Mathematical Model for Expediting the Execution of Projects under Uncertainty

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1 Intnational Jounal of Computational Engining & Managmnt, Vol. 4, Octob 20 ISSN (Onlin): Mathmatical Modl fo Expditing th Excution of Pojcts und Unctainty Ashok Mohanty, Biswajit Satpathy 2 and Jibitsh Misha 3 Rad, Dpt. of Mchanical Engg., Collg of Engining & Tchnology, BPUT, Bhubanswa, India amohanty0@yahoo.com 2 Pofsso, Dpt. of Businss Administation, Sambalpu Univsity, Sambalpu, Odisha, India 3 Rad, Dpt. of Comput Scinc & Application, Collg of Engining & Tchnology, BPUT, Bhubanswa, India Abstact Many pojct-basd oganizations manag a numb of simultanous pojcts that sha soucs fom a limitd pool. Conflicting intsts and comptition among pojcts fo limitd soucs is a majo poblms of managing multipl pojcts. Du to unctainty factos th pogss of wok of som pojcts may lag bhind its schdul. So xpditing som slct pojcts is an impotant contol action. Expdition of pojcts and souc allocation a basd on thi pioity lvl. Nomally, pioitization of pojcts is don by infomal and intuitiv mthod. But it is dsiabl to follow a fomal basis fo taking ths dcisions, fo which a simpl mathmatical modl has bn dvlopd and psntd in this pap. Application of modl has bn illustatd with a typical xampl. Kywods: Pojct Expditing, Pojct Pioitization, Monitoing and Contol,. Intoduction Succss of a pojct is dtmind by complting th sam within tim and cost limits and maintaining th quid quality standads. Maintaining th tim-schdul of pojct, facs challng du to two majo asons. i. Somtims allowabl pojct duation is shot than stimatd nomal duation. This may b du to ov optimistic incoct tim stimat o nfocd tim constaints du to makt focs. Excution of pojct in a shot tim fam, calls fo xpditing th wok of pojct to mt th tight tim schdul. ii. Pojcts a dynamic and a caid out in changing nvionmnts und unctainty. Among th factos liabl to chang th xisting plan a: th vision of activitis duation stimats, dlivy failus, changs in tchnical spcifications, tchnical difficultis, unxpctd wath conditions, and labo unst. Du to nvionmntal changs, it is vy difficult to xactly maintain th pojct schduls. So it is ncssay to hav a monitoing sys that gnats fdback fo xpditing th wok whv ncssay. 2. Monitoing, Contol and Expditing of Pojcts Fanian t al. (998) hav obsvd that whil much mphasis is givn to dvlopmnt of tactical and opational plans fo pojct implmntation; hadly any mphasis is givn to dvlopmnt of schduls fo monitoing and contolling pojct pogss. Monitoing is collcting infomation concning th timschdul pfomanc of th pojct. Dcisions lating to xpditing a takn in a dynamic nvionmnt basd on actual pogss data obtaind though monitoing (Mdith t al., 989). Nomally, monitoing of pojcts a hld at gula intvals; howv, oth possibilitis xist. Vaiabl viw piods povid sval altnativs: lss intnsiv monitoing in th aly stags of th pojct and mo viws as th pojct movs towad compltion; mo fqunt monitoing at th bginning and lss aftwad; viw of th pojct upon th compltion of ach activity o majo activitis; o pogss plotting (Schmidt, 988). Som common factos affcting th amount of monitoing in a pojct a th cost of monitoing, total duation of th pojct, avag tim span of th tasks involvd, th dg of compltion of th pojct s goundwok, th ugncy of th pojct, and xposu to dlays du to unfosn cicumstancs (Kupp, 984).

