Resource allocation in multi-server dynamic PERT networks using multi-objective programming and Markov process.

Transcription

1 IJST () A: -7 Irnin Journl of Science & Technology hp:// Resource llocion in uli-server dynic PERT neworks using uli-objecive progring nd Mrkov process S. Yghoubi, S. Noori nd M. Bgherpour * Depren of Indusril Engineering, Irn Universiy of Science nd Technology, Tehrn, Irn E-il: bgherpour@ius.c.ir Absrc In his reserch, boh resource llocion nd recive resource llocion probles in uli-server dynic PERT neworks re nlyiclly odeled, where new projecs re epeced o rrive ccording o Poisson process, nd civiy durions re lso known s independen rndo vribles wih eponenil disribuions. Such syse is represened s queuing nework, where uli servers ech service sion re lloced, nd lso ech civiy of projec is opered devoed service sion wih only one server loced node of he nework bsed on Firs Coe Firs Serve (FCFS) policy. In order o propose novel pproch for odeling of uli-server dynic PERT nework, iniilly he nework of queues is rnsfored ino sochsic nework. Then, differenil equions syse is orgnized o solve nd obin pproie copleion ie disribuion for ny priculr projec by pplying n pproprie finie-se coninuous-ie Mrkov odel. Finlly, uli-objecive odel including four confliced objecives is presened o opilly conrol he resources lloced o he service sions in uli-server dynic PERT nework, nd he gol inen ehod is furher eployed o solve discreeie pproiion of he priry uli-objecive proble. Keywords: Projec ngeen; Mrkov processes; uliple objecive progring; recive resource llocion. Inroducion Soe orgnizions re projec-oriened bsed nd opere heir civiies depending on projecs. In such siuions, he orgnizions y crry ou he uli projec concurrenly, wheres, Pyne [] reveled h up o 9% orgnizions eecue he projecs in uli-projec environen. Coonly, he liied resources re shred nd copeed ong uliple projecs for chieving heir own gols. Therefore, uli-projec ngeen syse is vil pproch in projec scheduling nd ngeen, wheres rdiionl projec scheduling hs been concerned osly wih single projec opiizion. Muli-projec resource consrined scheduling proble (MPRCSP) is he in opic of os invesigions on uli-projec scheduling considering sic nd deerinisic environens. Prisker e l. [] by using zero-one ineger progring pproch nd Wies [] by presening n heurisic odel, nlyzed he MPRCSP. Then, Kurulus nd Dvis [] nd Kurulus nd Nrul [] by pplying prioriy rules nd defining esures such s he re of uilizion of ech resource ype nd he pek of ol resource requireens, sudied he MPRCSP. *Corresponding uhor Received: April / Acceped: Augus Also, uli crieri nd uli objecive odeling is hen used in MPRCSP. For eple, gol progring is pplied by Chen [] in uliprojec resource-consrined scheduling for he inennce of inerl processes, nd leicogrphicly wo crieri is presened by Lov e l. [7]. Recenly, scheduling rules in he sic MPRCSP environen were presened by Kngsbphi e l. [8] considering perfornce esures involving en rdiness nd he iu rdiness of projecs involved. Moreover, heurisic & e-heurisic lgorihs for nlyzing MPRCSP were pplied [9-]. Recenly, MPRCSP ws eended by considering rnsfer ies nd is relevn coss by Kruger nd Scholl []. In he lierure, MPRCSP ws osly nlyzed on sic nd deerinisic environens nd few invesigions hve been focused on uli-projec scheduling under unceriny nd dynic condiions. A siulion odel for uli-projec resource llocion wih sochsic civiy, s uli-chnnel queuing, ws presened by Fei- Ghoi nd Ashjri []. Also, nonliner iedineger progring odel for opiizing he uli projec resource llocion ws proposed by Nozick e l. [7], wheres chnging resource llocions ffecs he probbiliy disribuion of civiy durion. An even-driven pproch ws

