl rqurs that th umbr of MUs b gratr tha th umbr of varabls. By th osd rasos th dual modl calld vlo bg asly solvd s rfrrd comard to th Multlrs modl. I th vlo modl th obctv s to dtrm th valus of mmzg as 3. M subct to : Y θx θ - Y X...s... r

3 Itratoal Joural f Idustral grg v BCC Modl dvlod by Bar Chars Ad Coor 984 allos varabl rtur of scal VRS avodg stg roblms mrfct comtto stuatos facal costrats tc. I ths cas th VRS frotr cosdrs crasg or dcrasg rturs th ffct frotr. To do ths ob t as troducd th CRS mod a covty costrat mag th sum qual to. Tryg to comar th CRS frotr to th VRS th Fgur that rrsts 5 MUs of o ut ad o outut shos th rlato bt ths frotrs for th to cass. Fgur. CRS ad VRS Frotrs. Y C R S Fgur. - CRS ad VRS Frotr VR S MU3 MU MU5 MU MU4 X I Fgur t s ossbl to vrfy that th MUs 4 ad 5 ar ffct. I th BCC modl that adots VRS ach MU s comard to th ffct MUs that orat th sam scal. So usg th ortato to uts vrfy that th otmum rocto of th MU 4 has a ot that rflcts th cov lar combato of MUs ad. Usg th ortato to uts vrfy that th otmum rocto of th sam MU 4 has a ot that rflcts th cov lar combato of MUs ad 3. Hovr for both ortato cass th lar combato valus ar gv by th s. Th vlo modl ortd to ut ad th rmal drvd modl multlrs ar gv by 4 ad 5. M θ subct to: Y θx - Y X...s... r 4

4 Itratoal Joural f Idustral grg v Ma h subct to : r s v X u Y u ad v s r u Y v X ad... urstrctd I th othr had th Artfcal Nural Ntors ANNs also o as coctost systms ar structurs srd th huma bra fuctoalty Rosblat 96. Th ANNs ar massvly arallld structurs basd o sml rocssg lmts P srd th bologcal uro ad dsly trcoctd. Th ma ANN charactrstcs ar: aralll sarch ad cott addrssg mtatg th bra that thr has mmory addrss or loos for a formato squtally; larg dog th tor th th caacty of larg crta oldg by succssv attr rstatos rc thout th cssty of lctg th algorthms to th cuto of a ob; assocato allog th tor to assocat dffrt attrs; gralzato ablg th tor to dal th os ad dstorto ad corrctly asr a vr s ut by smlarty th othr rvously rstd attrs; abstracto dog th tor th th ablty to abstract th ssc of a ut grou; ad fally th robustss allog thas to th aralllsm that v th th loss of Ps bad orato of th tor ll vr ha.. By scfc algorthms th ANNs lar som oldg trag st by th succssv rstato of amls attrs storg thm th coutlss stg coctos bt th Ps calld ghts. Th oldg s dstrbutd all ovr th tor avalabl ad rady to b usd cuto st may dffrt alcato aras as th ural lar rogrammg Zurada 99. I th cas of th Nuro-LP otmzato moduls art of th Nuro-A modl a structur smlar to th ANN s usd hr th syatc ghts obtad th trag st ar bascally formd by th coffct of th roblm costrat grous Rosblatt 96 ad Wassrma 993. Although thr ar may Ps modls as th Aalogcal lctroc modl ad th Fuushma modl hch s dly usd Fgur s basd o th modl roosd by McCulloch & Ptts. It trs to rroduc a sml ay th bologcal uro orato. Th P uts... or smlarly to th bologcal uro syass rcvg comg sgals of othr uro aos. 5 Fgur. Procssg lmts Modl. W W F W W bas As th bologcal uro th syatc coctos must caus ctato or hbto ractos th ghbor uros. So a good rocdur s to assocat to ach syatc cocto a ostv or a gatv ght valu W W... W dtrmg th ffct that a sourc P has ovr th dstato P. Cosqutly th P ould b abl to trggr h th odrd sum of th uts X ad th ghts W cd th thrshold valu W bas durg th latcy rod. Th P s dfd by a roagato rul that dtrms th ay th uts ll b comutd ad by a actvato fucto F that dtrms th valu of th actvato stat for th dstato P. Th mostly usd formula as roagato rul quato 6 dfs as th odrd sum of th valus rovdd as uts ad th ght valus of th P ut coctos Harvy 994.

