Multiple criteria network models for project management

Transcription

1 Mulple cea newo model fo pojec managemen Víceeální í oé modely pojeoém øízení T. ŠUBRT Czech Uney of Agculue, Pague, Czech Republc Abac: The am of he pape o peen one pobly of how o model and ole a eouce oened ccal pah poblem. A a ang pon, a ngle cea model fo ccal pah fndng holy menoned. Laely, moe cea funcon fo h model ae defned. If any pojec a ue moe eouce fo compleon, duaon uually depend on only one of hem ohe eouce ae no fully ued. In hee defned mulple cea appoach, hee dependence ae no aumed. Each cea funcon deed fom a heoecal a duaon baed on a numbe of un of only one eouce and on mpoance. Ung ehe lnea pogammng model wh aggegaed cea funcon o mple Excel calculaon wh Mcoof Pojec ofwae uppo, a o-called compome ccal pah can be found. On h pah, ome eouce ae oeallocaed and ome ae undeallocaed bu he oal um of all undeallocaon and all oeallocaon mnmzed. All eouce ae ued a effecely a poble and he pojec a ho a poble oo. Key wod: newo modelng, agculual pojec managemen, mul cea pogammng, compome ccal pah, eouce managemen, Excel Aba: Cílem ohoo pøípìu je pøeda jeden z poupù, øešících oázu anoení ompomní cé cey modelech pojeoého øízení zhledem opmálnímu yuží zdojù dponblní zdojoé záladny. Výchozím pøedpoladem modelu je nepøímá úmìno mez délou ání ènno a nenzou èepání zdoje, onanní nenza èepání po dobu ání ènno a onanní ohodnocení azeb mez jednolým ènnom. Záladní uua í oého modelu je popána yužím omezujících podmíne modelu celoèíelného pogamoání a zdojoé zabezpeèení e zahu èaoé náoèno jednolých ènnoí je fomalzoáno za pomoc eálních funcí. Poèe eálních funcí je oožný poèem zdojù zabezpeèujících poje. Každému zdoj pojeu je naíc možno pøøad uèou áhu, epezenující jeho dùležo, è ocenìní po pøílušnou ènno è po celý poje. Vznlou íceeální úlohu lze poom pøeé napøílad yužím ážené agegace eálních funcí na lacou maxmalzaèní úlohu celoèíelného lneáního pogamoání. Výledem øešení je ompomní cá cea, na eé je ouhn neyuží a pøeížení jednolých zdojù mnmální. Po aplac náledné analýzy opmálního øešení je možno ano opmální poèe zdojù nuný zabezpeèení daného pojeu zhledem ypoèené cé ceì. Analýza clo opmálního øešení dále umožòuje ano ìšnu lacých uazaelù ènnoí jao napøílad èaoé ezey è lhùoé uazaele. Klíèoá loa: í oé modely, pojeoé øízení zemìdìlí, íceeální pogamoání, ompomní cá cea, managemen zdojù, Excel INTRODUCTION Agculual pojec managemen he poce ha aue eey apec of poduc deelopmen and acon aen o mnmze boh me and co n bngng a poduc o he maeplace. A peen me, he moden pojec managemen ue a lo of ophcaed mahemacal mehod fo pojec duaon mnmzaon, fo co mnmzaon, fo opmal allocaon of eouce ec. Many of hee mehod ae baed on newo modelng echnque, epecally ccal pah mehod, and mahemacal pogammng algohm. Neehele, when wong wh eouce, many paccal appoache ae baed on empcal o heuc pocedue, becaue he exac mahemacal mehod ae que had o ue and o hey ae no ncluded n he majoy of ofwae andad. In h ex, I would le o peen one mple appoach, whch wll help u o effecely allocae and manage pojec eouce. METHODOLOGY Recenly, paccally all pojec managemen fomalzaon ool wee baed on he Acy on Ac (AOA) newo gaph whee all pojec a (ace) ae epeened by ac and node ae ued only fo mleone. Acually he majoy of pojec managemen ofwae ue acy on node (AON) newo gaph. I mean ha all pojec a (ace) ae epeened AGRIC. ECON. CZECH, 50, 2004 (2):