2 Intnational Jounal of Computational Engining & Managmnt, Vol. 4, Octob 20 ISSN (Onlin): Dcision fo xpditing is gnally basd on assssmnt of dlays. Shi t al. (200) hav psntd a mthod fo computing activity dlays and assssing thi contibutions to pojct dlay. Sycamo at l. (999), hav dscibd fou pimay status indicatos fo monitoing pogss of pojcts, namly, (i) Schdul (ii) Budgt (iii) Pcnt Complt and (iv) Quality. Th status infomation collctd on a pojct suggsts poblms quiing coctiv actions. Fo this th tool abstacts th contollabl lmnts into fou basic paamts, namly (i) Rsoucs, (ii) Wok Squnc, (iii) Pojct Scop and (iv) Poductivity. 3. Managing Multipl Pojcts Many pojct-basd oganizations manag a numb of simultanous pojcts that sha soucs fom a limitd pool. In a study of two cass that involvd 30 and 60 simultanous pojcts spctivly, Engwall and Jbant [2003] idntifid th following opational poblms in multi-pojct nvionmnts: i) Th snio manags sponsibl fo potfolio managmnt ( potfolio managmnt lvl ) w ovloadd with poblms. ii) Potfolio managmnt did pioity stting and souc -allocation on almost daily basis. iii) Th was a continuous ongoing gam of ngotiations concning accss to availabl soucs and th allocation of ctain individuals to spcific pojcts. iv) Th managmnt was pimaily ngagd in shot tm poblm solving. v) On pojct had ngativ ffcts on oth pojcts such as dlays and missd dadlins. Whn on pojct had poblms, oth pojcts w affctd dictly. vi) Th was tough comptition btwn pojcts and pojct manags kpt soucs woking on thi pojcts (unncssaily) in od not to los thm. vii) Pioitis of pojcts chang oftn. Th was no claity o guidlins concning pioitization of pojcts. Sval authos (Elonn and Atto, 2003; D Maio t al, 994; Platj t al, 994; Hndiks t al, 995) hav highlightd conflicting intsts and comptition among pojct manags fo limitd soucs as th main poblms of managing multipl pojcts. In od to satisfy th dmands of vy clint, wok is pushd though th sys. Th clint and th manag handling th pojct dmand that thi wok b xpditd on pioity. Pioitis a oftn st in an infomal and intuitiv mann. 4. Intnal Pioitizing Fulfilling th commints is an impotant issu with managing multipl pojcts. Th situation is bad whn pojct fim has accptd too many pojcts. Each custom thinks that fim is woking on his pojct activly and making good pogss on it. But in ality, th is no way th fim can wok on all of thm at th sam tim. So th pojct manags oftn do intnal o hiddn pioitizing. Thy choos to gt to som of th pojcts fist, laving th oths fo lat. In som oth cass, instad of doing intnal pioitizing, th pojct manags labl all wok as ugnt. Th staff is not abl to know which wok is ally ugnt and which on is not. Thy ty to wok on all pojcts at th sam tim. As a sult all th pojcts pogss too slowly. Th is nd fo a sys to dtmin th pioity of pojcts basd on cla guidlins. Th should b diffnt lvls of pioity so that pojcts a xpditd with appopiat statgy and intnsity cosponding to thi pioity lvls. 5. Expditing Statgis Th xpditing of pojcts may b don in many ways. Faiboz t al (993) in thi modl hav considd diffnt xpditing statgis. Summay of ths statgis is givn as und: i. Contol: Making popl wok had and mo fficintly by btt oganizing, clos monitoing and giving incntiv to popl fo high poductivity ii. Mo tim: Woking fo mo tim (without incasing pojct duation) by opating in shifts and ovtim iii. Rsoucs: Exta soucs (popl, quipmnt and matial) may b addd to complt th tasks fast iv. Chang contact: Off-load wok by sub-contacting activitis and changing tms of contact fo xpditious xcution at high cost v. Chang Changing th spcification of wok to spcification: nabl it to b don fast. vi. Abot: Giv up xpditing and lt a pojct ovun its schdul tim. If possibl it may b xpditd lat on to bing it pogss clos to schdul.