5 Itratoal Joural f Idustral grg v Th quato 7 shos th P outut. F Th actvato fucto tys mostly usd ar: th st; th symmtrcal st; th lar; th sudo lar; th sgmod; ad th tasgmod.thr ar bascally to tys of archtcturs usd ANNs: th fdforard tors thout fdbac; ad th fdbacard tors. Th mostly usd tor archtctur s th fdforard. Ths arragmt s comosd of a grou of Ps arragd layrs o or mor that trcoct thmslvs squc. Th most comlt cofgurato rsts o or mor trmdat or hdd layrs bt th ut ad th outut layr ad t s o as mult layr tor. Th fact of havg hdd layrs allo bttr rsults for crta roblms as ll as allog th soluto for mossbl roblms to b solvd th sgl layr tors Wassrma 993 ad Zurada 99. Th ural rocssg s accomlshd to ma hass: Trag ad cuto. Th trag has or larg s th udatg rocss for th cocto ghts. Its goal s to acqur formato ad stor t as a ght matr W Ms 969. I th cas of th Nuro LP ma cll of th Nuro A modl th roblm acoldgmt s rvously o by th LPP costrat coffcts ad lmats th d for ths has. Th cuto has rcall calculats th ANNs outut Y trms of th ctd stmulus th ut X ad th ghts obtad th trag has Y or mosd by th roblm tslf. I th Nuro LP cas th goal for ths has s to rcovr th formato hch mas dtrm th otmum valu for th LPP dcso varabls ad that th cas of th ata Wrag Aalyss may rrst th ffccy valu of a MU Bod. Ths rocss ll b do by th rsoluto of a dffrtal quato systm obtad by th trasformato of th orgal LPP a otmzato roblm thout costrats. Th umrcal mthod usd to solv th dffrtal quato systm ll b th dyamc gradt mthod drvd from th Nto mthod ad that s vry smlar to th ANNs trag mthod. Itally th ANN archtctur usd th Nuro LP modl ll b rstd as ll as th dvlomt of th trag algorthm basd o th mmzato of th sum squard rror th tor outut by th dcrasg gradt mthod ad ts varatos. I th Nuro LP cas th ANN s usd th cuto has ad t alrady has th oldg rfrrd to th LPP rrstd hr by th roblm costrat coffcts. Th mathmatcally rov th LPP trasformato comosd of a obctv fucto ad a costrat collcto a otmzato roblm thout costrats Bazaraa 993. A fucto calld sudo-cost as adotd hr a alty trm as addd causg a hgh cost vry tm a costrat s volatd. Th roblm ca b solvd by th gradt mthod turg t to a dffrtal quato systm hch ca b umrcally solvd. Fally a cas study s rstd. 7. LARNING PARAIGM F TH ARTIFICIAL NURAL WRKS Th rctro crdtd to Rosblatt as th frst larg mach th survsd trag ossblty. As sho Fgur 3 th tor currt outut s comard to ach trato th a dsrd valu T assocatd to th trag attrs gratg a rror sgal. Fally ths sgal suffrs a larg adatv rocss udatg th ghts ach trato Rosblatt 986 Rumlhart 986. Fgur 3. Prctro Archtctur.

6 Itratoal Joural f Idustral grg v Th Prctro s a tor that ca b rstd ust th a P sgl layr or th multl layrs MLP. Its orato s basd o th follog: th ut sgal X ut attr s multld by a ght grou W adustabl by a larg rul to grat a tral ottal calld that rrsts th odrd sum of th uts by th corrsodg ghts. Th sgal s th rocssd by a lmtr calld actvato fucto F sho fgur.4 rsultg a outut sgal {} or {-}. Th outut sgal s comard to th targt or dsrd valu T roducg th rror sgal usd th tractv adustmt of th ghts by a larg rul gv by hr th aramtr s a ostv valu calld larg rat rlatd to th sd ad th stablty of th ANN covrgc rocss. Th tractv rocss of attr rstato causs th rror dcras ad h ths rror rachs a stablshd valu ts sad that th ANN absorbd th dsrd oldg ad that th rocss covrgd Saura 996. I th frst rmts do by Rosblatt th Prctro trag algorthm as trly basd th tchqu dvlod by Wdro Hoff hr th rror sgal as obtad bfor th actvato fucto ad thrfor lar. v th mag of lally sarabl fuctos th tchqu fald. So Rosblatt dvlod a algorthm adustg th ghts by th mmzato of th sum-squard rror usg th dcrasg gradt mthod Rosblatt 986. Th th rror = T - s obtad by th dffrc bt ths valus tag aftr th actvato fucto. I ths cas du to th drvd mosd by th mthod; th actvato fucto must b drvabl through th hol doma as t has th th tasgmod. So = F = tah hr s a ostv scalar that rrsts th fucto clato Hay 994. P. rrst th P muts ad fo amout a sa.. hr. hr ma rror s Th squard th F T. so As. Alaygth cha rul. rlatd th ghts s gv by thdffrtal gradt Th stataos T T 4 tah

