Allocating Time and Resources in Project Management Under Uncertainty
Mark A. Turnquist
School of Civil and Environmental Eng.
Cornell University
Hollister Hall
Ithaca, NY 14853
Email: mat14@cornell.edu
Linda K. Nozick
School of Civil and Environmental Eng.
Cornell University
Hollister Hall
Ithaca, NY 14853
email: lkn3@cornell.edu
Abstract
We define and develop a solution approach for planning, scheduling and managing project efforts where there is significant uncertainty in the duration, resource requirements and outcomes of individual tasks. Our approach yields a nonlinear optimization model for allocation of resources and available time to tasks. This formulation represents a significantly different view of project planning from the one implied by traditional project scheduling, and focuses attention on important resource allocation decisions faced by project managers. The model can be used to maximize any of several possible performance measures for the project as a whole. We include a small computational example that focuses on maximizing the probability of successful completion of a project whose tasks have uncertain outcomes. The resource allocation problem formulated here has importance and direct application to the management of a wide variety of project-structured efforts where there is significant uncertainty.
I. Introduction
A wide variety of engineering and business activities are structured as projects: they have tasks, they require resources of various types, and they are constrained in both time and budget. Many types of projects are also subject to considerable uncertainty – uncertainty in time to complete specific tasks, in the resource requirements of those tasks, and in whether or not the effort will produce an outcome judged to be “successful.” Programs often have many such projects, and the program managers face critical decisions about what projects to pursue, how much time and money to invest in each one, and how to reach decisions to terminate individual projects (or parts of projects) if they do not seem promising.
The focus of this paper is on defining and developing a solution approach for planning, scheduling, and managing project efforts where there is significant uncertainty in the duration, resource requirements, and outcomes of individual tasks. We also focus on specific ways of defining relationships between resources and/or time allocated to individual tasks and their probability of successful completion. This provides a structure for the analysis of the core uncertainties in the system, and a means for making effective resource allocation decisions within an uncertain project management framework.
2. Perspectives on Uncertainty in Project
Management and Project Scheduling
The “management-oriented” literature contains considerable evidence of concern with uncertainty and risk in managing projects (see, for example, the recent articles by Haque and Pawar, 2001; MacCormack, 2001; Raz and Michael, 2001; DeMeyer, et al., 2002; and Elkington and Smallman, 2002). Much of this literature is very qualitative, focusing on the process of managing to ameliorate risks. On the other hand, the operations research literature is replete with articles on project scheduling, but very few of these articles deal with uncertainty. We believe this “disconnect” is the result of different collections of researchers defining problems at different levels of a hierarchy in a way that obscures their relationship to one another.Figure 1 portrays a nested view of the problem that highlights the potential connections between project scheduling and the larger issues of resource allocation and risk management. The innermost box represents the scheduling problem, where it is assumed that the available resources are fixed and specified, and the characteristics of individual tasks (duration and resource use) are given. With those
Determine what resources to make available, within overall budget limits
Change resource allocation to adjust task characteristics
Schedule tasks, given resources
and task characteristics
inputs specified, the scheduling problem is to determine a likely operational schedule for the tasks that “fits” within the available resources. The project scheduling literature focuses on this “inner” problem and generally does not deal with uncertainty.
PERT, developed in the 1950’s, represented the first consideration of uncertainty in project scheduling, focusing on uncertain task durations. This technique allowed an estimate of the overall duration of a project to be constructed. However, PERT has major weaknesses. It does not consider constraints on available resources and it assumes that all tasks will be completed successfully. Using a PERT framework, Valadares Tavares, et al. (1998) also consider uncertainty in the resource requirements of individual tasks and the resulting effect on overall project cost, but they do not incorporate resource constraints.
There is an enormous literature on resource-constrained project scheduling, but very little of that literature includes consideration of uncertainty. Hapke and Slowinski (1993; 1996), Yeh, et al. (1999) and Willis, et al. (1999) have proposed scheduling methods based on fuzzy number representations of task durations. Mori and Tseng (1997) consider projects in which task outcomes are uncertain. At the completion of a task (with a fixed duration and resource requirements), an evaluation is made of whether or not the result of the task is acceptable (a “success” or a “failure”). If the outcome is “failure,” the task is re-attempted, and another evaluation is performed. This cycle repeats until success is achieved. Thereby, a probability density function for the completion time of the task is developed.