3 Intnational Jounal of Computational Engining & Managmnt, Vol. 4, Octob 20 ISSN (Onlin): Effctivnss of xpditing will dpnd upon slction of xpditing statgis and th intnsity with which ths statgis a implmntd. An indx fo spcifying th ffctivnss of xpditing may b dfind as, A facto k, by which th oiginal duation of a pojct is ducd whn th pojct is xpditd. So if oiginal xpctd duation of a pojct is d with standad dviation σ, on xpditing its xpctd duation will bcom with standad dviation k. ( k)d Expditing a pojct will involv cost. A mo ffctiv statgy will nomally involv gat cost. So xpditing should b don only wh it is ncssay and appopiat xpditing statgy should b adoptd basd on cost citia. Th is also a limit to th amount of wok that can b xpditd. Fo this, pioity of individual pojct should b dtmind fo slcting appopiat xpditing statgy. Nomally, this is don by infomal and intuitiv mthod. But it is dsiabl to follow a scintific basis fo taking dcisions lating to xpditing of diffnt pojcts. 6. Mathmatical Modl In a multi-pojcts xcution stup, pojcts aiv at andom intval. Th schdul fo xcution of ths pojcts is ppad basd on pojct paamts such as wok contnt, xpctd duation, quid du dat fo compltion, valu and impotanc of ach pojct. Howv du to unctainty facto, xcution of pojcts dviats fom th schdul and coctiv contol actions a takn. Expditing is a fomost contol action fo binging th pojcts back to schdul. Fo ach pojct, xpctd (schduld) stat tim and maximum allowabl tim limit fo complting th pojct a spcifid. Som amount of magin duation may b allowd to th pojcts in th schduld to account fo unctainty factos and to povid flxibility to th pojct plan. Th amount of magin duation may vay dpnding upon citicality of pojct and oth factos. To fomulat th mathmatical modl fo xpditing th xcution of pojcts, lt us consid th wok pogss of a pojct with passag of tim. Th vaiabls usd in th modl a givn blow. t s = Expctd stat tim of pojct t as = Actual stat tim of pojct d = Expctd duation of pojct σ = Standad dviation in xpctd duation of pojct t m = Maximum allowabl finish tim = Faction of pojct compltd at tim of viw t α = Pobability (assuanc lvl) of complting th pojct within givn tim z = Valu cosponding to pobability p k = Effctivnss indx of xpditing Faction of wok compltd with passag of tim is shown in figu. Figu : Pogss of wok of a pojct shown against passag of tim If a pojct stats at xpctd stat tim t s and taks xpctd duation d, it is compltd at point A. Howv du to unctainty, th pojct may stat lat at tim t as. Th at of pogss may also b slow. Lt at tim t, th pojct has pogssd up to point B. If th pojct pogsss at this at, it may b compltd byond th maximum allowabl limit, as shown by point C. Howv if th pojct is xpditd, th at of wok pogss is impovd. So th pojct may b compltd bfo allowabl tim limit as shown by point D. Th abov figu is basd on xpctd at of wok pogss. But du to unctainty facto, th at of wok pogss may impov vn without xpditing. Howv th pobability of such occunc may b much lss. On th oth hand at of wok pogss may also bcom wos in spit of xpditing. So pobability aspcts should b considd fo taking dcisions about xpditing of pojcts. In an unctain nvionmnt, th assuanc lvl (pobability) of complting a pojct within xpctd duation d is only 0.5. But fo gat assuanc lvl, ith mo tim duation should b allowd to th pojct, o th pojct should b xpditd. As p pobability distibution, minimum tim duation within which a pojct is xpctd to b compltd with assuanc lvl α, is givn by: 2 x 2 d z wh z 2 dx (q. ) Fo a givn assuanc lvl α th valu of z can b calculatd o dictly ad fom nomal distibution tabl. If tim (t m t s ) allowabl to a pojct is mo than

4 Intnational Jounal of Computational Engining & Managmnt, Vol. 4, Octob 20 ISSN (Onlin): (d +zσ), th pojct can b compltd with quid assuanc α without any nd fo xpditing. i.. if (t m t s ) > (d + z σ) No xpditing is quid. Othwis th pojct should b xpditd. Suppos th pojct is xpditd with ffctivnss indx k, thn minimum tim quid to complt th pojct with assuanc lvl α is givn by (d +zσ)( k). So to nsu timly compltion of pojct with assuanc lvl α, th tim availabl fo xcution of pojct (t m t s ) should b mo than (d +zσ)( k). i.. (t m t s ) > (d +zσ)( k) So ffctivnss indx of xpditing k can b dtmind by th quation: ts k (q. 2) d z Basd on valu of k, appopiat xpditing statgy can b dcidd at th stat of th pojct. As th pojct pogsss, th actual pogss of wok may dviat fom th plan du to unctainty facto. So pogss of pojct is viwd piodically and appopiat dcision fo xpditing th pojct can b takn basd on status of pojct at that instanc of tim. Suppos th pojct is viwd at tim t and at that instanc stimatd faction of wok compltd is found to b. This amount of wok has bn don in tim duation (t t as ). As p calculation, xpctd duation of tim ndd to complt faction of wok is.d with standad dviation. So tim quid to complt faction of wok fo assuanc lvl α is givn by, ( d z d. So if actual tim takn (t t as ), is