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

2. 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
Chapter 13 - Project Management

3. 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.
Chapter 13 - Project Management

4. 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
Chapter 13 - Project Management

5. 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.
Chapter 13 - Project Management

6. Network for Building a House
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
Chapter 13 - Project Management

7. Network for Building a House with Activity Times
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
Chapter 13 - Project Management

8. Expanded Network for Building a House Showing Concurrent Activities
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
Chapter 13 - Project Management

9. Paths Through the House-Building Network
The Project Network
Paths Through a Network
Table 8.1
Paths Through the House-Building Network
Chapter 13 - Project Management

10. 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, = 9 months
Path B: 1  2  3  4  5  6  7, = 8 months
Path C: 1  2  4  6  7, = 8 months
Path D: 1  2  4  5  6  7, = 7 months
Chapter 13 - Project Management

11. Alternative Paths in the Network
The Project Network
The Critical Path (2 of 2)
Figure 13.6
Alternative Paths in the Network
Chapter 13 - Project Management

12. Earliest Activity Start and Finish Times
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
Chapter 13 - Project Management

13. Latest Activity Start and Finish Times
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
Chapter 13 - Project Management

14. Earliest and Latest Activity Start and Finish Times
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
Chapter 13 - Project Management

15. Calculating Activity Slack Time (1 of 2)
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
Chapter 13 - Project Management

16. Calculating Activity Slack Time (2 of 2)
The Project Network
Calculating Activity Slack Time (2 of 2)
Table 8.2
Activity Slack
Chapter 13 - Project Management

17. 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:
Chapter 13 - Project Management

18. Network for Installation Order Processing System
Probabilistic Activity Times
Example (1 of 3)
Figure 13.11
Network for Installation Order Processing System
Chapter 13 - Project Management

19. Activity Time Estimates for Figure 13.11
Probabilistic Activity Times
Example (2 of 3)
Table 8.3
Activity Time Estimates for Figure 13.11
Chapter 13 - Project Management

20. Network with Mean Activity Times and Variances
Probabilistic Activity Times
Example (3 of 3)
Figure 13.12
Network with Mean Activity Times and Variances
Chapter 13 - Project Management

21. Earliest and Latest Activity Times
Probabilistic Activity Times
Earliest and Latest Activity Times and Slack
Figure 13.13
Earliest and Latest Activity Times
Chapter 13 - Project Management

22. Activity Earliest and Latest Times and Slack
Probabilistic Activity Times
Earliest and Latest Activity Times and Slack
Table 8.4
Activity Earliest and Latest Times and Slack
Chapter 13 - Project Management

23. 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
Chapter 13 - Project Management

24. 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.
Chapter 13 - Project Management

25. Normal Distribution of Network Duration
Probability Analysis of a Project Network (2 of 2)
Figure 13.14
Normal Distribution of Network Duration
Chapter 13 - Project Management

26. Probability Analysis of a Project Network Example 1 (1 of 2)
Z value of 1.90 corresponds to probability of in Table A.1, Appendix A. Probability of completing project in 30 weeks or less: ( ) =
2 =  = 2.63
Z = (x-)/  = (30 -25)/2.63 = 1.90
Chapter 13 - Project Management

27. Probability the Network Will Be Completed in 30 Weeks or Less
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
Chapter 13 - Project Management

28. Probability Analysis of a Project Network Example 2 (1 of 2)
Z = ( )/2.63 = -1.14
Z value of 1.14 (ignore negative) corresponds to probability of in Table A.1, appendix A.
Probability that customer will be retained is .1271
Chapter 13 - Project Management

29. Probability the Network Will Be Completed in 22 Weeks or Less
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
Chapter 13 - Project Management

30. Probability Analysis of a Project Network
CPM/PERT Analysis with QM for Windows
Exhibit 13.1
Chapter 13 - Project Management

31. 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.
Chapter 13 - Project Management

