Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATION-BASED OPTIMIZATION Mchael E. Kuhl Radhamés A. Tolentno-Peña Industral & Systems Engneerng Department Rochester Insttute of Technology Rochester, NY 14623 USA ABSTRACT A dynamc smulaton-based crashng method s ntroduced n ths research to evaluate project networks and determne the optmum crashng confguraton that mnmzes the average project cost due to lateness penaltes and crashng costs. Ths dynamc approach wll let the user evaluate the project network to determne a crashng strategy at the begnnng of the project and also durng the lfe of the project. By reevaluatng the project network possble adjustments to the crashng strategy may be dentfed and mplemented. The output of the method ncludes a dstrbuton of the project completon tme, a dstrbuton of the project total cost, and the project cost savngs. 1 INTRODUCTION Project management s a tool that s used by many companes to help mprove performance and compettveness. Projects and ther executon, n general, requre resources. Project management, whch s characterzed by technques ntended to provde a better use of project resources (Kerzner 2003), can postvely mpact the proftablty of a company. An mportant aspect of project management s rsk management. Dfferent types of rsk are present n any gven project, but the emphass of ths research wll be focused on schedule/tme rsk and assocated costs. The schedule/tme rsk essentally mples not completng project actvtes on tme, resultng n a late completon of the project. Late project completon generally has negatve effects for the company such as penalty costs and customer dssatsfacton. If a project s runnng late project managers mght be able to brng the project back on track by ncorporatng addtonal resources (Esner, 2002). In project management, ths method of mtgatng rsk s known as crashng. Rosenau and Gthens (2005) state crashng s spend[ng] more money on the project n order to speed up accomplshment of scheduled actvtes. Snce crashng a project represents addtonal costs, crashng decsons need to be made n a cost-effectve way. When crashng a project the tradeoff between the crashng cost and the penalty cost needs to be evaluated. A typcal scenaro nvolvng a project that has potental for beng completed late (resultng n a penalty), and may beneft from crashng s llustrated n Fgure 1. As crashng of actvtes s mplemented, the total cost of crashng plus the penalty cost may ntally decrease. As the crashng amount s ncreased, dmnshng returns wll be realzed untl a pont where the total cost may begn to ncrease. The objectve s to determne the optmal crashng pont (ndcated by the arrow) where the total cost wll be mnmzed. The crashng method s focused on reducng the tme of the actvtes on the crtcal path. The crtcal path s the one that can cause a delay of the project because there s no slack on that path. The tradtonal method of crashng CPM/PERT networks only consders average actvty tmes for the calculaton of the crtcal path, gnorng the uncertanty related wth the duraton of the actvtes. Consequently, other paths that may have a hgh probablty of becomng crtcal are gnored. As a way to overcome ths ssue, smulaton can be used to model the stochastc nature of the duratons of the actvtes. Incorporatng stochastc duratons n the crashng process allows the generaton of the project completon tme dstrbuton and enables the analyss of the real effect that a specfc crashng confguraton may have on the project. Several smulaton based crashng methods are descrbed n the lterature (Bssr and Dunbar 1999, Haga and Marold 2004, Haga and Marold 2005). These methods are heurstcs that are developed to return satsfactory solutons but not necessarly an optmal soluton. The goal of ths research s to develop a method that allows project managers to make optmal dynamc, data drven crashng decsons that mnmze the average project 978-1-4244-2708-6/08/$25.00 2008 IEEE 2370
