PROGRAM PLANNING UNDER UNCERTAINTY

Transcription

1 Proceedings of the 7 Winter Simuation Conference S. G. Henderson, B. Bier, M.-H. Hsieh, J. Shorte, J. D. Tew, and R. R. Barton, eds. PROGRAM PLANNING UNDER UNCERTAINTY Kabeh Vaziri Pau Carr Linda Nozick Civi and Envir. Engr. Hoister Ha, Corne University Ithaca, N.Y. 4853, U.S.A. ABSTRACT Task durations are subject to uncertainty and they can be infuenced by the resources assigned. Commony there is the opportunity to assign workers to tasks, which require a secondary ski they have. When workers use secondary skis, they are not as efficient as workers for which that ski is their primary ski, however they can contribute to the competion of the task. This paper provides the means for managers to heuristicay optimize the aocation of their skied workers among individua tasks on severa competing projects when the task durations are uncertain and workers have mutipe skis. INTRODUCTION Projects are often undertaken in an environment of considerabe uncertainty. Uncertainty may be refected in the duration and outcomes of specific tasks, as we as in the effectiveness of resources appied to them. Project and program managers must decide, in these uncertain conditions, how to aocate and manage scarce resources across many projects that have competing needs. Whie the mode and methodoogy presented in this paper is appicabe to different industries, we have chosen to use construction projects as a means to iustrate these ideas. Each project is divided into a coection of tasks and there are precedence constraints between the tasks. A crew is a group of skied workers that are assigned to a task. The skied workers we focus on are carpenters, aborers, equipment operators and masons. Different categories of tasks require a different baance between the skied workers; hence there is a unique idea composition of workers for each task. When workers are assigned to a task in this idea configuration, the workers are % efficient eading to a typica or nomina crew size. Worker assignments to tasks which are mutipes of the nomina crew size, and for which the optima baance of skis across the workers is maintained, aso yied workers that are % efficient. We refer to a crew with the optima composition of workers as an idea crew. We refer to the crew s nomina size as the idea crew size with the understanding that mutipes of this size are equay efficient. By equay efficient we mean that if a crew is a mutipe of the nomina crew size the duration is scaed by that mutipe. If workers are added beyond the idea crew size but these workers do not create a mutipe of the idea crew size, the individua performance of the workers decreases. However, the tota output of the crew wi increase because of the additiona workers. This increase in the output of the crew resuts in a shorter duration for the corresponding task. One of the key eements of this probem is the deveopment of an understanding of when it is vauabe to operate crews that are not the idea crew size in order to produce a pan of shorter duration. Each worker has a primary ski at which he is % efficient. It may aso be possibe to assign the worker to a task using some secondary ski that is different than the primary. A worker using a secondary ski wi often not be as efficient as a worker for whom that ski is their primary ski. Hence a second key eement of this probem is the deveopment of an understanding of how to effectivey use the secondary skis of workers. The focus of this paper is the deveopment of a mode formuation and a soution procedure for aocating avaiabe resources across projects when there are significant uncertainties associated with the duration and resource needs of individua tasks. This paper makes four significant contributions to the iterature in project management for construction activities. First, it creates a mathematica formuation and soution procedure that captures the uncertainty associated with task duration and resource requirements. Second, it incudes an expicit mechanism to expore the consequences of cross-functioning personne. Third, it creates a mechanism to understand the vaue of repanning worker assignments as the projects unfod. Fourth, a compex case study is deveoped iustrating how to use the mode in practice. Section, describes the iterature that is most reevant to this research. Sections 3 and 4 discuss the mode and the soution procedure. Section 5 presents the assumptions for the case study incuding detais on worker skis and wages, and a method to estimate the probabiity distribu /7/$5. 7 IEEE 9

