Allocating Time and Resources in Project Management Under Uncertainty

Transcription

1 Proceedngs of the 36th Hawa Internatonal Conference on System Scences - 23 Allocatng Tme and Resources n Project Management Under Uncertanty Mark A. Turnqust School of Cvl and Envronmental Eng. Cornell Unversty Hollster Hall Ithaca, NY 4853 Emal: mat4@cornell.edu Lnda K. Nozck School of Cvl and Envronmental Eng. Cornell Unversty Hollster Hall Ithaca, NY 4853 emal: lkn3@cornell.edu Abstract We defne and develop a soluton approach for plannng, schedulng and managng project efforts where there s sgnfcant uncertanty n the duraton, resource requrements and outcomes of ndvdual tasks. Our approach yelds a nonlnear optmzaton model for allocaton of resources and avalable tme to tasks. Ths formulaton represents a sgnfcantly dfferent vew of project plannng from the one mpled by tradtonal project schedulng, and focuses attenton on mportant resource allocaton decsons faced by project managers. The model can be used to maxmze any of several possble performance measures for the project as a whole. We nclude a small computatonal example that focuses on maxmzng the probablty of successful completon of a project whose tasks have uncertan outcomes. The resource allocaton problem formulated here has mportance and drect applcaton to the management of a wde varety of project-structured efforts where there s sgnfcant uncertanty. I. Introducton A wde varety of engneerng and busness actvtes are structured as projects: they have tasks, they requre resources of varous types, and they are constraned n both tme and budget. Many types of projects are also subject to consderable uncertanty uncertanty n tme to complete specfc tasks, n the resource requrements of those tasks, and n whether or not the effort wll produce an outcome judged to be successful. Programs often have many such projects, and the program managers face crtcal decsons about what projects to pursue, how much tme and money to nvest n each one, and how to reach decsons to termnate ndvdual projects (or parts of projects) f they do not seem promsng. The focus of ths paper s on defnng and developng a soluton approach for plannng, schedulng, and managng project efforts where there s sgnfcant uncertanty n the duraton, resource requrements, and outcomes of ndvdual tasks. We also focus on specfc ways of defnng relatonshps between resources and/or tme allocated to ndvdual tasks and ther probablty of successful completon. Ths provdes a structure for the analyss of the core uncertantes n the system, and a means for makng effectve resource allocaton decsons wthn an uncertan project management framework. 2. Perspectves on Uncertanty n Project Management and Project Schedulng The management-orented lterature contans consderable evdence of concern wth uncertanty and rsk n managng projects (see, for example, the recent artcles by Haque and Pawar, 2; MacCormack, 2; Raz and Mchael, 2; DeMeyer, et al., 22; and Elkngton and Smallman, 22). Much of ths lterature s very qualtatve, focusng on the process of managng to amelorate rsks. On the other hand, the operatons research lterature s replete wth artcles on project schedulng, but very few of these artcles deal wth uncertanty. We beleve ths dsconnect s the result of dfferent collectons of researchers defnng problems at dfferent levels of a herarchy n a way that obscures ther relatonshp to one another. Fgure portrays a nested vew of the problem that hghlghts the potental connectons between project schedulng and the larger ssues of resource allocaton and rsk management. The nnermost box represents the schedulng problem, where t s assumed that the avalable resources are fxed and specfed, and the characterstcs of ndvdual tasks (duraton and resource use) are gven. Wth those /3 $7. (C) 23 IEEE

2 Proceedngs of the 36th Hawa Internatonal Conference on System Scences - 23 nputs specfed, the schedulng problem s to determne a lkely operatonal schedule for the tasks that fts wthn the avalable resources. The project schedulng lterature focuses on ths nner problem and generally does not deal wth uncertanty. Determne what resources to make avalable, wthn overall budget lmts Change resource allocaton to adjust task characterstcs Schedule tasks, gven resources and task characterstcs Fgure. A perspectve on problem levels wthn project management. PERT, developed n the 95 s, represented the frst consderaton of uncertanty n project schedulng, focusng on uncertan task duratons. Ths technque allowed an estmate of the overall duraton of a project to be constructed. However, PERT has