FUZZY PERT FOR PROJECT MANAGEMENT

Transcription

1 Itertol Jourl of dvces Egeerg & Techology Sept. 04. IJET ISSN: 96 FUZZY PERT FOR PROJECT MNGEMENT Ther hed Sdoo l S Rd M. Ro l Brhe ssst. Prof ssstt Lecturer College of dstrto d Ecoocs Mgeet Iforto Systes Deprtet Uversty of Mosul Irq BSTRCT Oe of the ost chllegg jobs tht y ger c tke o the geet of lrge scle project tht requres coordtg uerous ctvtes throughout the orgzto. yrd of detls ust be cosdered plg how to coordte ll these ctvtes developg relstc schedule d the otorg the progress of the project. Fortutely two closely relted opertos reserch techques PERT (progr evluto d revew techques) d CPM (crtcl pth ethod) were developed the 50's wth dfferet cotexts: the CPM ws developed for plg d cotrol of DuPot egeerg projects d the PERT ws developed for the geet of the producto cycle of the Polrs ssle. They shre the se objectves such s defg the project durto d the crtcl tsk. The PERT/CPM techque s bsed o two strght steps; forwrd propgto to defe the erlest strt d fsh dtes (d subsequetly the project durto d the free flots) d bckwrd proulgto for the ltest strt d fsh dtes (d the totl flots). Itlly the ctvty tes re sttc wth the CPM techque d probblstc wth the PERT techque. Over the lst few decdes both CPM d PERT techques h bee uversl to fuzzy d stochstc res. To trety wth ucertty project geet. Predotly Fuzzy PERT d CPM re to be delberted to trety prtculrly wth fuzzy plg. O the tgostc to PERT/CPM techque tht gores y cosderto of resources other Fortutously two closely relted opertos reserch techques PERT (progr evluto d revew techques) d CPM (crtcl pth ethod) re preseted to ssst the project ger crryg out these resposbltes PERT/cost s systetc produce (orlly coputerzed) to help the project ger pl schedule d cotrol projects cost. The PERT/cost produce begs wth the hrd work of developg estte of the cost ech ctvty whe t s perfored the plg wy (cludg y crshg). We propose to prove PERT by usg Fuzzy Delph for esttg T T d T for ech ctvty the fuzzy PERT s llustrted d gve cse study redyde fctory. The result of the proposed odel d te-cost trdeoff. I the ext secto proposed odel s ore thoroughly defed ths proposed odel s ore thoroughly defed. Ths s followed by the pplcto of the fuzzy logc to Pjs relty wer(clothg) fctory I Mosul. PERT/cost s systetc produce (orlly coputerzed) to help the project ger pl schedule d cotrol projects cost. The PERT/cost produce begs wth the hrd work of developg estte of the cost ech ctvty whe t s perfored the plg wy (cludg y crshg).we propose to prove PERT by usg Fuzzy Delph for esttg T T T for ech ctvty the fuzzy PERT s llustrted d gve cse study redyde fctory. KEY WORDS: PERT CPM FUZZY LOGIC PROJECT MNGEMENT I. INTRODUCTION My of dustrl projects ssg y of vlble physcl resource to types of ctvtes tht eeds y tes cheveet ther project relted to developet project buldg reserch producto vsble d defese whch eeds lrge cpblty plg d schedulg dfferet ctvtes (Rggs 997) [5]. 50 Vol. 7 Issue 4 pp

2 Itertol Jourl of dvces Egeerg & Techology Sept. 04. IJET ISSN: 96 I ths reserch two ethod used successfully plg for y lrge project whch s clled PERT d CPM especlly whe the project ctvtes h to be perfored specfed techologcl sequeces(rvrdr. et. l.987) [6]. Project Mgeet s coplcted eterprse volvg plg of vrous ctvty whch h to be perfored the process of developet of ew product or techology. Project h specfed begg d ed for coveece they re subdvded to ctvtes Whch lso h specfed beggs d eds. The ctvtes h to be perfored order Soe before others Soe sulteously the te requred for copleto of ech ctvty hs to be estted (George B. d Bojdzev M p78([]. The bsc for of PERT& CPM focus o detfyg the Logest te Cosug pth through etworks of tsks s bsc for plg d Cotrollg project (Mrk M. Dvs et. l. 00. p94) [9]. The Crtcl Pth Method s oe of the project schedulg specfctes. The jorty of the reserch o the project schedulg topc hs bee devoted to fuzzy PERT s expled before the PERT techque s coposed of two steps; the forwrd d the bckwrd propgtos.