Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
Chapter 13
Project Management
Chapter 13 - Project Management
1
Chapter Topics
The Elements of Project Management
The Project Network
Probabilistic Activity Times
Activity-on-Node Networks and Microsoft Project
Project Crashing and Time-Cost Trade-Off
Formulating the CPM/PERT Network as a Linear
Programming Model
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Overview
Uses networks for project analysis.
Networks show how projects are organized and are used to
determine time duration for completion.
Network techniques used are:
CPM (Critical Path Method)
PERT (Project Evaluation and Review Technique)
Developed during late 1950’s.
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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.
Primary elements of Project Management to be discussed:
Project Team
Project Planning
Project Control
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The Elements of Project Management
The Project Team
Project team typically consists of a group of individuals from
various areas in an organization and often includes outside
consultants.
Members of engineering staff often assigned to project
work.
Most important member of project team is the project
manager.
Project manager is often under great pressure because of
uncertainty inherent in project activities and possibility of
failure.
Project manager must be able to coordinate various skills of
team members into a single focused effort.
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The Elements of Project Management
The Project 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.
Figure 13.2
Network for Building a House
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The Project Network
Planning and Scheduling
Network aids in planning and scheduling.
Time duration of activities shown on branches:
Figure 13.3
Network for Building a House with Activity Times
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The Project Network
Concurrent Activities
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 13.4
Expanded Network for Building a House Showing Concurrent Activities
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The Project Network
Paths Through a Network
Table 8.1
Paths Through the House-Building Network
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The Project Network
The Critical Path (1 of 2)
The critical path is the longest path through the network;
the minimum time the network can be completed. In Figure
13.5:
Path A: 1  2  3  4  6  7, 3 + 2 + 0 + 3 + 1 = 9
months
Path B: 1  2  3  4  5  6  7, 3 + 2 + 0 + 1 + 1
+ 1 = 8 months
Path C: 1  2  4  6  7, 3 + 1 + 3 + 1 = 8 months
Path D: 1  2  4  5  6  7, 3 + 1 + 1 + 1 + 1 = 7
months
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The Project Network
The Critical Path (2 of 2)
Figure 13.6
Alternative Paths in the Network
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The Project Network
Activity Scheduling – Earliest Times
ES is the earliest time an activity can start. ESij = Maximum
(EFi)
EF is the earliest start time plus the activity time. EFij = ESij
+ tij
Figure 13.7
Earliest Activity Start and Finish Times
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The Project Network
Activity Scheduling – Earliest Times
LS is the latest time an activity can start without delaying
critical path time. LSij = LFij - tij
LF is the latest finish time. LFij = Minimum (LSj)
Figure 13.8
Latest Activity Start and Finish Times
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The Project Network
Activity Slack
Slack is the amount of time an activity can be delayed
without delaying the project.
Slack Time exists for those activities not on the critical path
for which the earliest and latest start times are not equal.
Shared Slack is slack available for a sequence of activities.
Figure 13.9
Earliest and Latest Activity Start and Finish Times
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The Project Network
Calculating Activity Slack Time (1 of 2)
Slack, Sij, computed as follows: Sij = LSij - ESij or Sij = LFij EFij
Figure 13.10
Activity Slack
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The Project Network
Calculating Activity Slack Time (2 of 2)
Table 8.2
Activity Slack
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Probabilistic Activity Times
Activity time estimates usually can not 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:
mean (expected time): t  a  4m  b
6

