Project Management
R
E
T
P
and
C
M
P
Part 1
Vincent F. Yu, IM, NTUST
Costco, 11/21/2008
I NTRODUCTION
The task of managing major projects is an ancient
and honorable art.
In about 2600 B.C., the Egyptians build the Great
Pyramid for King Khufu.
The Greek historian Herodotus claimed that 400,000
men worked for 20 years to build this structure.
Modern projects ranging from building a suburban
shopping center to putting a man on the moon are
amazingly large, complex, and costly.
Completing such projects on time and within budget
is not an easy task.
Indeed, the complicated problems of scheduling
such projects are often structured by the
interdependence of activities.
Typically, certain activities may not be initiated
before others have been completed.
Some key questions to be answered in project
management are:
1. What is the expected project completion date?
2. What is the potential “variability” in this date?
3. What are the scheduled start and completion
dates for each specific activity?
4. What activities are critical in the sense that
they must be completed exactly as scheduled
in order to meet the target for overall project
completion?
5. How long can noncritical activities be delayed
before a delay in the overall completion date
is incurred?
6. How might resources be concentrated most
effectively on activities in order to speed up
project completion?
7. What controls can be exercised on the flows
of expenditures for the various activities
throughout the duration of the project in
order that the overall budget can be adhered
to?
To answer these questions, we will use the methods
PERT (Program Evaluation Review Technique) and
CPM (Critical Path Method).
Both of these approaches to scheduling represents
a project as a network.
When a project involves uncertain elements, the
representation of the project requires a stochastic
network.
PERT was developed in the late 1950s by the Navy
Special Projects Office in cooperation with the
management consulting firm of Booz, Allen, and
Hamilton.
The technique was used in the engineering and
development program of the Polaris missile.
Many firms and government agencies today require
all contractors to use PERT.
CPM was developed in 1957 by J. E. Kelly of
Remington Rand and M. R. Walker of Du Pont.
CPM differs from PERT in the details of how time
and cost are treated.
The implementation of PERT and CPM had an
immediate impact on scheduling projects because it
allowed the practice of “management by exception.”
A TYPICAL PROJECT:
THE GLOBAL OIL CREDIT CARD OPERATION
Moving the Global Oil credit card operation to
Des Moines, Iowa, from the home office in Dallas is
an important project.
Global’s board of directors has set a firm deadline of
22 weeks for the move and has put the Operations
Analysis Group in charge of the move.
The move is difficult to coordinate because it
involves many different divisions within the
company.
Real estate must select one of three available
office sites.
Personnel has to determine which employees
from Dallas will move, how many new
employees to hire, and who will train them.
The systems group and the treasurer’s office
must organize and implement the operating
procedures and the financial arrangements for
the new operation.
The architects will have to design the interior
space and oversee needed structural
improvements.
Office partitions, computer facilities,
furnishings, and so on, must all be provided
for the existing building.
A second complicating factor is the interdependence of activities (i.e., some parts of the
project cannot be started until other parts are
completed).
Consider two obvious examples:
Global cannot construct the interior of an office
before it has been designed.
Global cannot hire new employees until it has
determined its personnel requirements.
THE ACTIVITY LIST
PERT and CPM are specifically designed for projects
of this sort.
The first step is to define the activities in the project
and to establish precedence relationships.
The first activity list prepared for the move is shown
below.
This is the most important part of any PERT or CPM
project and must be a group effort so that no
important activities are overlooked.
Note that the columns labeled Time and Resources
are indications of things to come.
Each activity (represented by a letter) is placed on a
separate line.
An activity’s immediate predecessors are recorded
on the same line.
An immediate predecessor is an activity that must
be completed prior to the start of the activity in
question (e.g., Global cannot start activity C until
activity B is completed).
THE GANTT CHART
Before discussing PERT and CPM, let’s look at
another graphical approach, the Gantt chart
(developed by Henry L. Gantt in 1918).
Activities
Time (weeks)
The horizontal axis is time while each activity is
listed on the vertical axis.
The beginning of the bar represents the earliest
possible starting time for the activity.
Activities
Each bar represents the
anticipated as well as
actual duration.