2 tion for task duration given the type of task, the number of peope assigned and their skis. The case study anaysis is presented in section 6. Section 7 describes key concusions and opportunities for future work. PRIOR RELATED WORK Emaghraby (977) and Moder et a. (983) describe and formuates severa modes that expore the time-cost tradeoff when that trade-off is continuous. See De et a. (995) for a review of the discrete time-cost tradeoff probem. Goenko-Ginzburg and Gonik (998) focus on deveoping a decision-making process to determine, as a project unfods, which tasks to start when and what resources to assign. Nozick et a. (4) uses the notion of resource mutipiers which determine the amount of a resource aocated to a task but in a panning context. Changing the resource mutipiers affects the probabiity distribution for the duration of the task. (5) uses the notion of resource mutipiers and buids a heuristic that combines simuated anneaing with a parae scheduing scheme to optimize the resource aocation poicies. We substantiay modify their genera mode to incude cross-functioning which contros for the baance between the workers skis assigned to a task. Further, this paper aso focuses on the vaue of re-panning as the project unfods. Our soution method is reated to agorithms for the resource-constrained project scheduing probem (RCPSP). Soving this probem has been a theoretica chaenge for researchers. Genera reviews of efforts to address the RCPSP can be found in Morton and Pentico (993), Özdamar and Uusoy (995), Herroeen et a. (998), Hartmann (998), W garz (999), Brucker, et a. (999), Hartmann and Koisch (4) and Koisch and Hartmann (6). This probem is aso reated to a variant on the RCPSP, namey the muti-mode resource constrained project scheduing probem. Reavant papers incude Acaraz, et a. (3), Jozefowska et a. () and Heimann () We focus on the use of priority rue scheduing and meta-heuristics. Koisch (996) has expored priority rue scheduing in great detai. Simuated anneaing (SA), genetic agorithms (GA), tabu search and greedy search have a been tested by practitioners to sove the RCPSP. Hartmann s (Hartmann, 998), GA, Boueimen and Lecocq s (998) SA approach and Debes et a. s (6) hybrid metaheuristic appear to be among the best currenty avaiabe methods. Vas et a. (5) shows the benefit of justification as an augmentation to severa existing soution procedures. Xu et a. (7) shows the benefit of incorporating roout into priority rue scheduing schemes. 3 MODEL FORMULATION A key eement of this formuation is the notion that by adding resources to a task we are abe to decrease the mean and variance of the duration of that task. We use two modeing mechanisms to contro the aocation of resources to tasks: resource mutipiers and duration mutipiers. The resource mutipier of a task determines how its resource aocation wi change compared to the defaut or nomina aocation. The resource mutipier that corresponds to the nomina resource aocation equas one. If the resource mutipier equas one, the task durations foows a nomina beta probabiity distribution. The nomina resource aocation and the probabiity distributions for the task durations are estimates based on expert judgment. If the resource mutipier of a task for some resource is greater than one then the resources aocated to that task are more than the nomina aocation and hence the probabiity distribution of the task duration shifts to the eft. On the other hand, a resource mutipier of ess than one wi cause the probabiity distribution of the task duration to shift to the right. The duration mutipier determines the amount by which the distribution shifts to the eft or right as the resource mutipier varies. M is the resource mutipier of task i in the th project and determines the crew size for task whie it is active. Once the resource mutipier of a task is known, the size of its crew can be cacuated from equation () where n is the nomina crew size for task and n is the crew size when the resource mutipier is set to M. We assume that there is ony one idea crew configuration for a given task hence no subscript is needed to indicate the crew type. ni n i M i i, () If there are k types of skied workers, r i k identifies the number of workers with ski k assigned to task. Function g (.) is the reationship between crew size and the number of peope in the crew with ski k. Therefore, equation () shows that the number of workers with each ski as a function of the crew size. rik g( n ) i i, k, () Equation (3) iustrates the functiona reationship between the resource mutipier and the duration mutipier for task i. Di Q( M ) i, (3) o p If i, i and are the nomina optimistic, i m pessimistic and most ikey durations for task, equations (4) to (6) indicate how these vaues change when the duration mutipier varies from its nomina vaue of one.