(msoudm Erw Hs d l Hït0) []. The geerlzto of the PERT techque to fuzzy preters s coplex tsk. The forwrd propgto s doe usg fuzzy rthetc ledg to fuzzy erlest dtes d fuzzy ed-of-project evet. Ufortutely bckwrd propgto s o loger pplcble becuse ucertty would be tke to ccout twce.( Chs et l. 00) [0].study the crtclty of tsks wth fuzzy project.( Dubos et l. 00) [6]. Show tht the boudres of soe fuzzy preters lke the tsks' ltest dtes d flots re reched extree cofgurtos Expdg the PERT ode to tke cogzce of the fct tht Fuzzy logc o project ctvtes s tkg plce t vryg rte dds coplexty to the odel. Now the te to coplete Ut s fucto of two vrble ely:. The uber of Te the vrous ctvtes h bee repeted o pror rus of the project.. Fuzzy Logc of ech te ctvty However ths coplexty c be redly by coputerzg the odel. The odel use uder the followg codto:. The te of ctvty hs bee copleted o project rus s ot se for ll ctvtes coprsg the etwork. Ths s the resultt of repetg prtlly repetg prtlly repettve project.. Fuzzy logc s tested for chrge project copleto tes whe fuzzy logc tke plce o ll ctvtes. The result of the proposed odel d te-cost trdeoff. I the ext secto proposed odel s ore thoroughly defed ths proposed odel s ore thoroughly defed. Ths s followed by the pplcto of the fuzzy logc to Pjs relty wer (clothg) fctory I Mosul. Fuzzy set through represet ttrctve tool to d reserch producto geet whe the Dyc of the producto Evroet lted the specfcto of odel objectves costrts d the precse esureet of odel preters. Ebled us to pply the proposed odel to detere the tes of the pleetto of the ew product Pjs fctory clothes Mosul d through the detfcto of ctvtes the costtuet of ths Delph project d reltoshps precedece betwee the vrous ctvtes d the te requred to pleet ech of the ctvtes whch h bee detered by experts usg the Delph ethod where bee detfed te optstc d pessstc d ost lkely te d were the used Fuzzy logc to detere these tes d the detere the cost d the we clculte the crtcl pth whch cludes ll the crtcl ctvtes tht cuse dely whch the dely the copleto of the fl product s well s detfy tes of ccelertg d ssocted costs h reched set of coclusos d recoedtos. II. RELTED WORK Trdtolly schedulg theory hs bee cocered wth llocto of resources to tsks or ctvtes (Prker 995) [0]. O the re of schedulg fst progress cocerg odels d ethods hs bee de. Two techques of resource geet ely: 5 Vol. 7 Issue 4 pp

3 Itertol Jourl of dvces Egeerg & Techology Sept. 04. IJET ISSN: 96 - Resource-costred project schedulg ubguously tkes to ccout costrts o resources d s t schedulg the ctvtes subject to the precedece costrts d the resource costrts order to ze the project durto. Resource-costred project schedulg proble s oe of the ost ttrctble clsscl probles prctce. Multple exct techques d heurstcs d uber of et-heurstcs h bee ppled to solve the RCPSP proble. - Resource levelg tkes to ccout the superorty costrts betwee the ctvtes d s t copletg the project wth ts due dte wth resource usge whch s s leveled s possble throughout the project durto.