As each
activity is
completed, the
appropriate bar
is shaded.
Time (weeks)
At any point in time, it is clear which activities are on
schedule and which are not.
As of week 13, activities D, E, and H are behind
schedule.
Activity G has
Activities
actually been
completed and
hence is ahead of
schedule.
Time (weeks)
The previous example shows how the Gantt chart is
mainly used as a record-keeping device for
following the progression in time of the subtasks of
a project.
With the Gantt chart, we can see which tasks are on,
behind or ahead of schedule.
It is important to note that in the context of the Gantt
chart, “on schedule” means “it has been completed
no later than the earliest possible completion time.”
However, this is too simple a concept for whether an
activity is on schedule.
The appropriate point of view should be whether the
overall project is being delayed in terms of a target
completion date.
The Gantt chart also fails to reveal which activities
are immediate predecessors of other activities.
This information is of vital importance in
determining project completion time.
We will now see that the network representation
contains the immediate predecessor information
that we need.
THE NETWORK DIAGRAM
In a PERT network diagram,
each activity is represented by an arrow
that is called a branch or arc.
the beginning and end of each activity (an
event) is indicated by a circle that is called
a node.
When an activity is completed, the event occurs.
Constructing the Network Diagram The following
network diagram shows activities A through C.
Note that the numbers assigned to
2
the nodes are arbitrary. They are
simply used to identify events and
A
do no imply anything about
precedence relationships.
1
4
In the network diagram,
each activity must start at
B
C
the node in which its
3
immediate predecessors
ended.
For example, activity C starts at node 3 because its
immediate predecessor, activity B, ended there.
Now we need to add activity D to the network.
However, note that activities A and C are both
immediate predecessors to activity D.
Therefore, nodes 2 and 4 must be combined so that
activity D can start from it.
5
4
E
D
3
A
1
C
B
2
Note that activity E, which
has only D as an immediate
predecessor, can be added
with no difficulty.
However, adding activity F creates new problems.
Since F has C as an immediate predecessor, it would
emanate from node 3 . However, this would imply
that A is also an immediate predecessor to F, which
is incorrect.
The Use of Dummy Activities This dilemma is
solved by introducing a dummy activity, which is
represented by a dashed line in the network diagram.
The dummy activity is fictitious in the sense that it
requires no time or resources.
It merely provides a pedagogical device that enables
us to draw a network representation that correctly
maintains the appropriate precedence relationships.
Here is the resulting network diagram with the
dummy activity.
2
1
D
5
A
F
B
4
C
3
6
E
7
The procedure is generalized as follows:
Suppose that we wish to add an activity A to the
network starting at node N, but not all of the
activities that enter node N are immediate
predecessors of activity A.
Create a new node M with a dummy activity running
from node M to node N.
Take those activities that are currently entering node
N and that are immediate predecessors of activity A
and reroute them to enter node M.
Now, make activity A start at node M.
This network diagram shows the complete activity
list and all of the precedence relationships.
Design
Select
site
2
D
5
E
A
1
F
Plan
4
B
C
3
Select
personnel
Personnel
requirements
6
Construct
Move
H
Hire
G
I
Financial
arrangements
7
Train
J
8
Some computer programs may have difficulty with
activities H and G since they both start at node 6
and end at node 7 .
These activities
may be read by the
2
5
computer program
D
as one instead of
E
A
two.
1
F
6
H
G
4
B
C
3
I
7
J
8
A dummy activity can be used to cure this condition.
Design [4]
Select
Site [3]
1
2
D
5
E
A
6
Select F
4
B
Personnel [2]
Construct [8]
Move
[2]
G
Hire [4]
Plan [5]
C
Personnel
Requirements [3]
3
Expected activity
completion times.
H
7
8
I
Financial
Arrangements [5]
Train [3]
J
9
An Activity-on-Nodes Example
The previous network was an example of an Activityon-Arc (or AOA) approach.
In the Activity-on-Nodes (AON) approach, the
activities are associated with the nodes of the
network while the arcs of the network display the
precedence relationships.
