PERT/CPM Models for Project Management
Irwin/McGraw-Hill
8.1
© The McGraw-Hill Companies, Inc., 2003
Project Management
•
Characteristics of Projects
–
–
–
–
•
Unique, one-time operations
Involve a large number of activities that must be planned and coordinated
Long time-horizon
Goals of meeting completion deadlines and budgets
Examples
– Building a house
– Planning a meeting
– Introducing a new product
•
PERT—Project Evaluation and Review Technique
CPM—Critical Path Method
– A graphical or network approach for planning and coordinating large-scale projects.
McGraw-Hill/Irwin
8.2
© The McGraw-Hill Companies, Inc., 2003
Example: Building a House
Time (Days)
Immediate
Predecessor
Foundation
4
—
Framing
10
Foundation
Plumbing
9
Framing
Electrical
6
Framing
Wall Board
8
Plumbing, Electrical
Siding
16
Framing
Paint Interior
5
Wall Board
Paint Exterior
9
Siding
Fixtures
6
Int. Paint, Ext. Paint
Activity
McGraw-Hill/Irwin
8.3
© The McGraw-Hill Companies, Inc., 2003
Gantt Chart
Start
Activity
5
10
Days After Start
15
20
25
30
5
10
15
35
40
45
50
35
40
45
50
Foundation
Framing
Plumbing
Electrical
Wall Board
Siding
Paint Interior
Paint Exterior
Fixtures
Start
McGraw-Hill/Irwin
20
25
30
Days After Start
8.4
© The McGraw-Hill Companies, Inc., 2003
PERT and CPM
•
Procedure
1. Determine the sequence of activities.
2. Construct the network or precedence diagram.
3. Starting from the left, compute the Early Start (ES) and Early Finish (EF) time for
each activity.
4. Starting from the right, compute the Late Finish (LF) and Late Start (LS) time for
each activity.
5. Find the slack for each activity.
6. Identify the Critical Path.
McGraw-Hill/Irwin
8.5
© The McGraw-Hill Companies, Inc., 2003
Notation
t
Duration of an activity
ES
The earliest time an activity can start
EF
The earliest time an activity can finish (EF = ES + t)
LS
The latest time an activity can start and not delay the project
LF
The latest time an activity can finish and not delay the project
Slack
The extra time that could be made available to an activity without
delaying the project (Slack = LS – ES)
Critical Path
The sequence(s) of activities with no slack
McGraw-Hill/Irwin
8.6
© The McGraw-Hill Companies, Inc., 2003
PERT/CPM Project Network
9
Plumbing
0
START
McGraw-Hill/Irwin
4
a
Foundation
10
b
Framing
c
6
d
Electrical
16
f
Siding
8.7
8
e
Wall
Board
5
g
6
0
FINISH
Paint
i
Interior
Fixtures
9
h
Paint
Exterior
© The McGraw-Hill Companies, Inc., 2003
Calculation of ES, EF, LF, LS, and Slack
GOING FORWARD
•
ES = Maximum of EF’s for all predecessors
•
EF = ES + t
GOING BACKWARD
•
LF = Minimum of LS for all successors
•
LS = LF – t
•
Slack = LS – ES = LF – EF
McGraw-Hill/Irwin
8.8
© The McGraw-Hill Companies, Inc., 2003
Building a House: ES, EF, LS, LF, Slack
Activity
ES
EF
LS
LF
Slack
(a) Foundation
0
4
0
4
0
(b) Framing
4
14
4
14
0
(c) Plumbing
14
23
17
26
3
(d) Electrical
14
20
20
26
6
(e) Wall Board
23
31
26
34
3
(f) Siding
14
30
14
30
0
(g) Paint Interior
31
36
34
39
3
(h) Paint Exterior
30
39
30
39
0
(i) Fixtures
39
45
39
45
0
McGraw-Hill/Irwin
8.9
© The McGraw-Hill Companies, Inc., 2003
PERT/CPM Project Network
8
e
4
a
4
d
5
g
5
i
0
START
0
FINISH
4
b
3
c
3
h
5
j
7
f
McGraw-Hill/Irwin
8.10
© The McGraw-Hill Companies, Inc., 2003
Example #2: ES, EF, LS, LF, Slack
Activity
ES
EF
LS
LF
Slack
a
0
4
0
4
0
b
0
4
1
5
1
c
4
7
5
8
1
d
4
8
4
8
0
e
4
12
5
13
1
f
4
11
6
13
2
g
8
13
8
13
0
h
8
11
10
13
2
i
13
18
13
18
0
j
11
16
13
18
2
McGraw-Hill/Irwin
8.11
© The McGraw-Hill Companies, Inc., 2003