Scheduling, however, is only one part of the overall problem of project management under uncertainty. From the perspective of a project manager, it is likely that adjusting resource allocation across tasks can change the parameters of the individual distributions. For example, increasing the resources allocated to a particular task is likely to reduce both the mean and variance of its duration. This reallocation of resources among tasks is a tool
that can be used to increase the likelihood of successful completion of the project within some available time window, and within the limits of resource availability. Thus, the middle box in Figure 1 represents this larger problem of resource allocation that is of greater interest to the project manager than just the scheduling problem in the inner box.
A program manager, who may have responsibility for a collection of projects, is concerned about a still-larger problem – determining what resources need to be made available or acquired to support the whole collection of project(s). He/she has an overall budget constraint, but can use that budget to acquire more of resource 1 and less of resource 2, for example, in an effort to give individual project managers more opportunity to successfully complete projects. This is represented by the outer box in Figure 1.
These resource allocation issues have been treated by a few previous authors (e.g., Repenning, 2000; Dickinson, et al., 2001), but in a way that is isolated from the “inner” problems. A primary objective of the work described in this paper is to begin making the connection between the various levels of interest represented in Figure 1, so that more effective and useful tools can be created. To accomplish that objective, we focus on an analytic formulation that relates resource allocation to the characteristics of task probability distributions, establishing the connection between the inner and middle levels of Figure 1. This formulation is directly extendable to consider the determination of overall resource levels, providing the connection to the outer level of Figure 1. This provides a more solid basis for overall consideration of risk management within the project(s).
3. Model Formulation
It is often important to reflect the fact that a task may not complete successfully, particularly in development environments. Ideas that have high initial promise may not work out; tests or experiments may result in failure, etc. Thus, it is also common in this kind of project environment to pursue multiple paths to achieving a goal. If one doesn’t work out successfully, an alternative approach may, allowing the overall effort to still succeed. The possibility that parallel tasks (or sequences of tasks) can be pursued simultaneously, and successful completion of any one path allows proceeding with a subsequent portion of the project, is not normally considered in project scheduling networks. We assume that alternate paths to project success may exist, and we are concerned with
Figure 1. A perspective on problem levels” within project management.
planning resource allocations in that type of environment.
It is also important to recognize that a variety of different performance criteria for the project may be relevant. Much of the project scheduling literature is focused on minimizing either makespan or discounted project costs, but other criteria may also be important. When significant uncertainty exists in the task outcomes, for example, we may be focused on allocating resources to tasks so as to maximize the probability of success for the project as a whole. We will focus particular attention on the “probability of success” measure, but most of the model structure applies equally as well to other performance metrics.
By focusing attention on the probability of success for the project, we are able to address two questions of particular interest:
1. How should available resources be allocated across multiple means of accomplishing a specific requirement, in order to maximize the overall probability of success?
2. If changes in allocated resources can change the probability distribution of time to successful completion of a task, how should a limited budget be allocated to maximize overall probability of successful completion of a project within a given amount of time?
We define an allowable duration, di, for task i, to
denote the length of the time window that will be created for completion of the task. If the distribution of duration for task i has a density function denoted by fi(t), and a cumulative distribution function
denoted by Fi(t), then the probability of successful
completion of task i within its allowed window is
Fi(di). Quite clearly, the probability of successful
completion of the task is a non-decreasing function of
di.
We also define a resource multiplier, ci, that
denotes a factor to be applied to all resources used by task i. The nominal value of this multiplier is 1.0; increases in resources applied to the task are denoted by ci > 1.0, and decreases by ci < 1.0. The effect of
changing the resource multiplier for a task is to shift its duration distribution, as illustrated in Figure 2. Increasing the value of ci shifts the distribution to the
left (in general, decreasing both mean and variance), and decreasing ci shifts the distribution to the right (in
general, increasing both mean and variance). In general, we can summarize the probability of successful task completion as Fi(di,ci) to emphasize
its dependence on both the allowable time window and the resource multiplier.