2 by node and he elaonhp by ac. When fndng ccal pah, each pojec a can ehe be ccal o nonccal. Le u emnd ha ccal pah ey mpoan fo ome pojec em and deadlne becaue he delayng ccal a wll caue delay of he pojec fnh dae. My fuhe appoach pmaly baed on mahemacal model of he AON newo model. Th model a mply modfed nege pogammng lnea model and all appoache below can be decbed ung h model. In he followng ex, no uch mahemacal model wll be fomalzed (fo moe deal abou h model ee Šub 2001) bu he whole dea and appoach wll be demonaed on a paccal example fom poduc maeng banch. The am of my appoach o how how o ue eouce n pojec managemen pojec n a bee way, moe effecely. Th appoach deal moly wh he eouce ype of wo (labo foce) becaue maeal eouce hae dffeen feaue no uable fo he ype of calculaon menoned below. A a ang pon, le u uppoe a pojec fomalzed n a newo gaph. Theoecally, no mae f we hall ue an Acy on Ac gaph o an Acy on Node gaph bu AON one moe wde-pead and allow o model moe complcaed pojec managemen uaon (fo example aou ype of a dependence ec.). All lae noaon wll aume AON gaph. Newo gaph n pojec managemen we uually ue o calculae lengh of he pojec, analyze ucue of ccal pah; o analyze nonccal a (pah) lac ec. Fo mo of h calculaon, we hae o aume fxed duaon of a a (uually deemnc), whch n eey eal uaon depend on eouce agned o and on he amoun of aalable eouce wo. (Noe ha no a n he unee can be compleed whou a eouce!) Le u aume hee eouce aalable fo he pojec and,, eouce agned o -h a. Clacal mehod aume ha he a duaon depend only on a ccal eouce, mean on a eouce whch amoun of wo and numbe of agned un deemne hee duaon. The duaon of a a can be defned a = 1,2,..., whee a -h a duaon, an amoun of wo fo one -h eouce un needed fo -h a compleon, a numbe of -h eouce un agned o -h a and a numbe of eouce agned o -h a. The -h eouce called ccal when = 1,2,..., All ohe eouce agned o h a ae no fully ued and we call hem undeallocaed. I mean ha paccally on each a n a pojec, one o moe eouce ae no effecely ued. Of coue afe compleon of he adequae pa of wo, hee eouce can be agned o anohe a and lag me can be mnmzed bu n pacce h polcy doe no wo o popely (fo moe nfomaon ee Ccal Chan Appoach Golda 1999). Calculang he pojec (ung mahemacal pogammng model o by modfed CPM mehod) we oban ccal pah, whee ccal eouce ae opmally (100%) allocaed and all ohe eouce ae undeallocaed. On nonccal pah ccal eouce ae 100% allocaed oo bu due o lac on nonccal a 100% allocaon of hee eouce canno be guaaneed on he whole lengh of any nonccal pah. Applyng co coeffcen eny analy (o a lac analy), a dealed eul of he poble nfecene of eouce uage can be done. Th clacal appoach can be called ngle cea becaue whle eachng ccal pah, he mahemacal pogammng model wh only one cea funcon n he fom of pojec duaon maxmzaon ued. Mulple cea appoach compome ccal pah In a mulple cea appoach le u aume heoecal dependence of a duaon on each eouce epaaely. I mean ha each a ha a many heoecal duaon a he numbe of eouce agned o and he whole model ha a many heoecal ccal pah a he numbe of all eouce defned. The dea of mehod menoned below o aggegae all hee nddual ccal pah no one compome ccal pah