major weaknesses. It does not consder constrants on avalable resources and t assumes that all tasks wll be completed successfully. Usng a PERT framework, Valadares Tavares, et al. (998) also consder uncertanty n the resource requrements of ndvdual tasks and the resultng effect on overall project cost, but they do not ncorporate resource constrants. There s an enormous lterature on resourceconstraned project schedulng, but very lttle of that lterature ncludes consderaton of uncertanty. Hapke and Slownsk (993; 996), Yeh, et al. (999) and Wlls, et al. (999) have proposed schedulng methods based on fuzzy number representatons of task duratons. Mor and Tseng (997) consder projects n whch task outcomes are uncertan. At the completon of a task (wth a fxed duraton and resource requrements), an evaluaton s made of whether or not the result of the task s acceptable (a success or a falure ). If the outcome s falure, the task s re-attempted, and another evaluaton s performed. Ths cycle repeats untl success s acheved. Thereby, a probablty densty functon for the completon tme of the task s developed. Schedulng, however, s only one part of the overall problem of project management under uncertanty. From the perspectve of a project manager, t s lkely that adjustng resource allocaton across tasks can change the parameters of the ndvdual dstrbutons. For example, ncreasng the resources allocated to a partcular task s lkely to reduce both the mean and varance of ts duraton. Ths reallocaton of resources among tasks s a tool that can be used to ncrease the lkelhood of successful completon of the project wthn some avalable tme wndow, and wthn the lmts of resource avalablty. Thus, the mddle box n Fgure represents ths larger problem of resource allocaton that s of greater nterest to the project manager than just the schedulng problem n the nner box. A program manager, who may have responsblty for a collecton of projects, s concerned about a stll-larger problem determnng what resources need to be made avalable or acqured to support the whole collecton of project(s). He/she has an overall budget constrant, but can use that budget to acqure more of resource and less of resource 2, for example, n an effort to gve ndvdual project managers more opportunty to successfully complete projects. Ths s represented by the outer box n Fgure. These resource allocaton ssues have been treated by a few prevous authors (e.g., Repennng, 2; Dcknson, et al., 2), but n a way that s solated from the nner problems. A prmary objectve of the work descrbed n ths paper s to begn makng the connecton between the varous levels of nterest represented n Fgure, so that more effectve and useful tools can be created. To accomplsh that objectve, we focus on an analytc formulaton that relates resource allocaton to the characterstcs of task probablty dstrbutons, establshng the connecton between the nner and mddle levels of Fgure. Ths formulaton s drectly extendable to consder the determnaton of overall resource levels, provdng the connecton to the outer level of Fgure. Ths provdes a more sold bass for overall consderaton of rsk management wthn the project(s). 3. Model Formulaton It s often mportant to reflect the fact that a task may not complete successfully, partcularly n development envronments. Ideas that have hgh ntal promse may not work out; tests or experments may result n falure, etc. Thus, t s also common n ths knd of project envronment to pursue multple paths to achevng a goal. If one doesn t work out successfully, an alternatve approach may, allowng the overall effort to stll succeed. The possblty that parallel tasks (or sequences of tasks) can be pursued smultaneously, and successful completon of any one path allows proceedng wth a subsequent porton of the project, s not normally consdered n project schedulng networks. We assume that alternate paths to project success may exst, and we are concerned wth /3 $7. (C) 23 IEEE 2

