MODELOVANJE NEIZVESNOSTI PRI UPRAVLJANJU PROJEKTIMA U POŠTI SRBIJE PRIMENOM BAJESOVSKIH MREŽA
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1 XXX Smpozjum o novm tehnologjama u poštanskom telekomunkaconom saobraćaju PosTel 2012, Beograd, decembar MODELOVANJE NEIZVESNOSTI PRI UPRAVLJANJU PROJEKTIMA U POŠTI SRBIJE PRIMENOM BAJESOVSKIH MREŽA Mlan Božć 1, Nataša Glšovć 2, Mloš Mlenkovć 3 1 Matematčk fakultet u Beogradu 2 Matematčk nsttut Srpske akademje nauka umetnost 3 Saobraćajn fakultet u Beogradu Sadržaj: Upravljanje projektma je vrlo složen proces s obzrom na to da uvek sadrž nezvesnost. Osnovn ulaz (vreme, trošak resurs potrebn za realzacju pojednh aktvnost) nsu determnstčk pod utcajem su razlčth zvora nezvesnost. Osm toga, postoj kauzaln odnos zmeđu pomenuth nezvesnost parametara projekta. U ovom radu prmenjen je koncept Bajesovskh mreža za modelovanje nezvesnost kauzalnost pr upravljanju projektma. Valdnost predloženog prstupa testrana je na prmeru projekta uvođenja franšznga u JP PTT saobraćaja Srbja. Ključne reč: upravljanje projektma, nezvesnost, Bajesovske mreže 1. Uvod Rzk predstavlja nemnovnost pr realzacj projekta. Upravljanje rzkom je postao važan deo upravljanja projektma. Premda u lteratur postoj velk broj tehnka, upravljanje rzkom projekta se ubrzano razvja upravljanje nezvesnostma u složenm projektma još uvek predstavlja zazov. Važna komponenta upravljanja rzkom projekta je analza rzka. Takođe poznata kao kvantfkacja rzka, analza rzka se zasnva na merenju rzka njhovh utcaja na razlčte parametre projekta (vreme, trošak kvaltet). Najranja stražvanja koja se odnose na nezvesnost/rzk u raspoređvanju projektnh aktvnost sprovedena su polovnom prošlog veka ([1], [2]). Od tad je velk broj tehnka uveden nekolko alata je razvjeno, pr čemu su mnog od njh prmenjen u praks. Međutm, ov alat često ne mogu adekvatno da tretraju nezvenost samm tm prozvode nepreczne, nekonzstentne nepouzdane rezultate. Nezvesnost u projektma ma nekolko aspekata od kojh samo nek mogu bt kategorsan posmatran kao rzc. Utvrđeno je ([3], [4]) da upravljanje rzkom projekata treba bt usmereno na upravljanje nezvesnostma razlčtm zvorma nezvesnost pre nego na naglašavanje skupa mogućh događaja koj mogu utcat na performanse projekta. Većna kvanttatvnh tehnka u aktuelnoj praks se zasnva na konceptu Utcaja verovatnoće koj poseduje ozbljne nedostatke [5]. Sa druge strane, Bajesovske mreže kao moćna tehnka za podršku odlučvanju u prsustvu nezvesnost prvlače mnogo pažnje u razlčtm oblastma. Bajesovske mreže
2 (poznate kao mreže uverenja l kauzalne probablstčke mreže) predstavljaju probablstčke grafčke modele koj se korste za nterpretacju saznanja rezonovanje u prsustvu nezvesnost. Postoj velka zanteresovanost za prmenu Bajesovskh mreža kao alata za modelovanje nezvesnost pr upravljanju projektma. Bajesovske mreže obezbeđuju metod za predstavljanje odnosa zmeđu promenljvh (čvorov u Bajesovskoj mrež) čak u slučajevma kada ov odnos sadrže nezvesnost. U ovom radu razmotren je problem obuhvatanja nezvesnost pr upravljanju projektma. Predložen prstup se zasnva na prmen koncepta Bajesovskh mreža pr modelovanju nezvesnost realzacje pojednh aktvnost sa aspekta vremena troškova. Predložen prstup je testran na prmeru projekta uvođenja franšznga u pošt Srbje. Rad je organzovan na sledeć načn. U drugom poglavlju razmotrena je prroda nezvesnost pr raspoređvanju projektnh aktvnost. U trećem poglavlju predstavljen je koncept Bajesovskh mreža. Četvrto poglavlje sadrž prstup za raspoređvanje projektnh aktvnost koj se zasnva na konceptu Bajesovskh mreža. U petom poglavlju zložen prstup je prmenjen na prmeru projekta uvođenja franšznga u JP PTT saobraćaja Srbja. Zaključna razmatranja data su u poslednjem šestom poglavlju. 