2 IJST () A: -7 represened by Ko e l. [8], nd lso, using Criicl Chin Projec Mngeen (CCPM) pproch, he unceriny in uli projec syse ws sudied by Byli nd Knnn [9]. MPRCSP is coonly nlysed by eiher connecing he ogeher ino lrge single projec by he ddiion of duy sr nd end civiies or considering he projecs s independen nd linking he by using n objecive funcion which conins ech projec individully (probbly wih pproprie weigh fcors) nd he corresponding resource consrins. In ny orgnizions, no only re he civiy durions uncerin, bu lso, new projecs dyniclly rrive o he projec bsed orgnizions over he ie horizon. Clerly, in his condiion, projec scheduling procedure would be ore difficul nd ore cople hn before. This proble, considered in projec-oriened orgnizions, ws sudied by Adler e l. [] by pplying siulion. In his invesigion, he orgnizion ws presened s sochsic processing nework wih collecion of service sions (work sions) or resources, where one or ore idenicl servers for serving projecs under pre-specified discipline, hs been seled ech sion. The represened orgnizion cn be epressed s queuing nework (dynic PERT nework), where ech civiy is geing he required services, queuing up for ccess o resource, or wiing o join predecessor civiy. Such proble is rcive for orgnizions wih siilr projecs, for eple, inennce projecs in which ypicl projec will be repeed. Also, he concep of CONWIP (consn work-inprocess) is eployed by Anvi-Iskow nd Golny [] in dynic PERT nework for conrolling projecs using siulion sudy. Auhors presened wo conrol echniss: CONPIP (COnsn Nuber of Projecs In Process) h liis he nuber of projecs, nd CONTIP (CONsn Tie of projecs In Process), h resriced he ol processing ie of ll cive projecs. A risk eleen ws considered in dynic PERT nework by Li nd Wng [] nd uli-objecive riskie-cos rde-off proble ws proposed bsed on generl projec risk eleen rnsission heory. Through resource llocion proble in dynic PERT nework, wo coonly used pproches eis. The firs pproch ws propounded by Cohen e l. [, ], where he resources y work in prllel, i.e., he nuber of servers nd resources lloced in every service sion re equl (e.g., elecricl work sion wih elecricins, echnicl work sion wih echnics, ec.) nd he oun of resources vilble o be lloced o ll service sions is consn. They presened ner opil resource lloced o he eniies h perfor he projecs in CONPIP syse by using Cross Enropy (CE) bsed on siulion. We denoine his pproch s resources s servers nd in his ricle, bsed on his pproch uli-objecive odel will be proposed. In he second pproch, invesiged by Azron nd Tvkkoli-Moghdd [], he nuber of servers in every service sion is fied nd resources lloced ffec he en of service ies. Auhors presened n nlyicl uli-objecive odel for he resource llocion proble in dynic PERT nework nd ssued he civiy durions re eponenilly disribued rndo vribles, he new projecs re genered ccording o Poisson process, he nuber of servers in every service sion is eiher one or infiniy nd he cpciy of he syse is infinie. Recenly, Yghoubi e l. [] odeled he resource llocion proble in dynic PERT neworks, where he cpciy of syse is finie nd projecs re genered ccording o Poisson process. We denoine his pproch s resources ffecing servers nd in his ricle, bsed on his pproch uli-objecive odel is proposed. In boh pproches, he unceriny is considered in he enrnce of projecs nd lso in he durion of service sions, wheres oher unceriny such s projec nework disrupion y hppen. In his reserch, for voiding projec nework disrupion, recive resource llocion is suggesed. Along wih he projec eecuion, projec y be disposed by considerble unforeseen disrupions, herefore, recive scheduling (rescheduling), wih revising or re-opiizing he iniil bseline schedule, is o djus he bseline schedule nd consequenly, overcoe he disrupions. Firsly, recive scheduling ws propounded in nufcuring environens nd hen i ws pplied hrough projec scheduling pproches. Coprehensive invesigions bou recive scheduling in nufcuring environens hve been sudied [7-]. Vieir e l. [9], bsed on wide vriey of eperienl nd prcicl invesigions, inroduced frework of sregies, policies nd ehods for recive scheduling nd Ayug e l. [] by defining differen ypes of uncerinies, proposed review of rescheduling bsed pproches. Herroelen nd Leus [] represened he bsic specs for scheduling under uncerin condiions: recive scheduling, sochsic projec scheduling, fuzzy projec scheduling, robus (procive) scheduling nd sensiiviy nlysis. Also, Vn de Vonder e l. [], bsed on prcicl design, nlysed severl predicive-recive resource-consrined projec scheduling procedures. Vrious pproches eis in he lierure of he recive scheduling probles. A siple nd iniil pproch is righ shif rule h is reoved hed