3 i, (4) i, (5) i, (6) o o i i Di p p i i Di m m i i Di S it is a binary variabe that equas one if task i is schedued to start in period t and zero if otherwise. Duration of task is a function of its resource mutipier for the type of crew it wi use, the most ikey, optimistic and m a b pessimistic durations, i.e., di f ( M,,, ) i i i i. Equation (7) simpy states that a task can ony start when a of its predecessors are finished. PRED j is the set of a immediate predecessors of task j of project. T T Si di S j i PRED j (7) Assume N is the competion task for project. This task is a dummy that has duration of zero with no resource requirements. If the panning horizon for a projects has the maximum possibe ength of T, then equation (8) represents the competion constraint for each project. T S N (8) Equation (9) represents the resource requirement constraints. In these constraints, r i k is the number of skied workers from category k aocated to task. Let R k represent the number of skied workers from category k avaiabe in each period during the panning horizon. Aso, et f () t be the probabiity density function for the duration of task, then F ( t ) shows the probabiity that this task woud compete at t had it started at time. The inner summation in equation (9) represent the probabiity that task i is active in period t. Equation (9) states that the expected use of skied workers in period t cannot be more than what is avaiabe. t r S [ F ( t )] R k, t (9) ik i i k i, Every worker has a primary ski for which they are % efficient. Based on the primary ski, the worker may have secondary skis for which they woud be at most % efficient. Since workers may have mutipe skis, it is usefu to generaize equation (9) to represent this crossfunctioning. This can be done by defining a new variabe X ikkt to represent the number of workers whose primary ski set is k but are using their secondary ski set k for task during period t. e i kk is the efficiency of worker k who is doing job k ( e i k k ). If crossfunctioning cannot exist, the efficiency equas zero. Equa- tions () and () represent a more generaized version of equation (9) that incudes the possibiity of using the secondary skis of workers. t ri [ ( )], k Si Fi t X ik k teik k k t () i, i,, k The tota number of workers used when crossfunctioning is aowed cannot exceed what is avaiabe. This notion is captured in equation (). X i, kkt Rk k t () i,, k Equation () creates the bounds on the resource mutipier for every task to prevent the assignment of too few or too many workers. M M M i, () min max i i i For the purpose of this anaysis we focus on minimizing the expected makespan of the coection of the projects subject to precedence constraints and resource imitations. If N represents the dummy node that marks the competion of a the projects, then the objective function is given in (3) and is accompished subject to equations () through (). T Min te [ S Nt ] (3) t This anaysis is easiy generaized to other objectives incuding minimizing a weighted average of the tardiness of each project and a weighted average of each project s makespan. 4 SOLUTION PROCEDURE The formuation that is presented in the previous section cannot be soved directy because the task durations are uncertain. The soution procedure is based on iterativey updating the resource mutipiers using simuated anneaing and evauating the impications of the new poicy on the duration of the projects using Monte Caro simuation based on a parae scheduing scheme. The Simuated Anneaing (SA) agorithm starts off with an initia set of randomy chosen resource mutipiers. After evauation of the current soution, a neighboring set of resource mutipiers is generated from the current soution. If the neighboring soution improves the performance measure, which is the expected makespan of the projects, the new set of resource mutipiers repaces the current soution. However, if the new soution does not improve the performance measure, there is sti some probabiity of accepting it over the current soution. The agorithm is composed of four key eements that are discussed in the foowing sections. 4. Initia Soution Generation The initia soution for each run is chosen by random samping. Each set of resource mutipiers is uniformy and independenty chosen from the soution space such that it fas between the stated minimum and maximum bounds