cost (the project cost n ths research s the sum of crashng costs and penalty costs). Crash + penalty cost Crashng amount Fgure 1: Relatonshp between crashng amount and the total cost (crash + penalty). 2 RELATED WORK 2.1 Determnng Project Completon Tmes The Crtcal Path Method (CPM) and the Project Evaluaton and Revew Technque (PERT) methods have been used snce the 1950s to estmate the completon tme of a project. CPM s a determnstc approach to calculate the duraton of a project, and PERT s a probablstc approach that enhances CPM by consderng uncertanty n actvty duratons by calculatng the probablty to complete the project by a gven tme (Lee and Ardt 2006). Hller and Leberman (2001) state that even when the orgnal versons of CPM and PERT have some sgnfcant dfferences, wth tme they have been consdered as one technque called CPM/PERT. The PERT method consders the mean and varance of each actvty to descrbe ts duraton and to represent the uncertanty assocated wth t; however, PERT only consders the mean tmes to calculate the crtcal path, gnorng the varances, thus makng a determnstc analyss (Ahuja et al. 1994). The research of Lu and AbouRzk (2000) presents a CPM/PERT smulaton model that ncorporates the dscrete event modelng approach and a smplfed crtcal actvty dentfcaton method. Lee (2005) presents a software tool, SPSS, whch can be used to determne the probablty assocated wth the completon of the project by a target date specfed by the user. Lee and Ardt (2006) descrbe a new smulaton system, S3, whch s an mprovement over SPSS. An advantage of S3 over SPSS s that S3 calculates a confdence nterval for the project mean duraton and also determnes the mnmum number of smulaton runs necessary to have a better estmator of the mean project duraton. Smmons (2002) and Prtsker (1986) also descrbe smulaton models that evaluate project networks. These smulaton models provde a hstogram of the project completon tme dstrbuton, whch can be used to perform rsk analyss. 2.2 Smulaton-based Crashng Methods Bssr and Dunbar (1999) present a method to crash a project network. They suggest the use of smulaton to obtan the average tme of each actvty, the crtcal path, and the near crtcal paths. A near crtcal path n ths model s a path whch length s smaller than the orgnal completon date but t s larger than the target completon date after crashng. After the path nformaton s collected a lnear program s appled to determne the crashng strategy. Haga (1998) along wth Haga and Marold (2004), and Haga and Marold (2005) present a seres of papers nvolvng heurstc crashng methods for project management utlzng smulaton. Haga and Marold (2004), propose a smulaton-based method that deals wth the tme-cost trade-off nvolved wth crashng a project. The authors state that the complete dstrbuton of project completon tme needs to be consdered when crashng. The method that they proposed s a two steps approach. The frst step s to apply the tradtonal PERT method to crash the project, and the second step conssts n testng each actvty that had not been crashed to the upper crashng lmt to determne f crashng that actvty further reduces the average total cost of the project. The authors consdered two sources that can ncrease the cost of the project, whch are crashng costs and overrun costs. Haga and Marold (2005) developed a method to montor and control a project. The output of ths method s a lst of dates at whch the project manager should revew the project to decde f actvtes need to be crashed. These dates are called crashng ponts, and they are determned by a backward run through the project network. The crashng ponts are establshed at the begnnng of the project and they reman fxed durng the entre project lfe. 3 METHODOLOGY The purpose of ths research s to develop a dynamc smulaton-based analyss method capable of evaluatng project networks to answer the followng questons: What actvtes should be crashed n order to mnmze the average project cost? To what extent should the actvtes be crashed? 2371
How often should the project network be reevaluated? The overall procedure s presented n two phases. Phase I consders the evaluaton of the project pror to the start of the project. Ths phase wll produce an optmal crashng strategy (wth respect to nformaton avalable pror to the start of the project) as well as recommendatons for reevaluaton durng the Phase II where the dynamc crashng porton of the method s mplemented. Ths procedure s desgned to evaluate the mpact that crashng each actvty (by nteger tme unts) has on the average project cost. The method s ntended to be robust and produce optmum results for analyzng project networks wth only one domnant crtcal path or multple crtcal paths. The output of the method ncludes a dstrbuton of the project completon tme, a dstrbuton of the project total cost, the actvtes to crash and the extent of the crashng, confdence and tolerance ntervals on the project completon tme, and the tme ponts at whch the project network mght be reevaluated. In the next sectons, Phase I and Phase II of the procedure are presented. Although the methods are desgned to be used together to maxmze the beneft of the method, Phase I can be appled ndependently from Phase II at the start of the project wth Phase II beng optonal. 