( Herroele 007) []. Severl studes h vestgted the cse where ctvty tes project re pproxtely kow d ore pproprtely represeted by fuzzy sets rther th crsp ubers. ( Lorterpog P. & Moselh O ) []. (D. Dubos H. Frger V. Glvgoo ) [6]. I specfc the probles of coputg the tervls of possble vlues of the ltest strtg tes d flots of ctvtes wth vgue durtos represeted by fuzzy or tervl ubers h fscted tesvely ttetos d y solutos ethods h bee suggested. (P. Zelsk ) [8].(D. Dubos H. Prde ) [5]. Most of the re strght forwrd postpoeets of deterstc CPM. They re ly bsed o the CPM wth foruls for the forwrd d bckwrd recursos whch the deterstc ctvty tes re replced wth the fuzzy ctvty tes. However s oted by Zelsk [] the bckwrd recurso fls to copute the sets of possble vlues of the ltest strtg tes d flots of ctvtes. Moreover for the se pth dfferet deftos of the fuzzy crtcl pth gve dfferet esttos of the grde of crtclty. Dubos et l. proposed severl heurstcs for coputg the sets of possble vlues of the ltest strtg tes d flots of ctvtes usg rgorous forulzto of fuzzy PERT. Zelsk [] developed ew polyol lgorths for deterg the tervls of the ltest strtg tes the geerl etwork. Chs d Zelsk [] dscussed the coplexty of crtclty Chs d Zelsk [] proposed turl geerlzto of the crtclty cocept for project etworks wth tervl d fuzzy ctvty tes whch two ethods of clcultg the degree of possble crtclty d soe results re provded. The dvtge of ths ethod s tht t prevets fuzzy ubers fro gettg lrger d lso the result of subtrcto of ech fuzzy uber fro tself s crsp zero []. (C.T. Che et. l.)[] proposed ethod to del wth copleto te geet d the crtcl degrees of ll ctvtes for project etwork[] []. Che d Hug[] lso proposed pproch usg postve trgulr fuzzy uber. Ths ethod however does ot support bckwrd pss clcultos drect er slr to tht used the forwrd pss. Ths s ly due to the fct tht fuzzy subtrcto s ot proportote to the verse of fuzzy ddto.therefore ths ethod s cpble of clcultg project chrcterstcs such s the ltest tes. (K. Ush Mdhur S. Sresh d N. Rv Shkr 0 0 4) [7]. V Drop d Kotoz clude the two sded power (TSP) Dstrbuto the PERT ethodology kg use of the dvtges tht ths four- preter dstrbuto offers. I order to be copletely detered dstrbuto of ths type eeds the se s the Bet Dstrbuto ew deprtet fro the three usul vlues optstc pessstc ost lkely (v drop d Kotoz 00 b 56([0] (J.Jssb d S. Kh Mohd ) [6] Itroduce ew pproch for predctg d lyss of the project durto usg fuzzy durtos d fuzzy possbltes the Bet probblty s chged to bet probblty dstrbuto fucto().(yo et l. ) [7].used sged dstce rkg of fuzzy ubers to fd crtcl pth fuzzy project Network. (Che et l.) []used defuzzfcto ethod to fd possble crtcl pths fuzzy project Network.(S. Chs d zelsk) []. ssue tht the cooperto te of ech ctvty c be represeted s crsp vlue tervl or fuzzy uber. (D. Dubos H. Frger V. Glvgoo) []. III. FUZZY PERT FOR TIME FORECSTING We propose to prove PERT by usg Fuzzy Delph ethod s geerlzto of the clsscl ethod for log rge forecstg geet scece kow s Delph ethod. It ws developed 5 Vol. 7 Issue 4 pp

4 Itertol Jourl of dvces Egeerg & Techology Sept. 04. IJET ISSN: 96 the sxtes by the Rd Corporto t St Moc Clfor. The e coes fro the cet Greek orcles of Delph who were fous for forecstg the future. The essece of Delph ethod c be descrbed s follows:. Experts wth hgh equlztos regrdg subject re requested to gve ther opo seprtely d depedetly of ech other bout the relzto dtes of cert evet sy scece techology or busess. They y be sked to forecst the geerl stte of the rket ecooy techologcl dvces etc.. The dt whch h subjectve chrcter re lyzed sttstclly by fdg