2. Nezvesnost pr raspoređvanju projektnh aktvnost Upravljanje rzkom predstavlja ključnu oblast upravljanja projektnm aktvnostma. Upravljanje projektnm rzkom obuhvata procese koj se odnose na sprovođenje planranja upravljanja rzkom, dentfkacju, analzu, reakcju, montorng kontrolu realzacje projekta. Aktueln proces upravljanja projektnm rzcma podstču ogrančen fokus na upravljanje nezvesnošću [3]. Termn rzk postao je povezan sa događajma pre nego opštjm zvorma nezvesnost [4]. U razlčtm procesma upravljanja projektma postoje razlčt aspekt nezvesnost. U ovom radu fokus je na nezvesnostma pr raspoređvanju projektnh aktvnost, odnosno na estmovanje dužne troškova projektnh aktvnost. Problem pr estmacj dužne troškova projektnh aktvnost može nastat usled nezvesnh posledca potencjalnh rzka l mogućnost. Ova nezvesnost potče od: Nvoa raspoložvh potrebnh resursa; Odnosa zmeđu resursa vremena; Mogućeg nastanka nezvesnh događaja (rzc) Kauzalnh faktora međuzavsnost koj obuhvataju opšte kauzalne faktore koj utču na vše aktvnost stovremeno; Nedostatka prethodnog skustva upotrebe subjektvnh umesto objektvnh podataka; Nekompletnh l neprecznh podataka l nedostatka podataka u celn. Metod krtčnog puta (Crtcal Path Method CPM) predstavlja najpoznatj metod za raspoređvanje projektnh aktvnost [8]. CPM je determnstčka tehnka koja koršćenjem mreže zavsnost zmeđu zadataka dath determnstčkh vrednost za dužne zadataka, računa najduž put na mrež, poznat kao krtčn put. Dužna krtčnog puta predstavlja najranje vreme završetka projekta. Neka je t vreme trajanja aktvnost. P ( S ) predstavlja skup prethodnh (narednh) aktvnost u odnosu na aktvnost. Clj je pronać mnmalno trajanje projekta T. Za dat projekat sledeća nformacja je raspoložva: ES : vreme najranjeg početka aktvnost ; LF : vreme najkasnjeg završetka aktvnost ; 98
3 Vremenska rezerva aktvnost ( Slack ): Maksmalno moguće vreme trajanja aktvnost bez povećanja vremena trajanja projekta. Predstavlja razlku zmeđu najkasnjeg završetka najranjeg početka aktvnost; Krtčna aktvnost ( Crt ): Aktvnost čje vremenske rezerve su jednake vremenu trajanja aktvnost. Drugm rečma, aktvnost je krtčna ako povećanje njenog trajanja drektno utče na trajanje projekta. U nastavku je dat detaljan ops metoda krtčnog puta [9]: Incjalzacja: ES = 0 : P = Ponovt: ES = Max ES + t : j P ES je određeno { } j j j P sve dok se ne odrede ES za = 1,2,..., N { } T = Max ES + t LF = T : S = Ponovt: LF = Mn LF t : j S LF je određeno { } j j j S sve dok se ne odrede LF za = 1,2,..., N Fnalzacja: Slack = LF EF = 1,2,..., N { } Crt = = 1,2,..., N Slack = t j j Za razlku od CPM metoda, Tehnka za ocenu pregled programa (Program Evaluaton and Revew Technque - PERT) obuhvata nezvesnost u jednom ogrančenom smslu koršćenjem raspodele verovatnoća za svaku aktvnost. Umesto jedne determnstčke vrednost, tr razlčte procene (pesmstčka, optmstčka najverovatnja) su aproksmrane. Zatm se određuje krtčn put vremena početka završetka aktvnost. rezultat dobjen PERT metodom su realstčnj u odnosu na rezultate CPM-a. PERT metod takođe razmatra samo dužnu trajanja aktvnost. Koršćenjem PERT okvra, takođe je razmatrena nezvesnost u troškovma pojednh zadataka [10]. Monte Carlo smulacja (Monte Carlo Smulaton MCS) predstavlja jednu od domnantnh tehnka za tretranje nezvesnost u projektnom raspoređvanju [11]. Softversk paket za upravljanje projektma, PertMater [12] korst podatke o raspoređvanju projektnh aktvnost z softvra kao što su MS-Project l Prmavera prmenjuje MCS u clju sprovođenja analze rzka projekta sa aspekta vremena troškova. Međutm, MCS metod takođe poseduje određene nedostatke. Pretpostavka o statstčkoj nezavsnost vremena trajanja pojednh aktvnost predstavlja jedan od nedostataka. Takođe, s