3.1 Phase I: Optmal Crashng Method Appled Pror to the Start of the Project The objectve of Phase I of the procedure s to obtan the optmal crashng confguraton pror to the start of the project that wll mnmze the total project cost wth respect to crashng costs and penalty costs. Phase I nvolves the followng steps: 1. Construct a smulaton model of the project network. 2. Identfy the potental/feasblty of crashng each actvty n the network and the related costs. 3. Utlze a stochastc smulaton optmzaton tool such as Industral Strength COMPASS (ISC) to determne the optmal project crashng confguraton. 4. Proceed to Phase II or mplement the optmal crashng soluton. The smulaton model s used to determne the project duraton and the addtonal project cost (crash + penalty costs). The frst step n usng the smulaton model s to nput the data that descrbes the project network, whch conssts of the probablty dstrbuton functons (PDF) that represent the actvty duratons, the crashng cost per tme unt for each actvty, the predecessors of each actvty, and the target completon tme. Although actvty tmes could follow any probablty dstrbuton f the approprate parameters that defne the PDF are known, usng the beta dstrbuton to represent actvty tmes s common n the feld of project management and we wll keep wth ths conventon n ths paper. The duraton of each actvty s defned by three estmates consstng of the optmstc, most lkely, and pessmstc duraton tmes; these estmates are used to estmate the parameters of the general beta dstrbuton, from whch the actvty duratons are sampled. Once the nformaton that descrbes the project network s defned n the smulaton model the actvty tmes wll be generated, and the startng and completon tmes of each actvty wll be calculated. The startng tme of each actvty wll be equal to the tme at whch all ts predecessors are completed. The completon tme of each actvty s represented by the followng expresson: ct = st + t x, wherect represents the actvty completon tme, st represents the actvty startng tme, t represents the actvty duraton, and x represents the number of tme unts by whch the actvty s crashed. After calculatng the completon tme for each actvty the project duraton s calculated; the project duraton s equal to the longest actvty completon tme. There s a penalty cost assocated wth a late completon of the project, and for some projects there s an addtonal proft assocated wth early completon. It s necessary to ncorporate n the smulaton model the functons that represent the penalty cost or addtonal proft. In ths phase of the research lnear functons are consdered. Fnally, the total cost s calculated, whch s equal to the crashng cost plus the penalty cost. (For the examples presented n ths paper, the smulaton model s developed usng the C++.) The smulaton model s used to generate the dstrbuton of the project cost and project completon tme when no crashng s appled to the project network; the dstrbuton of project cost s the baselne used to determne the level of rsk assocated wth penalty costs. The smulaton model uses nteger decson varables that represent the number of tme unts by whch an actvty s crashed; a partcular set of values for these nteger decson varables represents a crashng confguraton. The smulaton model nteracts wth an optmzaton engne wth the purpose of determnng the crashng confguraton wth the mnmum average total cost. The optmzaton engne used n ths step of the methodology s Industral Strength COMPASS (ISC) (Xu et al. 2007). ISC s a tool whch s derved from the COMPASS framework developed by Hong and Nelson (2006) for locally convergent, dscrete optmzaton-va-smulaton (DOvS). To utlze ISC the C++ smulaton model s ntegrated nto the ISC code. ISC requres nputs such as an n- 2372