ther rge d the results re coucted to the experts.. The experts revew the results d provde ew esttes whch re lyzed sttstclly d set g to the experts for estto. Ths process could be repeted g d g utl the outcoe coverges to resoble soluto fro the pot of vew of ger or goverg body. Usully two or three repettos re suffcet. The Fuzzy Delph Method s lytcl ethod bsed o the Delph Method tht drws o the des of the Fuzzy Theory. The Delph Method s type of collectve decso-kg ethod (Lstoe & Turoff 00) [0]. wth severl rouds of oyous wrtte questore surveys coducted to sk for experts opo. s drect predcto ethod bsed o the expert judget d expert eetg vestgto ethod t possesses the followg propertes:. oyty: The experts volved wth the predcto process do ot see ech other re oyous d do t kow how y experts re volved. Ths helps to prevet the fro fluecg d ecourges objectvty.. Feedbck: The survey feedbck gves the prtcpts de bout the des the group. They c the drw fro t forto relevt to the ke ew judget d the subt t to the group g.. Sttstcl: The expert opos re processed sttstclly d sples grph produced wth the expert opo frequeces rryed chroologclly. The top s the jorty cosesus (50% experts) represetg the predcto te s opo. The top d botto qurter percetle (ech represetg 5% of the experts) represet the predcto devto. 4. Covergece: Through ultple reverse feedbck ke the fl predcto results coverge. (Yu- Feg HoHso-L Wg) [8]. However log rge forecstg probles volve precse d -coplete dt forto. lso the decsos de by the experts rely o ther dvdul copetece d re subjectve. Therefore t s ore pproprte the dt to be preseted by fuzzy ubers sted of crsp ubers. Especlly trgulr ubers re very sutble for tht purpose sce they re costructed esly by specfyg three vlues the sllest the lrgest d the ost plusble. Isted of crsp rge the lyss wll be bsed o fuzzy rge. The Fuzzy Delph ethod ws troduced by Kuf d Gupt (988]. It cossts of the followg steps[9] [] : Step : Experts E.. re sked to provde the possble relzto dtes of cert evet scece techology or busess ely: the erlest dte the ost plusble dte d the ltest dte. The dt gve by the experts E re preseted the for of trgulr ubers ( )... () Step : Frst the rge (e) ( M ) of ll s coputed ( M ) ( 5 Vol. 7 Issue 4 pp ). () The for ech expert E the devto betwee d s coputed. It s trgulr uber defed by ( ) ( M M M M ) ()

5 Itertol Jourl of dvces Egeerg & Techology Sept. 04. IJET ISSN: 96 The devto - s set bck to the expert E for reexto. Step : Ech expert E presets ew trgulr uber : B ( b b b ) M... (4) Ths process strtg wth Step s repeted. The trgulr rge B s clculted ccordg to forul : ( ( ) wth the dfferece tht ow M c ( ( c 54 Vol. 7 Issue 4 pp ) M re substtuted correspodgly by c ) (5) b b M ecessry ew trgulr ubers re geerted d ther rge C s clculted. The process could be repeted g d g utl two successve es B C becoe resobly close. Step 4 : t lter te the forecstg y be reexed by the se process f there s portt forto vlble due to ew dscoveres. Fuzzy Delph ethod s typcl ult-experts forecstg procedure for cobg vews d opos. We propose to prove PERT by usg Fuzzy Delph For esttg t t M t for ech ctvty. Experts represet ech te for ctvty copleto by trgulr ubers of the type (t t M t ). For ech ctvty the trgulr rge uber s clculted. To defed crsp ctvty te vlue we h to use defuzzffcto Sply we y tke the xzg vlue (X x M) or resort to the rge foruls (5)()-(). () () () () () () Wth the dfferece. tht ow re substtuted Correspodgly by b b b. If ecessry ew trgulr ubers c () () () () ( c c c re geerted d ther rge C s clculted The process Could be repeled g d g utl two successve es Bre Bre Cre. Becoe resobly close X x M. (6) () M...