3 Proceedngs of the 36th Hawa Internatonal Conference on System Scences - 23 plannng resource allocatons n that type of envronment. It s also mportant to recognze that a varety of dfferent performance crtera for the project may be relevant. Much of the project schedulng lterature s focused on mnmzng ether makespan or dscounted project costs, but other crtera may also be mportant. When sgnfcant uncertanty exsts n the task outcomes, for example, we may be focused on allocatng resources to tasks so as to maxmze the probablty of success for the project as a whole. We wll focus partcular attenton on the probablty of success measure, but most of the model structure apples equally as well to other performance metrcs. By focusng attenton on the probablty of success for the project, we are able to address two questons of partcular nterest:. How should avalable resources be allocated across multple means of accomplshng a specfc requrement, n order to maxmze the overall probablty of success? 2. If changes n allocated resources can change the probablty dstrbuton of tme to successful completon of a task, how should a lmted budget be allocated to maxmze overall probablty of successful completon of a project wthn a gven amount of tme? We defne an allowable duraton, d, for task, to denote the length of the tme wndow that wll be created for completon of the task. If the dstrbuton of duraton for task has a densty functon denoted by f (t), and a cumulatve dstrbuton functon denoted by F (t), then the probablty of successful completon of task wthn ts allowed wndow s F (d ). Qute clearly, the probablty of successful completon of the task s a non-decreasng functon of d. We also defne a resource multpler, c, that denotes a factor to be appled to all resources used by task. The nomnal value of ths multpler s.; ncreases n resources appled to the task are denoted by c >., and decreases by c <.. The effect of changng the resource multpler for a task s to shft ts duraton dstrbuton, as llustrated n Fgure 2. Increasng the value of c shfts the dstrbuton to the left (n general, decreasng both mean and varance), and decreasng c shfts the dstrbuton to the rght (n general, ncreasng both mean and varance). In general, we can summarze the probablty of successful task completon as F (d,c ) to emphasze ts dependence on both the allowable tme wndow and the resource multpler. Prob. Densty Fgure 2. Illustraton of changng task duraton dstrbuton as resource multpler s changed Tm e Perods C =. C =.75 C =.25 d = 23 In Fgure 2, the vertcal lne at d = 23 serves as an example of the F (d,c ) calculaton. For each value of c, F (d,c ) s the ntegral of the probablty densty functon up to d. Fgure 2 provdes a vsual ndcaton of how F (d,c ) ncreases wth ncreasng c, and t should also be clear that for a gven value of c, ncreasng d (sldng the vertcal lne to the rght) ncreases the value of F (d,c ). We are assertng n ths formulaton a sngle c value for each task, to be appled to all resources used by that task. Ths s equvalent to assumng a proportonal use of resources n combnaton (for example, more people requre more materals, more computer resources and more budget). Ths formulaton precludes substtuton of resources on a task (.e., acheve the same overall task duraton dstrbuton by applyng more computer resources and fewer people, for example). Substtuton possbltes could be ncorporated by defnng the resource multplers as c k, wth separate multplers for each resource on each task. Ths s a very plausble extenson to the formulaton, but for the present we wll restrct our attenton to the proportonal model wth a sngle c value for each task. To mplement the concept of usng d and c as key decson varables, we must be able to express F (d,c ) n terms of d and c. One very useful way to do ths s to use a shfted Webull dstrbuton (sometmes called the three-parameter Webull), whch s characterzed by the shft (mnmum possble task duraton), d ; a scale parameter, α ; and a shape parameter, β. The Webull dstrbuton s a very flexble dstrbutonal form, for whch several other popular dstrbutons are specal cases (for specfc choces of α and β ). It s wdely used n relablty studes. For gven values of d, α and β we can wrte F (d ) as: /3 $7. (C) 23 IEEE 3

4 Proceedngs of the 36th Hawa Internatonal Conference on System Scences - 23 ( β ) d d α α β F( d ) = e d > d ;, > () We wll treat the shape parameter, β as a characterstc of the task,, but allow adjustment of the scale parameter, α, as a functon of resource multpler, c. Ths allows the effect depcted n Fgure, where the task tme dstrbuton shrnks or expands as the resource multpler changes. The specfc form of ths relatonshp s assumed to be: α = α c c > ; ε (2) ε α represents the nomnal scale factor for task (.e., to characterze the dstrbuton of tme to successful completon when c = ). ε can be thought of as an elastcty (n the sense that term s used by economsts) that s, the percentage change n α that results from a % change n c. The negatve sgn on ε mples that an ncrease n resources produces a decrease n the scale parameter (compressng the dstrbuton). Usng the Webull dstrbuton wth the resource multpler affectng only the scale parameter mples that the coeffcent of varaton n the task tme