Thinking in Project
Management Terms – Basic Methods and Calculations
By Bob Hugg
In Unit 1 Project Management (PM) was
discussed as a management philosophy. Within that philosophy there are many
methods and techniques; most are industry specific and are designed to provide
performance metrics meaningful to that industry. This unit will discus the
three chief methods in wide use with the most focus paid to the most commonly
used technique, PERT/CPM. PERT and CPM, though separate techniques, are
commonly used in tandem because together they provide a stronger tool and will
be discussed accordingly in this chapter.
A central weakness of both PERT and
CPM is the inability to deal with resource dependencies As discussed in chapter
1, resource dependencies are those that concern the availability of resources
whether they are human, mechanical or fiscal (PERT/CPM considers only causal
dependencies, the completion of a prior task). PERT/CPM also assumes that
additional resources can be shifted to a project as required. Because, in the
real world, all projects have finite resources to draw on the estimates and
expectations are frequently skewed.
Because of this weakness, a significant portion of the PM community
believes that PERT/CPM creates unrealistic expectations, at best. As a result,
management of projects using only PERT/CPM can be difficult and frustrating for
worker, Project Managers and stakeholders alike.
A newly emerging (within the last 10
years) methodology is Critical Chain
Project Management (CCPM), also referred to as Theory of Constraints. In
essence, CCPM focuses on managing constraints, the relationship between tasks
within a project and resources within project.
By actively managing these “hotspots” it is believed that
CCPM decreases project conflict and tension and provides a more balanced
expectation. Though an interesting theory, CCPM is largely unproven and appears
to be most applicable in projects concerning highly dynamic tasks that can be
grouped in modules. Module structure
groups tasks where the completion of a module delivers some degree of function
that can be used regardless of the status of the remainder of the project. An
example would be software development, where a subroutine that is common to
many applications can be completed and useful without the entire project is
completed. Because the relationship
between modules is not as critical, the modules themselves can be re-planned
and re-scheduled as necessary, adding a degree of efficiency and decreasing
conflict within a project or between projects. CCPM also focuses on overall
project progress instead of individual task progress. A perceived strength of
CCPM is that it is based on an absence of multi-tasking; a single resource is
only assigned to a single task/project. A relatively humanistic approach, CCPM
calculations also account for the inconsistent nature of human performance
(good days, bad days, sick time, training needed, etc). CCPM estimates are much
broader (50% probability, 90% probability, etc) and deal exclusively with a
single “normal” completion date of the project as a whole. As such,
it is believed that by identifying and grouping tasks and limiting constraints
the project becomes more manageable while providing incremental value. Critics
of CCPM argue that its assumptions (absence of multitasking, tasks may be
grouped into semi-independent yet value-filled groups) create unrealistic
expectations. In any event, CCPM seems applicable only in those industries
where incremental progress can deliver incremental value or function. Clearly,
only completing one wing of an airplane, 2 walls of a house or 1/3 of a
city-wide traffic risk assessment would provide little value, so CCPM has found
little acceptance outside of very specific hi-tech business areas.
The second method in use is a
variation of PERT called Earned Value
Method, introduced by the Department of Defense in the mid 60s. In the business world this method is
synonymous with ROI (Return On Investment).Simply put,
it examines the relationship between the cost of doing something and the value
received by doing it. Earned value does
not concentrate on probability of completion at a specific time, nor does it
deal with a specific time or range of times, though a by –product of the
analysis is a constantly moving completion projection. It tracks tasks and the
project as a whole in terms of money by analysis that answers 3 specific
questions:
1)
How does the cost of work performed compare to the value of the work
performed?
2)
What is the value (in dollars) of work performed so far?
3)
How does the amount of money spent so far on a project compare to what
should have been spent?