Figure 2. Illustration of changing task duration distribution as resource multiplier is changed
In Figure 2, the vertical line at di = 23 serves as
an example of the Fi(di,ci) calculation. For each value
of ci, Fi(di,ci)
is the integral of the probability density
function up to di. Figure 2 provides a visual
indication of how Fi(di,ci) increases with increasing ci, and it should also be clear that for a given value of ci, increasing di(sliding the vertical line to the right)
increases the value of Fi(di,ci).
We are asserting in this formulation a single ci
value for each task, to be applied to all resources used by that task. This is equivalent to assuming a proportional use of resources in combination (for example, more people require more materials, more computer resources and more budget). This formulation precludes substitution of resources on a task (i.e., achieve the same overall task duration distribution by applying more computer resources and fewer people, for example). Substitution possibilities could be incorporated by defining the resource multipliers as cik, with separate multipliers
for each resource on each task. This is a very plausible extension to the formulation, but for the present we will restrict our attention to the “proportional” model with a single ci value for each
task.
To implement the concept of using di and ci as
key decision variables, we must be able to express
Fi(di,ci) in terms of di and ci. One very useful way to
do this is to use a shifted Weibull distribution (sometimes called the three-parameter Weibull), which is characterized by the shift (minimum possible task duration),
d
i0; a scale parameter, αi;and a shape parameter, βi. The Weibull distribution is
a very flexible distributional form, for which several other popular distributions are special cases (for specific choices of αi and βi). It is widely used in
reliability studies. For given values of
d
i0, αi and βiwe can write Fi(di) as: 0 0.02 0.04 0.06 0.08 0. 1 0.12 0 10 20 30 40 Ti m e P e ri o ds Pr o b . D e n si ty C i = 1.0 C i = 0.75 C i = 1.25 di = 23
0 0 ( ) ( ) 1 0; , 0 i i i i i i i i i i d d F d e d d β α α β − − = − > ≥ > (1)
We will treat the shape parameter, βi as a
characteristic of the task, i, but allow adjustment of the scale parameter, αi, as a function of resource
multiplier, ci. This allows the effect depicted in
Figure 1, where the task time distribution shrinks or expands as the resource multiplier changes. The specific form of this relationship is assumed to be:
0 i
0;
0
i i
c
iεc
i iα α
=
−>
ε
≥
(2)0 i
α
represents the “nominal” scale factor for task i(i.e., to characterize the distribution of time to successful completion when ci = 1). εi can be thought
of as an “elasticity” (in the sense that term is used by economists) – that is, the percentage change in αi that
results from a 1% change in ci. The negative sign on
εi implies that an increase in resources produces a
decrease in the scale parameter (compressing the distribution). Using the Weibull distribution with the resource multiplier affecting only the scale parameter implies that the coefficient of variation in the task time distribution remains constant as ci changes. That
is, the mean and standard deviation of the distribution change in the same proportion.
Incorporating (2) into (1) allows us to write an expression for Fi(di,ci) that depends on four basic
input parameters for each task i:
d
i0,α
i0 , εi and βi . 0 0 0 ( ) ( ) 1 0; , , 0; 0 i i i i i i i i i i i i i i d d c F d e d d c β ε α−α β
ε
− − = − > ≥ > ≥ (3)The overall probability of success for the project will be some function of the collection of Fi(di,ci)
values for all tasks in the project (or collection of related projects). We will denote this function as
Z(F), where F represents the set of all Fi(di,ci) values.
We will assume that the objective of the project planning exercise is to determine diand ci values that
maximize Z(F), subject to constraints on resource availability, overall duration of the project, and precedence requirements among the tasks. The exact form of Z(F) depends on the specifics of a particular application, and what constitutes “success” in that context.
To represent the constraints, we must also determine the planned start times for the tasks, which we will denote by si. We will assume that the project
(or collection of projects) has a total of N tasks, and we will arbitrarily denote task N as a “completion”
task with zero duration and resource requirements. Then, if the entire effort has an available time frame of T time units, we can specify a completion constraint:
T
s
N≤
(4)Precedence constraints between tasks i and j (i is a predecessor of j) can be specified as:
i i
j
s
d
s
≥
+
(5)We denote the total requirement for resource k
(k = 1, …, K ) by task i (over its duration) as ciAik.