tal soluton, the range of possble values for each decson varable (crashng amounts), and the confdence level desred for the soluton; these nputs must be provded n a separate text fle from whch ISC reads them. (For a complete lst of the nputs requred to use ISC please refer to Xu et al. 2007). ISC searches the feasble regon defned by the potental actvtes that can be crashed and returns an optmal soluton wthn the specfed tolerance. Next we present an example llustratng the Phase I method. 3.2 Phase I: Example The followng project network, s bases on an example presented by Haga (1998), to llustrate Phase I of the crashng method. Table 1 shows the 36 actvtes n the network along wth the precedence relatonshps. The project network s depcted graphcally n Fgure 2. In ths example, the network contans only 1 possble crtcal path whch wll be the focus of ths example. The actvtes on ths crtcal path each have a potental of crashng up to 3 tme unts. The respectve parameters of the actvty tme dstrbutons and unt crashng costs are shown n Table 2. The target completon tme of the project s tme 180, and the equaton defnng the penalty cost for late completon s 0, P = 10( T 180), f T 180 f T > 180, where T s the resultng completon tme of the project. ISC was used to obtan the optmal crashng confguraton whch s to crash actvty 29 three tme unts. The orgnal project wthout crashng and the project wth the optmal crashng confguraton were each smulated 50,000 tmes to produce the dstrbuton of completon tme (Fgure 3) and the cumulatve dstrbuton of the total project cost (Fgure 4). In addton, Table 3 provdes the average and standard devaton of the project duraton and project cost. The optmal crashng confguraton generated by ISC s consstent wth the one presented by Haga (1998). 1 2 4 3 6 24 25 26 16 15 8 18 17 10 27 19 28 30 21 20 29 12 32 22 23 31 33 34 13 35 36 Table 1: Dependency relatonshps for the project network used for the example. Actvty Predecessors Actvty Predecessors 1-19 8, 15 2 1 20 10, 17 3 1 21 10, 17 4 1 22 12, 20 5 2 23 12,20 6 2 24 4 7 5 25 16, 24 8 5 26 18, 25 9 7 27 26 10 7 28 26 11 9 29 19, 27 12 9 30 19, 27 13 11 31 21, 22, 29 14 11 32 28, 30 15 3, 6 33 32 16 3, 6 34 23, 31, 33 17 8, 15 35 13, 34 18 8, 15 36 14, 35 Table 2: Mnmum (a), most lkely (ml), and maxmum (b) duraton and crashng cost for each actvty. Actvty a ml b Crash cost 1 10 20 30 9 4 12 14 16 8 24 14 18 22 4 25 12 18 30 6 26 10 20 30 9 27 8 12 16 9 29 18 25 32 1 31 10 20 30 8 34 15 20 25 9 35 6 12 18 4 5 7 9 11 14 Fgure 2: Project network used for example (based on Haga 1998). 2373
Table 3: Summarzed comparson between no crashng and optmal crashng. Duraton Cost Average Std. Dev. Average Std. Dev. Orgnal 179.997 7.63 30.51 44.88 Optmal 176.997 7.63 20.93 34.54 Probablty 16% 14% 12% 10% 8% 6% 4% 2% 0% 140 160 180 200 220 Duraton orgnal optmal Fgure 3: Project completon tmes comparson Orgnal versus Optmal. Cummulatve % 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 0 100 200 300 400 Cost orgnal optmal Fgure 4: Project costs comparson Orgnal versus Optmal. 3.3 Phase II: Dynamc Crashng In Phase I, pror to the start of the project an ntal optmal crashng confguraton s obtaned by analyzng the entre project network. Ths ntal optmal crashng confguraton consders the uncertanty assocated wth the duraton of all the actvtes of the project. As actvtes are completed the overall uncertanty about the project completon tme s reduced, and as a result the ntal optmal soluton mght change. In order to take nto account the effect that the uncertanty reducton has on the project completon tme and the crashng confguraton, the Phase II, dynamc crashng method, s used. The purpose of the dynamc method s to determne the optmal crashng confguraton for the remanng actvtes. After the project begns, the followng steps make up the dynamc method. We wll assume that the ntal reevaluaton ponts wll be the crashng ponts dentfed n Phase I. 1. As the project progresses, when the frst actvty that requres crashng as dentfed by Phase I s encountered, begn the reevaluaton process. 