Using answers to those questions,
Earned Value Method generates a variety of productivity indices that can be
used to forecast a project completion date. Because Earned Method focuses on
work performance in terms of cost and value, it is used extensively throughout
the Department of Defense in contracts administration and in industries where
significant amounts of work are performed either under contract or through
contractors. It is not commonly used in
Social and Behavioral sciences or technical production (software development,
healthcare, etc) because, in those disciplines, the tangible value of the process and result
is much more difficult to identify. Earned Value Method employs many
fundamentals of WBS and PERT and is commonly found as an analysis tool in most
mainstream PM software packages, including MS Project.
By far, the most common method used
is PERT/CPM. The remainder of this unit will focus on introducing basic methods
and calculations in use. As discussed in Unit 1, PERT is based on a beta
distribution that is useful in real-world planning because it accounts for a
degree of randomness (that all humans bring to the table). Based on its
theoretical model, PERT delivers a task or project completion estimate based on pessimistic,
optimistic and most likely estimates provided by the user. PERT also provides a
probability of completion on any date selected by the user. PERT calculations
are simple and straightforward, but tend to get lengthy when many tasks are
used. Before the task calculations can
be made, however, 2 steps must be taken in any project planning:
1)
Define the goal of the project and the tasks required to complete it
2)
Place tasks in a logical order and determine the critical path (it is helpful to diagram the tasks)
a.
The critical path is the
longest time path through the network of tasks
When these steps are complete,
generate a set of duration estimates for each task; each set should contain a
pessimistic, most likely and optimistic estimate. To keep the estimates straight, it is useful
to label pessimistic
estimates as TP,
optimistic estimates as TO and most likely estimates as TL
(any labeling system can be used, but these are fairly intuitive).For each task,
calculate the PERT derived expected duration (TE) based on a
formula, (TP
+ 4 TL + TO) / 6 = TE
1)
Read this formula as the sum of pessimistic plus 4 times likely plus
optimistic divided by 6 equals the expected duration
2)
Compete this calculation for all tasks; making sure to group tasks on
the critical path separately
a.
The critical path is the
longest time path through the network of tasks
b.
The sum of duration of tasks on the critical path will determine the
project duration
A second set of calculations are
necessary to determine information that will be useful later in the
process. These calculations will yield
the Standard Deviation (SD) and Variance (V) for each task
duration. The SD is the average
deviation form the estimated time; as a general rule, the higher the SD is the
greater amount of uncertainty exists. The V reflects the spread of a value over
a normal distribution. The SD and V will be useful in determining the
probability of the project meeting a desired completion date. The formulae for
calculating SD and V are:
1)
SD=(TP-T0)/6 {read as (pessimistic-optimistic)/6}
2)
V=SD2 (Standard Deviation squared)
3)
Compete this calculation for all tasks; making sure to group tasks on
the critical path separately
c.
The critical path is the longest
time path through the network of tasks
d.
The sum of duration of tasks on the critical path will determine the
project duration
Since
most projects involve several tasks, it is helpful to construct a table to stay
organized. A table might look like:
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CRITICAL
PATH TASKS (Longest Duration) |
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TASK |
TO |
TL |
TP |
TE |
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SD |
V |
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TOTAL |
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OTHER
PROJECT TASKS |
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TASK |
TO |
TL |
TP |
TE |
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SD |
V |
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TOTAL |
