This document outlines topics in project management including PERT/CPM, crashing, and other techniques. It provides an example of using PERT to analyze the critical path for a project to install air pollution control equipment within a 16 week deadline. It calculates the expected completion time as 15 weeks and probability of completing the project on time as 71.6%. The document also demonstrates how to crash the critical path by reducing activity times at added cost to meet a tighter 14 week deadline.

1. Project
Management
Agero, Venessa
Chua, Francisco
Erispe, Angelique
Lao, Inna
Santiago, Chiara 1

2. Chapter Outline
12.1 Introduction of Project Management
12.2 PERT/CPM
12.3 PERT/Cost
12.4 Project Crashing
12.5 Other Topics in Project Management
2

3. Things to remember
● The first step in planning and scheduling a project is
to develop the work breakdown structure.
● An activity is a job or task that is a part of a project.
● The beginning or end of an activity is called an
event.
● For more information, please check:
http://www.projectinsight.net/project-management-basics/basic-project-
management-phases
● http://www.prenhall.com/divisions/bp/app/russellcd/PROTECT/CHAPT
ERS/CHAP17/HEAD05.HTM
3

4. Quantitative analysis techniques
● The program evaluation
and review technique.
● PERT is probabilistic
● Developed by the Special
Projects Office of the U.S
Navy in 1958
● Used to plan and control
the Polaris missile
program.
● The critical path method.
● CPM is deterministic.
● Developed by J. E Kelly of
Remington Rand and M.R,
Walker of du Pont in 1957.
● Used to assist in the
building and maintenance
of chemical plants at du
Pont.
PERT CPM
4

5. Six steps of PERT/CPM
1. Define the project and all of its significant activities or tasks.
2. Develop the relationships among the activities. Decide which activities
must precede
others.
3. Draw the network connecting all of the activities.
4. Assign time and/or cost estimates to each activity.
5. Compute the longest time path through the network; this is called the
critical path.
6. Use the network to help plan, schedule, monitor, and control the project.
5

6. General Foundry Example
● General Foundry, Inc. has long been trying to avoid the
expense of installing air pollution control equipment.
● The local environmental protection group has recently given the
foundry 16 weeks to install a complex air filter system on its
main smokestack.
● General Foundry was warned that it will be forced to close
unless the device is installed in the allotted period.
● They want to make sure that installation of the filtering system
progresses smoothly and on time.
6

7. Drawing the PERT/CPM Network
● Activity-on-node (AON) where the nodes represent activities.
● Activity-on-arc (AOA) where the arcs are used to represent the
activities.
● The AON approach is easier and more commonly found in
software packages.
● One node represents the start of the project, one node for the
end of the project, and nodes for each of the activities.
● The arcs are used to show the predecessors for each activity.
7

8. Network of General Foundry
8

9. Activity Times
Times Estimates in PERT are:
Optimistic Time (a)=time activity will take if everything
goes as well as possible, small percentage happening
Pessimistic Time (b)=time activity will take assuming
unfavorable conditions, small percentage happening
Most likely Time (m)=most realistic time estimate to
complete activity
9

10. Activity Times
Beta Probability Distribution with Three Time Estimates
10

11. Activity Times Formulas
Expected Activity Time:
Dispersion/Variance of Activity Completion
Time:
11

12. Activity Times
Time Estimates (weeks) for General Fondry
12

13. How to Find the Critical Path
We accept the expected completion time for each task as the actual
time for now. The total of 25 weeks in Table 12.2 does not take into
account the obvious fact that some of the tasks could be taking place
at the same time. To find out how long the project will take we perform
the critical path analysis for the network.
The Critical Path is the longest path through the
network.
13

14. How to Find the Critical Path
General Fondry’s Network with Expected Activity Times
14

15. To find the critical path, we need to determine the following:
1. Earliest start time (ES): earliest time an activity can
begin
2. Earliest finish time (EF): the earliest time an activity can
end
3. Latest start time (LS): latest time an activity can begin
4. Latest finish time (LF): latest time an activity can begin
How to Find the Critical Path
15

16. In the nodes, the activity time and early and late
start finish times are represented like below:
How to Find the Critical Path
16

17. Earliest finish time = Earliest start time + Expected
activity TIme
EF = ES + t
Earliest start time = Largest of the earliest finish
times of immediate predecessors
ES= Largest EF of immediate predecessors
Computation of Earliest Times
17