Reliable Construction Company Project
•
The Reliable Construction Company has just made the winning bid of $5.4
million to construct a new plant for a major manufacturer.
•
The contract includes the following provisions:
–
–
A penalty of $300,000 if Reliable has not completed construction within 47 weeks.
A bonus of $150,000 if Reliable has completed the plant within 40 weeks.
Questions:
1.
2.
3.
4.
5.
6.
7.
8.
How can the project be displayed graphically to better visualize the activities?
What is the total time required to complete the project if no delays occur?
When do the individual activities need to start and finish?
What are the critical bottleneck activities?
For other activities, how much delay can be tolerated?
What is the probability the project can be completed in 47 weeks?
What is the least expensive way to complete the project within 40 weeks?
How should ongoing costs be monitored to try to keep the project within budget?
McGraw-Hill/Irwin
8.12
© The McGraw-Hill Companies, Inc., 2003
Activity List for Reliable Construction
Activity
Activity Description
Immediate
Predecessors
Estimated
Duration (Weeks)
A
Excavate
—
2
B
Lay the foundation
A
4
C
Put up the rough wall
B
10
D
Put up the roof
C
6
E
Install the exterior plumbing
C
4
F
Install the interior plumbing
E
5
G
Put up the exterior siding
D
7
H
Do the exterior painting
E, G
9
I
Do the electrical work
C
7
J
Put up the wallboard
F, I
8
K
Install the flooring
J
4
L
Do the interior painting
J
5
M
Install the exterior fixtures
H
2
N
Install the interior fixtures
K, L
6
McGraw-Hill/Irwin
8.13
© The McGraw-Hill Companies, Inc., 2003
Reliable Construction Project Network
START
A
Activity Code
0
A. Excavate
2
B. Foundation
C. Rough wall
B
D. Roof
4
E. Exterior plumbing
C
F. Interior plumbing
10
G. Exterior siding
H. Exterior painting
D
E
6
4
I
I. Electrical work
7
J. Wallboard
K. Flooring
L. Interior painting
G
F
7
5
M. Exterior fixtures
N. Interior fixtures
J
H
9
K
M
4
L
5
2
N
McGraw-Hill/Irwin
8
FINISH
0
8.14
6
© The McGraw-Hill Companies, Inc., 2003
The Critical Path
•
A path through a network is one of the routes following the arrows (arcs) from
the start node to the finish node.
•
The length of a path is the sum of the (estimated) durations of the activities
on the path.
•
The (estimated) project duration equals the length of the longest path through
the project network.
•
This longest path is called the critical path. (If more than one path tie for the
longest, they all are critical paths.)
McGraw-Hill/Irwin
8.15
© The McGraw-Hill Companies, Inc., 2003
The Paths for Reliable’s Project Network
Path
Length (Weeks)
StartA B C D G H M Finish
2 + 4 + 10 + 6 + 7 + 9 + 2 = 40
Start A B C E H M Finish
2 + 4 + 10 + 4 + 9 + 2 = 31
Start A B C E F J K N Finish
2 + 4 + 10 + 4 + 5 + 8 + 4 + 6 = 43
Start A B C E F J L N Finish
2 + 4 + 10 + 4 + 5 + 8 + 5 + 6 = 44
Start A B C I J K N Finish
2 + 4 + 10 + 7 + 8 + 4 + 6 = 41
Start A B C I J L N Finish
2 + 4 + 10 + 7 + 8 + 5 + 6 = 42
McGraw-Hill/Irwin
8.16
© The McGraw-Hill Companies, Inc., 2003
ES and EF Values for Reliable Construction
for Activities that have only a Single Predecessor
STA RT
D
6 ES = 16
EF = 22
G
ES = 22
7 EF = 29
H
0
A
2
ES = 0
EF = 2
B
4
ES = 2
EF = 6
C
10
ES = 6
EF = 16
E
4
ES = 16
EF = 20
F
I
ES = 16
7 EF = 23
J
8
ES = 20
EF = 25
5
9
K
4
L
5
M 2
N
6
FINISH 0
McGraw-Hill/Irwin
8.17
© The McGraw-Hill Companies, Inc., 2003
ES and EF Times for Reliable Construction
START
D
G
6 ES = 16
EF = 22
ES = 0
EF = 0
A
2
