taylor introms10 ppt 08 MEF
Project Management
Chapter 8
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-1
Chapter Topics
■ The Elements of Project Management
■ CPM/PERT Networks
■ Probabilistic Activity Times
■ Microsoft Project
■ Project Crashing and Time-Cost Trade-Off
■ Formulating the CPM/PERT Network as a Linear
Programming Model
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-2
Overview
■ Network representation is useful for project analysis.
■ 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.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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.
■ Primary elements of Project Management to be discussed:
Project Planning
Project Team
Project Control
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8-4
Elements of Project Management
Project Planning
■ Objectives
■ Project Scope
■ Contract Requirements
■ Schedules
■ Resources
■ Personnel
■ Control
■ Risk and Problem Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-5
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.
■ Project team may include workers.
■ 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. Potential
rewards, however, can be substantial.
■ Project manager must be able to coordinate various skills of team
members into a single focused effort.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-6
The Project Management Process
Figure 8.1
The project management process
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8-7
Elements of Project Management
Scope Statement
■
Document providing common understanding of project.
■
Justification describing the factors giving rise to need for project.
■
Expected results and what constitutes success.
■
List of necessary documents and planning reports.
■
Statement of work (SOW) - a planning document for
individuals, team members, groups, departments, subcontractors
and suppliers, describing what are required for successful
completion on time.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-8
Elements of Project Management
Work Breakdown Structure (WBS) (1 of 2)
■
WBS breaks down project into major components (modules).
■
Modules are further broken down into activities and, finally, into
individual tasks.
■
Identifies activities, tasks, resource requirements and relationships
between modules and activities.
■
Helps avoid duplication of effort.
■
Basis for project development, management , schedule, resources and
modifications.
■
Approaches for WBS development:
1. Top down process 2. Brainstorm entire project
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8-9
Elements of Project Management
Work Breakdown Structure (2 of 2)
Figure 8.2 WBS for Computer Order-processing System Project
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8-10
Elements of Project Management
Responsibility Assignment Matrix (1 of 2)
■ Project manager assigns work elements to organizational units,
departments, groups, individuals or subcontractors.
■ Uses an organizational breakdown structure (OBS).
■ OBS is a table or a chart showing which organizational units are
responsible for work items.
■ OBS leads to the responsibility assignment matrix (RAM)
■ RAM shows who is responsible for doing the necessary work in
the project
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8-11
Elements of Project Management
Responsibility Assignment Matrix (2 of 2)
Figure 8.3
A responsibility assignment matrix
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8-12
Elements of Project Management
Project Scheduling
■ Project Schedule evolves from planning documents, with focus on
timely completion.
■ Critical element in project management – source of most conflicts and
problems.
■ Schedule development steps:
1. Define activities,
3. Estimate activity times,
2. Sequence activities,
4. Construct schedule.
■ Gantt chart and CPM/PERT techniques can be useful.
■ Computer software packages available, e.g. Microsoft Project.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-13
Elements of Project Management
Gantt Chart (1 of 2)
■ Popular, traditional technique, also known as a bar chart -developed
by Henry Gantt (1914).
■ Direct precursor of 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.
■ Drawback: precedence relationships are not always discernible.
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8-14
Elements of Project Management
Gantt Chart (2 of 2)
Figure 8.4 A Gantt chart
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8-15
Elements of Project Management
Project Control
■ Process of ensuring progress toward successful completion.
■ Monitoring project to minimize deviations from project plan and
schedule.
■ Corrective actions necessary if deviations occur.
■ Key elements of project control
Time management
Cost management
Performance management
Earned value analysis.
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8-16
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.
Figure 8.5
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Nodes and Branches
8-17
The Project Network
Concurrent Activities
■ Network aids in planning and scheduling.
■ 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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8-18
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
7. Finish work
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1
4, 6
1
8-19
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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8-20
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
Figure 8.8
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8-21
The Project Network
Paths Through a Network
Path
A
B
C
D
Events
1247
12567
1347
13567
Table 8.1
Paths Through the House-Building Network
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8-22
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
Path B: 1 2 5 6 7
Path C: 1 3 4 7
Path D: 1 3 5 6 7
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3 + 2 + 3 + 1 = 9 months
3 + 2 + 1 + 1 + 1= 8 months
3 + 1 + 3 + 1 = 8 months
3 + 1 + 1 + 1 + 1 = 7 months
8-23
The Project Network
Activity Start Times
Figure 8.9 Activity start time
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8-24
The Project Network
Activity-on-Node Configuration
Figure 8.10 Activity-on-Node Configuration
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8-25
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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8-26
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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8-27
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.
Shared Slack is slack available for a sequence of activities.