obzrom na to da je događajno orjentsan (pretpostavlja projektne rzke kao nezavsne događaje) MCS ne dentfkuje zvore nezvesnost. Upravljanje nezvesnostma u projektma ne treba da se zasnva samo na varjablnost raspoložvm podacma. Neophodno je razmotrt sve nedoumce prment strukturu znanje [13]. U clju pravlnog merenja analzranja nezvesnost, potrebno je modelovat relacje zmeđu uzroka (zvora) rzka utcaja (posledca). Najzastupljenj prstup za tretranje nezvesnost u ovm okolnostma je Bajesovsk prstup ([14], [15]). U slučajevma kada su kauzaln odnos prsutn, Bajesovsk prstup je prošren koršćenjem Bajesovskh mreža. 99
4 3. Bajesovske mreže Bajesovske mreže su razvjene na Stanford unverztetu tokom 70-th godna prošlog veka [16]. Ovm mrežama se opsuju uzročno-posledčn odnos zmeđu promenljvh. Tpčna Bajesovska mreža se sastoj od orjentsanog acklčnog grafa čvorova lukova koj konceptualzuje jedan sstem. Vrednost čvorova su defnsane preko razlčth međusobno sključvh stanja [17]. Odnos među njma su opsan uslovnm raspodelama verovatnoće koje predstavljaju zavsnost zmeđu promenljvh. Ukolko postoj luk zmeđu čvorova A C, tada se čvor A posmatra kao rodteljsk čvor čvora C, dok se za C kaže da je potomak čvora A. Na slc 1(a), rodteljsk čvorov A B čvora C predstavljaju kauzalne faktore čvora C. Stanja svh čvorova su predstavljena na slc 1(b). Čvor A može pretpostavt dskretna stanja vsoko l nsko, dok čvor B može pretpostavt dskretna stanja tačno netačno. Stanja promenljvh A B odredće da l je promenljva C u stanju vsoko, srednje l nsko. Uslovn odnos zmeđu rodteljskh čvorova A B čvora C je defnsan tabelom uslovne verovatnoće (Slka 1(c)). Ova tabela se može nterpretrat kao verovatnoća da će C bt u stanjma vsoko, srednje, nsko za data stanja čvorova A B [18]. Slka 1. Prmer strukture Bajesovske mreže [18] Razlčte vrste čvorova se mogu obuhvatt Bajesovskom mrežom: čvorov prrode, čvorov odluke čvorov vrednost. Čvorov prrode predstavljaju promenljve na koje se može utcat akcjama od strane donosoca odluke. Čvorov prrode se korste za predstavljanje emprjskh l zračunath parametara verovatnoća nastanka razlčth stanja. Ulazn čvorov (čvorov bez rodtelja) mogu bt strukturran kao konstante l kao stanja sa odnosnm margnalnm raspodelama verovatnoća. Čvorov odluke predstavljaju upravljačke promenljve (događaje) koje mogu bt drektno mplementrane od strane donosoca odluke. Ov čvorov predstavljaju skup raspoložvh upravljačkh akcja. Čvorov odluke su uvek povezan sa čvorovma vrednost koj se korste za procenu pravla optmalne odluke na mrež koja će maksmzrat sumu očekvanh vrednost čvorova vrednost. Bajesovske mreže se zasnvaju na Bajesovoj teorem teorje verovatnoće za propagranje nformacje zmeđu čvorova. Bajesova teorema opsuje kako je prethodno znanje o hpotezama H ažurrano realzovanm događajem E. Teorema povezuje uslovne margnalne verovatnoće H E na sledeć načn: P( H ) P( E / H ) P( H / E) = (1) P( H ) P( E / H) de gde P( H ) predstavlja početnu (a pror) verovatnoću hpoteze (verovatnoća da će H bt u određenom stanju pre razmatranja blo kakvog dokaza). P( E / H ) je uslovna verovatnoća (verovatnoća dokaza za datu hpotezu koja će bt testrana). P( H / E) predstavlja krajnju (a posteror) verovatnoću hpoteze (verovatnoća da je H u datom stanju, pod uslovom da 100