2. Determne whch actvtes are completed or n process when the reevaluaton pont s reached. For those n process, estmate the remanng processng tme. 3. Reevaluate the remanng project network usng the smulaton model and ISC as descrbed n Phase I. 4. Implement the project network under the new crashng confguraton and contnue untl ether a) the next actvty that requres crashng s encountered and go to step 2; or b) the project s complete. For each teraton of steps 2-4 of the dynamc crashng procedure, a new crashng confguraton for the remander of the project wll be dentfed that takes nto account the sunk actvty tmes and costs assocated wth the actvtes n progress and the actvtes that have been completed. 3.4 Dynamc Crashng Example (Phases I and II) To llustrate the dynamc crashng method the followng project network shown n Fgure 5 s used. The actvty duratons are represented by beta dstrbutons; the mnmum (a), most lkely (b), and maxmum (b) duraton for each actvty, as well as randomly generated crashng cost are shown n Table 4. The penalty for late completon s equal to 40 cost unts per tme unt. The target completon tme s set to 70 tme unts. The project network s evaluated when no crashng s appled (by runnng 10,000 replcatons), and the average cost s 21.21 cost unts wth a varance of 1556.38. The optmal crashng confguraton provded by ISC s that actvty 2 should be crashed two tme unts, and actvty 9 should be crashed one tme unt. In ths case the average cost s 14.39 cost unts, and the varance s 417.87. To llustrate Phase II, the dynamc crashng nvolvng reevaluaton durng the project, we have conducted 10 trals. Each tral represents a sngle realzaton of the project 2374
evaluated wthout crashng, wth Phase I crashng only, and wth the Phase II crashng. The detaled descrpton for Tral 1s as follows. In Tral 1, a realzaton of the project actvty tmes s generated (va smulaton) and applyng the dynamc method algorthm, the frst reevaluaton pont wll be the start tme of actvty 2. By the start tme of actvty 2, actvty 1 s completed, and there are no actvtes n process. The duraton of actvty 1, whch s equal to the start tme of actvty 2, s equal to 10.21 tme unts. The network s reevaluated assumng the duraton of actvty one as beng determnstc (a sunk cost); the new optmal crashng confguraton s that actvtes 2 and 9 should be crashed two tme unts each. Ths new soluton confrms that actvty 2 should be crashed, and suggests that actvty 9 should also be crashed but by two tme unts nstead of by one tme unt as ntally suggested. The new average cost s 14.53 wth a varance of 278.51. After crashng actvty 2, the project proceeds. The next reevaluaton pont s the start tme of actvty 9. Just before the start of actvty 9, actvtes 1 to 4 have been completed and actvty 5 s n process. The network s reevaluated consderng the duraton of actvtes 1 to 5 as determnstc, and crashng actvty 2 two tme unts. The new optmal crashng confguraton ndcates that actvty 9 shouldn t be crashed, nor any other actvty. The average cost s 6 cost unts wth a varance of 0; that cost s the result of crashng actvty 2 twce and resultng n an on-tme project. Smlarly, a total of ten trals of the dynamc method were performed. The results of each tral are shown n Table 5 and Table 6. Over these 10 trals, mplementng Phase I alone provded an average cost savngs of 36% over not crashng at all. The dynamc crashng method provded a cost reducton of 69% over not crashng at all and an addtonal 52% reducton over usng the Phase I crashng method alone. These results demonstrate the types of benefts that can be obtaned when the project network s dynamcally crashed durng the project lfe. 1 3 2 4 5 6 7 8 9 10 11 Fgure 5: Project network used to evaluate the dynamc method. Table 4: Mnmum (a), most lkely (ml), and maxmum (b) duraton and crashng cost for each actvty. Crash Actvty a ml b Cost A1 8 10 12 6 A2 6 10 14 3 A3 6 8 10 5 A4 10 15 20 4 A5 12 17 22 5 A6 3 5 7 8 A7 6 9 12 5 A8 4 6 8 5 A9 11 13 15 2 A10 13 15 17 8 A11 5 7 9 7 Table 5: Summary of results of the dynamc method mplementaton. Project Cost Tral No Crashng Statc Crashng Dynamc Crashng 1 0.0 8.0 6.0 2 18.2 8.0 7.0 3 0.0 8.0 0.0 4 82.0 50.0 43.1 5 61.8 29.8 17.0 6 0.0 8.0 6.0 7 85.6 53.6 17.0 8 45.9 8.0 9.0 9 104.8 72.8 12.0 10 0.0 8.0 6.0 Average 39.8 25.4 12.3 Table 6: Actvtes crashed n each tral of the dynamc method. Crashng Amount Tral A2 A9 A11 1 2 0 0 2 1 2 0 3 0 0 0 4 2 2 0 5 2 2 1 6 2 0 0 7 2 2 1 8 1 0 0 9 2 3 0 10 2 0 0 2375