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Table 1, Sample table of estimates
Consider a sample project, planting flowers
and trees. This project could involve 8 tasks; when diagramed it would look
like:
Figure 1, PERT Diagram for sample project
For this sample project a table
would be helpful in getting organized, and would yield more usable information
than the PERT diagram.. This sample project, with 8
tasks, complete with optimistic, pessimistic and likely estimates could then
populate the table. When PERT expected durations and SD and V are added using
the formulae, the table would look like:
|
CRITICAL
PATH TASKS (Longest Duration) |
|||||||||||
|
TASK |
TO |
TL |
TP |
TE |
|
|
|
|
|
SD |
V |
|
1 |
1 |
3 |
5 |
3 |
|
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|
|
|
.67 |
.44 |
|
2 |
2 |
4 |
7 |
4.17 |
|
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|
|
|
.83 |
.69 |
|
5 |
1 |
3 |
6 |
3.17 |
|
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|
.83 |
.69 |
|
6 |
1 |
3 |
5 |
3 |
|
|
|
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|
.67 |
.44 |
|
8 |
1 |
2 |
4 |
2.17 |
|
|
|
|
|
.5 |
.25 |
|
TOTAL |
7 |
15 |
28 |
15.51 |
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|
3.5 |
2.51 |
|
OTHER
PROJECT TASKS |
|||||||||||
|
TASK |
TO |
TL |
TP |
TE |
|
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|
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|
SD |
V |
|
3 |
.5 |
1 |
3 |
1.25 |
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|
.42 |
.17 |
|
4 |
.5 |
1 |
3 |
1.25 |
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|
.42 |
.17 |
|
7 |
.5 |
1 |
3 |
1.25 |
|
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|
.42 |
.17 |
|
TOTAL |
1.5 |
3 |
9 |
3.75 |
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|
1.26 |
.51 |
Table 2, Sample Project populating a table of estimates
This table now provides a wealth of
information. It contains a list of required tasks and separates those tasks in
critical path and non-critical path tasks.
It also lists the best and worst case estimates and the expected
duration for each task and the project as a whole (the sum of the expected
critical path tasks). This table also lists the Standard Deviations and
Variances for each task and the project as a whole, a valuable (but not
intuitive) indicator of the probability of project completion by a desired
date. A manual method to calculate the probability of meeting a desired date is
somewhat more complicated than the other formulae used so far, but would look
like:
1)
Denote the sum of all expected durations on the critical path as S
2)
Denote the sum of all variances on the critical path as V
3)
Select a desired completion time, denote this as D
4)
COMPUTE: (D-S)/square root (V) = Z ( Read as the result of D minus S
divided by the square root of V equals Z)
5)
Enter a standard normal table to find a probability that corresponds
with Z or go online to:
a.
http://math.uc.edu/statistics/statbook/tables.html)
to enter a z number - the application will retrieve the probability from the
very lengthy table
For our sample project, figure a
probability based on a desired time,
15 days: ((D-S)/sqrt {V} =Z)
a.
(15-15.51)/square root(2.51) = (15-15.51)/1.59=-.321 (Z) (Rounded)
b.
A corresponding probability is 37.7% (Rounded)
c.
In other word, there is a 37.7% probability that the project will be
completed within 15 days of the start date
It is also helpful to determine the
earliest and latest dates a task can start to highlight areas that may be
improved upon. For each task, determine
the latest allowable time for moving to the next task. Think of these tasks as
flexible tasks that can be started earlier or later in the process with no
effect on the project duration. The difference between latest time and expected
time is called slack time; Tasks with zero slack time are on the critical
path. For our sample project, these
dates would look like:
|
CRITICAL
PATH TASKS (Longest Duration) |
|||||||||||
|
TASK |
TO |
TL |
TP |
TE |
ES |
EF |
LS |
LF |
Slack |
SD |
V |
|
1 |
1 |
3 |
5 |
3 |
0 |
3 |
0 |
3 |
0 |
.67 |
.44 |
|
2 |
2 |
4 |
7 |
4.17 |
3 |
7.17 |
3 |
7.17 |
0 |
.83 |
.69 |
|
5 |
1 |
3 |
6 |
3.17 |
7 |
10.17 |
7 |
10.17 |
0 |
.83 |
.69 |
|
6 |
1 |
3 |
5 |
3 |
10 |
13 |
10 |
13 |