The Global Oil network would be represented as
follows:
A
D
J
E
G
B
C
F
H
I
THE CRITICAL PATH MEETING THE BOARD’S DEADLINE
The activity list and an appropriate network diagram
are useful devices for representing the precedence
relationships among the activities in a project.
Recall that the board has set a firm goal of 22 weeks
for the overall project to be completed.
The time estimates must first be incorporated before
we can tell if this goal can be reached.
The PERT-CPM procedure requires management to
produce an estimate of the expected time it will take
to complete each activity on the activity list.
These time estimates are given below:
THE CRITICAL PATH CALCULATION
If you added up all of the expected activity times in
the above table, the total working time required to
complete all the individual activities would be
3 + 5 + 3 + 4 + 8 + 2 + 4 + 2 + 5 + 3 = 39 weeks
However, this does not take into account those
activities which can be performed simultaneously.
For example, activities A (3 weeks) and B (5 weeks)
can be initiated at the same time. Therefore, the
total time to complete both activities would be 5
weeks.
To obtain a prediction of the minimum calendar time
required for overall project duration, we must find
the critical path in the network.
A path is a sequence of connected activities that
leads from the starting node 1 to the completion
node 9 (e.g., path B-C-D-E-J).
To complete the project, the activities on all paths
must be completed (all paths must be traversed).
The task is to analyze the total amount of calendar
time required for all paths to be traversed and to find
the longest path from start to finish.
The longest path through the network is called the
critical path and will determine the overall project
duration (because no other path will be longer).
The activities on the critical path are called critical
activities of the project since if they are delayed,
then the entire project will be delayed.
It is this subset of activities that must be kept on
schedule.
Earliest Start and Earliest Finish Times Now let’s
look at the steps employed in finding a critical path.
Fundamental in this process is the earliest start time
for each activity.
To illustrate this idea, consider activity D, “design
facility.” Now assume that the project starts at time
zero and ask yourself:
“What is the earliest time at which activity
D can start?”
Clearly, it cannot start until activity A is complete (3
weeks). However, it also cannot start before the
dummy activity is complete (0 weeks).
Since the dummy cannot start until B and C are
complete (a total of 8 weeks), we see that D cannot
start until 8 weeks have passed.
In this calculation, it is crucial to note that activities
A and B both start at time 0.
A is complete
C is
after
complete
3 weeks.
after 5 + 3 = 8 weeks
B requires another 2 weeks
Design [4]
B is complete after
5 weeks and C can start
Select
2
5
D
Site [3]
1
E
A
Select
4
B
6
F
Personnel [2]
Construct [8]
Move
[2]
G
Hire [4]
Plan [5]
C
3
Personnel
Requirements [3]
H
7
8
I
Financial
Arrangements [5]
Train [3]
J
9
Thus, after 8 weeks, both A and C are complete and
D can start. In other words,
earliest start for activity D = 8 weeks
Another important concept is earliest finish time for
each activity. If we let
ES = earliest start time for a given activity
EF = earliest finish time for a given activity
t = expected activity time for a given activity
Then, for a given activity, the relation between
earliest start time and earliest finish time is
EF = ES + t
Consider the following rule:
Earliest Start Time Rule
The ES time for an activity leaving a particular node
is the largest of the EF times for all activities
entering the node.
Looking at nodes 1 , 2 , 3 , and 4 of the previous
network, the result is shown in brackets below
[ES, EF].
The ES rule says that
the ES for activity D
is equal to the larger
D [8,12]
2
EF
for
all
activities
A [0,3]
entering node 2.
1
Continuing to each
4
B [0,5]
F [8,10]
C [5,8]
3
I [5,10]
node in a forward
pass through the
entire network, the
values [ES, EF] are
then computed for
each activity.
The network with the ES and EF values is given
below:
D [8,12]
2
5
A [0,3]
E [12,20]
1
F [8,10]
4
6
H
[12,12]
B [0,5]
C [5,8]
3
8
7
J [20,23]
9
I [5,10]
Note that the earliest finish time for J is 23 weeks.
Therefore, the earliest completion time for the entire
project is 23 weeks. We have now answered the 1st
question.