dstrbuton remans constant as c changes. That s, the mean and standard devaton of the dstrbuton change n the same proporton. Incorporatng (2) nto () allows us to wrte an expresson for F (d,c ) that depends on four basc nput parameters for each task : d, α, ε and β. β ( d d) ε αc αβ Fd () = e d> d ;,, c> ; ε (3) The overall probablty of success for the project wll be some functon of the collecton of F (d,c ) values for all tasks n the project (or collecton of related projects). We wll denote ths functon as Z(F), where F represents the set of all F (d,c ) values. We wll assume that the objectve of the project plannng exercse s to determne d and c values that maxmze Z(F), subject to constrants on resource avalablty, overall duraton of the project, and precedence requrements among the tasks. The exact form of Z(F) depends on the specfcs of a partcular applcaton, and what consttutes success n that context. To represent the constrants, we must also determne the planned start tmes for the tasks, whch we wll denote by s. We wll assume that the project (or collecton of projects) has a total of N tasks, and we wll arbtrarly denote task N as a completon task wth zero duraton and resource requrements. Then, f the entre effort has an avalable tme frame of T tme unts, we can specfy a completon constrant: s N T (4) Precedence constrants between tasks and j ( s a predecessor of j) can be specfed as: s s + d (5) j We denote the total requrement for resource k (k =,, K ) by task (over ts duraton) as c A k. Ths reflects both a nomnal requrement, A k, and the effect of the resource multpler, c. If the allowable duraton for task s specfed as d, then we wll assume that the resource consumpton of resource k occurs at a unform rate, c A k /d, over the duraton of the task. We can then summarze the consumpton rate of resource k by task as a functon of tme, denoted as r k (t): c Ak / d f s t s + d rk ( t) = (6) otherwse Equaton (6) s a convenent way to express the resource consumpton rates n terms of the core decson varables, d and c. An ncrease (decrease) n the resource multpler causes a correspondng ncrease (decrease) n the usage rate (as well as the total resource requrement) of all resources used by task. Conversely, the resource usage rate for task can be reduced (wthout changng ts total resource requrement) by ncreasng the duraton, d. Increasng the task s allowable duraton stretches out the resource requrement over more tme, reducng the usage rate. The tme wndow over whch the resources are requred for task s shfted by changng s. Thus, the three key varables for each task are used n conjuncton to adjust the resource consumpton and allow the overall project schedule to adjust to resource constrants (subject, of course, to constrants (4) and (5)). From a computatonal standpont, (6) s problematc because t s dscontnuous and the decson varables s and d appear n the condtonng statement that swtches the rate between and a postve value. Our goal s to create a formulaton wth contnuous varables, so that we are not lmted by any arbtrary dscrete defntons of tme perods. Ths has led us to construct a representaton of r k (t) that s a contnuous functon of s, c and d, and that /3 $7. (C) 23 IEEE 4

5 Proceedngs of the 36th Hawa Internatonal Conference on System Scences - 23 approxmates the step-functon n (6). One way of dong that s wth the functon shown n (7): t s t s t s d t s d wd wd wd wd ca k e e e e (7) rk () t = 2d t s t s t s d t s d wd wd wd wd e + e e + e where w s a constant. The functon specfed n (7) has a value that s approxmately zero, except for the range s t s + d, where t rses to approxmately the value c A k /d. It s contnuous (and contnuously dfferentable) over the entre real lne. The sze of the scalng constant w determnes the sharpness of the rse and fall of the functon. It also has a closedform ntegral, whch s very convenent for wrtng the constrants on total resource consumpton. The functon (7) s llustrated n Fgure 3 (for a case n whch s = and d = ). At the begnnng and end of the task, the functon changes from 2% of the nomnal usage rate (c A k /d ) to 98% of that rate over a tme nterval of approxmately 4w. In the llustraton shown n Fgure 3, the value of w has been set to.25. Thus, by settng w to a small value (e.g., a small fracton of a day), the functon (7) rses and falls qute abruptly, and approxmates the stepfuncton (6) qute closely. In realty, the step functon may be smply a convenence, and the real resource usage rate may rse and fall more gradually at the begnnng and end of a task. In ths case, w can be set to a larger value