7 Itratoal Joural f Idustral grg v ] - [ : Fally ] [- Thrfor 4 - So As Fgur 4 shos th mrovmt of th Prctro trag mthod roosd by Rosblatt Hay th Kogthat. th Kogthat. ] [ Thrfor Th rror mmzato rocss rqur that th ght varato occurs th drcto of th gradt gatv. So ] [. Aftr th trag has hr th valus of th syatc ghts ar dtrmd th ANN ll gt rady to b cutd. I ths has cuto th ANN rcvs sgals th ut hch dd ot ta art th trag has ad rsts th rsult th outut accordg to th oldg acqurd durg th trag has ad stord th ght matr. I th Nuro LP cas th ANN hr th ghts ar alrady o ad that rrst th roblm costrat coffcts th dtrmato of th outut s th t st hch dcats th valu of th LPP dcso varabls Bod. To th mult layr Prctro a algorthm smlar to th o dvlod ad calld bac-roagato s usd th trag has. Th oly dffrc s th calculato of th rror sgal th trmdat layrs of th Ps. Ths sgal must allo th rror roagato to th rvous layrs bac-roagato utl t rachs th frst o. Although du to th fact that ths subct s ot drctly rlatd to th dvlomt of th Nuro-LP t ll ot b vstgatd Wassrma 993. Fgur 4. Imrovmt of th Prctro Trag. 3. MATHMATICAL BASICS

8 Itratoal Joural f Idustral grg v Cosdr a otmzato roblm thout costrats hr sh to fd th valu  that mmzs a scalar fucto calld Psudo-cost rgy or bctv fucto. Accordg to Cchoc 996 th ot * ll b th global mmum of f * <= for all  ad a local mmum f th rlatosh * <= s t for a crta trval >.If th frst ad th scod drvatvs st th ot * ll b a local mmum f th gradt * = ad th Hssa matr * >. Th cssary ad suffct codtos for th stc of a local mmum ar: For o sgular for th ot * so * ll b <= for all < - * < > f th gradt * = ad th Hssa matr s symmtrc ad ostv * >. Th yamc Gradt Mthod s th mostly srad mthod from th os srd th Stst sct ad th Nto Mthod Bazaraa 993. Its basd th trasformato of th otmzato roblm thout costrats a frst ordr ordary dffrtal quato systm rrstd by 8. d dt hr ar a talcodtos ad s a larg matr. 8 So to fd th valu * that tas to a mmum ts cssary to solv or smulat th soluto of a dffrtal quato systm subctd to tal codtos. Its ossbl to coclud that * ca b dtrmd by th soluto ath or tractory curv of th roosd systm 9. * lm t t 9 Accordg to TAHA 99 gv a cost fucto hch must b mmzd hr = rrsts th ot hr th rocdur starts ad; bg * th gradt of for th th ot th ath th; th da s to dtrm a artcular ath hr d/d s mmzd ach ot of th ath. Th umrcal soluto ca b obtad cosdrg th follog: If succssv ots + ar dtrmd obyg th follog soluto ath: + = - hr <= <= ma s calld larg rat or tgrato st. Th valu of s dtrmd a ay that + alays rsults th mrovmt of th obctv fucto + <. Th rocdur ds h to coscutv ots + ad ar aromatly th sam. As so =. 4. NUR-LP MLING A LPP s classfd as a otmzato roblm th costrats. To solv t usg th ANN hlosohy ts cssary to buld a fucto calld sudo-cost or rgy fucto hch global mmum s th otmum soluto of th LPP. To buld th fucto ts cororatd a fucto or alty trm P [R ] to th orgal obctv fucto Kdy 994 Ch 99 Bargla 996 Zhu 99. Th alty trm must caus a hgh cost alty to th fucto vry tm a costrat s volatd R < ad zro cost f th costrat s satsfd R >. I ths ay th LPP s trasformd to a otmzato roblm thout costrat hr ts dsrabl to fd *  that mmzs th fucto. Th alty trm must alz bg for th cass of o fasbl solutos ad hbt for vabl solutos of th LPP s 996 Pa 995 ad Wrr 998. Th otmzato roblm thout costrat th alty trm ca b solvd smlarly to th ANN trag has alyg th dcrasg gradt mthod. So t must b rtt as a ordary dffrtal quato systm ad solvd umrcally. I ths cas th soluto ath quato + = + hr =... rrsts th umbr of varabls dcats th covrgc th valu of th roblm dcso varabls ad th roduct of th cost fucto gradt rlato to by a costat factor rlatd to th covrgc sd ad th stablty of th mthod. To sur accuracy of th mthod th alty aramtr must b vry hgh. Th ractc shos that valus trmly hgh ar ot covt from th comutg ot of v. I ths cas hghr valus of ar ot cssary for th corrct covrgc of th rocss. So th rasoabl valus th mmum of th sudo-cost fucto s quvalt to th otmum soluto of th orgal LPP. Accordg to Cchoc 996 a grat choc s to cosdr th sudo-cost fucto.