Latest Start and Latest Finish Times We now
proceed with a backward pass calculation in order
to identify possible start and completion dates, the
activities on the critical path, and how long
noncritical activities may be delayed without
affecting the overall completion date (answering the
3rd, 4th, and 5th questions).
Now that we have the target completion date of 23
weeks, we can work backward from this date,
determining the latest date each activity can finish
without delaying the entire project.
The backward pass begins at the completion node,
node 9 . Then, trace back through the network
computing the latest start time and latest finish time
for each activity.
LS = latest start time for a given activity
LF = latest finish time for a given activity
The relation between these quantities is
LS = LF - t
The general rule is
Latest Finish Time Rule
The LF time for an activity entering a particular node
is the smallest of the LS times for all activities
leaving that node.
The complete network with the LS and LF values
(given below the [ES,EF] values) is shown below:
A [0,3]
5
[5,8]
6
4
C [5,8]
1
D [8,12]
2 [8,12]
3
H
[12,12]
[18,20] 7
8
9
Slack and the Critical Path The next step of the
algorithm is to identify the amount of slack, or free
time, associated with each activity.
Slack is the amount of time an activity can be
delayed without affecting the completion date for the
overall project.
Slack is the same concept covered in linear
programming and is the extra time that could be
spent on that path without affecting the length of the
critical path.
For each activity, the slack is computed as:
Slack = LS – ES = LF – EF
For example, the slack for activity G is:
Slack for G = LS for G – ES for G
= 16 – 10 = 6 weeks
Every activity on the critical path should have a
slack of 0. This means that the activity cannot be
delayed without affecting the entire project.
Therefore, any activity with a slack of 0 is a critical
activity and is on the critical path.
The critical path activities are those with 0 slack.
Spreadsheet Approach for the Network The
spreadsheet solution of this problem is most easily
done with =D2+C2
an activity-on-the-node
(AON) approach.
=MIN(F5)
=F2-D2
=IF(H2=0,”YES”,”NO”)
=G2-C2
=MIN(F4,F10)
=MAX(E3)
=MIN(F5,F7)
=MIN(F6)
=MAX(E2,E4)
=MAX(E5)
=MIN(F11)
=MAX(E4)
=MIN(F8,F9)
=MAX(E7)
=MIN(F11)
=MIN(F11)
=MAX(E7)
=E13
=MAX(E3)
=MAX(E6,E8,E9)=E13
=MAX(E2:E11)
We have now answered the following questions:
1. What is the expected project completion date?
Answer: 23 weeks
3. What are the scheduled start and completion
dates for each specific activity?
Answer: An activity may be scheduled to
start at any date between “earliest
start” and “latest start.” The
scheduled completion date will be
“start date + expected activity time.”
4. What activities are critical in the sense that
they must be completed exactly as scheduled
in order to meet the target for overall project
completion?
Answer: The activities on the critical path are:
B, C, D, E, J.
5. How long can noncritical activities be delayed
before a delay in the overall completion date
is incurred?
Answer: Any activity may be started as late
as the “latest start” date without
delaying the overall project
completion.
The remaining questions will be answered later.
It is clear from the critical path analysis that we have
a problem. The board of directors wants to start
operating in Des Moines in 22 weeks, and with the
current plan 23 weeks are required.
WAYS OF REDUCING PROJECT DURATION
There are two basic approaches to reducing the time
required to complete a project:
1. A strategic analysis: Here the analyst asks:
“Does this project have to be done the way it
is currently diagrammed?”
In particular, “Do all of the activities on the
critical path have to be done in the specified
order?” Can we make arrangements to
accomplish some of these activities in a
different way not on the critical path?
2. A tactical approach: In this approach, the
analyst assumes that the current diagram is
appropriate and works at reducing the time of
certain activities on the critical path by
devoting more resources to them.
The current expected times assume a certain
allocation of resources. For example, the 8
weeks for construction (activity E) assumes a
regular 8-hour workday. The contractor can
complete the job more rapidly by working
overtime, but at increased costs.
The tactical approach takes into consideration CPM
models, which will be discussed later. For now, let’s
focus on the so-called strategic questions.
A Strategic Analysis This analysis is analogous to
“What if?” analysis done with spreadsheets.