lss than ( z, th pogss of wok may b considd as satisfactoy. To valuat how fast th wok on pojct has pogssd, wok pogss indx η, may b computd as und. d z η (q. 3) t t as So if η > th of pogss of pojct may b considd as satisfactoy And if d z d fast than xpctd. η th of pogss of pojct is Howv it is to b dtmind if maining potion of wok can b compltd with quid assuanc lvl α within th du dat. Th tim ndd to complt maining ( faction of wok fo a givn assuanc lvl α is givn by, ( d z. So if allowabl tim fo complting th maining potion of pojct (t m t), is lss than ( d z, th pojct may nd to b xpditd. Suppos th maining potion of pojct is xpditd with ffctivnss indx k, so that th tim quid fo doing th wok is ducd by a facto ( k). To nsu compltion of pojct at quid assuanc lvl α, this ducd tim should b lss than th maining allowabl tim. i.. [( d z ]( k) ( t) ( t) (q. 4) o k ( d z So basd on abov quation, valu of k can b dtmind and a suitabl statgy fo xpditing th pojct can b slctd accodingly. If maining allowabl tim (t m t), is gat than ( d z, th valu of k bcoms ngativ. This indicats that xpditing is not ncssay. Th modl quis that th "faction of pojct compltd" should b stimatd at ach viw. In many cass, it may b quit difficult to objctivly assss th faction of wok that has bn compltd. In such cass assssmnt can b don subjctivly. Fo xampl, consid wok of witing pogam cod fo a softwa modul. It is difficult to objctivly stimat what faction of pogam cod has bn compltd unlss th coding is fully compltd. But h th pogamm can mak som subjctiv assssmnt of what amount of his wok has bn compltd. Subjctiv assssmnt may not b that accuat, but it is btt than no assssmnt. Th actual tim lapsd in xcution of pojct (t t as ) should also b significant nough fo dawing any maningful conclusion. Whn a pojct is xcutd, th pogss is not visibl immdiatly. Pogss in wok is gnally potd in multipl of som fixd amount, say 5% o 0%. In pactic, if pogss of wok is lss than this amount, it is somtims ignod and potd as zo. So tim lapsd in xcution of pojct should b asonabl nough to daw any maningful conclusion about pogss of wok.

5 Intnational Jounal of Computational Engining & Managmnt, Vol. 4, Octob 20 ISSN (Onlin): Illustation of th Modl To illustat th modl, som pojcts of a fictitious pojct oganization a considd. Suppos it is quid that th pojcts should b compltd within maximum allowabl tim limit at assuanc lvl of 95% (i.. α = 0.95). Fo this quid assuanc lvl α, th valu of z takn fom nomal distibution tabl is.645. So ffctivnss indx of xpditing k, quid fo timly compltion of ach pojct is dtmind by quation.2. Th pojcts with fictitious data a shown in tabl. If fo a pojct th valu of k is ngativ, it indicats that th pojct is not citical and it nd not b xpditd. So fom abov data, only th pojcts P2 and P4 nd to b xpditd. Tabl : Dcision fo xpditing th fictitious at stat of pojct (t=0) Pojct Expctd duation d Standad dviation σ Schduld stat tim ts Maximum allowabl tim Effctivnss indx of xpditing k P P P P P A pojct can b xpditd with vaying intnsity fom vy low to vy high. Fo ou illustation pupos, fictitious valus of ffctivnss indx of diffnt pioity lvls a givn in tabl 2. Pioity lvl S S2 S3 S4 S5 Tabl 2: Hypothtical xpditing statgis Intnsity xpditing Vy low Low Modat High Vy high of Effctivnss indx of xpditing (k) Minimum valu of ffctivnss indx quid fo xpditing pojcts P2 and P4 a and 0.80 spctivly. So S and S2 a th appopiat pioity lvl fo pojcts P2 and P4 spctivly. (s tabl 2). Howv as tim pogsss, som pojcts may b dlayd and som may pogss fast. Suppos th pojcts a viwd at tim t=5, and potion of wok compltd in ach pojct is stimatd. Wok pogss indx and minimum valu of ffctivnss indx of xpditing fo pojcts can thn b dtmind by applying quation-3 and quation-4 spctivly. This is tabulatd in tabl 3. Fom th tabl 3, it is sn that th pojct P which was not citical ali, now nds to b xpditd with statgy having minimum ffctivnss indx But no statgy having such high ffctivnss indx of xpditing is availabl. So pojct can b xpditd with pioity lvl S5 that has th highst ffctivnss indx of 0.5. Howv complting this pojct at 95% assuanc lvl (α=0.95) cannot b nsud. Th pojct P2 which was ali xpditd at vy low intnsity (pioity lvl S), now nds to b xpditd with high intnsity (pioity lvl S4). Tabl 3: Status of pojcts at tim (t = 5) Pojct P P2 P3 P4 P5 Expctd duation d Schduld stat tim t s Maximum allowabl tim t m Minimum valu of k at tim of viw Actual stat tim t as Faction compltd Wok Pogss Indx η Minimum valu of k Whn th pojcts a viwd again say at tim t=0, th sam pocdu is followd to dtmin wok pogss, wok pogss indx and ffctivnss indx of xpditing. Fo illustation pupos hypothtical data is tabulatd in tabl 4. Fom th tabl 4 it is sn that pojct P4 nds to b xpditd with highst possibl intnsity. So xpditing with highst intnsity (pioity lvl S5) should b adoptd fo this pojct. Th pojct P3 should b xpditd with high intnsity (pioity lvl S4). Expditing nd not b don fo pojct P5.