Activity LS
ES
LF
EF Slack, S
Table 8.2
*1
0
0
3
3
0
*2
3
*4
3
4
5
3
3
5
5
5
8
5
4
8
0
1
0
5
6
6
7
5
6
7
8
6
7
1
1
*7
8
8
9
9
0
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*Critical path
8-28
The Project Network
Activity Slack Time (2 of 2)
Figure 8.13 Activity slack
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8-29
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
(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:
= 25 weeks
2 = 62/9
= 6.9 (weeks)2
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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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8-38
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 .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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8-40
CPM/PERT Analysis with
QM for Windows & Excel QM (1 of 2)
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CPM/PERT Analysis with
QM for Windows & Excel QM (2 of 2)
Exhibit 8.2
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Analysis with Microsoft Project (1 of 13)
Microsoft Project handles only AON networks.
Exhibit 8.3
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Analysis with Microsoft Project (2 of 13)
Exhibit 8.4
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Analysis with Microsoft Project (3 of 13)
Exhibit 8.5
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Analysis with Microsoft Project (4 of 13)
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Exhibit 8.6
8-46
Analysis with Microsoft Project (5 of 13)
Figure 8.7
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Analysis with Microsoft Project (6 of 13)
Figure 8.8
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Analysis with Microsoft Project (7 of 13)
Exhibit 8.9
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Analysis with Microsoft Project (8 of 13)
Exhibit 8.10
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Analysis with Microsoft Project (9 of 13)
Exhibit 8.11
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Analysis with Microsoft Project (10 of 13)
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Figure 8.12
8-52
Analysis with Microsoft Project (11 of 13)
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Figure 8.13
8-53
Analysis with Microsoft Project (12 of 13)
Exhibit 8.14
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Analysis with Microsoft Project (13 of 13)
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Exhibit 8.15
8-55
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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8-56
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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8-57
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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8-60
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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8-61
Project Crashing and Time-Cost Trade-Off
Project Crashing with QM for Windows
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Exhibit 8.16
8-62
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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8-63
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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8-64
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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8-66
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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8-67
The CPM/PERT Network
Example Problem Solution with Excel (1 of 4)
B6:B12
Exhibit 8.17
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The CPM/PERT Network
Example Problem Solution with Excel (2 of 4)
Exhibit 8.18
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The CPM/PERT Network
Example Problem Solution with Excel (3 of 4)
Exhibit 8.19
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The CPM/PERT Network
Example Problem Solution with Excel (4 of 4)
Exhibit 8.20
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8-71
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 30
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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8-72
Project Crashing with Linear Programming
Excel Solution (1 of 3)
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Exhibit 8.21
8-73
Project Crashing with Linear Programming
Excel Solution (2 of 3)
Exhibit 8.22
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8-74
Project Crashing with Linear Programming
Excel Solution (3 of 3)
Exhibit 8.23
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8-75
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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8-76
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
8-78
Example Problem Solution (2 of 4)
Step 2: Determine the earliest and latest activity times & slacks
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8-79
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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8-80
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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8-81
Chapter 8
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-1
Chapter Topics
■ The Elements of Project Management
■ CPM/PERT Networks
■ Probabilistic Activity Times
■ Microsoft Project
■ Project Crashing and Time-Cost Trade-Off
■ Formulating the CPM/PERT Network as a Linear
Programming Model
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-2
Overview
■ Network representation is useful for project analysis.
■ 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.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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.
■ Primary elements of Project Management to be discussed:
Project Planning
Project Team
Project Control
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-4
Elements of Project Management
Project Planning
■ Objectives
■ Project Scope
■ Contract Requirements
■ Schedules
■ Resources
■ Personnel
■ Control
■ Risk and Problem Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-5
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.
■ Project team may include workers.
■ 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. Potential
rewards, however, can be substantial.
■ Project manager must be able to coordinate various skills of team
members into a single focused effort.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-6
The Project Management Process
Figure 8.1
The project management process
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-7
Elements of Project Management
Scope Statement
■
Document providing common understanding of project.
■
Justification describing the factors giving rise to need for project.
■
Expected results and what constitutes success.
■
List of necessary documents and planning reports.
■
Statement of work (SOW) - a planning document for
individuals, team members, groups, departments, subcontractors
and suppliers, describing what are required for successful
completion on time.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-8
Elements of Project Management
Work Breakdown Structure (WBS) (1 of 2)
■
WBS breaks down project into major components (modules).
■
Modules are further broken down into activities and, finally, into
individual tasks.
■
Identifies activities, tasks, resource requirements and relationships
between modules and activities.
■
Helps avoid duplication of effort.
■
Basis for project development, management , schedule, resources and
modifications.
■
Approaches for WBS development:
1. Top down process 2. Brainstorm entire project
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-9
Elements of Project Management
Work Breakdown Structure (2 of 2)
Figure 8.2 WBS for Computer Order-processing System Project
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-10
Elements of Project Management
Responsibility Assignment Matrix (1 of 2)
■ Project manager assigns work elements to organizational units,
departments, groups, individuals or subcontractors.