5 postoj određen dokaz). Integral u (1) predstavlja verovatnoću da će dokaz bt ostvaren, za datu raspodelu verovatnoća. 4. Predloženo rešenje za upravljanje projektnm aktvnostma zasnovano na konceptu Bajesovskh mreža Bajesovska mreža predstavlja orjentsan graf zajedno sa odgovarajućm skupom tabela verovatnoća. Graf se sastoj od čvorova lukova. Na Slc 2. data je Bajesovska mreža koja modeluje uzrok kašnjenja određene aktvnost projekta. Čvorov predstavljaju nezvesne promenljve, koje mogu (ne moraju) bt obzervablne. Svak čvor ma skup stanja. Lukov predstavljaju kauzalne l utcajne odnose zmeđu promenljvh. Postoj tabela verovatnoće za svak čvor koja sadrž verovatnoće za svako stanje promenljve. Za promenljve bez rodtelja, tabela sadrž samo margnalne verovatnoće. Ovo je poznato kao aprorno uverenje o promenljvoj. Za svaku promenljvu sa rodteljma, tabela verovatnoća ma uslovne verovatnoće za svaku kombnacju stanja rodtelja. Slka 2. Model Bajesovske mreže za kašnjenje određene aktvnost projekta Prema Bajesovoj teorem, ako postoje dva događaja A B njhove verovatnoće, pr čemu je verovatnoća događaja B razlčta od nule važ: P( B / A) P( A) P( A / B) = (2) P( B) Prmenom teoreme na Bajesovsk mrežn model zaključvanja, za čvorove A1, A2,..., A n : n P( A,..., A ) P( A / A,..., A ) = (3) 1 n 1 n = 1 Pretpostavmo da je potrebno zračunat verovatnoću događaja D, koj predstavlja kašnjenje u realzacj aktvnost. Neka je promenljva podugovor označena sa SC ( SC1 - kašnjenje, SC2 - na vreme), dok je promenljva skustvo angažovanh zvršlaca označeno kao SQ ( SQ1 - velko, SQ2 - malo). Ako je d kašnjenje u realzacj aktvnost ( d 1 je D, d 2 je D ), sled da je verovatnoća da će doć do kašnjenja u realzacj aktvnost: n n P( D = d ) = P( D / SC, SQ, D = d ) P( SC ) P( SQ ) = (4) 1 1 = 1 = = Prmena koncepta Bajesovskh mreža za modelovanje nezvesnost pr upravljanju projektma u Pošt Srbje U clju praktčne nterpretacje koncepta Bajesovskh mreža razmotren je projekat mplementacje franšznga u JP PTT saobraćaja Srbja. Projekat sadrž 23 aktvnost. Mrežn djagram projekta sa odgovarajućm vremenma trajanja troškovma realzacje 101
6 aktvnost, t j ( C j ) predstavljen je na Slc 3. Krtčn put projekta je B-H-M-O-P-Q-S-W. S obzrom na to da nema nkakvog značaja za demonstracju Bajesovskh mreža, detaljna specfkacja aktvnost je zostavljena. Pored brojnh komercjalnh alata koj se korste (AgenaRsk [19]), za potrebe ovog rada model Bajesovskh mreža za modelovanje nezvesnost u upravljanju projektma je mplementran u programskom jezku C#. Slka 3. Mrežn djagram projekta sa vremenma trajanja troškovma sprovođenja aktvnost Na osnovu ekspertske procene verovatnoća uspeha realzacje pojednačnh aktvnost projekta formran je Bajesovsk model (Slka 4.). Za svaku aktvnost postoje dva stanja, za svako od njh određene su verovatnoće realzacje. U analzu je uključen samo rzk vremena. Slka 4. Bajesovska mreža kojom su modelovane nezvesnost u vremenu trajanja projektnh aktvnost Rezultat ukazuju na to da će projekat bt zvršen bez kašnjenja sa verovatnoćom od 82%. Takođe, analzom Bajesove mreže, može se uočt da realzacju aktvnost W prat već rzk, samm tm, neophodno je detaljnje razmotrt sve nezvesnost rzke ove aktvnost. Ukolko se analzom obuhvat rzk troškova realzacje projektnh aktvnost na osnovu Bajesovske mreže za tako formulsan problem verovatnoća realzacje projekta na vreme se smanjuje sa 82% na 80.9% (Slka 5.). 102
7 Slka 5. Bajesova mreža kojom su modelovane nezvesnost u vremenu trajanja troškovma projektnh aktvnost Moguće je takođe odredt optmstčko, realno pesmstčko vreme realzacje projekta. Tabela 2. sadrž optmstčke, realne pesmstčke procene za svaku aktvnost na krtčnom putu projekta. Nezvesnost realzacje pojednh aktvnost utče na nezvesnost realzacje projekta u celn. Verovatnoća realzacje aktvnost na krtčnom putu varra u zavsnost od pesmstčkog, realnog l optmstčkog posmatranja nezvesnost (na prmer, P se menja od 72.9%, 78.9% do 66.9% pr čemu rok projekta varra u opsegu 18 do 22 dana. Pesmstčko gledšte je da se projekat okonča za 22 dana, realno je 20 dana, a optmstčko 18 dana. Tabela 2. Procena vremena realzacje projekta prmenom koncepta Bajesovskh mreža Verovatnoća realzacje aktvnost na krtčnom putu [%] Aktvnost Pesmstčka Optmstčka na krtčnom putu Realna procena procena procena B H M O P Q S W Zaključak Razmatranje rzka nezvesnost predstavlja jednu od osnovnh komponenata upravljanja projektma. Većna raspoložvh tehnka za upravljanje rzkom nezvesnošću pr raspoređvanju projektnh aktvnost su događajno orjentsane zasnvaju se na modelovanju mogućh utcaja na performance projekta. U ovom radu predložen je prstup Bajesovskh mreža kojm je moguće uključt rzk, nezvesnost kauzalnost u raspoređvanju projektnh aktvnost. Prstup je testran na prmeru projekta uvođenja franšznga u Pošt Srbje. Reference [1] D. Malcolm, J. Roseboom, C. Clark and W. Fazer, Applcaton of a Technque for Research and Development Program Evaluaton, Operatons Research, vol. 7, pp ,
8 [2] R. Mller, How to Plan and Control wth Pert, Harvard Busness Revew: pp , [3] S. Ward and C. Chapman, Transformng Project Rsk Management nto Project Uncertanty Management, Internatonal Journal of Project Management, vol. 21, pp , [4] R. Atknson, L. Crawford and S. Ward, Fundamental Uncertantes n Projects and the Scope of Project Management, Internatonal Journal of Project Management, vol. 24, pp , [5] V. Khodakaram, N. Fenton and M. Nel, Project Schedulng: Improved Approach to Incorporate Uncertanty Usng Bayesan Networks, Project Management Journal, pp , [6] A. H. Taha, Operaton Research: An Introducton. Prentce-Hall, U.S.A, [7] B. W. Taylor, Introducton to Management Scence. Vrgna Polytechnc Insttute and State Unversty, Prentce Hall, [8] V. Khodakaram, Applyng Bayesan Networks to model Uncertanty n Project Schedulng. PhD dssertaton, Queen Mary, Unversty of London, [9] B. A. Hafızoğlu, Dscrete tme cost trade-off problem n project schedulng. Master Thess, Mddle East Techncal Unversty, Turkey, [10] L. Valadares Tavares, J.A. Antunes Ferrera and J. Slva Coelho On the Optmal M anagement of Project Rsk, European Journal of Operatonal Research, vol. 107, pp , [11] Cook M. S, Real-world Monte Carlo Analyss, Proceedng of PMI Annual Semnars and Symposum, Nashvlle, [12] Pertmaster Project Rsk Software, [13] C. Chapman and S. Ward Estmaton and Evaluaton of Uncertanty: A Mnmalst Frst Pass Approach, Internatonal Journal of Project Management, vol. 18, pp , [14] B. Efron, Bayesans, Frequentsts, and Scentsts, Journal of the Amercan Statstcal Assocaton, vol. 100, pp. 1-5, [15] Goldsten, M. Subjectve Bayesan Analyss: Prncple and Practce, Bayesan analyss, vol. 1, pp , [16] B. McCabe, S. AbouRzk and R. Goebel, Belef networks for constructon performance dagnostcs, Journal of Computng n Cvl Engneerng, vol. 12, pp , [17] R. K. McCann, B. G. Marcot and R. Ells, Bayesan belef networks: applcatons n ecology and natural resource management, Canadan Journal of Forest Research, vol. 36, pp , [18] M. Kragt, A begnners gude to Bayesan network modelng for ntegrated catchment management. Australan Government Department of the Envronment, Water, Hertage and the Arts (DEWHA), [19] Abstract. Project management represents a very complex process because t always contans uncertanty. Basc nputs (tme, costs and requred resources) are not determnstc and they are nfluenced by a broad set of uncertantes. Besdes that, there s a causal relatonshp between these uncertantes and project parameters. In ths paper, a concept of Bayesan Networks has been appled for uncertanty and causalty modelng n project management. Valdty of proposed approach s tested on a project of franchsng mplementaton n PE of PTT Communcatons Srbja. Keywords: project management, uncertanty, Bayesan networks UNCERTAINTY MODELING IN PROJECT SCHEDULING USING BAYESIAN NETWORKS: THE CASE OF POST OF SERBIA Mlan Bozc, Natasa Glsovc, Mlos Mlenkovc 104
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