6 Intnational Jounal of Computational Engining & Managmnt, Vol. 4, Octob 20 ISSN (Onlin): Tabl 4: Status of pojcts at tim (t = 0) Pojct P P2 P3 P4 P5 Expctd duation d Schduld stat tim t s Maximum allowabl tim t m Minimum valu of k at tim of viw Actual stat tim t as Faction compltd Wok Pogss Indx η * Minimum valu of k * Indicats that tim lapsd in xcution of pojct (t t as ) is too shot to dtmin η 8. Coction Facto fo Non-Citical Activitis Pioity lvl of pojct is basd xpctd pojct duation and unctainty in duation spcifid by standad dviation in duation. Th pojct duation is th sum of duation of activitis of th citical path. So whn pioity lvl of pojct is spcifid, it is applicabl to th citical activitis and not to all activitis of a pojct. Fo non-citical activitis ncssay coction may b don fo dtmining th pioity lvl. A pojct can hav many paths having diffnt duations. Suppos xpctd duation and standad dviation in duation of a paticula non-citical path is d and σ spctivly. Lt us dfin citicality indx of path c as atio btwn d and d. d d z c d d z Whn d = d, citicality indx of path c = o th path is th citical path. Fo non-citical paths, th valu of citicality indx is lss than. If activitis of citical paths a xpditd with intnsity k thn activitis of noncitical paths may b xpditd with low intnsity k. As p quation 2, th minimum valu of ffctivnss indx of xpditing fo activitis of citical path, k is: ts t k o k d z d z Similaly minimum valu of ffctivnss indx of xpditing fo activitis of non-citical path, k is: ts t k o k d z c ( d z ) So ffctivnss indx of xpditing fo non-citical activitis k may b dtmind by using th quation: k = ( k)/c (q. 5) Sinc c is lss than, th valu of k is lss than k. So th activitis of non-citical paths should b xpditd with lss intnsity than that is quid fo activitis of citical path. If valu of k is lss than zo, th activitis of th path nd not b xpditd ispctiv of pioity lvl of pojct. 9. Conclusion Expditing of pojcts is vy impotant fo minimizing pojct dlays. But whn pojcts a xpditd basd only on intuitiv judgmnt, it somtims sults in incuing xpnditu on xpditing pojcts that a not waantd. Dtmination of objctiv masus such as ffctivnss indx of xpditing is usful in slcting pioity lvl without any subjctiv bias. Whn numbs of pojcts a bing xcutd simultanously, ach pojct is compting with th oth fo utilizing maximum sha of oganization soucs. Dtmination of pioity lvl of pojcts povids a basis fo distibution of soucs among pojcts. It also maks it asi fo th pojct manag to incu xta xpnditu fo xpditing a pojct. Th ida of xpditing can also b xtndd to oth aas such as: o Invntoy Managmnt: fo xpditing supplis fom vndos o Makting Managmnt: fo xpditing shipmnt o of goods and alization of paymnts Poduction planning: fo xpditing jobs at diffnt wok stations Expditing is an intgal pat of contol. It has wid applicability and scop fo futh sach in pojct managmnt and in oth aas. Rfncs [] D Maio A, Vganti R and Coso M A, (994), Multipojct Managmnt Famwok fo Poduct Dvlopmnt, Euopan Jounal of Opational Rsach, 78 pp [2] Elonn S & Atto K A, (2003), Poblms in Managing Intnal Dvlopmnt Pojcts in Multi-pojct Envionmnts, Intnational Jounal of Pojct Managmnt, 2, pp [3] Engwall M and Jbant A, (2003), Th Rsouc Allocation Syndom: th Pim Challng of Multi-Pojct

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