3 IJST () A: -7 in ie of ll he ffeced civiies []. Full rescheduling, he oher pproch, considers he reinder of civiies h re o be copleed. The oher iporn pproch is iniu perurbion sregy which pplies he ec nd subopil ehod for iniizing he difference beween he revised schedule nd he priry schedule [-]. Recenly, Liu nd Shih [7], bsed on priry schedule nd cul progress, sudied resource-consrined consrucion rescheduling nd suggesed new rescheduling opiizion odel using consrin progring, lso, Novs nd Henning [8] inroduced he repirbsed recive scheduling of indusril bch plns. Reviewing he bove enioned reserches indiced no closely reled work ws found o nlyiclly nlyze uli-server dynic PERT neworks. As he in conribuion of his reserch, we develop novel pproch for he resource llocion proble (or ie-cos rde off proble) nd lso recive resource llocion proble in uli-server dynic PERT neworks by ens of uli- objecive progring nd Mrkov process. Through his invesigion, we consider uliple environen nd concurren projecs including new projecs, conining ll he civiies h rrive he syse ccording o n independen Poisson process. I is lso ssued ech civiy of he projecs is perfored devoed service sion loced node of he nework wih FCFS policy. I is furher ssued differen servers re lloced in ech service sion, while he services processing ies (or civiy durions) re followed s independenly rndo vribles wih eponenil disribuions. For odeling uli-server dynic PERT nework, firsly, for obining he ses of syse, he nework of queues is rnsfored ino sochsic nework. Then, syse of differenil equions is orgnized o solve nd obin he pproie copleion ie disribuion for ny priculr projec by creing n pproprie finiese coninuous-ie Mrkov odel. Finlly, uli-objecive odel wih four confliced objecives is presened o opilly conrol he resources lloced o service sions in uliserver dynic PERT nework, nd he gol inen ehod is finlly eployed o solve discree-ie pproiion of he priry uliobjecive proble. This pper is coprised of five secions. The reinder is orgnized s follows. In Secion, we odel he uli-server dynic PERT nework by eploying finie-se coninuous-ie Mrkov process nd propose uli-objecive odel o opilly conrol he resources lloced o service sions in uli-server dynic PERT nework. In Secion, recive resource llocion in he uli-server dynic PERT nework is discussed. We solve n illusrive cse in Secion, nd he conclusion is given in Secion.. Muli-server dynic PERT nework In his secion, he uli-server dynic PERT nework is odeled o opilly conrol he resources lloced o he corresponding civiies. Also, n nlyicl ehod o copue he pproie disribuion funcion of projec copleion nd uli-objecive odel in uliserver dynic PERT nework re presened... Coninuous-ie Mrkov process For odeling he uli-server dynic PERT nework, we use he ehod presened by Kulkrni nd Adlkh [9]. This is he reson his ehod is n nlyicl pproch, siple, esy o ipleen hrough copuer, nd is copuionlly sble. I is ssued h projec is represened s n Aciviy-on-Node (AoN) srucure, lso new projecs, conining ll he civiies, rrive he uli projec syse ccording o Poisson process he re of. Furherore, ech civiy of he projec is eecued devoed service sion seled in node of he nework bsed on FCFS policy, where he service ie (civiy durion) is eponenilly disribued. Such syse cn be considered s nework of queues, where he rrivl sre of projecs o ech service sion is followed ccording o Poisson process wih he re of, nd he service ies re he durions of he corresponding civiies. I is ssued h service processing ies in service sion re eponenilly disribued re of nd he nuber of servers in node is. So he node is reed s n M / M / odel. The flow chr of our proposed ehod for uliserver dynic PERT nework is presened in Fig.. To coninue he seps of our proposed ehod for uli-server dynic PERT nework s coninuousie Mrkov process is eensively eplined: Sep. Copue he densiy funcion of he sojourn ie (wiing ie plus civiy durion) in ech service sion. (see ppendi A) Sep.. If, hen he queueing syse would be n M / M / queue, nd he densiy funcion of ie spen he service sion ( w ( )) would be eponenilly epressed wih preer, herefore, w () is clculed s follows: ( ) w () ( ). e, if ()

4 IJST () A: -7 Sep.. If, hen he queueing syse is M / M /, nd he densiy funcion of ie spen in he service sion would be eponenilly epressed wih preer, herefore, w () is clculed s follows: w (). e, if () Sep.. If, hen he queueing syse is M / M /, nd he densiy funcion of ie spen in he service sion would be pproiely wo series eponenil wih preers ( ) nd ( ),where. Therefore, w () is pproiely clculed s follows: ( ) w () ( )( ). e ( ) ( ) ( ) ( ) ( )( ) e ( ) ( ) ( ) () Sep. Conver he dynic PERT nework s n Aciviy-on-Node (AoN) srucure ino subsiue clssicl PERT nework represened s n Aciviyon-Arc (AoA) grph. Sep.. By considering he AoN grph, subsiue ech node wih sochsic rc (civiy) whose lengh is equl o he sojourn ie in he corresponding service sion. For his purpose, node in he AoN grph should be replced wih sochsic civiy. Assue b, b,..., b n re he incoing rcs o node nd d, d,..., d re he ougoing rcs fro i. Then, node is subsiued by civiy ( vw, ), whose lengh is equl o he sojourn ie in he service sion. Furherore, ll rcs b, b,..., b n erine wih node v while ll rcs d, d,..., d begin fro node w. (for ore deils, see []) Sep.. Trnsfor he PERT nework, obined in sep., ino new PERT nework wih eponenilly disribued rc lengh. Resources s servers pproch: in his pproch in which he nuber of servers nd resources lloced in every service sion re equl, every rc would be subsiued wih wo series of eponenil rc wih he preers ( ) nd ( ). Afer replcing ll rcs wih he proper eponenil wo series rc, he PERT nework obined in sep., is rnsfored ino new PERT nework. Resources ffecing servers pproch: in his pproch, he nuber of servers in every service sion is fied nd resources lloced ffec he en of service ies. As enioned in seps. nd., If one or infinie severs be in he work sion, hen he lengh of rc would be eponenil wih preers nd, respecively, nd he corresponding rc would no be chnged. Bu, if severl servers ( ) be in he work sion, hen he corresponding rc would be subsiued wih wo series of eponenil rc wih he preers ( ) nd ( ). Afer replcing ll such rcs wih he proper eponenil wo series rc, he PERT nework obined in sep., is rnsfored ino new PERT nework. Sep. Deerine coninuous-ie Mrkov process wih finie ses. Sep.. Deerine he ses spce of syse. For his purpose, le G ( V, A) be he PERT nework, obined in sep., wih single source nd single sink, in which V represens he se of nodes nd A represens he se of rcs of he nework in he AoA nework. Also, le G ( V, A) be new PERT nework, obined in sep., in which V represens he se of nodes nd A represens he se of rcs of he nework in new AoA grph. Le s nd be he source nd sink nodes in he new PERT nework, respecively, nd he lengh of rc A be rndo vrible h is eponenilly disribued wih preer. For A, he sring node nd he ending node of rc, re denoed s () nd (), respecively. Henceforh in his secion, we nlyze he new PERT nework o deerine coninuous-ie Mrkov process wih finie se spce. Definiion. Le I (v) be he se of rcs ending node v nd O (v) be he se of rcs sring node v in he new PERT nework, which re defined s follows: (see [9]) I (v) A : () v ( v V ), O(v) A : () v ( v V ). ()