ES = 0
EF = 2
B
4
ES = 2
EF = 6
C
10
ES = 6
EF = 16
E
4
ES = 22
7 EF = 29
H
0
ES = 16
EF = 20
F
5
I
ES = 16
7 EF = 23
J
8
ES = 20
EF = 25
ES = 29
9 EF = 38
K
M
4 ES = 33
EF = 37
L
2 ES = 38
EF = 40
N
McGraw-Hill/Irwin
ES = 25
EF = 33
FINISH
0 ES = 44
EF = 44
8.18
6
5
ES = 33
EF = 38
ES = 38
EF = 44
© The McGraw-Hill Companies, Inc., 2003
LS and LF Times for Reliable’s Project
START
D
G
6 LS = 20
LF = 26
=0
0 LS
LF = 0
LS = 0
LF = 2
A
2
B
4
LS = 2
LF = 6
C
10
LS = 6
LF = 16
E
4
LS = 16
LF = 20
LS = 26
7 LF = 33
H
9
F
5
I
LS = 18
7 LF = 25
J
8
LS = 20
LF = 25
LS = 33
LF = 42
K
M
4 LS = 34
LF = 38
2 LS = 42
LF = 44
N
McGraw-Hill/Irwin
LS = 25
LF = 33
FINISH
0
LS = 44
LF = 44
8.19
L
= 33
5 LS
LF = 38
= 38
6 LS
LF = 44
© The McGraw-Hill Companies, Inc., 2003
The Complete Project Network
START
D
G
6 S = (16, 20)
F = (22, 26)
9
S = (0, 0)
F = (0, 0)
S = (0, 0)
F = (2, 2)
A
2
B
4
S = (2, 2)
F = (6, 6)
C
10
S = (6, 6)
F = (16, 16)
E
4
S = (22, 26)
7 F = (29, 33)
H
0
S = (16, 16)
F = (20, 20)
F
5
I
7
J
8
S = (20, 20)
F = (25, 25)
S = (25, 25)
F = (33, 33)
S = (29, 33)
F = (38, 42)
K
M
4 S = (33, 34)
F = (37, 38)
2 S = (38, 42)
F = (40, 44)
N
FINISH
McGraw-Hill/Irwin
S = (16, 18)
F = (23, 25)
0 S = (44, 44)
F = (44, 44)
8.20
6
L
5
S = (33, 33)
F = (38, 38)
S = (38, 38)
F = (44, 44)
© The McGraw-Hill Companies, Inc., 2003
Slack for Reliable’s Activities
McGraw-Hill/Irwin
Activity
Slack (LF–EF)
On Critical Path?
A
0
Yes
B
0
Yes
C
0
Yes
D
4
No
E
0
Yes
F
0
Yes
G
4
No
H
4
No
I
2
No
J
0
Yes
K
1
No
L
0
Yes
M
4
No
N
0
Yes
8.21
© The McGraw-Hill Companies, Inc., 2003
Spreadsheet to Calculate ES, EF, LS, LF, Slack
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
B
Activity
A
B
C
D
E
F
G
H
I
J
K
L
M
N
C
Description
Excavate
Foundation
Rough Wall
Roof
Exterior Plumbing
Interior Plumbing
Exterior Siding
Exterior Painting
Electrical Work
Wallboard
Flooring
Interior Painting
Exterior Fixtures
Interior Fixtures
McGraw-Hill/Irwin
D
Time
2
4
10
6
4
5
7
9
7
8
4
5
2
6
E
ES
0
2
6
16
16
20
22
29
16
25
33
33
38
38
Project Duration
8.22
F
EF
2
6
16
22
20
25
29
38
23
33
37
38
40
44
G
LS
0
2
6
20
16
20
26
33
18
25
34
33
42
38
H
LF
2
6
16
26
20
25
33
42
25
33
38
38
44
44
I
Slack
0
0
0
4
0
0
4
4
2
0
1
0
4
0
J
Critical?
Yes
Yes
Yes
No
Yes
Yes
No
No
No
Yes
No
Yes
No
Yes
44
© The McGraw-Hill Companies, Inc., 2003
PERT with Uncertain Activity Durations
•
If the activity times are not known with certainty, PERT/CPM can be used to
calculate the probability that the project will complete by time t.
•
For each activity, make three time estimates:
– Optimistic time: o
– Pessimistic time: p
– Most-likely time: m
McGraw-Hill/Irwin
8.23
© The McGraw-Hill Companies, Inc., 2003
Beta Distribution
Assumption: The variability of the time estimates follows the beta distribution.
Beta distribution
0
o
m
p
Elapsed time
McGraw-Hill/Irwin
8.24
© The McGraw-Hill Companies, Inc., 2003
PERT with Uncertain Activity Durations
Goal: Calculate the probability that the project is completed by time t.
Procedure:
1. Calculate the expected duration and variance for each activity.
2. Calculate the expected length of each path. Determine which path is the mean
critical path.
3. Calculate the standard deviation of the mean critical path.
4. Find the probability that the mean critical path completes by time t.
McGraw-Hill/Irwin
8.25
© The McGraw-Hill Companies, Inc., 2003
Expected Duration and Variance for Activities (Step #1)
•
The expected duration of each activity can be approximated as follows:

•
o 4m p
6
The variance of the duration for each activity can be approximated as follows:
2
 p  o
 2  

 6 
McGraw-Hill/Irwin
8.26
© The McGraw-Hill Companies, Inc., 2003
Expected Length of Each Path (Step #2)
•
The expected length of each path is equal to the sum of the expected durations
of all the activities on each path.
•
The mean critical path is the path with the longest expected length.
McGraw-Hill/Irwin
8.27
© The McGraw-Hill Companies, Inc., 2003
Standard Deviation of Mean Critical Path (Step #3)
•
The variance of the length of the path is the sum of the variances of all the
activities on the path.
2path = ∑ all activities on path 2
•
The standard deviation of the length of the path is the square root of the
variance.
 path   2path
McGraw-Hill/Irwin
8.28
© The McGraw-Hill Companies, Inc., 2003
Probability Mean-Critical Path Completes by t (Step #4)
•
What is the probability that the mean critical path (with expected length tpath and standard
deviation path) has duration ≤ t?
z
•
t  (t path)
 path
Use Normal Tables (Appendix A)
Probability
Density
Function
t
Ğ3
Ğ2

tpath
+
+2
+3
Path duration
McGraw-Hill/Irwin
8.29
© The McGraw-Hill Companies, Inc., 2003
Example
2- 4-5
b
0
START
3- 4-6
d
2- 3-4
a
0
FINISH
1- 3-7
c
2- 3-8
e
Question: What is the probability that the project will be finished by day 12?
McGraw-Hill/Irwin
8.30
© The McGraw-Hill Companies, Inc., 2003
Expected Duration and Variance of Activities (Step #1)
2

o 4m p
6
 p  o
  

 6 
2
Activity
o
m
p
a
2
3
4
3.00
1/
9
b
2
4
5
3.83
1/
4
c
1
3
7
3.33
1
d
3
4
6
4.17
1/
4
e
2
3
8
3.67
1
McGraw-Hill/Irwin
8.31
© The McGraw-Hill Companies, Inc., 2003
Expected Length of Each Path (Step #2)
Path
Expected Length of Path
a-b-d
3.00 + 3.83 + 4.17 = 11
a-c-e
3.00 + 3.33 + 3.67 = 10
The mean-critical path is a - b - d.
McGraw-Hill/Irwin
8.32
© The McGraw-Hill Companies, Inc., 2003
Standard Deviation of Mean-Critical Path (Step #3)
•
The variance of the length of the path is the sum of the variances of all the
activities on the path.
2path = ∑ all activities on path 2 = 1/9 + 1/4 + 1/4 = 0.61
•
The standard deviation of the length of the path is the square root of the
variance.
 path   2path  0.61 0.78
McGraw-Hill/Irwin
8.33
© The McGraw-Hill Companies, Inc., 2003
Probability Mean-Critical Path Completes by t=12 (Step #4)
•
The probability that the mean critical path (with expected length 11 and standard
deviation 0.71) has duration ≤ 12?
t  (t path) 1211
z