(7) () M (8) () X MX 4 () () X M () X MX 6 c M 4 ) (9) The project schedulg reserch d developet of ew product (Pjs) redy wer fctory Mosul. I order to coplete ths project we wll eed forto fro the fctory reserch d developet product testg ufcturg cost esttg d rket reserch groups. Tble () Descrpto of the Pjs producto process ctvty Descrpto Iedte predecessors Optstc Most probble Pessstc R& D Product desg B Pl rket reserch C Routg 4 D Buld prototype odel 4 E Prepre rketg 4 brochure F Cost estte C.5.5 G Prelry product testg D b If

6 Itertol Jourl of dvces Egeerg & Techology Sept. 04. IJET ISSN: 96 H Mrket Surrey BE I Prcg d Fr cost H.5 report J Fl report FGI Tble () Iproved etwork Plg odel usg Fuzzy PERT ctvty Norl te Crsh te Norl Cost c x Crsh cost C C x Cost slope E H I J Tble () Norl d crsh te d cost of crtcl ctvty E H I J The cost slope coeffcet clculted for ctvty gves K ctvty Norl te Crsh te Norl Cost $ Crsh cost C T x x ctvty te C C c x c x ctvty Tble (4) Pjs Network PERT clculto verge ctvty T T T B B B C T C T D D D E T E F T F G Optstc te t Most lkely te t Pessstc te T T.5 5 C T 4 T 4 E T T G G H T H I F T.5.5 T H T T I I J T J T J T Vol. 7 Issue 4 pp

7 Itertol Jourl of dvces Egeerg & Techology Sept. 04. IJET ISSN: 96 C F B 5.5 E 4 D.5 H G Fg () Pjs etwork Ech trgulr uber represetg the rge ctvty te ( the secod colu lbel 4 ) hs to be defuzzfed to produce crsp uber expressg the ctvty copleto toe. These Trgulr uber re lost cetrl fro hece we c pply forul ( 6 ) for defuzzcto whch produces the uber the fourth colu lbeled T M the use of foruls (789) gves close results. The defuzzfed tes c be preseted proved etwork plg odel ( fg ) The totl te for project copleto expressed by the trgulr uber T s the te for copletos the ctvtes o the Crtcl pth. ddg the ubers o the three Colus Tble 4 desgted by t t M t excludg those belogg to ctvtes B d D gves T T T E T H T I T J (5.59) Hece the project durto wll be betwee dys d 9 dys ost lkely 5.5 dys. The lst uber 5.5 dys s the result of defzzcto geertes the Crsp Nubers : () T x () T x () T x I They re close to ( 7.75 ) s cocluso the copleto te for the project s forecsted to be ( 7.75 ) dys If the trgulr uber re s close to cetrl trgulr uber eg tht s lost the ddle of ( ) the (6 ) gves good crsp vlue X x. the the three rge foruls () () (789 ) lso produce uber ( xzg vlues ) close to hece there s o eed to be used usully pplctos the trgulr rge ubers pper to be cetrl for. however the experts delg wth gve stuto h to use ther judget whe selectg xzg vlue Followg PERT we troduce the ottos t-orl te for copletg ctvty. C orl cost for copletg ctvty C c Crsh cost ( cresed cost ) for copletg ctvty crsh te. We llustrte here Fuzzy PERT for Shorteg project legth o the troduce ew product. To shorte project legth es to shorte the te for copleto the crtcl pth.e to shorte the totl te T x 7 dys.shorteg durto te of ctvtes ot o the crtcl pth d ( D G J) ( B H I J ). Wll ot reduce T x However soe resources llocted to (CFJ) (DGJ) ( B H I J ) could be rellocted to ctvtes ( E H I J ) order to shorte ther copleto te ( terl rellocto ). Here we cosder shorteg ctvtes te o the crtcl pth wthout terl rellocto of recourses. The orl te t for ech ctvty s lredy estted t s the te T x t show tble.7 the fourth colu. J 56 Vol. 7 Issue 4 pp