whee no eouce can be defned a ccal and whee he um of eouce oeallocaon mnmzed and he um of eouce undeallocaon mnmzed oo. Analyzng hee oe (unde) allocaon by he applcaon of eny analy mehod, each a agnmen can be mnmally modfed o oban a oluon wh eouce agnmen a good a ge. Fo olng h nd of mulple cea model, many algohm ae aalable e.g. ngle cea opmzaon wh ohe a conan, weghed goal pogammng, cea funcon aggegaon, ec. The mo uable appoach eem o be he la one cea funcon aggegaon. F wo ae no ey effcen becaue n fac only one ccal pah ex and he decon mae ha no eaon o lm he heoecal duaon of nonexen pah. Thee ae many way of how o aggegae hee cea funcon. The mo effece one fo h pupoe eem o be a conex lnea combnaon of cea funcon coeffcen. In fac hee conex lnea combnaon of coeffcen n he meanng of cea funcon nomalzed wegh epeen a elae mpoance of each eouce fo he pojec o fo he nddual a whee he concee eouce agned. Th elae mpoance of a eouce can be expeed fo nance by co, by cedbly, by of ue ec. Solng mahemacal pogammng model, we oban a compome ccal pah and lengh deemnng pojec duaon. On h pah (and on ome ohe pah oo), ome eouce ae oeallocaed (ome a hae 72 AGRIC. ECON. CZECH, 50, 2004 (2): 71 75

3 no enough eouce fo he ealzaon) and ome of hem ae undeallocaed (hee one hae fee capace) bu he oal um of unde allocaon and oe allocaon mnmzed and elaely balanced. The hghe a eouce wegh wa e he loe oe (unde) allocaon. The heoecal duaon of each ccal a = w = 1 whee w a wegh of -h eouce. Fom h fomula, he heoecal amoun of eouce needed fo a compleon could be defned a w = If he dffeence = poe hen he -h eouce undeallocaed and of un fee. If he dffeence negae hen he -h eouce would be oeallocaed and he numbe of un agned mu nceae by o complee he a n me. If he majoy of eouce fom eouce pool ae agned o all a, we can ue global eouce wegh ( w ) fo he whole pojec,.e. = 1 w = 1 When hee ae many dffeen eouce wong on epaae a bee o ue nddual wegh fo each a ( ), mean o ue wegh whee = 1 w = 1, = 1,2,..., n RESULTS w Le u hae a mall pa of lage maeng pojec called Ealuae Bune Appoach, Poenal R and Rewad. Th pojec pa ha been fomalzed n an AON newo. I ha 8 a (node) and 6 eouce agned o hem. Each node n he gaph nclude he followng nfomaon: Ta ID, Ta Name and Agned Reouce Name. Table 1 how he amoun of eouce wo needed fo a compleon. Fly le u apply clacal appoach whee each a duaon defned a = 1,2,..., o 1 = 12, 2 = 10, 3 = 12, 4 = 24, 5 = 24, 6 = 24, 7 = 8 and 8 = 12. Ung h appoach, we hae o eep n mnd ha Pae agned only o 66.7% of f a duaon. I hould mean ehe ha be ule only 66.7% of h capacy, ha h neny of wo how 33.3% deceae, o ha he can a 4 hou lae han Johnon, o ha he can be fnhed 4 hou befoe Johnon ec. Applyng ehe CPM mehod o mahemacal pogammng model wh ngle cea funcon max we oban ccal pah and lengh (pojec duaon) 92h. Analyzng eouce on h pah, we can ee ha ha he effcency of uage elaely poo bu no eouce oeallocaed (ee Table 3). Now le u apply mulple cea appoach. Theoecally we can defne a many nddual ccal pah (cea funcon) a he numbe of eouce n he pojec and a mulple cea-pogammng poblem ae. Fo nance CP fo Smh can be decbed by cea funcon max ucue wll be 2 4 and lengh 28h. Le u aume each eouce ha nddual wegh dependng on mpoance fo he concee a (ee Table 2). Fo effece olng of h poblem, we hae o fnd he way of how o aggegae hee nddual ccal pah no one. 1 2 Idenfy on-gong bune puchae oppouny Johnon; Lew; Smh 5 Reeach fanche appoach Johnon; Lew; Smh 6 Summaze bune appoach Pae; Lew 9 Defne new eny equemen Johnon; Pae Deemne fnancal equemen Pae; Lew Ae mae ze and ably Pae; Hamlon Emae he compeon Pae; Hamlon Ae needed eouce aalably Pae; Hamlon Fgue 1. Coepondng pa of he maeng pojec AON gaph AGRIC. ECON. CZECH, 50, 2004 (2):