9 Itratoal Joural f Idustral grg v hr } R m{ C m m... If If curvs ar Th soluto gatvst a s ad ushmt If hbt If... ad As b a b a a C R R a C a R m m 5. NUR-LP ML IMPLMNTATIN Th roosd modlg 4. suggsts th mlmtd archtctur usg th Smul tool cororatd Matlab 5.3 vrso ad rrstd Fgur 5. Fgur 5. Imlmtato utl 5 varabls.

10 Itratoal Joural f Idustral grg v NUR-A ML IMPLMNTATIN Cosdrg th data stc for P MUs th R uts ad S oututs ad that th th MU s rrstd by a colum vctor X uts ad a Y oututs. Th rlatosh bt all uts ad oututs s obtad to ach MU: u Y / v X hr u ad v ar outut ght vctors ad ut ght vctors rsctvly. Th otmum valus for ths ghts ar obtad solvg a LPP for ach MU Norma 99 ad Kallrath 997. Th Nuro-A archtctur modl s totally basd th Nuro-LP modl rstd ths ar Bod. So for P MUs ll hav P Nuro-LP moduls. ach LPP of th Nuro-A modl ll rrst a LPP th Nuro-LP modl ad ll b abl to dtrm th rlatv ffccy of o MU of P MUs that comos th systm. Fgur 6 a shos roosd bloc dagram of th Nuro-A modul ad 6 b th adotd A modl to ach MU. Th mlmtato as do usg th CRS vlo modl ut ortd. Th raso for ths choc s du to th fact that ca rduc th umbr of costrats bcaus th vlo modl hav a costrat for ach ut/outut. Som data of a cas volvg 5MUs th to uts ad o outut ar sho Fgur 7 a. Fgur 7 b shos a CRS dagram that s rfrrg to th roblm hr ts ossbl to vrfy that MUs 3 ad 4 ar ot ffct ad MUs ad 5 ar ffct. Fgur 7 c shos th 3 frotr ad fally Fgur 7 d thr s a tabl that comars th obtad rsults usg to commrcal ad coscratd softars Ldo ad Frotr Aalyst th th rsults of th modl roosd ths ar ad calculatg th rror rctag. Fgur 6. Bloc agram ad Adotd Modl.

11 Itratoal Joural f Idustral grg v Fgur 7. a ata b CRS agram c 3 Frotr ad d MUs ffccy. 7. CNCLUSINS Th rstd cas as slctd amog hudrds of accomlshd tsts shog that th rstd roosal s cosstt. Th valus obtad th our rototy r valdatd comarg thmslvs th rsults obtad by coscratd softars as LIN to saratly solv th LPPs rfrrg to th MUs ad th FRNTIR to drctly solv th A. I ths cas th obsrvd rror dd ot surass.5% th th ossblty of dcrasg th som mrovmts th accuracy of th mthod ad th umrcal tolrac. Actually a study usg Lagrag Multlrs s bg dvlod hch bascally ll otmz th st fucto sho fgur 5. ad dtfd th otmzator by. Th soluto mthod for th ordary dffrtal quato systm usd th Nuro-LP modl s smlar to th tchqu usd ANN trag has bcaus thy us th dcrasg gradt mthod. Th voluto of th soluto mthod for th dffrtal quato systm rrstd by th soluto ath curv dcats th covrgc th valu of th dcso varabls of th roblm. Fally ts mortat to hghlght that th covrgc sd ca gt trmly hgh f th roosd moduls ar tgratd a ch ad coctd to a fr slot a comutr.

