Project Management

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Introduction to Management Science 8th
Edition by Bernard W. Taylor III
Chapter 13 Project Management
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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 1950s.

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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)
• variance

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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

• 25 weeks
• ?2 6.9 weeks

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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 y12 x2 -
x1 ? 12 x7 ? 30 y23 ? 3 y23 x3 - x2 ? 8
y67 ? 1 y24 ? 1 y24 x4 - x2 ? 4 x67 x7 -
x6 ? 4 y34 ? 0 y34 x4 - x3 ? 0 xj, yij ? 0
y45 ? 3 y45 x5 - x4 ? 4 y46 ? 3 y46 x6 -
x4 ? 12 y56 ? 3 y56 x6 - x5 ? 4
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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)
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PERT Project Management Example Problem Solution
(1 of 4)
Step 1 Compute the expected activity times and
variances.
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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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