0 |
.67 |
.44 |
|
8 |
1 |
2 |
4 |
2.17 |
13 |
15.17 |
13 |
15.17 |
0 |
.5 |
.25 |
|
TOTAL |
7 |
15 |
28 |
15.51 |
|
|
|
|
|
3.5 |
2.51 |
|
OTHER
PROJECT TASKS |
|||||||||||
|
TASK |
TO |
TL |
TP |
TE |
ES |
EF |
LS |
LF |
Slack |
SD |
V |
|
3 |
.5 |
1 |
3 |
1.25 |
0 |
1.25 |
3 |
4.25 |
3 |
.42 |
.17 |
|
4 |
.5 |
1 |
3 |
1.25 |
0 |
1.25 |
3 |
4.25 |
3 |
.42 |
.17 |
|
7 |
.5 |
1 |
3 |
1.25 |
1.25 |
2.50 |
4.25 |
5.50 |
3 |
.42 |
.17 |
|
TOTAL |
1.5 |
3 |
9 |
3.75 |
|
|
|
|
|
1.26 |
.51 |
|
ES=Earliest Start
EF= Earliest Finish LS=Latest Start LF=Latest Finish |
|||||||||||
Table
3, Sample Project populating a table of estimates with start dates
The table is now complete and is a treasure
trove of project information, but has proven labor intensive due to the number
of manual calculations (imagine a project with dozens or hundreds of
tasks!). The same results can be
obtained in much less time with much less effort using MS Excel.
Open a new workbook in Excel and
structure a spreadsheet to resemble the table for the sample project. This spreadsheet can become a template for
future project calculations and, in effect becomes a PERT calculator. It may
look like:
Figure 2, PERT Analysis Calculator layout
Notice how the spreadsheet resembles
the table used so far, with a notable addition. This calculator will include a function
to calculate the probability of completing a project on a desired date. Use the
dame formulae discussed so far to write equations for each cell address. To
start, begin writing equations for the cell addresses for calculating the PERT expected duration.
Figure 3, PERT Analysis
Calculator layout – PERT Expected
duration equations
1)
For each task cell: (Optimistic + 4x Typical + Pessimistic)/6
2)
Adjust cell address for each task
Next, write equations to calculate the Variances for each task:
Figure 4, PERT Analysis Calculator
layout – PERT Variance
equations
1)
For each task cell: ((Pessimistic-Optimistic)/6)2
2)
Adjust cell address for each
task
Next, write equations to calculate the Standard Deviations for each task:
Figure 5, PERT Analysis Calculator
layout – PERT Standard Deviation
equations
1)
For each task cell: sqrt (V) ( the square
root of V for that task)
2)
Adjust cell address for each
task
Next, write an equation to sum the PERT Expected Date
for the project:
Figure 6, PERT Analysis Calculator
layout – Summing Pert Expected Dates
Next, write an equation to sum the Variances for the project:
Figure 7, PERT Analysis
Calculator layout – Summing Variances
Finally, write the equation to calculate the Probability of Completion for a desired
project date:
Figure 8, PERT Analysis Calculator
layout – Probability of Completion
Excel
uses a formula designed to compute the probability of placement of a
combination of elements in a normal distribution that is very accurate for use
in real-world situations. The equation is NORMDIST(x, mean, standard_dev,
cumulative) in which:
1)
X is the value for which you want
the distribution (desired date)
2)
Mean is the arithmetic mean of the
distribution (summed PERT expected durations)
3)
Standard_dev is the standard deviation of the distribution (square root
of the summed variances)
4)
Cumulative is a logical value that
determines the form of the function. If cumulative is TRUE, NORMDIST returns
the cumulative distribution function (probability of completion on the date
entered)
Some
things to consider when setting up this equation:
1)
Be sure to adjust formulae as necessary when adding additional tasks
a.
If a error message shows up check cell addresses in the formulae first
– formulae must reflect intent
2)
This set of formulae mirrors the manual calculations but takes less
time for the user
3)
Because PERT is a probabilistic approach, these formulae can deliver a
100% probability – but no plan is perfect – these are always estimates
4)
Never feel there is a 100% probability of a project completing on the
estimated date
Armed with a substantial tool to
compute PERT expectations and probabilities, all that remains is to complete a
few simple CPM calculations. As
discussed in Unit 1, and earlier in this unit, CPM deals with a single expected
date and anticipates that a project may be crashed. Crashing a project is reducing the project to
its shortest duration by adding resources.