 1.41
 path
0.71
•
Then, from Normal Table: Prob(Project ≤ 12) = Prob(z ≤ 1.41) = 0.92
McGraw-Hill/Irwin
8.34
© The McGraw-Hill Companies, Inc., 2003
Reliable Construction Project Network
START
A
Activity Code
0
A. Excavate
2
B. Foundation
C. Rough wall
B
D. Roof
4
E. Exterior plumbing
C
F. Interior plumbing
10
G. Exterior siding
H. Exterior painting
D
E
6
4
I
I. Electrical work
7
J. Wallboard
K. Flooring
L. Interior painting
G
F
7
5
M. Exterior fixtures
N. Interior fixtures
J
H
9
K
M
4
L
5
2
N
McGraw-Hill/Irwin
8
FINISH
0
8.35
6
© The McGraw-Hill Companies, Inc., 2003
Reliable Problem: Time Estimates for Reliable’s
Project
Activity
o
m
p
Mean
Variance
A
1
2
3
2
1/
B
2
3.5
8
4
1
C
6
9
18
10
4
D
4
5.5
10
6
1
E
1
4.5
5
4
4/
F
4
4
10
5
1
G
5
6.5
11
7
1
H
5
8
17
9
4
I
3
7.5
9
7
1
J
3
9
9
8
1
K
4
4
4
4
0
L
1
5.5
7
5
1
M
1
2
3
2
1/
9
N
5
5.5
9
6
4/
9
McGraw-Hill/Irwin
8.36
9
9
© The McGraw-Hill Companies, Inc., 2003
Pessimistic Path Lengths for Reliable’s Project
Path
Pessimistic Length (Weeks)
StartA B C D G H M Finish
3 + 8 + 18 + 10 + 11 + 17 + 3 = 70
Start A B C E H M Finish
3 + 8 + 18 + 5 + 17 + 3 = 54
Start A B C E F J K N Finish
3 + 8 + 18 + 5 + 10 + 9 + 4 + 9 = 66
Start A B C E F J L N Finish
3 + 8 + 18 + 5 + 10 + 9 + 7 + 9 = 69
Start A B C I J K N Finish
3 + 8 + 18 + 9 + 9 + 4 + 9 = 60
Start A B C I J L N Finish
3 + 8 + 18 + 9 + 9 + 7 + 9 = 63
McGraw-Hill/Irwin
8.37
© The McGraw-Hill Companies, Inc., 2003
Three Simplifying Approximations of PERT/CPM
1. The mean critical path will turn out to be the longest path through the project
network.
2. The durations of the activities on the mean critical path are statistically
independent. Thus, the three estimates of the duration of an activity would
never change after learning the durations of some of the other activities.
3. The form of the probability distribution of project duration is the normal
distribution. By using simplifying approximations 1 and 2, there is some
statistical theory (one version of the central limit theorem) that justifies this as
being a reasonable approximation if the number of activities on the mean
critical path is not too small.
McGraw-Hill/Irwin
8.38
© The McGraw-Hill Companies, Inc., 2003
Calculation of Project Mean and Variance
Activities on Mean Critical Path
Mean
Variance
A
2
1/
B
4
1
C
10
4
E
4
4/
F
5
1
J
8
1
L
5
1
N
6
4/
Project duration
p = 44
2p = 9
McGraw-Hill/Irwin
8.39
9
9
9
© The McGraw-Hill Companies, Inc., 2003
Spreadsheet for PERT Three-Estimate Approach
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Activity
A
B
C
D
E
F
G
H
I
J
K
L
M
N
McGraw-Hill/Irwin
C
D
E
Time Estimates
o
m
p
1
2
3
2
3.5
8
6
9
18
4
5.5
10
1
4.5
5
4
4
10
5
6.5
11
5
8
17
3
7.5
9
3
9
9
4
4
4
1
5.5
7
1
2
3
5
5.5
9
F
On Mean
Critical Path
*
*
*
*
*
*
*
*
8.40
G
H

2
4
10
6
4
5
7
9
7
8
4
5
2
6

0.1111
1
4
1
0.4444
1
1
4
1
1
0
1
0.1111
0.4444
I
J
K
Mean Critical
Path

44

9
P(T<=d) =
where
d=
0.8413
47
© The McGraw-Hill Companies, Inc., 2003