4 indicated by equation (). Once the initia resource mutipiers have been identified, they can be evauated. 4. Evauation of a Set of Resource Mutipiers Once the resource mutipiers are known, we use simuation to evauate the expected makespan of the coection of the projects resuting from that resource aocation poicy. Each repication of the simuation creates a feasibe schedue using samped task durations. The information coected from the simuation is used to update the resource mutipiers and generate a neighboring soution to the current one. Assume that the resource mutipiers are known. The duration mutipier for each task can be cacuated from equation (3). The optimistic, pessimistic and most ikey durations for each task can be cacuated from equations (4), (5) and (6). The scheduing is done using the Parae Scheduing Scheme (PSS) where tasks are randomy seected from the decision with a bias based on the tota number of successors. 4.3 Seection Poicy Once the initia set of resource mutipiers is evauated, meaning that the expected makespan that corresponds 5. to 5. Resources that set of resource mutipiers is known, a new set of resource mutipiers is generated. This new set is produced using the neighborhood definition discussed in the next subsection. The new soution is samped ony enough times, that the Wicoxon rank-sum test can determine which of the soutions is better. If the new soution is better, it wi be accepted. However, if the soutions are of the same quaity, meaning that they produce the same expected makespan vaue, or if the new soution is worse than current, there is sti some probabiity of accepting the new soution. This procedure is repeated each time a new set of resource mutipiers is generated. In the ast iteration, doube justification introduced in Vas et a. (5) is used to improve our estimate of the project duration. 4.4 Neighborhood Definition The neighboring soution to the current one is identified by changing the resource aocation poicy. The key point is to increase the resources assigned for the tasks that are on the critica path most of the time by reeasing resources from those tasks that are not on the critica path. Let CP i be the percent of the time that task is on the critica path. In this discussion we consider the critica path for the coection of projects not each of them individuay. If task has been on the critica path for at east one repication, then we increase the resource mutipier of that task randomy between zero and CP i (-u)(-n i /N) where u is the average utiization of peope in the current soution and n i /N is the size of crew c of task for peope when N is the tota number of skied workers avaiabe. If task has never been part of the critica path in any repications, the resource mutipier of that task for a of the resources wi be decreased by a random number between zero and ( E( LFTi EST )/ ( )) ( / ) i E TD u n i N where TD stands for tota duration of the project and EST and LFT represent the eariest start time and atest finish time of task. A changes in the resource mutipiers occur subject to the avaiabiity of the peope during the panning horizon. 5CASE STUDY To iustrate these ideas, we focus on a sma contractor with about five miion doars annuay in construction projects that was currenty executing three projects. The first project (Victim s Assistance Center) is made up of two identica subprojects of tasks each. The second project (Cerebra Pasy Center) consists of 73 tasks and the third one (Methodist Church) has 77 tasks. Incuding dummies that mark the singe beginning and singe end of the projects, the tota number of tasks is 93. We assume that the tasks are numbered based on their precedence, i.e., if task is an immediate predecessor to task j, then i j. There are four genera types of peope avaiabe throughout the panning horizon: carpenters, aborers, masons and equipment operators. The contractor has a poo of 8 aborers, 7 carpenters, 6 masons and 6 equipment operators that have to be aocated to projects efficienty. We aso assume that the resource requirements of the tasks are constant for the time they are active. Scarcity of materia and equipment is not considered in this anaysis because abor is the driving concern. There are three categories of tasks. The miestones are assumed to have no resource requirements and no duration. The tasks that are sub-contracted require no resources from the contractor poos but do require a fixed period of time to compete. The remaining crew types are defined by the number of aborers, carpenters, equipment operators and masons, respectivey. Each task is mapped to ony one crew type. The crew types are assigned to the tasks based on expert judgment. Tabe shows the combination of workers in each of these crew types. For exampe if a task is of type 3, the nomina crew composition is two aborers, one carpenter and two equipment operators and the nomina crew size is 5. If the nomina crew is assigned to the task whie it is active, its task duration is described by its nomina distribution. The houry wages of aborers, carpenters, operators and masons are assumed to be $8.83, $33.67, $39.7, $34.94, respectivey. These wages incude workers compensation at 8%, socia security at 7.65%, and other taxes at 5.5%. As is typica in construction, a worker is assumed to be paid the higher rate between his primary ski and the ski used. For exampe if a carpenter is asked to work as a