to reflect that buld-up and close-out effect. Fracton of Nomnal Rate Fgure 3. Illustraton of the contnuous representaton of resource usage rate for a task, as a fracton of the nomnal rate (c A k/d ). The avalablty of resource k s defned over a set of M k contguous tme ntervals, and we wll defne mk as the start tme of the m th nterval for resource k, where m =, 2,,M k +. The start tme of the (M k +) th nterval defnes the end of the m th nterval. R mk defnes the amount of resource k avalable n nterval m, where m =, 2,,M k. Then the resource constrants can be wrtten by ntegratng the usage rates: Tme N τ m+, k r ( t) dt R k =,..., K; m=,..., M k mk k = t= τmk (8) If we use the representaton n (7) for r k (t), the ntegrals n (8) can be wrtten as: τ m, k v v2 ( e + )( e + ) 3 4 ( e + )( e + ) + ca kw rk () t dt = ln (9) v v 2 mk τ where: 2 ( τ m+, k s ) v = wd () 2( mk s d) v2 = τ wd () 2( mk s ) v3 = τ wd (2) 2 ( τ m+, k s d) v4 = wd (3) The non-lnear optmzaton problem s then to determne c, d, and s for =,, N so as to maxmze Z(F), subject to (4), (5) and (8), wth (7) and (9)-(3) used to calculate (8). Ths non-lnear optmzaton s complcated because the constrants expressed n (8) are nonconvex, and we also have no assurances that the objectve functon Z(F) s concave n the decson varables (at least not wthout further specfcaton of that functon). These dffcultes mply that there may be multple local maxma for the optmzaton. It s clear from (3) that allowng more tme to complete a task always ncreases ts probablty of successful completon, so we can be sure that constrants (4) and (5) wll always be bndng n an optmal soluton. However, the same asserton cannot be made wth respect to the resource constrants (8), and t s n those constrants that the non-convextes appear. We have compled some prelmnary computatonal experence wth the model, usng the llustratve problem descrbed n the next secton. However, there remans sgnfcant work to do n explorng the computatonal characterstcs of the problem. Parametrc studes that vary the overall tme constrant, T, and trace out changes n the resultng probablty of project success, are hghly useful. Ths /3 $7. (C) 23 IEEE 5

6 Proceedngs of the 36th Hawa Internatonal Conference on System Scences - 23 s also dscussed (n the context of a specfc example) n the followng secton. Ths formulaton s easly extended to represent the outer level of Fg. by treatng R mk as a varable to be determned endogenously by the model, and addng an overall budget constrant wth unt costs for the varous resources. 4. An Illustratve Example As an llustraton of the modelng approach outlned n the prevous secton, consder the small example project network shown n Fgure 4. Task represents buldng a physcal prototype of a new product to test specfc functons. Tasks 2 and 3 represent steps n an alternatve means of evaluatng the functonalty, usng computer smulaton. Successful completon of ether task or tasks 2 and 3 can lead to a prelmnary desgn revew (task 4), and ths or structure s ndcated by the use of the vertcal lne n front of task 4. Followng that desgn revew, parallel efforts on product testng (task 5) and manufacturng analyss (task 6) can proceed, and both of those must be completed successfully pror to a second desgn revew (task 7), whch sgnfes the end of the project. For analyss of ths example project, we focus on the probablty of successfully reachng the second desgn revew (task 7) n one fscal quarter (63 workng days), and our formulaton of the objectve functon s Z(F) = (F + F 2 F 3 - F F 2 F 3 )F 4 F 5 F 6. The dependence of each F (d, c ) term on d and c has been suppressed to smplfy the notaton Fgure 4. Small project network for llustratve analyss. Each task requres both people and money; and those resources are avalable n constraned amounts over varyng perods. The total number of persondays avalable s specfed by month over a threemonth (one fscal quarter) plannng horzon, and the total dollars avalable for non-labor drect costs s specfed over the fscal quarter. For the calculatons shown here, we have assumed the personnel avalablty s 47 person-days n each month (derved from 2 days per month for 7 people), and the budget avalablty s $25, over the quarter. There are three key decsons n ths example problem: ) How should resources be allocated between the alternatve means of reachng the frst desgn revew? Should we focus on Task, on the combnaton of Tasks 2 and 3, or dvde our resources across the two strateges? 