5 IJST () A: -7 Definiion. For X V such h s X nd X V X, n (s,) cu is defined s follows: Fig.. Srucure of he proposed ehod (X,X ) A : () X, () X. () An (s, ) cu (X, X ) is denoined uniforly direced cu (UDC), if ( X,X), i.e. no wo rcs in he cu belong o he se ph in he projec nework. Ech UDC is clerly se of rcs, in which he sring node of ech rc belongs o X nd he ending node of ech rc belongs o X. Eple. Consider he nework shown in Fig. ken fro []. According o he definiion, he UDCs of his nework re (, ), (, ), (,, ), (,, ) nd (, ). Definiion. An ( E, F), subses of A, is defined s dissible -priion of UDC D if D E F nd E F, nd lso I( ()) F for ny F. s Fig.. The eple nework Agin consider Eple. As enioned (,, ) is UDC. For eple, his cu cn be decoposed ino E, nd F. In his cse, he cu is n dissible -priion, becuse I( ()), E nd F. Furherore, if, priion, becuse I( ()) F,. F, hen he cu is no n dissible -

6 IJST () A: -7 Definiion. Along wih he projec eecuion ie, ech civiy (rc) cn be in one nd only one of he cive, dorn or idle ses, which re defined s follows: (i) Acive: n civiy is cive ie if i is being perfored ie. (ii) Dorn: n civiy is clled dorn ie if i hs copleed bu here is les one unfinished civiy in I( ()) ie. (iii) Idle: n civiy is denoined idle ie if i is neiher cive nor dorn ie. Also, Y() nd Z() re defined s follow: Y () A : is cive ie, Z () A : is dorn ie, () nd X() (Y(), Z()). All dissible -priion cus of he nework of Fig. re presened in Tble. A superscrip sr is pplied o denoe dorn civiy nd ll ohers re cive. E nd F conin ll cive nd ll dorn civiies, respecively. The se of ll dissible -priion cus for he nework re defined s S nd lso S S (, ). Noe h X ( ) (, ) presens h he ll civiies re idle ie nd herefore he projec is finished by ie. I is deonsred h X ( ), is finie-se bsorbing coninuousie Mrkov process. (for ore deil, see [8]) Sep.. Obin he syse of differenil equions. As previously enioned, UDC is divided ino E nd F h conin cive nd dorn civiies, respecively. If civiy erines (wih he re of ), nd I( ( )) F, here is les one unfinished civiy in I ( ( )), hen E E, FF. Furherore, if by copleing civiy, ll civiies in I ( ( )) becoe idle ( I( ( )) F ), hen E ( E ) O( ( )), F F I( ( )). Nely, ll civiies in I( ( )) will becoe idle nd lso he successor civiies of his civiy, O( ( )), will becoe cive. Therefore, he coponens of he infiniesil generor ri Q q( E, F),( E, F), ( E, F) nd ( E, F) S re obined s follows: q( E, F),( E, F) (7) E if : E, I( ( )) F, E E, F F if : E, I ( ( )) F, E( E ) O( ( )), FFI( ( )) if : E E, F F oherwise X(), is coninuous-ie Mrkov process wih finie se spce S nd since q(, ),(, ), he projec is copleed. In his Mrkov process ll of he ses ecep X ( ) (, ) which is n bsorbing se, re rnsien. Furherore, he ses in S should be nubered such h his Q ri be n upper ringulr one. I is ssued h he ses re nubered s,,..., N S so h X ( ) ( O( s), ) nd X ( ) (, ) re se (iniil se) nd se N (bsorbing se), respecively. Le T be he lengh of he longes ph or he projec copleion ie in he new PERT nework, obined in sep.. Obviously, T in : X() N X(). Chpn Kologorov bckwrd equions cn be used o clcule F() P(T ). If i is defined P ( ) P ( X() N X() i ) i,,..., N, (8) i hen F() P (). The syse of liner differenil equions for he vecor P() P T ( ) P ( )... PN ( ) is presened s follows: dp () P() Q. P() d T P()..., (9) where P () nd Q represen he derivion of he se vecor P () nd he infiniesil generor ri of he sochsic process X(),, respecively... Muli-objecive resource llocion In his pper, following he reserch presened by Azron nd Tvkkoli-Moghdd [], we propose uli-objecive odel o opilly