8 Itertol Jourl of dvces Egeerg & Techology Sept. 04. IJET ISSN: 96 The crsh te t c the orl cost C d the crsh Cost C c for ech ctvty could be forecsted Slrly to the orl te t pplyg Fuzzy Delph The defuzzfed vlues bsed o forul ( 6 ) wll be deoted by t c x d C c x correspodgly. Here estto s preseted for the orl Cost C c c be estted slrly. Three experts re sked to estte the orl Cost for copleto ctvty the for of trgulr uber C C ( C C C ) where C s Cost d C s the hghest cost. ssue the experts esttes re those tble 5. ( M ) ( () X x M () X x () X x () X x 4 4 Expert 6 Tble 5 Experts estte for copleto ctvty t orl Cost C Lowest cost C Most lkely cost C Hghest cost C N E E E totl Usg forul ( step ) gves the rge orl cost three dgts to or 000 gves c c ( ) re the decls d roudg off the lst Further groups of experts forecst t c C c d C c for the other ctvtes o the crtcl pth the defuzzuffy d roud off s bove. ssue tht the defuzzfed results for the ctvtes o the crtcl pth re those preseted tble (6). To select ctvtes for shorteg durto te PERT uses the oto of cost slope wth our ottos t s preseted s ( see fg ) K cost slop C t x C x t c c x x. () Fg () shows tht s orl te t x decreses pprochg the crsh te t c x the orl cost C x creses pprochg the crsh cost C c x. The cost slope fg () clculted for ctvty gve : 57 Vol. 7 Issue 4 pp

9 Itertol Jourl of dvces Egeerg & Techology Sept. 04. IJET ISSN: 96 ctvty cost Crsh pot C c x.. C x t c x Norl pot T x ctvty durto Fg() Cost Slope Tble 6 Defuzzfed orl d crsh tes d cost for ctvtes New product plg ctvty Norl Te t x Te t c x Cost C x Crsh cost C c x Cost slope $ per dy E H I J The cost slope coeffcet for the other ctvtes re Clculted slrly. the results re dsplyed the lst colu of tble 6. I geerl ddtol resources should be ppled frst to ctvtes wth sllest cost slope. The ctvtes tble 6 re rked tble 7 ccordg to ther cost slopes fro the sllest to be the lrgest. Tble: 7 Rked ctvtes ccordg to cost slope. Rk ctvty Reduced te T x - t c x ddtol cost C x-c x Cost slope $ per dy H J I E ssue tht the geet wts to reduce the legth of the project fro 7 dys to 07 dys reducto of (0) dys of the ctvtes o the crtcl pth ctvty H rked frst ( Tble 7 ) hs the 58 Vol. 7 Issue 4 pp

10 Itertol Jourl of dvces Egeerg & Techology Sept. 04. IJET ISSN: 96 sllest K $ 000 per dy By vestg $ 0000 the te durto for ctvty H c be reduced by 0 dys. further reducto of ( 0 ) dys ust be foud good cddte s ctvty rked wll cost o Tble 7. 0 dy reducto wll cost dollrs However f there re soe reso gst shorteg the ctvty toe for H or for or for both other Optos ust be exed. IV. CONCLUSION PERT ethodology s to estte the e d the vrce of rdo vrble of whch oly the vlues (pessstc) (ost lkely) d p (optstc) suppled by expert re Kow d for whch uderlyg bet dstrbuto s ssued.to obt the estte t the e t s oly requred tht ths (stdrdzed bet ) dstrbuto s esokurtc (B ) or of costt vrce(-/6).the gol of ths pper s to get optl PERT susg Fuzzy Logc. The ssues cosdered whle selectg PERT s totl u cost tke for project to coplete power requred to coplete the project fucto pots requred to coplete t.ll three costrts re crshed te bss.the costrts re crshed by cosderg the crshed te oly.now the optl PERT s to be selected usg Fuzzy Logc. We developed fuzzy expert syste whch s used select optl PERT chrt. optlty s descrbed here wth the rules of Fuzzy Logc. We h used ew defuzzfcto forul for trpezodl fuzzy uber d ppled to the flot te for ech ctvty the fuzzy project etwork to fd the crtcl pth. Cosderg the proble of Pjs' project geet the redyde fctory Mosul ths reserch s lso devoted o the fuzzy PERT/cost lyss of ctvty durtos. the the crtcl pth of the fuzzy PERT/cost wll be cheved. REFERENCES [] buj H. N. Dozz S.p. d bourzk S. M. 994 project Mgeet Wley New York. [] C.T. Che S.F. Hug pplyg fuzzy ethod for esurg crtclty Project etwork Iforto sceces 77 (007) [] Chs S. P. Zelsk 00crtcl pth lyss the etwork wth fuzzy tsk te Fuzzy sets d systes0-6) [4] Chs S. Dubos D. Zelsk P. 00. Necessry crtclty the etwork wth precse ctvty tes. IEEE Trsctos o Systes M d Cyberetcs [5] Che S. M. d Chg T.H. 00 Fdg Multple possble crtcl pths usg Fuzzy PERT IEEE Trsctos o systes d cyberetcs- prt :systes d hus vol.o.6. [6] Dubos D. Frger V vgoo H.V. 00 o ltest strtg tes d flots tsk etworks wth ll-kow durtos. Europe Jourl of opertol reserch 450 t [7] Dubos D. d Prde H. 988 possblty theory : pproch to coputerzed processg of ucertty ple N.4 [8] D. Dubos H. Prde (978) Opertos o fuzzy ubers Itertol Jourl of systes scece [9] Dvs Mrk M. Nchols J qulo d Rchrd B Chse 00 Fudetl of Operto Mgeet 4th McGrw Hll Irw P 94. [0] Drop JR V Kotoz S (00b) the stdrd two sded power dstrbuto d ts propertes :wth pplctos fcl egeerg the erc sttstc :90-99 p56. [] N. Rv Shkr V. Sreesh d P. Ph Bush Ro lytcl Method for Fdg Crtcl Pth Fuzzy Project Network. [] George Bojdzev & Mr Bojdzev 007 dvces Fuzzy syste : pplcto d Theory Fuzzy logc for Busess fce d geet d ed. vol.. World Scetfc publsher Co. Pte. Ltd. p78. [] Guffrd lfred L. d Ng Rkesh. Fuzzy set theory pplctos producto Mgeet Reserch Lterture Surrey p.-. [4] Hpke M. d Slowsk R. 996 Fuzzy prorty heurstcs for project schedulg fuzzy sets d systes vol. 8 o.pp [5] Jes L.Rggs 987 producto syste : plg lyss d cotrol 4th ed joh Wley & Sos New York. [6] Jssb J. d Mohd S.Kh ew pproch for predctg project durto usg Bet shpe ebershp fucto sd sulto. 59 Vol. 7 Issue 4 pp

11 Itertol Jourl of dvces Egeerg & Techology Sept. 04. IJET ISSN: 96 [7] K. Ush Mdhur S. Sresh d N. Rv Shkr 0 New pproch for Solvg Fuzzy Crtcl Pth Proble Usg L-L Fuzzy Nubers [8] Kuf d Gupt M.M (985) Fuzzy thetcl Models Egeerg d Mgeet Scece. [9] Kuf.d Gupt M.M 988 Itroducto to Fuzzy rthetc : theory d pplcto v o strd Rehold New York. [0] Lstoe H.. & Turoff M. ed 00 The Delph Method: Techques d pplctos ISBN [] Loots F.989. Stochstc d fuzzy PERT. Europe Jourl of Opertol Reserch [] Lorterpog P. & MoselhO. (996) Project-etwork lyss usg fuzzy sets theory. Jourl of Costructo Egeerg Mgeet [] Mlek Erw Hs d l Hït. Fuzzy tctcl project plg: pplcto to helcopter tece. I the 6th IEEE Itertol Coferece o Eergg Techologes d Fctory utoto ETF'0 Toulouse Frce Septeber 0. (Cted o pges ). [4] Mo D. L. Chegs CH. d Lu H. C. 995 pplcto of fuzzy dstrbuto o project geet fuzzy sets d systes vol. 7 o.pp7-4. [5] P. Zelsk (005) O coputg the ltest strtg tes d flots of ctvtes etwork wth precse durtos Fuzzy Sets d Systes [6] Rvdr Phlps Do T. & Jes J Solberg 987 opertos reserch Prcples d ed. Joh Wley & Sos M.Y. [7] YO J.S. d L F.T. 000 Fuzzy crtcl pth ethod Bsed o Sged Dstce rkg of fuzzy ubers JIEEE Trsctos o systes d cyberetcs prt. Systes d Hus Vol.0Nol 0:76-8. [8] Yu-Feg HoHso-L Wg pplyg Fuzzy Delph Method to Select the Vrbles of Sustble Urb Syste Dycs Model. [9] Zdeh. L. (98) the Role of Fuzzy logc the geet of ucertty Expert systes Fuzzy sets d syste pp UTHORS THEIR HMED SDOON L SMMN ws bor MOSUL IRQ. He receved the Bchelor 984 degree fro the uversty of MOSUL IRQ d the ster yer 987 degree fro the uversty of MOSULIRQ both busess geet d the Ph.D. yer 008 degree fro the uversty of MOSULIRQ.He s curretly ss. Prof. geet forto syste the college of Busess dstrto uversty of MOSUL. Hs reserch terests clude operto geet d opertos reserch. 60 Vol. 7 Issue 4 pp