5 aborer, he wi be compensated as a carpenter because the wage of a carpenter is more than a aborer but if a carpenter is asked to work as an equipment operator, he wi be paid as an equipment operator. Further, if a worker is not caed to work for some period of time, he is not paid. Tabe : Crew Types Crew Type Laborer Carpenter Operator Mason Cross-Functioning The workers in this case study are categorized based on their primary ski set as carpenters, aborers, operators or masons. However, it is possibe to make use of their secondary speciaties. Based on expert input, we have assigned efficiency factors for the use of secondary skis given the primary ski. These efficiency factors are as foows. Laborers can do carpenter tasks with 7% efficiency if it invoves framing and forming. Aso since a of the masonry in this case study invoves bock-work, the aborers can do that work with 7% of the efficiency of a mason. On the other hand, masons and carpenters can do aborer tasks with 9% efficiency. The equipment operators cannot be cross-functioned to do any of the other jobs, however, since these projects are fairy sma and they do not need compicated equipment, carpenters, aborers and masons can do operator tasks with % of efficiency (the same efficiency of an operator). 5.3 Impact of Resource Assignments on Task Duration As more skied workers are added to the crew, or eiminated from the crew, the average worker efficiency as we as the duration mutipier for that task changes. Figure shows a potentia reationship of duration mutipier of a task of crew type 8 and the crew size for that task. The crew size is determined once the resource mutipier is known. If the crew size is increased, the duration mutipier of the task decreases, and vice versa. There is a simiar reationship between the crew size and duration mutipier for other tasks. It is important to notice that in this appication, the increments and decrements to the crew composition can ony happen in specific ways. These rues must be specified for each crew type and are thus important input data for the mode. Tabe shows how changes in the crew size of a task that uses crew type of 8 wi change the crew composition. Duration Mutipier Crew Size Figure : Reationship of the Duration Mutipier and the Resouce Mutipier (Function Q(.)). Tabe : Cacuation of Function g(.) for a Crew of Type 8 Size Laborer Carpenter RESULTS 6. Program Panning In this section three cases are discussed and they are caed Nomina, Optimized and CrossF Optimized. Nomina is when the defaut resource aocation to the tasks is used. In this case, the resource mutipier for each task is equa to one. This yieds a probabiity distribution of tota duration for a projects with a mean of 93 days with standard deviation of 4 days. The standard deviation of this distribution is quite sma in comparison to the mean indicating that there is reativey itte schedue risk. The mean abor cost of the projects is about $765, with a standard deviation of about $,. Optimized is when the resource mutipiers are optimized but cross-functioning is not aowed and therefore each worker performs tasks based on his primary ski ony. In this case, the mean and standard deviation of abor cost are about $765, and $, and the mean and standard deviation of duration are about 63 and 4 days. With no change in the average direct abor cost, it is possibe to achieve more than a 5% reduction in the average 3

6 duration. Again the standard deviation of this distribution is quite sma in comparison to the mean hence the schedue risk has not increased even though the duration has shrunk. CrossF Optimized is when the resource aocation is optimized and cross-functioning is aowed. The mean and standard deviation of the tota duration are about 5 and 5 days respectivey. The average cost increases to about $8, with an increase in the standard deviation to about $6,. With a 4% increase in average tota abor cost, the mean duration drops by 6% compared to Optimized and more than % compared to Nomina. In this case study the tota vaue of the projects incuding abor, materias, subcontractors and profit is about $.6 miion and the performance period is about 93 days. As mentioned previousy, about.3% of tota construction project cost is overhead hence the daiy overhead rate for these projects based on the Eichey formua (Levin, 998) is about $,4. In this exampe, speeding up the projects using cross-functioning is not economicay effective if ony overhead benefits are considered. The overhead savings from Optimized soution is about $7, but the additiona abor costs are about $36,. In this case study the schedue risk as measured by the standard deviation of the duration distribution to the mean of that distribution is quite sma indicating reativey itte schedue risk. The Optimized soution provides substantia reduction in the distribution for duration at effectivey no additiona cost. The CrossF Optimized soution provides for a further reduction in duration but at a higher cost. If just the overhead savings resuting from the schedue acceeration is considered against the higher abor costs it does not appear