This reflects both a nominal requirement, Aik, and the
effect of the resource multiplier, ci. If the allowable
duration for task i is specified as di, then we will
assume that the resource consumption of resource k
occurs at a uniform rate, ciAik/di, over the duration of
the task. We can then summarize the consumption rate of resource k by task i as a function of time, denoted as rik(t): ≤ ≤ + = cA d if s otherwiset s d t r i ik i i i i ik 0 / ) ( (6)
Equation (6) is a convenient way to express the resource consumption rates in terms of the core decision variables, di and ci. An increase (decrease) in
the resource multiplier causes a corresponding increase (decrease) in the usage rate (as well as the total resource requirement) of all resources used by task i. Conversely, the resource usage rate for task i
can be reduced (without changing its total resource requirement) by increasing the duration, di.
Increasing the task’s allowable duration “stretches out” the resource requirement over more time, reducing the usage rate. The time window over which the resources are required for task i is shifted by changing si. Thus, the three key variables for each
task are used in conjunction to adjust the resource consumption and allow the overall project schedule to adjust to resource constraints (subject, of course, to constraints (4) and (5)).
From a computational standpoint, (6) is problematic because it is discontinuous and the decision variables si and di appear in the conditioning
statement that “switches” the rate between 0 and a positive value. Our goal is to create a formulation with continuous variables, so that we are not limited by any arbitrary discrete definitions of time periods. This has led us to construct a representation of rik(t)
0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 Time F rac ti on of N o m in al R at e
approximates the step-function in (6). One way of doing that is with the function shown in (7):
( ) 2 i i i i i i i i i i i i i i i i i i i i i ik ik i t s t s t s d t s d wd wd wd wd t s t s t s d t s d wd wd wd wd c A r t d e e e e e e e e − − − − − − − − − − − − − − − − = − − − + + (7)
where w is a constant. The function specified in (7) has a value that is approximately zero, except for the range si≤ t ≤ si + di, where it rises to approximately
the value ciAik/di. It is continuous (and continuously
differentiable) over the entire real line. The size of the scaling constant w determines the sharpness of the rise and fall of the function. It also has a closed-form integral, which is very convenient for writing the constraints on total resource consumption.
The function (7) is illustrated in Figure 3 (for a case in which si= 1 and di = 1). At the beginning and
end of the task, the function changes from 2% of the nominal usage rate (ciAik/di) to 98% of that rate over a
time interval of approximately 4w. In the illustration shown in Figure 3, the value of w has been set to 0.025. Thus, by setting w to a small value (e.g., a small fraction of a day), the function (7) rises and falls quite abruptly, and approximates the step-function (6) quite closely. In reality, the step step-function may be simply a convenience, and the real resource usage rate may rise and fall more gradually at the beginning and end of a task. In this case, w can be set to a larger value to reflect that build-up and close-out effect.
Figure 3. Illustration of the continuous representation of resource usage rate for a task, as a fraction of the
nominal rate (ciAik/di).
The availability of resource k is defined over a set of Mk contiguous time intervals, and we will
define mk as the start time of the mth interval for
resource k, where m = 1, 2,…,Mk+1. The start time of
the (Mk+1)th interval defines the end of the mth
interval. Rmk defines the amount of resource k
available in interval m, where m = 1, 2,…,Mk. Then
the resource constraints can be written by integrating the usage rates:
1, 1 ( ) 1,..., ; 1,..., m k mk N ik mk k i t r t dt R k K m M τ τ + = = ≤ = =
∑ ∫
(8)If we use the representation in (7) for rik(t), the
integrals in (8) can be written as:
(
)(
)
(
)(
)
1 2 1, 3 41
1
( )
ln
2
1
1
m k mk v v ik i ik v ve
e
c A w
r t dt
e
e
τ τ +
+
+
=
+
+
∫
(9) where: i i k mwd
s
v
2
(
1,)
1−
=
τ
+ (10) i i i mkwd
d
s
v
2=
2
(
τ
−
−
)
(11) i i mkwd
s
v
3=
2
(
τ
−
)
(12) i i i k mwd
d
s
v
2
(
1,)
4−
−
=
τ
+ (13)The non-linear optimization problem is then to determine ci, di, and si for i = 1,…, N so as to
maximize Z(F), subject to (4), (5) and (8), with (7) and (9)-(13) used to calculate (8).
