Project Management
Chapter 8
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8-1
Chapter Topics
■ The Elements of Project Management
■ CPM/PERT Networks
■ Probabilistic Activity Times
■ Formulating the CPM/PERT Network as a Linear
Programming Model
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8-2
Overview
■ Network representation is useful for project analysis.
■ Project examples: Construction of a building, organization
of a conference, installation of a computer system etc.
■ Networks show how project activities are organized and are
used to determine time duration of projects.
■ Network techniques used are:
▪ CPM (Critical Path Method)
▪ PERT (Project Evaluation and Review Technique)
■ Developed independently during late 1950’s.
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8-3
Elements of Project Management
■ Management is generally perceived as concerned with
planning, organizing, and control of an ongoing process
or activity.
■ Project Management is concerned with control of an
activity for a relatively short period of time after which
management effort ends.
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Project Planning
■ Objectives
■ Project Scope
■ Contract Requirements
■ Schedules
■ Resources
■ Control
■ Risk and Problem Analysis
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Project Planning
■ All activities (steps) of the project should be identified.
■ The sequential relationships of the activities (which
activity comes first, which follows, etc.) is identified by
precedence relationships.
■ Steps of project planning:
■ Make time estimates for activities, determine project
completion time.
■ Compare project schedule objectives, determine
resource requirements.
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8-6
Elements of Project Management
Project Scheduling
■ Project Schedule : Timely completion of project.
■ Schedule development steps:
1. Define activities,
2. Sequence activities,
3. Estimate activity times,
4. Construct schedule.
■ Gantt chart and CPM/PERT techniques can be useful.
■ Computer software packages available, e.g. Microsoft Project.
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8-7
Elements of Project Management
Gantt Chart (1 of 2)
■ Popular, traditional technique, also known as a bar chart -developed
by Henry Gantt (1914).
■ Used in CPM/PERT for monitoring work progress.
■ A visual display of project schedule showing activity start and finish
times and where extra time is available.
■ Suitable for projects with few activities and precedence relationships.
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8-8
Elements of Project Management
Gantt Chart (2 of 2)
Figure 8.4 A Gantt chart
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8-9
The Project Network
CPM/PERT
Activity-on-Arc (AOA) Network
■ A branch reflects an activity of a project.
■ A node represents the beginning and end of activities, referred to as
events.
■ Branches in the network indicate precedence relationships.
■ When an activity is completed at a node, it has been realized.
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The Project Network
Concurrent Activities
■ Time duration of activities shown on branches.
■ Activities can occur at the same time (concurrently).
■ A dummy activity shows a precedence relationship but reflects
no passage of time.
■ Two or more activities cannot share the same start and end nodes.
Figure 8. 7 A Dummy Activity
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8-11
The Project Network
House Building Project Data
No. Activity
Activity Predecessor
Duration (Months)
1. Design house and
obtain financing
-
3
2. Lay foundation
1
2
3. Order Materials
1
1
4. Build house
2, 3
3
5. Select paint
2, 3
1
6. Select carpet
5
1
7. Finish work
4, 6
1
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The Project Network
AOA Network for House Building Project
Figure 8.6
Expanded Network for Building a
House Showing Concurrent Activities
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8-13
The Project Network
AON Network for House Building Project
Activity-on-Node (AON) Network
 A node represents an activity, with its label and time shown on the node
 The branches show the precedence relationships
 Convention used in Microsoft Project software
Label
Duration
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Figure 8.8
8-14
The Project Network
Paths Through a Network
Path
A
B
C
D
Events
1247
12567
1347
13567
Table 8.1
Paths Through the House-Building Network
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8-15
The Project Network
The Critical Path
The critical path is the longest path through the network; the
minimum time the network can be completed. From Figure 8.8:
Path A: 1  2  4  7
3 + 2 + 3 + 1 = 9 months
Path B: 1  2  5  6  7
3 + 2 + 1 + 1 + 1= 8 months
Path C: 1  3  4  7
3 + 1 + 3 + 1 = 8 months
Path D: 1  3  5  6  7
3 + 1 + 1 + 1 + 1 = 7 months
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8-16
The Project Network
Activity Start Times
Figure 8.9 Activity start time
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8-17
The Project Network
Activity-on-Node Configuration
Figure 8.10 Activity-on-Node Configuration
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8-18
The Project Network
Activity Scheduling : Earliest Times
■ ES is the earliest time an activity can start: ES = Maximum (EF)
■ EF is the earliest start time plus the activity time: EF = ES + t
Figure 8.11 Earliest activity start and finish times
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8-19
The Project Network
Activity Scheduling : Latest Times
■ LS is the latest time an activity can start without delaying critical path time:
LS = LF - t
■ LF is the latest finish time. LF = Minimum (LS)
Figure 8.12 Latest activity start and finish times
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8-20
The Project Network
Activity Slack Time (1 of 2)
 Slack is the amount of time an activity can be delayed without
delaying the project: S = LS – ES = LF - EF
 Slack Time exists for those activities not on the critical path for
which the earliest and latest start times are not equal.
Activity
*1
*2
3
LS
0
3
4
ES
0
3
3
LF
3
5
5
EF
3
5
4
Slack, S
0
0
1
*4
5
6
5
6
7
5
5
6
8
7
8
8
6
7
0
1
1
*7
8
8
9
9
0
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Table 8.2
*Critical path
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The Project Network
Activity Slack Time (2 of 2)
Figure 8.13 Activity slack
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8-22
Probabilistic Activity Times
■ Activity time estimates usually cannot be made with
certainty.
■ PERT used for probabilistic activity times.
■ In PERT, three time estimates are used: most likely time
(m), the optimistic time (a), and the pessimistic time (b).
■ These provide an estimate of the mean and variance of
a beta distribution:





variance: v  b - a
6
2





mean (expected time): t  a  4m  b
6
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Probabilistic Activity Times
Example (1 of 3)
Figure 8.14 Network for Installation Order Processing System
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Probabilistic Activity Times
Example (2 of 3)
Table 8.3
Activity Time Estimates for Figure 8.14
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Probabilistic Activity Times
Example (3 of 3)
Figure 8.15 Earliest and Latest Activity Times
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Probabilistic Activity Times
Expected Project Time and Variance
■ Expected project time is the sum of the expected times of the
critical path activities.
■ Project variance is the sum of the critical path activities’
variances
■ The expected project time is assumed to be normally distributed.
■ Epected project time (tp) and variance (vp) interpreted as the mean ()
and variance (2) of a normal distribution:
 = 25 weeks
2 = 62/9
= 6.9 (weeks)2
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8-27
Probability Analysis of a Project Network (1 of 2)
■ Using the normal distribution, probabilities are determined
by computing the number of standard deviations (Z) a
value is from the mean.
■ The Z value is used to find corresponding probability in
Table A.1, Appendix A.
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Probability Analysis of a Project Network (2 of 2)
Figure 8.16 Normal Distribution of Network Duration
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Probability Analysis of a Project Network
Example 1 (1 of 2)
What is the probability that the new order processing system will be
ready by 30 weeks?
µ = 25 weeks
2 = 6.9  = 2.63 weeks
Z = (x-)/  = (30 -25)/2.63 = 1.90
Z value of 1.90 corresponds to probability of .4713 in Table A.1,
Appendix A. Probability of completing project in 30 weeks or less:
(.5000 + .4713) = .9713.
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Probability Analysis of a Project Network
Example 1 (2 of 2)
Figure 8.17 Probability the Network Will Be Completed in 30 Weeks or Less
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Probability Analysis of a Project Network
Example 2 (1 of 2)
■ A customer will trade elsewhere if the new ordering system is not
working within 22 weeks. What is the probability that she will be
retained?
Z = (22 - 25)/2.63 = -1.14
■ Z value of 1.14 (ignore negative) corresponds to probability of
.3729 in Table A.1, appendix A.
■ Probability that customer will be retained is (0.5 - .3729) = .1271
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Probability Analysis of a Project Network
Example 2 (2 of 2)
Figure 8.18
Probability the Network Will Be Completed in 22 Weeks or Less
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Example Problem
Problem Statement and Data (1 of 2)
Given this network and the data on the following slide, determine the
expected project completion time and variance, and the probability
that the project will be completed in 28 days or less.
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Example Problem
Problem Statement and Data (2 of 2)
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Example Problem Solution (1 of 4)
Step 1: Compute the expected activity times and variances.
t  a  4m  b
6
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




v b-a
6
2





8-36
Example Problem Solution (2 of 4)
Step 2: Determine the earliest and latest activity times & slacks
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Example Problem Solution (3 of 4)
Step 3: Identify the critical path and compute expected
completion time and variance.
 Critical path (activities with no slack): 1  3  5  7
 Expected project completion time: tp = 9+5+6+4 = 24 days
 Variance: vp = 4 + 4/9 + 4/9 + 1/9 = 5 (days)2
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Example Problem Solution (4 of 4)
Step 4: Determine the Probability That the Project Will be
Completed in 28 days or less (µ = 24,  = 5)
Z = (x - )/ = (28 -24)/5 = 1.79
Corresponding probability from Table A.1, Appendix A, is .4633 and
P(x  28) = .4633 + .5 = .9633.
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Project Crashing and
Time-Cost Trade-Off Overview
■ Project duration can be reduced by assigning more resources to
project activities.
■ However, doing this increases project cost.
■ Decision is based on analysis of trade-off between time and
cost.
■ Project crashing is a method for shortening project duration by
reducing one or more critical activities to a time less than normal
activity time.
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Project Crashing and Time-Cost Trade-Off
Example Problem (1 of 5)
Figure 8.19 The Project Network for Building a House
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Project Crashing and Time-Cost Trade-Off
Example Problem (2 of 5)
Crash cost & crash time have a linear relationship:
Total Crash Cost
$2000