■ Uses an organizational breakdown structure (OBS).
■ OBS is a table or a chart showing which organizational units are
responsible for work items.
■ OBS leads to the responsibility assignment matrix (RAM)
■ RAM shows who is responsible for doing the necessary work in
the project
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-11
Elements of Project Management
Responsibility Assignment Matrix (2 of 2)
Figure 8.3
A responsibility assignment matrix
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8-12
Elements of Project Management
Project Scheduling
■ Project Schedule evolves from planning documents, with focus on
timely completion.
■ Critical element in project management – source of most conflicts and
problems.
■ Schedule development steps:
1. Define activities,
3. Estimate activity times,
2. Sequence activities,
4. Construct schedule.
■ Gantt chart and CPM/PERT techniques can be useful.
■ Computer software packages available, e.g. Microsoft Project.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-13
Elements of Project Management
Gantt Chart (1 of 2)
■ Popular, traditional technique, also known as a bar chart -developed
by Henry Gantt (1914).
■ Direct precursor of 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.
■ Drawback: precedence relationships are not always discernible.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-14
Elements of Project Management
Gantt Chart (2 of 2)
Figure 8.4 A Gantt chart
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-15
Elements of Project Management
Project Control
■ Process of ensuring progress toward successful completion.
■ Monitoring project to minimize deviations from project plan and
schedule.
■ Corrective actions necessary if deviations occur.
■ Key elements of project control
Time management
Cost management
Performance management
Earned value analysis.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
8-16
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.
Figure 8.5
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Nodes and Branches
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The Project Network
Concurrent Activities
■ Network aids in planning and scheduling.
■ 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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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
7. Finish work
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1
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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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
Figure 8.8
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The Project Network
Paths Through a Network
Path
A
B
C
D
Events
1247
12567
1347
13567
Table 8.1
Paths Through the House-Building Network
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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
Path B: 1 2 5 6 7
Path C: 1 3 4 7
Path D: 1 3 5 6 7
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3 + 2 + 3 + 1 = 9 months
3 + 2 + 1 + 1 + 1= 8 months
3 + 1 + 3 + 1 = 8 months
3 + 1 + 1 + 1 + 1 = 7 months
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The Project Network
Activity Start Times
Figure 8.9 Activity start time
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The Project Network
Activity-on-Node Configuration
Figure 8.10 Activity-on-Node Configuration
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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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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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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.
Shared Slack is slack available for a sequence of activities.
Activity LS
ES
LF
EF Slack, S
Table 8.2
*1
0
0
3
3
0
*2
3
*4
3
4
5
3
3
5
5
5
8
5
4
8
0
1
0
5
6
6
7
5
6
7
8
6
7
1
1
*7
8
8
9
9
0
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*Critical path
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The Project Network
Activity Slack Time (2 of 2)
Figure 8.13 Activity slack
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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
(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:
= 25 weeks
2 = 62/9
= 6.9 (weeks)2
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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 .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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CPM/PERT Analysis with
QM for Windows & Excel QM (1 of 2)
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CPM/PERT Analysis with
QM for Windows & Excel QM (2 of 2)
Exhibit 8.2
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Analysis with Microsoft Project (1 of 13)
Microsoft Project handles only AON networks.
Exhibit 8.3
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Analysis with Microsoft Project (2 of 13)
Exhibit 8.4
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Analysis with Microsoft Project (3 of 13)
Exhibit 8.5
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Analysis with Microsoft Project (4 of 13)
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Exhibit 8.6
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Analysis with Microsoft Project (5 of 13)
Figure 8.7
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Analysis with Microsoft Project (6 of 13)
Figure 8.8
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Analysis with Microsoft Project (7 of 13)
Exhibit 8.9
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Analysis with Microsoft Project (8 of 13)
Exhibit 8.10
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Analysis with Microsoft Project (9 of 13)
Exhibit 8.11
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Analysis with Microsoft Project (10 of 13)
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Figure 8.12
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Analysis with Microsoft Project (11 of 13)
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Figure 8.13
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Analysis with Microsoft Project (12 of 13)
Exhibit 8.14
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Analysis with Microsoft Project (13 of 13)
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Exhibit 8.15
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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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The CPM/PERT Network
Example Problem Solution with Excel (1 of 4)
B6:B12
Exhibit 8.17
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The CPM/PERT Network
Example Problem Solution with Excel (2 of 4)
Exhibit 8.18
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The CPM/PERT Network
Example Problem Solution with Excel (3 of 4)
Exhibit 8.19
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The CPM/PERT Network
Example Problem Solution with Excel (4 of 4)
Exhibit 8.20
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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 30
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
Excel Solution (1 of 3)
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Exhibit 8.21
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Project Crashing with Linear Programming
Excel Solution (2 of 3)
Exhibit 8.22
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Project Crashing with Linear Programming
Excel Solution (3 of 3)
Exhibit 8.23
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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
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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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