7 7 IJST () A: -7 conrol he resources lloced o he service sions in dynic PERT nework bsed on he enioned wo pproches.... Resources s servers We propose uli-objecive odel o opilly conrol he servers lloced (s resources) o he service sions in dynic PERT nework, represened s nework of queues, where we lloce ore servers o he service sion, he en ie spen (sojourn ie) in he service sion will be decresed nd direc cos will be incresed, wheres he direc cos of ech civiy is non-decresing funcion of he oun of he lloced server. Noe h he en of civiy durion is consn vlue. If we decrese he oun of resource lloced (servers) o he service sions, he projec direc cos will herefore be decresed. Conversely, he en projec copleion ie will hen be incresed, becuse hese objecives re in conflic wih ech oher. Therefore, he ol direc coss nd he en projec copleion ie re dependen on ech oher nd n pproprie rde-off beween he is required. Anoher effecive objecive h should lso be included in he odel, is he vrince of he projec copleion ie, becuse he en nd he vrince re wo copleenry conceps. The ls objecive h should lso be considered is he probbiliy h he projec copleion ie does no eceed cerin hreshold for on-ie delivery perfornce. Le d ( ) be he direc cos of civiy A in he PERT nework, obined in secion. sep., in which he oun of servers ws lloced o i nd i is ssued o be non-decresing funcion. Therefore, he projec direc cos (PDC) would be equl o PDC d ( ). Le A U be he iu oun of server (resource) vilble o be lloced o he civiy ( A), L be he iniu oun of server needed o eecue he civiy, T : A nd M represen he oun of servers vilble o be lloced o ll civiies. Moreover, we define u s hreshold vlue in which projec copleion ie does no eceed he vlue. Therefore, his is uli-objecive sochsic progring proble. The objecive funcions re given s follows:. Miniizing he projec direc cos Min f ( ) ( ) d () A. Miniizing he en of projec copleion ie ( P ( ) is he derivion of densiy funcion of projec copleion ie.) Min f ( ) E ( T ) ( P ( )) d P( )) d Tble. All dissible -priion cus for he eple nework (). Miniizing he vrince of projec copleion ie Min f ( ) Vr ( T ) P ()) d P ()) d (). Miizing he probbiliy h he projec copleion ie does no eceed cerin hreshold M f ( ) P ( u ) P(T u) () The infiniesil generor ri Q would be funcion of he conrol vecor T : A. Therefore, he non-liner dynic odel is P() Q( ). P() Pi () i,,..., N P () N () The ne consrin should be regrded o gurnee hving response in he sedy-se.. A. () In he heicl progring, we do no use such consrins. Hence, following he esblishen of consrin... A A () *. (,). (,,). (,) *. (,, ) *. (, ) * * 7. (,, ). (,,) 8.,,) 9. ( *,,). (,, ) 7. (, ). (, *,). (,) *. (,, ) * * (. (,, ). ( *,). (, * )

8 IJST () A: -7 8 Consequenly, he pproprie uli-objecive opil conrol proble is epressed s below: Min f ( ) ( ) d A Min f ( ) P ( )) d Min f ( ) P ( )) d P ( )) d M f ( ) P( u ) P() Q( ). P() Pi () i,,..., N PN (). A.. A L A U A M A is ineger A (7) This coninuous-ie sochsic progring is ipossible o solve (for ore deils see []), herefore, bsed on he definiion of inegrl hinking, we divide he ie inervl ino R equl porions wih he lengh of. Indeed, we rnsfor he differenil equions ino difference equions. Thus, he corresponding discree se odel cn be given s follows: Min f ( ) d ( ) A R r Min f ( ) r( P ( r ) P ( r)) R r R r ( P( r ) P( r)) r Min f ( ) ( r ) ( P ( r ) P ( r)) u M f ( ) P s. : P ( r ) P( r) Q( ) P( r) r,,,..., R Pi () i,,..., N P ( r) N r,,..., R (8) Pi ( r) i,,..., N, r,,..., R. A.. A L A U A A M is in eger A... Resources ffecing servers In his secion, we propose uli-objecive o opilly conrol he resources lloced o he service sions bsed on resources ffecing servers pproch in uli-server dynic PERT nework, represened s nework of queues. The direc cos of ech civiy is non-decresing funcion nd he en service ie in ech service sion is nonincresing funcion of he oun of resource lloced o i. Le be he resource lloced in service sion ( A), lso d ( ) be he direc cos of civiy A in he PERT nework, obined in secion. sep., while i is ssued o be nondecresing funcion of oun of resources lloced o i. Thus, he projec direc cos (PDC) would be PDC ) d ( A /. Also, he en service ie in he service sion A, g ( ) is ssued o be non-incresing funcion of he oun of resource lloced o i h would be equl o g ( ) A (9) Le U be he iu oun of resource vilble o be lloced o he civiy ( A), L be he iniu oun of resource needed o eecue he civiy, T : A nd J represen he oun of resource vilble o be lloced o ll of civiies. Moreover, we define u s hreshold ie h projec copleion ie does no eceed. Le B be he se of rcs in he PERT nework, obined in secion. sep., where here re infinie servers seled on he corresponding service sion. The ne consrin should be sisfied o keep he response in he sedy-se.... A B A B B () In prcice, d ( ) nd g ( ) cn be obined by eploying liner regression bsed on he previous siilr civiies or pplying he judgens of epers in his re. Consequenly, he pproprie uli-objecive opil conrol proble is:

9 9 IJST () A: -7 Min f ( ) ( ) d A Min f ( ) P ( )) d Min f ( ) P ( )) d P ( )) d M f ( ) P( u ) s. : P() Q( ). P() Pi () i,,..., N PN () g ( ) A (). A B.. A B B L A U A J A We divide he ie inervl ino R equl porions wih he lengh of. The corresponding discree se odel s follows Min f ( ) ( ) d A R r Min f ( ) r ( P( r ) P( r)) R r R r ( P( r ) P( r)) r Min f ( ) ( r ) ( P( r ) P( r)) u M f ( ) P () s. : P ( r ) P( r) Q( ) P( r) r,,,..., R Pi () i,,..., N PN ( r) r,,..., R P ( r) i,,..., N, r,,..., R i g ( ) A. A B.. A B B L A U A J A.. Gol inen ehod We now need o pply uli-objecive ehod o solve he proposed odels, nd we cully pply gol inen echnique for his purpose. Assue here is uli-objecive progring wih n objecives, see (), where f j () nd X re j h objecive nd fesible region of he proble, respecively. Min f ( ), f ( ),..., f n ( ) s. : X () The gol inen ehod requires deerining gol, b j, nd weigh, c j, for every objecive. c s reflec he ipornce of objecives, wheres if j n objecive hs he slles c, hen i will be he j os iporn objecive. c j s ( j,,..., n) re n coonly norlized such h c. j Therefore, he pproprie gol inen forulion of he uli-objecive proble is given by Min z s. : f j( ) c jz bj j,,..., n X j () For solving he uli-objecive odels proposed in secion. wih he gol inen ehod, he gols, b s, nd weighs, c s ( j,,, ), should j j be deerined for every objecive, nely, he projec direc coss, he en of projec copleion ie, he vrince of projec copleion ie nd he probbiliy h he projec copleion ie does no eceed cerin hreshold. Then, by pplying () he pproprie gol inen forulion of he uli-objecive proble should be fored.. Recive resource llocion in uli-server dynic PERT neworks In he previous secion, uli-objecive odel for he resource llocion in uli-server dynic PERT nework ws proposed. In he presened odel, he unceriny ws considered in he enrnce of projecs nd lso in he durion of service sions, wheres, oher unceriny s projec nework disrupion cn occur. In his secion, projec nework disrupion such s: insering new civiy (service sion), deleing n civiy nd chnges in precedence relions of projec re considered. For coping wih projec nework disrupion, recive resource llocion

10 IJST () A: -7 is lso suggesed. Along wih he projec eecuion, projec y be disposed he considerble unforeseen disrupions, herefore, recive scheduling (rescheduling), wih revising or reopiizing he bseline schedule, is o repir he bseline schedule nd consequenly, overcoe disrupions. For overcoing hese disrupions, firsly he revised PERT nework is obined by considering he chnges in he projec nework. Then, by using secion., he chnged PERT nework is rnsfored ino new PERT nework wih n eponenilly disribued rc lengh. Finlly, by dding he new objecive, nely, iniizing he suion of cos of chnges, nd pplying he odels represened in secions.. nd.., respecively, for resources s servers pproch nd resources ffecing servers pproch, he recovery odel is consruced... Resources s servers Le : A T nd : A T be he resource lloced o service sions in priry nd recive resource llocion, respecively. Also, le c c T : A be he chnge cos of resource lloced per every uni in service sions. The in seps of our proposed ehod for he recive resource llocion in resources s servers pproch re s follows: Sep. Cree revised PERT nework by considering he required chnges in he projec nework. Sep. Trnsfor he chnged PERT nework ino new PERT nework wih eponenilly disribued rc lengh by using secion.. Sep. Apply he odel represened in secion.. for he nework obined in sep, by dding new objecive s Min. c, where A A represens he se of rcs of he nework in AoA nework, obined in sep... Resources ffecing servers Le T : A nd T : A be he resource lloced o service sions in priry nd recive resource llocion, respecively. Also, le c c : A T be he chnge cos of resource lloced per every uni in service sions. The in seps of our suggesed ehod for he recive resource llocion in resources ffecing servers pproch re s follows: Sep. Cree revised PERT nework by considering he required chnges in he projec nework. Sep. Trnsfor he chnged PERT nework ino new PERT nework wih eponenilly disribued rc lengh by using secion.. Sep. Apply he odel represened in secion.. for he nework obined in sep, by dding new objecive s Min. c, where A A represens he se of rcs of he nework in AoA nework, obined in sep.. An illusrive cse To illusre he nlyicl proposed ehod, we solve nuericl eple o presen he resource llocion in uli-server dynic PERT neworks, which is presened s he nework of queue. I is ssued we hve syse wih he si service sions depiced s he AoN grph in Fig.. We wn o deerine he opil resource llocion in uli-server dynic PERT nework for boh pproches, nely, resources ffecing servers pproch nd resources s servers pproch. For solving his eple, we lso pply he gol inen ehod. Fig.. The AON nework of he projec under sudy.. Resources ffecing servers The ssupions of his eple for he resources ffecing servers pproch re: The new projecs, conining ll heir civiies, rrived he syse ccording o Poisson process wih he re of per yer. The civiy durions (service processing ies) re independen rndo vribles wih eponenil disribuions. The hreshold ie, u, h projec copleion ie does no eceed is yers. The oun of resources vilble o be lloced o ll service sions is. In ll eperiens, he vlue of is equl o..