18. At the start of the project, we set the time to
zero, so ES=0 for both A and B.
How to Find the Critical Path
18

19. General Fondry’s Earliest Start (ES) and Earliest Finish
(EF) Times
How to Find the Critical Path
19

20. Finding the Critical Path
Latest start time = Latest finish time - Activity
time (LS = LF - t)
Latest finish time = Smallest of latest start
times for following activities.
LF = Smallest LS of following activities.
Slack = LS - ES, or Slack = LF - EF
20

21. General Foundry’s Critical Path
TABLE 12.3
General Foundry’s
Schedule and Slack
Times
FIGURE 12.5
General Foundry’s
Latest Start (LS)
and Latest Finish
(LF) Times
21

22. General Foundry’s Critical Path
● From Table 12.3 we see activities A, C, E, G, and H have no slack
time.
● These are called critical activities and they are said to be on the critical
path.
● The total project completion time is 15 weeks.
● Industrial managers call this a boundary timetable.
22

23. Probability of Project Completion
● The critical path analysis helped us determine that the foundry’s
expected project completion time is 15 weeks.
● If the project is not complete in 16, the foundry will have to close.
● PERT uses the variance of critical path activities to help determine the
variance of the overall project.
23

24. (cont.)
*Project variance = ∑ variances of activities on the critical path
24

25. Results
Project standard deviation
= σT = √Project Variance
= √3.11 = 1.76 weeks
25

26. Probability of Project Completion
26
The probability that the project
can be completed in 16 weeks is
.716

27. Results
● The project’s expected completion date is 15
weeks.
● There is a 71.6% chance that the equipment will
be in place within the 16-week deadline.
● Five activities (A, C, E, G, H) are on the critical
path.
● Three activities (B, D, F) are not critical but have
some slack time built in.
27

28. Using Excel QM in the
General Foundry Example
28
Excel QM Initialization
Screen for General
Foundry Example
with Three Time
Estimates

29. Using Excel QM in the
General Foundry Example
29
Excel QM Input
Screen and Solution
for General Foundry
Example with Three
Time Estimates

30. Project Crashing
● Projects have deadlines that are impossible to meet
through normal procedures.
● Through exceptional methods, it’s possible to finish
the project in less time than normally required at a
greater cost.
● Reducing a project’s completion time is called
crashing.
30

31. Project Crashing
Crashing a project start with using the normal
time to create the critical path. The normal cost is
the cost for completing the activity using normal
procedures. If the project will not meet the required
deadline, extraordinary measures must be taken.
The crash time is the shortest possible activity
time and will require additional resources. The crash
cost is the price of completing the activity in the
earlier-than- normal time.
31

32. Four Steps to Project Crashing
1. Find the normal critical path and identify the critical
activities.
2. Get the crash cost per time period for all activities in
the network with this formula:
Crash cost/Time period = Crash cost - Normal
cost
Normal time - Crash time
32

33. Four Steps to Project Crashing
33
3. Select smallest crash cost per week on the critical
path and crash this to the maximum extent possible
or to the point at which your desired deadline has
been reached.
4. Check to be sure that the critical path you were
crashing is still critical. If it is still the longest path
through the network, return to step 3. If not, find the
new critical path and return to step 2.

34. General Foundry
Instead of 16 weeks, General Foundry has been given 14
weeks to install the new equipment. The critical path is 15
weeks. What options does the firm have?
The normal crash times and costs are shown in table 12.9. Crash costs
are assumed to be linear and Figure 12.11 shows the crash cost for activity
B.
Crashing activities B and A will shorten the completion time
to 14 but it creates a second critical path. Any further
crashing must be done to both critical paths.
34

35. General Foundry
Normal and Crash Data for General Foundry, Inc.
Table 12.9
Time (Weeks) Cost ($)
Activity Normal Crash Normal Crash Crash Cost per
Week ($)
Critical Path
A 2 1 22000 23000 1000 Yes
B 3 1 30000 34000 2000 No
C 2 1 26000 27000 1000 Yes
D 4 3 48000 49000 1000 No
E 4 2 56000 58000 1000 Yes
F 3 2 30000 30500 500 No
G 5 2 80000 86000 2000 Yes
H 2 1 16000 19000 3000 Yes
35

36. General Foundry
Crash and Normal Times and Costs for Activity B
Figure 12.11
36

37. Project Crashing with Linear
Programming
•Another approach to finding the best project crashing
schedule.
•The data needed are derived from the normal and
crash data for General Foundry and the project network
with activity times.
37