2
variance: v   b - a 
 6 
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Probabilistic Activity Times
Example (1 of 3)
Figure 13.11
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 13.11
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Probabilistic Activity Times
Example (3 of 3)
Figure 13.12
Network with Mean Activity Times and Variances
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Probabilistic Activity Times
Earliest and Latest Activity Times and Slack
Figure 13.13
Earliest and Latest Activity Times
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Probabilistic Activity Times
Earliest and Latest Activity Times and Slack
Table 8.4
Activity Earliest and Latest Times and Slack
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Probabilistic Activity Times
Expected Project Time and Variance
The expected project time is the sum of the expected times
of the critical path activities.
The project variance is the sum of the variances of the
critical path activities.
The expected project time is assumed to be normally
distributed (based on central limit theorem).
In example, expected project time (tp) and variance (vp)
interpreted as the mean () and variance (2) of a normal
distribution:
Critical Path Activity
 = 25 weeks
2 = 6.9 weeks
Variance
13
35
57
79
total
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1/9
16/9
4
62/9
23
Probability Analysis of a Project Network (1 of 2)
Using normal distribution, probabilities are determined by
computing number of standard deviations (Z) a value is from
the mean.
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 13.14
Normal Distribution of Network Duration
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Probability Analysis of a Project Network
Example 1 (1 of 2)
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.
2 = 6.9  = 2.63
Z = (x-)/  = (30 -25)/2.63 = 1.90
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Probability Analysis of a Project Network
Example 1 (2 of 2)
Figure 13.15
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)
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 .1271
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Probability Analysis of a Project Network
Example 2 (2 of 2)
Figure 13.16
Probability the Network Will Be Completed in 22 Weeks or Less
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Probability Analysis of a Project Network
CPM/PERT Analysis with QM for Windows
Exhibit 13.1
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Activity-on-Node Networks and Microsoft Project
The project networks developed so far have used the
“activity-on-arrow” (AOA) convention.
“Activity-on-node” (AON) is another method of creating a
network diagram.
The two different conventions accomplish the same thing,
but there are a few differences.
An AON diagram will often require more nodes than an
AOA diagram.
An AON diagram does not require dummy activities
because two “activities” will never have the same start and
end nodes.
Microsoft Project handles only AON networks.
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Activity-on-Node Networks and Microsoft Project
Node Structure
This node includes the
activity number in the upper
left-hand corner, the activity
duration in the lower left-hand
corner, and the earliest start
and finish times, and latest
start and finish times in the
four boxes on the right side of
the node.
Figure 13.17
Activity-on-Node Configuration
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Activity-on-Node Networks and Microsoft Project
AON Network Diagram
Figure 13.18
House-Building Network with AON
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Activity-on-Node Networks and Microsoft Project
Microsoft Project (1 of 4)
Exhibit 13.2
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Activity-on-Node Networks and Microsoft Project
Microsoft Project (2 of 4)
Exhibit 13.3
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Activity-on-Node Networks and Microsoft Project
Microsoft Project (3 of 4)
Exhibit 13.4
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Activity-on-Node Networks and Microsoft Project
Microsoft Project (4 of 4)
Exhibit 13.5
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Project Crashing and Time-Cost Trade-Off
Definition
Project duration can be reduced by assigning more
resources to project activities.
Doing this however 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.
Crashing achieved by devoting more resources to crashed
activities.
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Project Crashing and Time-Cost Trade-Off
Example Problem (1 of 5)
Figure 13.19
Network for Constructing a House
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Project Crashing and Time-Cost Trade-Off
Example Problem (2 of 5)
Crash cost and crash time
have linear relationship:
total crash cost/total crash
time = $2000/5 = $400/wk
Figure 13.20
Time-Cost Relationship for Crashing Activity 12
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Project Crashing and Time-Cost Trade-Off
Example Problem (3 of 5)
Table 8.5
Normal Activity and Crash Data for the Network in Figure 13.19
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Project Crashing and Time-Cost Trade-Off
Example Problem (4 of 5)
Figure 13.21
Network with Normal Activity Times and Weekly Activity 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 13.22
Revised Network with Activity 12 Crashed
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Project Crashing and Time-Cost Trade-Off
Project Crashing with QM for Windows
Exhibit 13.6
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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 13.23
A Time-Cost Trade-Off
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The CPM/PERT Network
Formulating as a Linear Programming Model
The objective is to determine the earliest time the project
can be completed (i.e., the critical path time).
General linear programming model:
Minimize Z = xi
subject to:
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)
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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The CPM/PERT Network
Example Problem Formulation and Data (2 of 2)
Figure 13.24
CPM/PERT Network for the House-Building Project with Earliest Event Times
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The CPM/PERT Network
Example Problem Solution with Excel (1 of 4)
Exhibit 13.7
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The CPM/PERT Network
Example Problem Solution with Excel (2 of 4)
Exhibit 13.8
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The CPM/PERT Network
Example Problem Solution with Excel (3 of 4)
Exhibit 13.9
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The CPM/PERT Network
Example Problem Solution with Excel (4 of 4)
Exhibit 13.10
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Probability Analysis of a Project Network
Example Problem – Model Formulation
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
Minimize Z = $400y12 + 500y23 + 3000y24 + 200y45 + 7000y46 +
200y56 + 7000y67
subject to:
y12  5
y23  3
y24  1
4
y34  0
y45  3
y46  3
y56  3
Chapter 13 - Project Management
y12 + x2 - x1  12
y23 + x3 - x2  8
y24 + x4 - x2  4
y34 + x4 - x3  0
y45 + x5 - x4  4
y46 + x6 - x4  12
y56 + x6 - x5  4
x7  30
y67  1
x67 + x7 - x6 
xj, yij  0
54
Probability Analysis of a Project Network
Example Problem – Excel Solution (1 of 3)
Exhibit 13.11
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Probability Analysis of a Project Network
Example Problem – Excel Solution (2 of 3)
Exhibit 13.12
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Probability Analysis of a Project Network
Example Problem – Excel Solution (3 of 3)
Exhibit 13.13
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PERT Project Management Example Problem
Problem Statement and Data (1 of 2)
Given the following data 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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PERT Project Management Example Problem
Problem Statement and Data (2 of 2)
Activity
12
13
23
24
34
35
45
Chapter 13 - Project Management
Time Estimates (weeks)
a
m
b
5
7
3
1
4
3
3
8
10
5
3
6
3
4
17
13
7
5
8
3
5
59
PERT Project Management Example Problem
Solution (1 of 4)
Step 1: Compute the expected activity times and variances.
t  a  4m  b
6





v b-a
6
Activity
12
13
23
24
34
35
44
Chapter 13 - Project Management
t
v
9
10
5
3
6
3
4
4
1
4/9
4/9
4/9
0
1/9
2





60
PERT Project Management Example Problem
Solution (2 of 4)
Step 2: Determine the earliest and latest times at each node.
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PERT Project Management 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  2  3  4  5
Expected project completion time (tp): 24 days
Variance: v = 4 + 4/9 + 4/9 + 1/9 = 5 days
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PERT Project Management Example Problem
Solution (4 of 4)
Step 4: Determine the Probability That the Project Will be
Completed in 28 days or less.
Z = (x - )/ = (28 -24)/5 = 1.79
Corresponding probability from Table A.1, Appendix A, is .4633
and P(x  28) = .9633.
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