It is important to note that the effort required to
complete a task or project remains the same, only the duration may be
shortened. By its nature, crashing a
project is disruptive (it pulls resources from other tasks) and increases the project
cost. Careful consideration must be made whether the effects of crashing a
project are worth completing a project earlier.
The answer will vary from project to project and situation to situation.
Again, using a table to get organized, a very simple example might look like:
The basic steps in completing a CPM
analysis are:
1)
Develop time
and cost data ("normal" and "crashed") for all tasks
2)
Develop
cost-per-week for crashing (crashed costs divided by time saved)
3)
Develop
project network (PERT)
4)
Crash the
activity on the critical path with the lowest cost-for-crashing
5)
Recalculate
the project network (the critical path might change!)
6)
Repeat steps
4 & 5 until all the paths have been crashed.
7)
Ease up on
all non-critical paths, just to the point that all paths are critical
|
Activity |
Begin |
End |
Time (Crashed) |
Time ( |
Cost (Crashed) |
Cost ( |
Time Saved |
Cost Increase |
Cost / Week |
|
Foundation |
1 |
2 |
1 |
2 |
4000 |
3000 |
1 |
1000 |
1000 |
|
Frame |
2 |
3 |
1 |
4 |
8000 |
4000 |
3 |
4000 |
1333 |
|
cost-per-week for crashing =
crashed costs divided by time saved |
|||||||||
Table
4, Sample Project populating a table of CPM estimates of time and costs
In this example the time saved (4 weeks)
is substantial but the cost increase is also substantial ($5,000). At an
aggregate increase of $2,333 per week for crashing this project, this would be
a course of action not lightly taken. Sometimes crashing a project is
unavoidable but a serious consideration of the tangible and intangible costs
must be undertaken.
When used together, PERT and CPM can
provide:
1)
A range of time estimates (PERT)
2)
Likely time estimates (PERT and CPM)
3)
Cost estimates (CPM)
4)
Time and costs if crashed (CPM)
5)
Probabilities of completion on time for a range of times (PERT)
6)
A clear path of tasks that are critical to the project (PERT and CPM)
7)
A central focus for solid communications on project issues (PERT and
CPM)
All plans are estimates and should
be viewed as such. When used together, PERT and CPM provide a valuable tool for
organizing and tracking projects as well as providing a usable “what
if” forum. Care must be taken in collecting estimates used in planning
– any plan is only as good as the most unrealistic estimate. By using
simple but complementary formulae, managers at all levels can get, and keep, a
good handle on a project or its tasks.
A
step-by-step tutorial for using both Microsoft Project and setting up an Excel
spreadsheet for estimating the times on a PERT analysis, refer to the powerpoint tutorials
Resources Used in This
Unit
Bonini, Charles, et al,
Quantitative Analysis for Management, Columbus: McGraw Hill, 1997.
Goldratt, Eli, Dr., The Goal: A Process of Ongoing Improvement, Great
Mednick, Barry, PERT-CPM on
Excel,
MS Project, by
Microsoft Corporation.
MS Excel, by Microsoft
Corporation.
PM Body of Knowledge (PMBOK),
Project
Management Institute (PMI)
<Project Management Institute Website>.
ProjeX,
by WAA, Inc .
Systema, Sid, Probabilistic Solutions
to Project Scheduling,
US
National Performance Survey, The Standish Group, 1998.
Verma,
Vijay K., Managing the Project Team: The Human Aspects of Project Management,
Wiest,
Jerome D., and Levy, Ferdinand K., A Management Guide to PERT/CPM,
© 1996 A.J.Filipovitch
Revised 11 March 2005