32. Activity-on-Node Configuration
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
Chapter 13 - Project Management

33. House-Building Network with AON
Activity-on-Node Networks and Microsoft Project
AON Network Diagram
Figure 13.18
House-Building Network with AON
Chapter 13 - Project Management

34. Activity-on-Node Networks and Microsoft Project
Microsoft Project (1 of 4)
Exhibit 13.2
Chapter 13 - Project Management

35. Activity-on-Node Networks and Microsoft Project
Microsoft Project (2 of 4)
Exhibit 13.3
Chapter 13 - Project Management

36. Activity-on-Node Networks and Microsoft Project
Microsoft Project (3 of 4)
Exhibit 13.4
Chapter 13 - Project Management

37. Activity-on-Node Networks and Microsoft Project
Microsoft Project (4 of 4)
Exhibit 13.5
Chapter 13 - Project Management

38. 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.
Chapter 13 - Project Management

39. Network for Constructing a House
Project Crashing and Time-Cost Trade-Off
Example Problem (1 of 5)
Figure 13.19
Network for Constructing a House
Chapter 13 - Project Management

40. Time-Cost Relationship for Crashing Activity 12
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
Chapter 13 - Project Management

41. Normal Activity and Crash Data for the Network in Figure 13.19
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
Chapter 13 - Project Management

42. Network with Normal Activity Times and Weekly Activity Crashing Costs
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
Chapter 13 - Project Management

43. Revised Network with Activity 12 Crashed
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
Chapter 13 - Project Management

44. Project Crashing and Time-Cost Trade-Off
Project Crashing with QM for Windows
Exhibit 13.6
Chapter 13 - Project Management

45. 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.
Chapter 13 - Project Management

46. Project Crashing and Time-Cost Trade-Off
General Relationship of Time and Cost (2 of 2)
Figure 13.23
A Time-Cost Trade-Off
Chapter 13 - Project Management

47. Formulating as a Linear Programming Model
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
Chapter 13 - Project Management

48. Example Problem Formulation and Data (1 of 2)
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
Chapter 13 - Project Management

49. Example Problem Formulation and Data (2 of 2)
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
Chapter 13 - Project Management

50. Example Problem Solution with Excel (1 of 4)
The CPM/PERT Network
Example Problem Solution with Excel (1 of 4)
Exhibit 13.7
Chapter 13 - Project Management

51. Example Problem Solution with Excel (2 of 4)
The CPM/PERT Network
Example Problem Solution with Excel (2 of 4)
Exhibit 13.8
Chapter 13 - Project Management

52. Example Problem Solution with Excel (3 of 4)
The CPM/PERT Network
Example Problem Solution with Excel (3 of 4)
Exhibit 13.9
Chapter 13 - Project Management

53. Example Problem Solution with Excel (4 of 4)
The CPM/PERT Network
Example Problem Solution with Excel (4 of 4)
Exhibit 13.10
Chapter 13 - Project Management

54. 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 = $400y y y y y y y67
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
Chapter 13 - Project Management

55. Probability Analysis of a Project Network
Example Problem – Excel Solution (1 of 3)
Exhibit 13.11
Chapter 13 - Project Management

56. Probability Analysis of a Project Network
Example Problem – Excel Solution (2 of 3)
Exhibit 13.12
Chapter 13 - Project Management

57. Probability Analysis of a Project Network
Example Problem – Excel Solution (3 of 3)
Exhibit 13.13
Chapter 13 - Project Management

58. 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.
Chapter 13 - Project Management

59. PERT Project Management Example Problem
Problem Statement and Data (2 of 2)
Chapter 13 - Project Management

60. PERT Project Management Example Problem Solution (1 of 4)
Step 1: Compute the expected activity times and variances.
Chapter 13 - Project Management

61. PERT Project Management Example Problem Solution (2 of 4)
Step 2: Determine the earliest and latest times at each node.
Chapter 13 - Project Management

62. 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
Chapter 13 - Project Management

63. 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 and P(x  28) =
Chapter 13 - Project Management

64. Chapter 13 - Project Management