2) When should the frst desgn revew (Task 4) be scheduled? How should we dvde the avalable tme between the actvtes leadng up to that revew and the actvtes followng t? 3) How should we allocate resources between Tasks 5 and 6 to best nsure that both are successful wthn the allotted tme between the frst desgn revew and the second desgn revew? As we look at the solutons for ths example, we want to focus our attenton on those three key decsons. Table summarzes the nput data for the seven tasks. The mnmum duraton s the value of d for each task. The mean and standard devaton of the tme to complete each tasks successfully are the bass for specfyng the two parameters of the Webull dstrbuton for each task; gven those two values, we can solve for α and β for each task. The resultng parameter values are shown n Table 2. No dstrbuton parameters are estmated for tasks 4 and 7 because the standard devatons of both tasks are specfed n the nput data as zero. The elastcty values (ε ) n Table defne the percentage reducton n the scale parameter ( α ) of the dstrbuton of tme to successful completon for each task, resultng from a one percent ncrease n resources appled to the task. The two columns labeled Nomnal Person-Days and Nomnal Budget specfy the A k values for the two resources for each task. No resource requrements or elastcty are shown for task 7 (fnal desgn revew) because we are usng that task as the desgnaton of project completon. Project success (at least for ths smple example) s defned as the ablty to reach that desgn revew wthn the tmeframe of 63 workng days /3 $7. (C) 23 IEEE 6

7 Proceedngs of the 36th Hawa Internatonal Conference on System Scences - 23 Task Mnmum Duraton (Days) Table. Input data for example tasks. Mean Tme to Complete (Days) Std. Dev. of Completon Tme (Days) Nomnal Person- Days Requred Nomnal Budget Requred ($) Elastcty of Dstrbuton to Resources Table 2. Estmated Webull dstrbuton parameters for the tasks. Task Scale Parameter α Shape Parameter β N/A N/A N/A N/A For ths small example, the nonlnear optmzaton can be performed usng Excel s Solver, whch uses a generalzed reduced gradent algorthm (Fylstra, et al., 998). The optmzed results for task start tmes, allowable duratons, and resource multplers, as well as the resultng probabltes for successful completon of each task, are shown n Table 3. Ths set of values results n a probablty of success for the project as a whole of.95. We notce frst that the resource multpler for task s very small, and the multplers for tasks 2 and 3 are relatvely large. Ths mples that n the frst porton of the project, resources are beng allocated to the smulaton opton for reachng the frst desgn revew, and the physcal prototype opton s beng gnored. The value of. for the resource multpler s actually a lower bound that was set n the computaton of a soluton, to avod potental numercal problems n evaluatng expresson (3), so we can nterpret the soluton as reducng the resources to task as much as possble. The resource multplers for tasks 2 and 3 are relatvely dfferent, as a result of the dfferng elastctes for those two tasks. Task 2 has a much larger multpler because t has a larger elastcty, and the model takes advantage of the ablty to use resources to compress ts dstrbuton, allowng more tme for task 3. Table 3. Results of optmzaton for example project. Task Start Tme, s (Day) Allowable Duraton, d (Days) Resource Multpler, c Resultng Probablty of Successful Completon N/A N/A N/A Fgure 5 shows the effect that the large resource multpler on task 2 has on the dstrbuton of tme to successful completon. The mean and standard devaton nputs for ths task are such that the resultng Webull dstrbuton reduces to the specal case of an exponental dstrbuton (β 2 = ), wth a mnmum possble value of 5. Under nomnal condtons (.e., resource multpler = ), the tme to complete ths task has very large varance. However, the elastcty of the scale parameter wth respect to resource changes s large (-.), so the addton of resources to ths task compresses the dstrbuton very sgnfcantly. At the allowed duraton for ths task (9. days), the probablty of successful completon s.988. The effect of the combnaton of c 2 and d 2 (contnung to focus on task 2) on total resource use s also worth notng. Fgure 6 shows the rate of consumpton of the personnel resource for task 2 Prob. Densty Days Nomnal Optmzed Fgure 5. Result of applyng addtonal resources to task 2 (dstrbuton of tme to successful completon). under the optmzed plan and two alternatve possbltes. In the optmzed plan, the resource multpler s 5.34 and the duraton s 9. days. Ths mples a usage rate for resources as ndcated by the Optmzed curve n Fgure 6. The ntegral under ths curve s 