4 Table 1. Amoun of eouce wo needed fo each a compleon (hou) Ta ID Reouce name Pae Hamlon Lew Smh Johnon Table 2. Inddual eouce wegh Ta ID Reouce name Pae Hamlon Lew Smh Johnon Table 3. Impoan a chaacec obaned by clacal appoach Ta ID Ta duaon (h) Toal lac (h) % of eouce allocaon Pae Hamlon Lew Smh Johnon Ccal pah lengh (h) 92 Table 4. Impoan a chaacec obaned by compome ccal pah appoach Ta ID Ta duaon (h) Toal lac (h) % of eouce allocaon Pae Hamlon Lew Smh Johnon Ccal pah lengh (h) 83.4 Accodng o he algohm menoned eale, we apply he weghed adde aggegaon, whee a duaon wll be defned a = w = 1 and o 1 = = 10.4; 2 = 9.8, 3 = 8.8, 4 = 22.2, 5 = 21, 6 = 20.4, 7 = 8 and 8 = 12. On he f a, Johnon now abou 15% oeallocaed and Pae abou 23% undeallocaed (ee Table 4). Inceang Pae wegh, h oeallocaon wll deceae and deceang Pae wegh, h oeallocaon wll nceae. 74 AGRIC. ECON. CZECH, 50, 2004 (2): 71 75

5 Changng wegh, he pojec manage can effecely conol eouce uage on a a. Paccal expeence demonae ha 10 15% of oe allocaon oleable (epecally when moe han one eouce un agned o a a fo nance n cae of manual woe) and many eouce can allow uch nceae neny of wo. Applyng mahemacal pogammng model wh aggegaed cea funcon max we oban dffeen ccal pah and lengh (pojec duaon) 83.4h. On h ccal pah, he um of oe- and undeallocaon balanced (dependen on eouce wegh) and on all nonccal pah, we can addonally manage oeallocaon by conumng a lac. Table 3 and 4 compae boh appoache. F one conan chaacec acqued by clacal appoach, econd one by compome ccal pah appoach. CONCLUSION Thee ae eeal mehod of ncopoang mulple objece n pojec managemen poblem. One of hem deal wh he pobly of ccal pah fndng ung mahemacal pogammng model. Applyng mul objece appoach, a compome ccal pah wh opmum eouce allocaon can be found. If each cea funcon epeen one ngle eouce, heoecally a many ccal pah a he numbe of pojec eouce can be defned. Afe agnng a elae eouce mpoance numbe o each eouce, one aggegaed objece funcon ae and he model can be oled ung ngle objece pogammng mehod. Remembe ha h appoach doe no ubue eolng eouce conflc ung eouce-leelng algohm. I moly focued on eouce managemen whn a ngle a and conequenly on fndng compome ccal pah. Shang eouce among wo o moe paallel a an objece of ubequen algohm. REFERENCES Golda E. (1999): Ccal Chan. Nohe Pe, Gea Bengon, MA. Keze H. (2000): Pojec Managemen: A yem Appoach o Plannng, Schedulng and Conollng. John Wley & Son, New Yo. Šub T. (2001): Mul Cea Balen Pogammng Model fo Acy on Node Newo. In: Poceedng of he 19 h Inenaonal Confeence Mahemacal Mehod n Economc 2001 Confeence, UHK, Hadec Káloé. Taha H.A. (1992): Opeaon Reeach, Macmllan, New Yo. Aed on 21 Januay 2004 Conac adde: Ing. Tomáš Šub, PhD., Èeá zemìdìlá uneza Paze, Kamýcá 129, Paha 6-Suchdol, Èeá epubla e-mal: ub@pef.czu.cz AGRIC. ECON. CZECH, 50, 2004 (2):