to be financiay attractive to acceerate the schedue beyond that indicated by the Optimized soution. Figure shows a part of the network of Cerebra Pasy Center to iustrate how Optimized and CrossF Optimized soutions affect the schedue. Each box represents a task. For exampe task ID 7 is Frost Wa. Frost wa has to be competed before vertica insuation and form and prep was can begin. The notation of (X,O)=(,) in the box indicates that in CrossF Optimized (X), the resource mutipier for this task is set to and in Optimized (O) the resource mutipier is set to. the resource mutipiers of many of the tasks on the critica paths increase. However, in CrossF Optimized soution more of these mutipiers increase further because the resources are now avaiabe to make this feasibe. For exampe, tasks 7, 74, 76, 79, 8, 8 and 84 are iustrated in Fig.. 74, 76, 8 and 84 are more frequenty found on the critica hence their mutipiers are set to. However, tasks 7, 79 and 8 are sometimes found on the critica path but there are insufficient resources avaiabe when crossfunctioning is not aowed to increase these resource mutipiers to the point where they have a meaningfu positive impact on the durations of the projects. Much of the schedue benefit achieved in the cross-functioned soution stems from the abiity of aborers to perform many of the tasks of carpenters. Aso, as a further resut of crossfunctioning, some of the tasks associated with the Methodist church can aso be done about days earier and concurrent with the tasks in Figure. These tasks incude excavate shaow footer, insta rebar and forms for shaow footer, pace concrete for shaow footer, insta rebar and forms at shaow wa, etc. Figure 3 and Figure 4 show the impacts of Optimized and CrossF Optimized soutions on the distribution of makespan and abor costs for the projects. Both shift the probabiity distribution of the tota duration to the eft. Notice that the cost distribution for Optimized soution is the same as for Nomina, however; the duration distribution is shifted substantiay to the eft and is therefore stochasticay smaer. CrossF Optimized soution has a stochasticay arger distribution for tota abor cost but a stochasticay smaer distribution for tota duration. Probabiity Distribution CrossF Optimized Optimized Nomina Tota Duration (Days) Figure 3: Effect of Change in Resource Mutipiers on the Duration Probabiity Distribution Figure : Portion of the Project Network for the Cerebra Pasey Center In the Nomina soution a the resource mutipiers are set to one. In Optimized and CrossF Optimized soutions, As mentioned previousy, the poo of skied workers of this sma contractor consists of 7 carpenters, 8 aborers, 6 masons and 6 equipment operators. Figure 5 to Figure 8 show the expected utiization of the workers during the makespan of the projects. 4

7 Probabiity Distribution CrossF Optimized Tota Labor Cost (Doars) Nomina & Optimized Figure 4: Effect of Change in Resource Mutipiers on the Labor Cost Probabiity Distribution Carpenter Use Laborer Use 5 5 CrossF Optimized Nomina Optimized 5 5 Duration (Days) 5 5 Mason Use Figure 5: Average Carpenter Use over Time CrossF Optimized Nomina Optimized 5 5 Duration (Days) Figure 6: Average Laborer Use over Time CrossF Optimized Nomina Optimized 5 5 Duration (Days) Figure 7: Average Mason Use over Time As Figure 5 shows, the period of proonged high utiization for carpenter is onger with both Optimized and CrossF Optimized soutions compared to Nomina. It is desirabe that in the CrossF Optimized soution the use profie for carpenters is high and then becomes very ow quicky because it indicated that the carpenters can essentiay move on to other work. Equipment Operators Use CrossF Optimized Nomina Optimized Duration Figure 8: Average Equipment Operator Use over Time Figure 6 iustrates that the aborer use increases when cross-functioning is aowed because aborers are now performing some carpenter tasks. The drop in abor use is quite gradua over time in contrast to that for carpenters. In these projects, ess than 5% of the tasks require masons. In Figure 7, the distribution of use of the masons over time is shown for a three soutions. Ceary there is an issue with masons in the contractor s abor poo during this time period. The contractor reay does not have any need for actua masons across these projects because a of the masonry work can be done by aborers. If the contractor beieves this condition is ikey to exist into the future it is more reaistic to eiminate masons from the resource poo and simpy empoy aborers to do these tasks. At the beginning of the panning horizon, there are some tasks that require equipment operators, e.g., rough grade or seect fi at the basement sab. Since the carpenters and aborers are aready busy with their own tasks, and that these tasks are time critica, in the CrossF Optimized soution the operator tasks are done by operators. Since the tasks are time critica, resource mutipiers of two are assigned to these tasks so they use a 6 equipment operators for a short period of time. Aside from this period, note that in Figure 8 at east two-thirds of the equipment operators are ide as the projects unfod. This is due to the fact that the equipment operators do not have a secondary ski which can be utiized. 6. Program Contro The method discussed in the previous sections and appied in the case study focuses on the assignment of workers to tasks (optimization of the resource mutipiers) from a panning point of view. That is, it focuses on the creation of an up-front pan and estimates the impact of that pan if 5