5.34*2 person-days, ndcatng that the total resource usage s 5.34 tmes the nomnal value (2 person-days, as ndcated n Table ) for ths task. Alternatvely, f we dd not ncrease the resource usage rate and kept the same allowable duraton (9. days), the resource usage would be as ndcated by the Nomnal Resources curve. The ntegral under ths curve s 2 person-days, the nomnal requrement for the task. Ths s clearly a lower overall consumpton of resources, but at the nomnal resource rate the probablty dstrbuton of tme to successful completon of the task s the Nomnal dstrbuton shown n Fgure 5, so n 9. days the probablty of success for ths task s only.56, as compared to.988 under the optmzed plan. We could ncrease the probablty of success under the nomnal resource allocaton by extendng the allowable tme for the task. Wth the nomnal tmeto-success dstrbuton shown n Fgure 5, we would have to allow 27. days for ths task at the nomnal resource allocaton to acheve a probablty of success /3 $7. (C) 23 IEEE 7

8 Proceedngs of the 36th Hawa Internatonal Conference on System Scences - 23 equal to.988. Ths would mply resource usage at a much lower rate over a much longer tme, as shown by the Extended Duraton curve n Fgure 6. The ntegral under ths curve s also 2 person-days, ndcatng that the total resource usage s unchanged. In the scheme of the entre project, however, t s not practcal to allow 27. days for task 2, so the optmzaton uses avalable resources more ntensvely to acheve a hgh probablty of success n task 2 n a much shorter tme. Planned Personnel Use Rate Days from Start of Task Optmzed Nomnal Resources Extended Duraton Fgure 6. Resource usage rate curves for task 2 under three alternatve plans. The usage rate mpled by the optmzed schedule would requre approxmately 2 people over the 9.-day duraton of the task, and ths may not be truly feasble snce we sad at the outset of the example that the person-days of avalablty were based on a staff of seven. The resource constrants (8) constran only the aggregate person-days of use wthn a specfed perod they do not constran the maxmum rates at whch tasks can use resources wthn that perod. Addtonal constrants on the rates of use (as well as the ntegrals) mght be approprate, and they could be added to the formulaton relatvely easly. The frst desgn revew has been scheduled for tme Ths s the second key decson n the project plan. Its tmng balances (wthn the avalable resources) the probablty of success for the frst tasks (2 and 3) leadng up to the desgn revew, wth the probablty of success for tasks 5 and 6 followng the revew. Tasks 5 and 6 are allotted the same allowable duraton (38.5 days), but task 6 s allocated sgnfcantly more resources. Ths s the thrd key decson n the project plan, and t llustrates the character of resource allocaton among two parallel tasks whch both must be completed successfully. Task 6 has a lower elastcty, so more resources must be allocated to t to keep ts dstrbuton comparable to that for task 5. Ths s n contrast to the soluton for tasks n sequence, lke tasks 2 and 3, where more resources are allocated to the task wth hgher elastcty to preserve as much tme as possble for the other task. At the soluton values shown n Table 3, the usage of the personnel resource n the three months s 47 person-days, 38 person-days, and 44 persondays, respectvely. Thus, the personnel avalablty constrant has a small amount of slack n the last two months (avalablty s 47 person-days per month), but the budget constrant for non-labor drect costs over the fscal quarter s bndng, whch precludes expandng the resource multplers for tasks n those two months. The Lagrange multpler for the budget constrant s approxmately.7, and the multpler for the frst-month personnel constrant s.3, so we have some ndcaton that t would be advantageous to have a lttle more money and a lttle less labor, f resources could be reallocated overall. Ths nsght helps to create the connecton between the mddle box and the outer box n Fgure, and provdes nformaton useful at the program manager level. As mentoned n secton 3, the overall tme constrant (4) wll always be bndng on the soluton, so t s of nterest to explore the senstvty of the soluton to changes n the allowable tme for project completon. Fgure 7 shows the relatonshp of overall probablty of success n the project to the allowable tme, for solutons that change T n unts of weeks (5 workng days). The uncertanty n the tme to successful completon of the varous tasks n the project means that attempts to compress the schedule for the overall effort can result n a very sgnfcant