This non-linear optimization is complicated because the constraints expressed in (8) are non-convex, and we also have no assurances that the objective function Z(F) is concave in the decision variables (at least not without further specification of that function). These difficulties imply that there may be multiple local maxima for the optimization. It is clear from (3) that allowing more time to complete a task always increases its probability of successful completion, so we can be sure that constraints (4) and (5) will always be binding in an optimal solution. However, the same assertion cannot be made with respect to the resource constraints (8), and it is in those constraints that the non-convexities appear.
We have compiled some preliminary computational experience with the model, using the illustrative problem described in the next section. However, there remains significant work to do in exploring the computational characteristics of the problem.
Parametric studies that vary the overall time constraint, T, and trace out changes in the resulting probability of project success, are highly useful. This
is also discussed (in the context of a specific example) in the following section.
This formulation is easily extended to represent the outer level of Fig. 1 by treating Rmk as a variable
to be determined endogenously by the model, and adding an overall budget constraint with unit costs for the various resources.
4. An Illustrative Example
As an illustration of the modeling approach outlined in the previous section, consider the small example project network shown in Figure 4. Task 1 represents building a physical prototype of a new product to test specific functions. Tasks 2 and 3 represent steps in an alternative means of evaluating the functionality, using computer simulation. Successful completion of either task 1 or tasks 2 and 3 can lead to a preliminary design review (task 4), and this “or” structure is indicated by the use of the vertical line in front of task 4. Following that design review, parallel efforts on product testing (task 5) and manufacturing analysis (task 6) can proceed, and both of those must be completed successfully prior to a second design review (task 7), which signifies the end of the project. For analysis of this example project, we focus on the probability of successfully reaching the second design review (task 7) in one fiscal quarter (63 working days), and our formulation of the objective function is Z(F) = (F1 + F2F3 - F1F2F3)F4F5F6. The dependence of each Fi(di, ci)
term on di and ci has been suppressed to simplify the
notation.
Figure 4. Small project network for illustrative analysis.
Each task requires both people and money; and those resources are available in constrained amounts over varying periods. The total number of person-days available is specified by month over a three-month (one fiscal quarter) planning horizon, and the total dollars available for non-labor direct costs is specified over the fiscal quarter. For the calculations shown here, we have assumed the personnel availability is 147 person-days in each month
(derived from 21 days per month for 7 people), and the budget availability is $250,000 over the quarter.
There are three key decisions in this example problem:
1) How should resources be allocated between the alternative means of reaching the first design review? Should we focus on Task 1, on the combination of Tasks 2 and 3, or divide our resources across the two strategies?
2) When should the first design review (Task 4) be scheduled? How should we divide the available time between the activities leading up to that review and the activities following it?
3) How should we allocate resources between Tasks 5 and 6 to best insure that both are successful within the allotted time between the first design review and the second design review?
As we look at the solutions for this example, we want to focus our attention on those three key decisions.
Table 1 summarizes the input data for the seven tasks. The minimum duration is the value of
d
i0for each task. The mean and standard deviation of the time to complete each tasks successfully are the basis for specifying the two parameters of the Weibull distribution for each task; given those two values, we can solve forα
i0 and βi for each task. The resultingparameter values are shown in Table 2. No distribution parameters are estimated for tasks 4 and 7 because the standard deviations of both tasks are specified in the input data as zero.
The elasticity values (εi) in Table 1 define the
percentage reduction in the scale parameter (
α
i0) of the distribution of time to successful completion for each task, resulting from a one percent increase in resources applied to the task. The two columns labeled Nominal Person-Days and Nominal Budget specify the Aik values for the two resources for eachtask. No resource requirements or elasticity are shown for task 7 (final design review) because we are using that task as the designation of project completion. Project success (at least for this simple example) is defined as the ability to reach that design review within the timeframe of 63 working days.
1 2 3 4 5 6 7
Table 1. Input data for example tasks. Task Minimum Duration (Days) Mean Time to Complete (Days) Std. Dev. of Completion Time (Days) Nominal Person-Days Required Nominal Budget Required ($000) Elasticity of Distribution to Resources 1 10 21 5 63 75 0.8 2 5 10 5 20 5 1.0 3 7 10 2 20 5 0.2 4 1 1 0 8 0 0 5 20 30 5 126 150 0.8 6 20 30 5 42 10 0.2 7 0 0 0 0 0 0
Table 2. Estimated Weibull distribution parameters for the tasks.