4 CONCLUSION We have presented a smulaton-based methodology to evaluate project networks and determne an optmal crashng strategy. The methodology has two phases: Phase I, crashng appled pror to the start of the project, and Phase II, dynamc crashng appled durng the project lfe to update the crashng strategy. Applyng Phase I to a project network reduces the average cost, and n certan cases the acheved average cost reducton mght be enough for the decson makers; however, when Phase II s appled all the uncertanty that has been elmnated s taken nto account to produce an updated crashng strategy, whch generally yelds lowest project costs. These methods utlze a proven stochastc optmzaton procedure that provdes asymptotcally optmal results whch provdes a sgnfcant contrbuton to the lterature that currently conssts prmarly of heurstc methods. The future research efforts wll be focused on conductng a rgorous expermental performance evaluaton, nvestgatng alternatve methods for determnng reevaluaton ponts for Phase II, and nvestgatng the scalablty of the computatonal methods for large project networks. In addton, we ntend to generalze the approach to nclude alternatve probablty dstrbutons for crashed actvty tmes as opposed to the standard assumpton n the lterature of nteger reductons n actvty tmes for crashed actvtes. REFERENCES Ahuja, H. N., S. P. Dozz, and S. M. AbouRsk. 1994. Project Management Technques n Plannng and Controllng Constructon Projects. 2nd Ed., Wley, New York. Bssr, Y., and S. Dunbar. 1999. Resource allocaton model for a fast-tracked project. Internatonal Conference on Intellgent Processng and Manufacturng of Materals, 635-640. Esner, H. 2002. Essentals of Project and Systems Engneerng Management. 2nd Ed., Wley, New York. Haga, W. A. 1998. Crashng PERT networks. Ph.D. Dssertaton, Unversty of Northern Colorado, Colorado. Haga, W. A., and K. A. Marold. 2004. A smulaton approach to the PERT CPM tme-cost trade-off problem. Project Management Journal, 35(2): 31-37. Haga, W. A., and K. A. Marold. 2005. Montorng and control of PERT networks. The Busness Revew, 3(2): 240-245. Hller, F. S., and G. J. Leberman. 2001. Introducton to Operatons Research. 7th Ed., McGraw-Hll, New York. Kerzner, H. 2003. Project management: a systems approach to plannng, schedulng, and controllng. 8th Ed., Wley, Hoboken, NJ. Lee, D. 2005. Probablty of project completon usng stochastc project schedulng smulaton. Journal of Constructon Engneerng and Management, 131(3): 310-318. Lee, D., and D. Ardt. 2006. Automated statstcal analyss n stochastc project schedulng smulaton. Journal of Constructon Engneerng and Management, 132(3): 268-277. Lu, M., and S. M. AbouRzk. 2000. Smplfed CPM/PERT smulaton model. Journal of Constructon Engneerng and Management, 126(3): 219-226. Nelson, L.J. and B. L. Nelson. 2006. Dscrete optmzaton va smulaton usng COMPASS. Operatons Research, 54:115-129. Prtsker, A. A. B. 1986. Introducton to Smulaton and SLAM II. 3rd Ed., Wley & Sons, Inc, New York. Rosenau, M. D. and G. D. Gthens. 2005. Successful Project Management : a Step-by-step Approach wth Practcal Examples. 4th Ed., Wley, Hoboken, N.J. Smmons, L. F. 2002. Project management - crtcal path method (CPM) and PERT smulated wth Process Model. Proceedngs of the 2002 Wnter Smulaton Conference, Dec 8-11 2002: 1786-1788. Xu, J., B. L. Nelson, and L. J. Hong. 2007. Industral Strength COMPASS: A Comprehensve Algorthm and Software for Optmzaton va Smulaton. Webste <http://users.ems.northwestern.edu/ ~nelsonb/isc/>. AUTHOR BIOGRAPHIES MICHAEL E. KUHL s an Assocate Professor n the Industral and Systems Engneerng Department at Rochester Insttute of Technology. He has a Ph.D. n Industral Engneerng from North Carolna State Unversty (1997). Hs research nterests nclude smulaton modelng and analyss wth applcaton to nput modelng, healthcare, project management, and semconductor manufacturng. He served as Proceedngs Edtor for the 2005 Wnter Smulaton Conference. He s currently presdent of the INFORMS Smulaton Socety, and a member of IIE and ASEE. Hs e-mal address s <Mchael.Kuhl@rt.edu> and hs web address s <people.rt.edu/mekee>. RADHAMÉS A. TOLENTINO-PEÑA s a Master of Scence canddate n Industral Engneerng n the Industral and Systems Engneerng Department at Rochester Insttute of Technology. Hs research nterests nclude the applcaton of smulaton and operatons research methods to the areas of project management and logstcs. He s a member of IIE, APICS, and SHPE. Hs e-mal address s <radhames.tolentno@mal.rt.edu>. 2376