reducton n the probablty of project success. A reducton of just two weeks, from 63 workng days to 53, reduces the probablty of successful completon from.95 to.55. In the computatons for ths example, the reducton n probablty of success s resultng from two prmary sources. Frst, compressng the schedule forces the allowable duraton for each task to be reduced, and ths reduces the probablty of successful completon for that task. Secondly, the avalablty of personnel resources for the project s based on a staff of seven people (n ths example), and f the allowable tme for the effort s reduced, the total avalable person-days of that resource are also reduced. Ths reducton n resources that can be appled to the project also lowers the lkelhood of project success. Prob(Success) Allowed Tme (Days) Fgure 7. Effects of changng the allowed tme for project completon on the probablty of success n the project /3 $7. (C) 23 IEEE 8

9 Proceedngs of the 36th Hawa Internatonal Conference on System Scences Conclusons Ths paper focuses on defnng and developng a soluton approach for plannng, schedulng, and managng project efforts where there s sgnfcant uncertanty n the duraton, resource requrements, and outcomes of ndvdual tasks. Under such uncertanty, one may allocate resources to multple methods of accomplshng a specfc requrement to maxmze the probablty of success, for example, and ths dffers n a fundamental way from the vew of projects represented n most of the schedulng lterature. The paper also focuses on specfc ways of defnng relatonshps between resources and/or tme allocated to ndvdual tasks and ther probablty of successful completon. The problem formulaton yelds a model for plannng allocaton of tme and resources to tasks under uncertanty. The model can be expressed as a nonlnear constraned optmzaton. We have ncluded an example that uses the objectve of maxmzng the probablty of success for the project, but other relevant objectves could be used nstead. The example llustrates the knds of key decsons that must be made by a project manager allocatng resources across multple means of accomplshng a specfc mlestone, determnng tme allocaton to dfferent subsets of tasks, and allocatng resources across tasks n sequence as well as n parallel. Ths formulaton represents a new and dfferent way of lookng at project management decsons, and t has mportant and drect applcaton to the management of many dfferent types of projects and programs. 6. References. DeMeyer, A., C.H. Loch and M.T. Pch (22). Managng Project Uncertanty: From Varaton to Chaos, MIT Sloan Management Revew, Wnter, Dcknson, M.W., A.C. Thornton and S. Graves (2). Technology Portfolo Management: Optmzng Interdependent Projects Over Multple Tme Perods, IEEE Transactons on Engneerng Management, 48:4, Hapke, M. and R. Slownsk (993). A DSS for Resource-Constraned Project Schedulng under Uncertanty, Journal of Decson Systems, 2:2, Hapke, M. and R. Slownsk (996). Fuzzy Prorty Heurstcs for Project Schedulng, Fuzzy Sets and Systems, 83:3, Haque, B. and K.S. Pawar (2). Improvng the Management of Concurrent New Product Development Usng Process Modellng and Analyss, R&D Management, 3:, MacCormack, A. (2). Product Development Practces That Work: How Internet Companes Buld Software, MIT Sloan Management Revew, Wnter, Mor, M. and C.-C. Tseng (997). A Resource Constraned Project Schedulng Problem wth Reattempt at Falure: A Heurstc Approach, Journal of the Operatons Research Socety of Japan, 4:, Raz, T. and E. Mchael (2). Use and Benefts of Tools for Project Rsk Management, Internatonal Journal of Project Management, 9, Repennng, N.P. (2). A Dynamc Model of Resource Allocaton n Mult-Project Research and Development Systems, System Dynamcs Revew, 6:3, Valadares Tavares, L., J.A. Antunes Ferrera and J. Slva Coelho (998). On the Optmal Management of Project Rsk, European Journal of Operatonal Research, 7, Wlls, R.J., H. Pan and C.-H. Yeh (999). Resource- Constraned Project Schedulng under Uncertan Actvty Duraton, n Computatonal Intellgence for Modellng, Control & Automaton, M. Mohammadan (ed.), IOS Press, Amsterdam, pp Yeh, C.-H., H. Pan and R.J. Wlls (999). A Heurstc Approach to Fuzzy Resource-Constraned Project Schedulng, n Computatonal Intellgence for Modellng, Control & Automaton, M. Mohammadan (ed.), IOS Press, Amsterdam, pp Elkngton, P. and C. Smallman (22). Managng Project Rsks: A Case Study from the Utltes Sector, Internatonal Journal of Project Management, 2, Fylstra, D., L. Lasdon, J. Watson and A. Waren (998). Desgn and Use of the Mcrosoft Excel Solver, Interfaces, 28:5, /3 $7. (C) 23 IEEE 9