11 IJST () A: -7 Tble. Chrcerisics of he civiies Aciviy ( ) d ) ) ( g (..8 L U Tble shows he chrcerisics of he civiies, where he ie uni nd he cos uni re, respecively, in yer nd in housnd dollrs. Now, we subsiue he nodes of,,, in Fig. wih wo series of eponenil node, see Fig., nd hen we deerine he syse ses nd rnsiion res, depiced in Tble nd Fig., Tble. All dissible -priion cus of he projec where ( )., for,,, nd for,. We orgnize he infiniesil generor ri Q () ccording o (7). Tble presens he infiniesil generor ri Q () (digonl coponens re equl o inus su of he oher coponens he se row).. (,). (,) 9. (,). (, * ) 7. ( ). (,). (, ) *. (, ). (,) 8. ( ). (,) * 7. (, ). (,) * (, ) 9. ( ). (,) 8. (,). (, ). ( ). (, ) Tble. Mri Q ()

12 IJST () A: -7 Tble. The copuionl resuls R z f f f f CT : : : : : : : Fig.. The subsiued AON nework of he projec Fig.. Re digr for he coninuous-ie Mrkov chin in he projec under sudy The objecive is o obin he opil resources lloced o he differen civiies by solving (9). For his purpose, we consider he gols, b, b., b., b. 9 nd he weigh, c., c., c., c., for he four objecives, nd lso he vrious cobinions of R nd for ie inervl yers. To do so, we eploy LINGO 8 on PC Peniu, CPU GHz. The opil lloced resources, he copuionl ies, CT (:ss), nd lso he vlues of ll objecives for he differen cobinions of R nd re shown in Tble. So, he opil lloced resources re:.,,.9,.,.8,.9, nd he objecive funcion vlues re: Sr f 7., f., f.8, f.797 ( z.9 ). Bsed on Tble, if he lengh of is decresed, he ccurcy of he soluion is incresed i.e. he vlue of z is decresed nd he copuionl ie, CT, is lso incresed (for ore deils, see []). Noe h he siulion resuls si si si re:.,.,.8, si si si.97,.,.7, while he siuled en of projec copleion ie is.... Resources s servers Finish The ssupions of his eple for he resources s servers pproch re: The new projecs, conining ll heir civiies, rrived he syse ccording o Poisson process wih he re of per yer. The civiy durions (service ies) re independen rndo vribles wih eponenil disribuions. The oun of resource (server) vilble o be lloced o ll service sions is. The hreshold ie, u, h projec copleion ie does no eceed is yers. In ll eperiens, he vlue of is equl o.. Tble shows he chrcerisics of he civiies, where he ie uni nd he cos uni re, respecively, in yer nd in housnd dollrs.

13 IJST () A: -7 Tble. Chrcerisics of he civiies Aciviy ( ) ) d ( L U Now, we subsiue every node in Fig. wih wo series of eponenil node, see Fig., nd s noed in secion.., we deerine he syse ses, rnsiion res nd he infiniesil generor ri Q (). The objecive is o obin he opil servers lloced o he differen service sions by solving (8). For his purpose, we consider he gols, b, b, b., b. 9 nd he weigh, c., c., c., c., for he four objecives, nd lso he vrious cobinions of R nd for ie inervl yers. To do so, we eploy LINGO 8 on PC Peniu, CPU GHz. The opil lloced resources, he copuionl ies, CT (:ss), nd lso he vlues of ll objecives for he differen cobinions of R nd re shown in Tble 7. So, he opil lloced servers re:,,,,,, nd he objecive funcion vlues re: f., f.8, f., f. 8 ( z. 8 ). si Noe h he siulion resuls re:, si si si si si,,,,, while he siuled en of projec copleion ie is.8. Sr. Conclusion In his ricle, we proposed uli-objecive odel o opilly conrol he resources lloced o he service sions in uli-server dynic PERT nework for boh pproches, nely resources s servers nd resources ffecing servers, using Mrkov process nd uli objecive progring. This dynic PERT nework ws represened s nework of queues, where severl servers re in ech service sion nd he cpciy of he syse is infinie. In boh pproches, for odeling uli-server dynic PERT nework, firsly he nework of queues ws rnsfored ino sochsic nework nd hen, he ses of he syse were defined. Noe h he nuber of syse ses grows cobinorilly wih he nuber of UDCs. Ne, syse of differenil equions ws fored o solve nd obin he pproie copleion ie disribuion for ny priculr projec by creing n pproprie finie-se coninuous-ie Mrkov odel. A uli-objecive odel wih four confliced objecives ws presened o opilly conrol he resources lloced o service sions in uli-server dynic PERT nework. Finish Tble 7. The copuionl resuls Fig.. The subsiued AON nework of he projec R z f f f f CT : : : : : :