Task Scale Parameter 0 i α Shape Parameter 1 β i 1 12.4 2.5 2 5.0 1.0 3 3.3 1.5 4 N/A N/A 5 11.3 2.2 6 11.3 2.2 7 N/A N/A
For this small example, the nonlinear optimization can be performed using Excel’s Solver, which uses a generalized reduced gradient algorithm (Fylstra, et al., 1998). The optimized results for task start times, allowable durations, and resource multipliers, as well as the resulting probabilities for successful completion of each task, are shown in Table 3. This set of values results in a probability of success for the project as a whole of 0.95. We notice first that the resource multiplier for task 1 is very small, and the multipliers for tasks 2 and 3 are relatively large. This implies that in the first portion of the project, resources are being allocated to the “simulation” option for reaching the first design review, and the “physical prototype” option is being ignored. The value of 0.1 for the resource multiplier is actually a lower bound that was set in the computation of a solution, to avoid potential numerical problems in evaluating expression (3), so we can interpret the solution as reducing the resources to task 1 as much as possible. The resource multipliers for tasks 2 and 3 are relatively different, as a result of the differing elasticities for those two tasks. Task 2 has a much larger multiplier because it has a larger elasticity, and the model takes advantage of the ability to use resources to “compress” its distribution, allowing more time for task 3.
Table 3. Results of optimization for example project.
Task Start Time, si (Day) Allowable Duration, di (Days) Resource Multiplier, ci Resulting Probability of Successful Completion 1 0 23.5 0.1 0.012 2 0 9.1 5.34 0.988 3 9.1 14.4 2.16 0.985 4 23.5 1 1 1.0 5 24.5 38.5 1.19 0.982 6 24.5 38.5 2.79 0.990
7 63 N/A N/A N/A
Figure 5 shows the effect that the large resource multiplier on task 2 has on the distribution of time to successful completion. The mean and standard deviation inputs for this task are such that the resulting Weibull distribution reduces to the special case of an exponential distribution (β2 = 1), with a
minimum possible value of 5. Under nominal conditions (i.e., resource multiplier = 1), the time to complete this task has very large variance. However, the elasticity of the scale parameter with respect to resource changes is large (-1.0), so the addition of resources to this task compresses the distribution very significantly. At the allowed duration for this task (9.1 days), the probability of successful completion is 0.988.
The effect of the combination of c2 and d2
(continuing to focus on task 2) on total resource use is also worth noting. Figure 6 shows the rate of consumption of the personnel resource for task 2
Figure 5. Result of applying additional resources to task 2 (distribution of time to successful completion).
under the optimized plan and two alternative possibilities. In the optimized plan, the resource multiplier is 5.34 and the duration is 9.1 days. This implies a usage rate for resources as indicated by the “Optimized” curve in Figure 6. The integral under this curve is 5.34*20 person-days, indicating that the total resource usage is 5.34 times the nominal value (20 person-days, as indicated in Table 1) for this task. Alternatively, if we did not increase the resource usage rate and kept the same allowable duration (9.1 days), the resource usage would be as indicated by the “Nominal Resources” curve. The integral under this curve is 20 person-days, the nominal requirement for the task. This is clearly a lower overall consumption of resources, but at the nominal resource rate the probability distribution of time to successful completion of the task is the “Nominal” distribution shown in Figure 5, so in 9.1 days the probability of success for this task is only 0.56, as compared to 0.988 under the optimized plan. We could increase the probability of success under the nominal resource allocation by extending the allowable time for the task. With the nominal time-to-success distribution shown in Figure 5, we would have to allow 27.1 days for this task at the nominal resource allocation to achieve a probability of success
0 0.2 0.4 0.6 0.8 1 1.2 0 5 10 15 Days Pr ob . D en si ty Nominal Optimized
equal to 0.988. This would imply resource usage at a much lower rate over a much longer time, as shown by the “Extended Duration” curve in Figure 6. The integral under this curve is also 20 person-days, indicating that the total resource usage is unchanged. In the scheme of the entire project, however, it is not practical to allow 27.1 days for task 2, so the optimization uses available resources more intensively to achieve a high probability of success in task 2 in a much shorter time.