12 Itratoal Joural f Idustral grg v RFRNCS. Abraham Chars Wlla W. Coor Ar Y. L ad Larc M. Sford 996. ata vlomt Aalyss: Thory Mthodology ad Alcato Bosto Klur Acadmc Publshrs.. Agulo Mza L. 998 ata vlomt Aalyss A a trmação da fcêca dos Programas d Pós- Graduação do CPP/UFRJ ssrtação d Mstrado Uvrsdad Fdral o Ro d Jaro CPP RJ Brasl. 3. Bar R. Chars A. Coor W. W.984. Som Modls for stmatg Tchcal a Scal Iffccs ata vlomt Aalyss Maagmt Scc v Bazaraa M. S. Shral H.. Shtty C. M. 993 Nolar Programmg Thory ad Algorthms N Yor USA Joh Wly & Sos Ic. 5. Bod L. N. stllta Ls t al. Nuro-A: Novo Paradgma ara trmação da fcêca Rlatva d Udads Tomadoras d csão 9 º Cogrsso da Assocação Portugusa d Ivstgação racoal º Chars A. Coor W. W. 96 Programmg th lar fractoal fuctoal Naval Rsarch Logstcs Quartrly v.9 3/ Chars A. Coor W. W. Rhods Masurg th ffccy of cso-mag Uts uroa Joural of ratoal Rsarch v Ch J. Shablatt M. Ad Maa C. 99. Imrovd Nural Ntor for Lar ad Nolar Programmg Itratoal Joural of Nural Systms o Cchoc A. Ad Bargla A Nural Ntors for Solvg Lar Iqualty Systms Joural of Paralll Comutg dsoívl m htt://.b.r.go./absl.. Cchoc A. Ad Ubhau R Nural Ntors for tmzato ad Sgal Procssg N Yor USA Joh Wly & Sos Ic.. avd M. Saura 996. Buldg Nural Ntors N Yor Addso-Wsly Publshg Comay.. s J.. Ad Schabl R.B.996. Numrcal Mthods for Ucostrad tmzato ad Nolar quatos glood Clffs N.J. USA Prtc Hall Ic. 3. stllta L. M. Lída Agulo t Al. Aáls d voltóra d ados Prsctvas d Ambt d Aoo à csão Ro d Jaro Brasl CPP/UFRJ. 4. stllta L. M.. Morra 998. Modlo I Sts ara slção d varávs m Modlos d Aáls d voltóra d ados Rvsta Psqusa racoalv.9 o Hamdy A. Taha 99. ratos Rsarch: A Itroducto sth dto N Jrsy Prtc Hall. 6. Htor Pa 995. Métodos Numércos Lsboa Portugal McGra-Hll. 7. Jac M. Zurada 99. Itroducto to Artfcal Nural Systms Lodo Wst Publshg Comay. 8. J.Moscs Zbg goos 995. Advacd Cotrol th MatLab Smul Lodo lls Horood Lmtd. 9. Kallrath J. ad W. J Busss tmzato usg Mathmatcal Programmg Lodo Macmlla Prss Ltd.. Kdy M. P. S Chua L Nural Ntors for Nolar Programmg I Trasactos o Crcuts ad Systms o M. Ms Ad S. Part 969. Prctros Cambrdg MIT Prss.

13 Itratoal Joural f Idustral grg v Math Wors 995. Smul yamc Systm Smulato Usr s Gud Massachustts USA Math Wors Ic. 3. Phl. Wassrma 993. Advacd Mthods Nural Comutg Lodo Va Nostrad Rhold. 4. Robrt L. Harvy 994. Nural Ntor Prcls N Yor Prtc Hall Itratoal dtos. 5. Rosblatt F. 96. Prcls of Nurodyamcs N Yor Sarta dtos. 6. Rumlhart.. Hto G.. Ad Wlla R. J Larg Itral Rrstato by rror Proagato. I Paralll strbutd Procssg Vol Cambrdg M. A... Rumlhart ad J. L. McClllad ds MIT Prss. 7. Smo Hay 994. Nural Ntors a Comrhsv Foudato Lodo Macmlla Collg Publshg Co.. 8. Tm Coll. S. Prasada Rao ad Gorg. Batts 998. A Itroducto to ffccy ad Productvty Aalyss Bosto Klur Acadmc Publshrs. 9. Wrr C. Rhboldt 998. Mthods for Solvg Systms of Nolar quatos Phladlha SIAM dtors. 3. Zhu X. Zhag S. ad Costatds A. G. 99. Lagrag ural Ntors to lar rogrammg Joural of Paralll strbutd Comutg o