14 IJST () A: -7 In our odel, he ol projec direc cos ws considered s n objecive o be iniized nd he en projec copleion ie s noher effecive objecive, should lso be ccouned o be iniized. The vrince of he projec copleion ie ws noher effecive objecive in he odel, becuse he en nd he vrince re wo copleenry conceps. The probbiliy h he projec copleion ie does no eceed cerin hreshold ws considered s he ls objecive. Finlly, he gol inen ehod ws eployed o solve discree-ie pproiion of he priry uli-objecive proble. For obining he bes opil lloced resource, we considered he vrious cobinions of porions for specific ie inervl. Bsed on he presened eple, if he lengh of every porion, is decresed, he ccurcy of he soluion is incresed i.e., he vlue of he objecive is decresed nd he copuionl ie, CT, is lso incresed. Appendi A: If here is server in he service sion seled in he h node, hen he queueing syse is M / M / nd herefore, he densiy funcion of sojourn ie (civiy durion plus wiing ie in queue) is clculed s follow []: q w () w () ( ) e ( ) w () q ( ( ) )( ). e ( ) (A-) Where nd re, respecively, he rrivl re of new projec nd he service re of service sion. Also, w q (), he probbiliy of being zero queue lengh, nd P re obined s follow: w q nd: P ( ) () P!( ) n n ( ) ( ) ( ) n!! (A-) (A-) As is observed, obining w () in n M / M / is very hrd. We cn rewrie he w () s follow, which is siilr o wo series of eponenil disribuion wih preers ( ) nd : q w () ( w ())( )( ). e ( ) q w () ( ) e ( ) ( ) (A-) I sees h we cn pproie densiy funcion of ie spen in service sion wih wo series eponenil. Therefore, our pproie for densiy funcion of ie spen in service sion would be wo series eponenil wih preers ( ) nd ( ), where ( ). This pproiion is quie siple nd esy nd here is no need o clcule P nd w q () which is boring, especilly when is lrge. Moreover, our pproiion cn be used in heic progring proble nd Mrkov chin, convenienly. We wn o evlue he en of sojourn ie, nd herefore, he epeced nuber of projecs, nd lso he cuulive disribuion funcion of sojourn ie. In Fig. A-, he en nuber of projecs in node s chnges, is presened. In he lierure, soe pproiions for he sojourn ie in M / M / hve been inroduced. Sksegw [] proposed closed-for pproiion for he epeced wiing ie in queue, nd herefore, he pproiion of epeced sojourn ie W in work sion ws clculed s follows: ( ) W ( ) (A-) Also, Hlfin nd Whi [] developed closedfor pproiion for he w q (), nd herefore, he pproiion of epeced sojourn ie in work sion ws obined s follows: W ( ). ( ). e (A-) Where ( ) nd () is he cuulive disribuion funcion of sndrd norl disribuion hving en nd vrince. On he oher hnd, our pproiion for he epeced sojourn ie in work sion would be:

15 IJST () A: -7 W ( ) ( ) (A-7) Tble A-. Nuericl vlue of ec nd pproiion epeced nuber of projecs in work sion L Our pp. Error(%) L Our pp. Error(%) ) b) c) d) Fig. A-. The ec nd our pproiion of epeced nuber of projecs Obviously, our pproiion is very siple nd is coponens esily, wihou boring copuions, re conrollble, i.e., he copleiy of our pproiion is less hn oher pproiions. Furherore, s enioned before, our pproiion cn be used in heic progring proble nd Mrkov chin, convenienly. In Tble A-, he epeced nuber of projecs in work sion ( L ) nd he error of our pproiion re represened. As nd., our pproiion for epeced nuber of projecs in work sions is very poor, herefore, for coping wih his shorge, we consider.. Also, in Fig. A-, he ec cuulive disribuion funcion nd our pproie cuulive disribuion funcion for nd wih vrious uilizion fcors, is shown. Moreover, in Tble A-, he iu difference (MD) beween he ec cuulive disribuion funcion nd our pproie cuulive disribuion funcion for vrious nuber of server nd uilizion fcors, is shown. Consequenly, we hve:. If, hen he queueing syse would be n M / M / queue, nd he densiy funcion of ie spen in he service sion, w (), would be eponenilly wih preer, herefore, w () is clculed s follows: w e if (A-8) ( ) () ( ).,. If, hen he queueing syse is M / M /, nd he densiy funcion of ie spen in he service sion would be eponenilly epressed wih preer, herefore, w () is clculed s follows:

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