8 it is foowed without modification as the projects unfod. However, it may be possibe to achieve better outcomes if the pan is modified as the projects unfod. From an operationa perspective, this means modifying the resource mutipiers as the projects progress. The key question is then, how much of a benefit in performance is possibe if the contractor goes through the effort to re-pan on a reguar basis. To understand what the answer to this question might be, we can appy the agorithm describes above on a roing horizon basis by sequentiay drawing an observation from each of the task duration distribution as those tasks are executed and as each task competes, reoptimizing the resource mutipiers for a the tasks which have yet to be started. Figure 9 iustrates the distributions for tota duration and cost with and without reguar re-panning of the resource assignments when cross-functioning is aowed. The key insight from this figure is that by re-panning the contractor can reduce his schedue risk. Effectivey the upper tai is removed from the distribution of duration. In these projects there is reativey itte schedue risk hence the benefit is somewhat modest however it does iustrate that benefits are possibe. Probabiity Distribution Roing Horizon CrossF Optimized Tota Duration (days) Figure 9: Impact of Re-Optimizing the Resource Mutipiers as the Project Unfods Overa, updating the resource assignments throughout the project provides a better outcome because it uses information as it becomes avaiabe throughout the course of the project. In genera, the resource mutipiers are not modified substantiay over the course of the projects. Most changes in the resource mutipiers are associated with tasks that require a ot of carpentry, i.e. in tasks that map to crew types, 4, 8 and. Since carpenters are highy utiized, when a task takes onger that requires substantia support from them, there is a risk that other tasks that aso require substantia support from carpenters wi be deayed. By carefuy managing these tasks and hence workers with skis in high demand, the overa duration can be better managed. 7 CONCLUSIONS In this paper, we deveop a mode and soution procedure that aocates avaiabe workers among competing projects for a singe contractor to decrease the expected duration of the projects. This formuation incudes an expicit mechanism to refect the impact of changes in the resources assigned to a task on the probabiity distribution for task duration and cost. Further this formuation aso incudes opportunities to capitaize on the secondary skis of workers through cross-functioning. These opportunities provide for the acceeration of work abeit at an increase in cost. This mode aso incudes an expicit mechanism based on expert opinion to refect the impact of crew sizes and compositions on crew productivity and therefore on task duration and cost. This mode and soution procedure has been appied to a reaistic case study based on a sma contractor. The case study has iustrated the foowing. First, carefu assignment of resources to tasks can substantiay decrease project duration. Second, cross-functioning can create opportunities to finish projects faster, however since crossfunctioned workers are not as efficient and because the workers are paid based on what they do not their primary speciaty, cross-functioning increases the abor costs. Third, cross-functioning may provide an opportunity to substantiay reduce the size of abor poos that are not in high demand based on the nature of the work the contractor engages in and for which there is another type of worker that can effectivey do much of that work. This aows the contractor to potentiay offer more work to empoyees that are fet to be more critica hence making it easier to retain those workers. Fourth, reconsidering the assignment of workers to future tasks as projects unfod can improve schedue performance. Much of this benefit stems from carefuy managing the work assignments of individuas with skis in high demand. ACKNOWLEDGEMENTS This work was supported by the Nationa Science Foundation under Grant No This support is gratefuy acknowedged but it impies no endorsement of the findings. This work aso benefited greaty from the opinions and comments of Robert Reddick, President of Con Tech Buiding Systems in Gouverneur, New York. REFERENCES Acaraz, J., C. Maroto and R. Ruiz, Soving the mutimode resource-constrained project scheduing probem with genetic agorithms, Journa of the Operationa Research Society, vo. 54:6: 64-66, 3. Boueimen, K. and H. Lecocq, A new efficient simuated anneaing agorithm for the resource-constrained project scheduing probem and its mutipe-mode ver- 6