38. Project Crashing with Linear
Programming
General Foundry’s Network With Activity Times
Figure 12.12
38

39. Project Crashing with Linear
Programming
Objective function:
Minimize crash cost = 1,000YA+ 2,000YB+ 1,000YC+ 1,000YD+ 1,000YE+ 500YF+ 2,000YG+
3,000YH
Subject to these constraints:
YA≤1 YG≤3
YB≤2 YH≤1
YC≤1 Xfinish≤ 12
YD≤1 Xfinish≥ XH
YE≤2
YF≤1
39

40. Project Crashing with Linear
Programming
where,
XA= EF for activity A
XB= EF for activity B
XC= EF for activity C
XD= EF for activity D
XE= EF for activity E
XF= EF for activity F
XG= EF for activity G
XH= EF for activity H
Xstart= start time for project (usually 0)
Xfinish= earliest finish time for the project
40

41. Project Crashing with Linear
Programming
Constraints describing the network have the form:
EF time≥ EF time for predecessor + Activity time
EF≥ EFpredecessor+ (t–Y), or
X≥ Xpredecessor+ (t–Y)
For activity A,XA≥ Xstart+ (2 –YA) or XA–Xstart+ YA≥ 2
For activity B,XB≥ Xstart+ (3 –YB) or XB–Xstart+ YB≥ 3
For activity C,XC≥ XA+ (2 –YC) or XC–XA+ YC≥ 2
For activity D,XD≥ XB+ (4 –YD) or XD–XB+ YD≥ 4
For activity E,XE≥ XC+ (4 –YE) or XE–XC+ YE≥ 4
For activity F,XF≥ XC+ (3 –YF) or XF–XC+ YF≥ 3
For activity G,XG≥ XD+ (5 –YG) or XG–XD+ YG≥ 5
For activity G,XG≥ XE+ (5 –YG) or XG–XE+ YG≥ 5
For activity H,XH≥ XF+ (2 –YH) or XH–XF+ YH≥ 2
For activity H,XH≥ XG+ (2 –YH) or XH–XG+ YH≥ 2
41

42. Solution to Crashing Problem Using
Solver in Excel
Program 12.2 42

43. Other Topics in Project Management
Subprojects Milestones Resources Leveling Software
•Extremely large
projects
•Creation of several
small sub-activities
which is subproject
of the original
•Major events in a
project
•Gantt and PERT
charts to show
importance of
reaching these
events
•Adjusting at the
start of the event
•Even distribution
of resources
•Creates PERT and
Gantt charts
•Budget schedules,
adjust future start
times, and level
resource utilization
43

44. 1. The earliest start time for an activity is equal to 3. If activity A is not on the critical path, then the
a. the largest EF of the immediate predecessors. slack for A will equal
b. the smallest EF of the immediate predecessors. a. LF - EF.
c. the largest ES of the immediate predecessors. b. EF - ES.
d. the smallest ES of the immediate predecessors c. 0
2. The standard deviation for the PERT project is d. all of the above
approximately 4. The critical
path is the
a. the square root of the sum of the variances along thea.shortest path in a network.
critical path. b. longest
path in a network.
b. the sum of the critical path activity standard deviations. c. path with the smallest variance.
c. the square root of the sum of the variances of the project d. path with the largest variance.
activities. e. none of
the above.
d. all of the above. 5. PERT stands for
______________.
e. none of the above. 6. Project crashing can be
performed using a _____________.
Sample Exercises
44

45. 1. A
2. A
3. A
4. B
5. Program Evaluation and Review Techniques
6. Linear Programming model
Answers to sample exercises
45

46. 1. Sid Davidson is the personnel director of Babson and Willcount, a
company that specializes in consulting and research. One of the
training programs that Sid is considering for the middle-level managers
of Babson and Willcount is leadership training. Sid has listed a number
of activities that must be completed before a training program of this
nature could be conducted. The activities and immediate predecessors
appear in the following table.
Develop a network for this problem
Sample Problem
46

47. Answers to sample problem
47

48. http://www.projectinsight.net/project-management-basics/basic-project-
management-phases
https://www.youtube.com/watch?v=BjOLdukjAck
http://www.prenhall.com/divisions/bp/app/russellcd/PROTECT/CHAPTERS/
CHAP17/HEAD05.HTM
http://www.prenhall.com/divisions/bp/app/russellcd/PROTECT/CHAPTERS/
CHAP17/HEAD05.HTM
References
48