After reviewing the network, it is discovered that the
current network assumes that activity J, the training
of new employees, must be carried out in the new
building (after E is complete), and after records and
key personnel have been moved (after H is
complete).
Perhaps these requirements can be changed. First,
J can be accomplished independently of H.
Moreover, an alternative training facility can be
secured by arranging to use surplus classroom
space in Des Moines at a minimal cost. The new
employees can be trained and ready to start the
moment that construction ends.
On the other hand, a new activity will have to be
added to the activity list:
secure a training facility (activity K).
All of these changes to the network may have
created a new critical path with a still unsatisfactory
minimum time (i.e., one greater than 22 weeks).
Spreadsheet Output for the Redefined Network This
redefined activity list is shown in the form of the
activity-on-the-arc (AOA) diagram.
Design [4]
2
D
A
5
E
Select
Site [3]
1
Select F
Plan [5]
4
6
H
Personnel [2]
B
7
C
Personnel
Requirements [3]
8
I
3
Financial
Arrangements [5]
J
9
Here is the same network using the Activity-on-theNode (AON) approach.
A
D
E
B
C
F
G
J
I
H
K
The modified spreadsheet is shown below:
=F2-D2
=IF(H2=0,”YES”,”NO”)
=D2+C2
=G2-C2
=MIN(F5)
=MIN(F4,F10)
=MAX(E3)
=MIN(F5,F7)
=MAX(E2,E4)
=MIN(F6)
=MAX(E5)
=E14
=MAX(E4)
=MIN(F8,F9,F12)
=MAX(E4)
=MIN(F11)
=MAX(E7)
=E14
=MAX(E3)
=E14
=MAX(E8,E12) =E14
=MAX(E7)
=MIN(F11)
=MAX(E2:E12)
Note that the redefined project completion time is 20
weeks and the new critical path is B-C-D-E.
Ex. A purchase manager at Costco is ordering coffee
beans from three suppliers in the United States.
The products needs to be on shelf in 3 months
(90 days). Supplier 1, 2 are existing suppliers,
while supplier 3 is a new supplier.
Before actually placing the orders, the manager
needs to first request samples from each
supplier. It takes 10 days, 10 days, and 14 days
to receive samples from supplier 1, 2, and 3,
respectively.
After all the samples arrive, it takes 5 days to
examine the samples, and decide the product
mix and order quantity. To save cost, it is desired
that the total order fills up a container.
The manager will then negotiate price and
contract terms with the supplier which normally
takes 10 days for existing suppliers, and 15 days
for new suppliers.
To assure the quality and safety of the products,
new suppliers need to get certified by the FDA
before a supply contract agreement can be
reached. It takes 20 days to obtain the
certification from FDA.
The purchase manager also needs to make
arrangements for the transportation of goods
from the supplier to the seaport in the U.S.,
secure a container from U.S. to Taiwan, and
transportation from seaport to Costco’s
warehouse, and warehouse to local stores.
It takes 5 days to make transportation
arrangements from the suppliers’ sites to
seaport, 2 days to book a container from liners,
and 2 days to arrange the transportation in
Taiwan.
The transportation time from suppliers to the
seaport in the U.S. is 10 days, 20 days, and 10
days for supplier 1, 2, and 3, respectively. The
shipping time from the U.S. to Taiwan is 21 days.
After the goods arrive at the seaport in Taiwan, it
takes 7 days for the goods to pass the custom.
It takes 1 day to transport the products from the
seaport to Costco’s new warehouse. The
products is then packaged/labeled at the
warehouse, which normally takes 5 days for a
container load.
The products are then shipped from the
warehouse to Costco’s stores in Taipei,
Taichung, and Kaohsiung. The transportation
time is 1 day, 2 days and 3 days respectively.
The products are placed on the stores’ shelves
immediately after they arrive at stores.
1. Draw an AOA project network for this
purchase project.
2. Draw an AOA project network for this
purchase project.
3. Determine the duration of this purchase
process using CPM and the project networks
obtained in part 1 or part 2.
4. Implement the CPM in Excel to determine the
project duration.
End of Part 1
Please continue to Part 2