9 sion, European Journa of Operationa Research, vo. 49, pp. 68-8, 3. Brucker P., A. Drex, R. Mohring, K. Neumann, and E. Pesch. Resource-constrained project scheduing: notation, cassification, modes, and methods, European Journa of Operationa Research 999;: 3-4. De, P. Dunne, E.J., Ghosh, J.B., Wes, C. The discrete time-cost trade-off probem revisisted,, European Journa of Operationa Research, vo. 8, Debes, D., B. DeReyck, R. Leus, and M. Vanhoucke, A hybrid scatter search/eectromagnetic meta-heuristic for project scheduing, European Journa of Operationa Research, vo. 69, pp , 6. Emaghraby, S. Activity networks: project panning and contro by network modes, New York, Wiey, 977. Feszar K., and K. Hindi. Soving the resource-constrained project scheduing probem by a variabe neighborhood search. European Journa of Operationa Research vo. 55: 4-43, 4. Goenko-Ginzburg, D. and A. Gonik, A heuristic for network project scheduing with random activity durations depending on the resource aocation, Int. J. Production Economics, vo. 55, 49-6, 998. Hartmann, S., Project scheduing under imited resources, modes, methods, and appications. Berin, Germany: Springer-Verag, 998. Hartmann, S., A competitive genetic agorithm for the resource-constrained project scheduing, Nava Research Logistics, vo. 456, , 998. Hartmann, S., and Koisch, R. Experimenta evauation of state-of-the-art heuristics for resource-constrained project scheduing probem. European Journa of Operationa Research vo. 7: ,. Heimann, R. Resouce-constrained project scheduing: a heuristic for the muti-mode case, OR Spektrum, vo. 3:3, ,. Herroeen, W., B. De Reyck, and E. Demeuemeester, Resource-constrained project scheduing: A survey of recent deveopments, Computers & Operations Research, vo. 5, no. 4: 79-3, 998. Jozefowska, J., M. Mika, R. Rozycki, G. Waigora, and J. Wegarz. Simuated Anneaing for muti-mode resource-constrained project scheduing, Annas of Operations Research, vo. :37-55,. Koisch, R. Efficient priority rues for the resourceconstrained project scheduing probem, Journa of Operations Management, vo. 4: 79-9, 996. Koisch R. and Hartmann S. Experimenta investigation of heuristics for resource-constrained project scheduing: An update. European Journa of Operationa Research, vo. 74: 3-37, 6. Moder, J.J., Phiips, C.R., and Davis, E.W. Project management with CPM, PERT and Precedence Diagaming, Van Nostrand Reinhod, New York, 983. Morton T. E., and Pentico D. W. Heuristic scheduing systems. John Wiey and Sons; 993. Nozick, L. K., M. A. Turnquist, and N. Xu., Managing portfoios of projects under uncertainty, Annas of Operationa Research, vo. 3, pp , 4. Özdamar L., and Uusoy G. A survey on the resourceconstrained project scheduing probem. IIE Transactions;7: , 995. Vas, V., F. Baestín and S. Quintania, Justification and RCPSP: A technique that pays, European Journa of Operationa Research, vo. 65, , 5. Vaziri, K., L. K. Nozick, and M. A. Turnquist, Resource Aocation and Panning for Program Management Winter Simuation Conference, 7-35, 5. W garz J. (Ed.) Project scheduing: recent modes, agorithms and appications. Kuwer, Dordrecht; 999. Xu, N., McKee, S., Nozick, L. and R. Ufomata, Augmenting priority rue heuristics with justification and roout to sove the resource-constrained project scheduing probem, accepted for pubication in Computers and Operations Research, 7. AUTHOR BIOGRAPHIES KABEH VAZIRI is a strategic research consutant at the Corporate Executive Board Company where she is focused on case-based best practice research in project management. Her research interests incude ogistics and program risk management. She has a Ph.D, from Corne University in transportation systems engineering. Her emai address is kv37@corne.edu. PAUL CARR is an adjunct associate professor in the Schoo of Civi and Environmenta Engineering. Previousy he was Chief Executive Officer of the Bernier Carr and Associates, P.C., a muti-discipined engineering, architecture and construction management firm. He is currenty a consutant focused on dispute resoution in the buiding industry. He has a Ph.D. in Construction Engineering from Virginia Tech. His web page can be found at < His emai address is pgc3@corne.edu. LINDA NOZICK is a professor in the Schoo of Civi and Environmenta Engineering at Corne University. Her research interests are in the deveopment of mathematica modes for use in the management of compex systems. She has a particuar interest in systems that can be represented mathematicay as networks, incuding transportation and ogistics systems, civi infrastructure networks and project networks. Her web page can be found at < Her emai address is kn3@corne.edu. 7