Total Crash Time 5 weeks
 $400 / wk
Figure 8.20
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Project Crashing and Time-Cost Trade-Off
Example Problem (3 of 5)
Table 8.4
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Project Crashing and Time-Cost Trade-Off
Example Problem (4 of 5)
Figure 8.21 Network with Normal Activity Times and Weekly Crashing Costs
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Project Crashing and Time-Cost Trade-Off
Example Problem (5 of 5)
As activities are crashed, the critical path may change and
several paths may become critical.
Figure 8.22
Revised Network with
Activity 1 Crashed
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Project Crashing and Time-Cost Trade-Off
Project Crashing with QM for Windows
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Exhibit 8.16
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Project Crashing and Time-Cost Trade-Off
General Relationship of Time and Cost (1 of 2)
■ Project crashing costs and indirect costs have an inverse
relationship.
■ Crashing costs are highest when the project is shortened.
■ Indirect costs increase as the project duration increases.
■ Optimal project time is at minimum point on the total
cost curve.
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Project Crashing and Time-Cost Trade-Off
General Relationship of Time and Cost (2 of 2)
Figure 8.23
The Time-Cost Trade-Off
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The CPM/PERT Network
Formulating as a Linear Programming Model
The objective is to minimize the project duration (critical path time).
General linear programming model with AOA convention:
Minimize Z = xi
subject to: i
xj - xi  tij for all activities i  j
xi, xj  0
Where:
xi = earliest event time of node i
xj = earliest event time of node j
tij = time of activity i  j
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The CPM/PERT Network
Example Problem Formulation and Data (1 of 2)
Figure 8.24
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The CPM/PERT Network
Example Problem Formulation and Data (2 of 2)
Minimize Z = x1 + x2 + x3 + x4 + x5 + x6 + x7
subject to:
x2 - x1  12
x3 - x2  8
x4 - x2  4
x4 - x3  0
x5 - x4  4
x6 - x4  12
x6 - x5  4
x7 - x6  4
xi, xj  0
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Project Crashing with Linear Programming
Example Problem – Model Formulation
Minimize Z = $400y12 + 500y23 + 3000y24 + 200y45 + 7000y46
+ 200y56 + 7000y67
subject to:
y12  5
y12 + x2 - x1  12 x7  36
y23  3
y23 + x3 - x2  8
xi, yij ≥ 0
y24  1
y24 + x4 - x2  4
Objective is to
y34  0
y34 + x4 - x3  0
minimize the
y45  3
y45 + x5 - x4  4
cost of crashing
y46  3
y46 + x6 - x4  12
y56  3
y56 + x6 - x5  4
y67  1
x67 + x7 - x6  4
xi = earliest event time of node I
xj = earliest event time of node j
yij = amount of time by which activity i  j is crashed
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Project Crashing with Linear Programming
Example Problem – Model Formulation
Minimize Z = $400y12 + 500y23 + 3000y24 + 200y45 + 7000y46
+ 200y56 + 7000y67
Objective is to
subject to:
y12  5
x2 - x1  12-y12
x7  36 minimize the
y 3
x - x  8-y
x , y ≥ 0 cost of crashing
23
3
2
23
i
ij
y24  1
x4 - x2  4-y24
y34  0
x4 - x3  0-y34
y45  3
x5 - x4  4-y45
y46  3
x6 - x4  12-y46
y56  3
x6 - x5  4-y56
y67  1
x7 - x6  4-x67
xi = earliest event time of node I
xj = earliest event time of node j
yij = amount of time by which activity i  j is crashed
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A Comparison of AON and AOA
Network Conventions
Activity on
Node (AON)
(a)
A
C
B
A
(b)
C
B
B
(c)
Activity
Meaning
A
C
A comes before B,
which comes
before C
A and B must both be
completed before C
can start
B and C cannot
begin until A is
completed
Activity on
Arc (AOA)
A
B
C
A
B
C
B
A
C
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A Comparison of AON and AOA
Network Conventions
Activity on
Node (AON)
A
C
B
D
(d)
A
C
(e)
B
D
Activity
Meaning
C and D cannot
begin until both A
and B are
completed
C cannot begin until
both A and B are
completed; D cannot
begin until B is
completed. A dummy
activity is introduced
in AOA
Activity on
Arc (AOA)
A
C
B
D
A
C
Dummy activity
B
D
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A Comparison of AON and AOA
Network Conventions
Activity on
Node (AON)
A
B
(f)
C
Activity
Meaning
D
B and C cannot
begin until A is
completed. D
cannot begin until
both B and C are
completed. A
dummy activity is
again introduced in
AOA.
Activity on
Arc (AOA)
A
Dummy
activity
B
D
C
8-56