Figure 6. Resource usage rate curves for task 2 under three alternative plans.
The usage rate implied by the optimized schedule would require approximately 12 people over the 9.1-day duration of the task, and this may not be truly feasible since we said at the outset of the example that the person-days of availability were based on a staff of seven. The resource constraints (8) constrain only the aggregate person-days of use within a specified period – they do not constrain the maximum rates at which tasks can use resources within that period. Additional constraints on the rates of use (as well as the integrals) might be appropriate, and they could be added to the formulation relatively easily.
The first design review has been scheduled for time 23.5. This is the second key decision in the project plan. Its timing balances (within the available resources) the probability of success for the first tasks (2 and 3) leading up to the design review, with the probability of success for tasks 5 and 6 following the review.
Tasks 5 and 6 are allotted the same allowable duration (38.5 days), but task 6 is allocated significantly more resources. This is the third key decision in the project plan, and it illustrates the character of resource allocation among two parallel tasks which both must be completed successfully. Task 6 has a lower elasticity, so more resources must be allocated to it to keep its distribution comparable to that for task 5. This is in contrast to the solution for tasks in sequence, like tasks 2 and 3, where more resources are allocated to the task with higher elasticity to preserve as much time as possible for the other task.
At the solution values shown in Table 3, the usage of the personnel resource in the three months is
147 days, 138 days, and 144 person-days, respectively. Thus, the personnel availability constraint has a small amount of slack in the last two months (availability is 147 person-days per month), but the budget constraint for non-labor direct costs over the fiscal quarter is binding, which precludes expanding the resource multipliers for tasks in those two months. The Lagrange multiplier for the budget constraint is approximately .0007, and the multiplier for the first-month personnel constraint is .0003, so we have some indication that it would be advantageous to have a little more money and a little less labor, if resources could be reallocated overall. This insight helps to create the connection between the middle box and the outer box in Figure 1, and provides information useful at the program manager level.
As mentioned in section 3, the overall time constraint (4) will always be binding on the solution, so it is of interest to explore the sensitivity of the solution to changes in the allowable time for project completion. Figure 7 shows the relationship of overall probability of success in the project to the allowable time, for solutions that change T in units of weeks (5 working days). The uncertainty in the time to successful completion of the various tasks in the project means that attempts to compress the schedule for the overall effort can result in a very significant reduction in the probability of project success. A reduction of just two weeks, from 63 working days to 53, reduces the probability of successful completion from 0.95 to 0.55.
In the computations for this example, the reduction in probability of success is resulting from two primary sources. First, compressing the schedule forces the allowable duration for each task to be reduced, and this reduces the probability of successful completion for that task. Secondly, the availability of personnel resources for the project is based on a staff of seven people (in this example), and if the allowable time for the effort is reduced, the total available person-days of that resource are also reduced. This reduction in resources that can be applied to the project also lowers the likelihood of project success.
Figure 7. Effects of changing the allowed time for project completion on the probability of success in the
project. 0 2 4 6 8 10 12 14 0 10 20 30
Days from Start of Task
Pl an ne d Per son ne l U se R at e Optimized Nominal Resources Extended Duration 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 50 55 60 65 70
Allowed Time (Days)
P
rob
(S
uc
5. Conclusions
This paper focuses on defining and developing a solution approach for planning, scheduling, and managing project efforts where there is significant uncertainty in the duration, resource requirements, and outcomes of individual tasks. Under such uncertainty, one may allocate resources to multiple methods of accomplishing a specific requirement to maximize the probability of success, for example, and this differs in a fundamental way from the view of projects represented in most of the scheduling literature. The paper also focuses on specific ways of defining relationships between resources and/or time allocated to individual tasks and their probability of successful completion.
The problem formulation yields a model for planning allocation of time and resources to tasks under uncertainty. The model can be expressed as a nonlinear constrained optimization. We have included an example that uses the objective of maximizing the probability of success for the project, but other relevant objectives could be used instead. The example illustrates the kinds of key decisions that must be made by a project manager – allocating resources across multiple means of accomplishing a specific milestone, determining time allocation to different subsets of tasks, and allocating resources across tasks in sequence as well as in parallel.
This formulation represents a new and different way of looking at project management decisions, and it has important and direct application to the management of many different types of projects and programs.
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