Project Management
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Projects are unique, one-time operations
designed to accomplish a specific set of
objectives in a limited timeframe
Project managers are responsible for
managing the cost and timely completion of
the project
PERT/CPM are tools to support project
managers
Schware Foundry Inc.
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Schware Foundry, Inc., a metalworks plant,
has been given 16 weeks by the local
environmental protection agency to install a
complex air-filter system on its main
smokestack.
The project manager has identified the 8 key
activities that are necessary to complete the
smokestack modification as well as the
estimated completion times for each activity.
(These activities are shown on the next slide)
Immediate
Expected
Activity
Description
A
Build internal components
B
Modify roof and floor
C
Construct collection stack
D
Pour concrete & install
frame
E
Build high temp. burner
F
Install control system
Predecessors
--A
Time
2 weeks
3 weeks
2 weeks
B
C
C
4 weeks
4 weeks
3 weeks
G
Install air pollution device
D,E
5 weeks
H
Inspection & Testing
F,G
2 weeks
Project Management
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Determine all specific activities associated
with the project along with estimates for the
amount of time required to complete each
activity. Identify the interdependencies of
the activities
Create a network to depict the sequential
relationships between the activities
Perform PERT or CPM to determine the
critical activities and expected project
completion time.
Critical Path Analysis Definitions
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A path is a sequence of activities that leads
from the starting node of the project to the
finishing node of the project
The length of the path is the sum of the
activity times of the activities comprising the
path
The critical path is the path with the longest
length of time. It represents the minimum
project completion time.
Critical Activity Definitions
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Critical activities are the activities which make
up the critical path. Any delays with these
activities will hurt the project completion
time.
Slack time is the delay an activity can
experience without hurting the project
completion time, assuming that the other
activities stay on track with their schedule.
Information Recorded for Each Node
ESTi
i
EFTi
ti
LSTi LFTi
ti = time required to perform activity i
ESTi = earliest possible start time for activity i
EFTi = earliest possible finish time for activity i
LSTi = latest possible start time for activity i
LFTi = latest possible finish time for activity i
CPM Analysis Algorithm
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Step 1: Make a forward pass through the
network as follows: For each of these
activities, beginning at node i, compute:
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Earliest Start Time = the maximum of earliest
finish times for all of its immediate predecessors
Earliest Finish Time = (Earliest Start Time) +
(Time to complete activity i).
The project completion time is the
maximum of the Earliest Finish Times at the
completion node.
CPM Analysis Algorithm
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Step 2: Make a backwards pass through
the network as follows: Move sequentially
backwards from the last node, N, to node
N-1, to node N-2, etc. At a given node, j,
compute:
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Latest Finish Time = the minimum of the latest
start times for all activities that immediately
follow the activity (For node N, this is the
project completion time.)
Latest Start Time = (Latest Finish Time) - (Time
to complete activity j).
CPM Analysis Algorithm
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Step 3: Calculate the slack time for
each activity by:
Slack = (Latest Start) - (Earliest Start) or
= (Latest Finish) - (Earliest Finish).
A critical path is a path of activities, from node
1 to N, with 0 slack times.
Note that the slack for each noncritical activity is shared
with other noncritical activities on the same path.
PERT
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In the three-time estimate approach, the time to
complete an activity is assumed to follow a Beta
distribution.
An activity’s mean completion time is:
t = (a + 4m + b)/6
An activity’s completion time variance is:
((b-a)/6)2
 a = the optimistic completion time estimate
 b = the pessimistic completion time estimate
 m = the most likely completion time estimate
Immediate
Activity
A
B
C
D
E
F
G
H
Optimistic
Time
1 week
2 weeks
1 week
2 weeks
3 weeks
2 weeks
2 weeks
1 week
Most Likely
Pessimistic
Time
Time
2 weeks
3 weeks
2 weeks
4 weeks
4 weeks
3 weeks
5 weeks
2 weeks
3
4
4
5
6
4
7
3
weeks
weeks
weeks
weeks
weeks
weeks
weeks
weeks
PERT
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In the three-time estimate approach, the
critical path is determined as if the mean
times for the activities were fixed times.
The overall project completion time is
assumed to have a normal distribution with
mean equal to the sum of the means along
the critical path and variance equal to the
sum of the variances along the critical path.
Example: ABC Associates
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Consider the following project:
Immed. Optimistic Most Likely Pessimistic
Activity Predec. Time (Hr.) Time (Hr.) Time (Hr.)
A
—
4
6
8
B
—
1
4.5
5
C
A
3
3
3
D
A
4
5
6
E
A
0.5
1
1.5
F
B,C
3
4
5
G
B,C
1
1.5
5
H
E,F
5
6
7
I
E,F
2
5
8
J
D,H
2.5
2.75
4.5
K
G,I
3
5
7
Example: ABC Associates
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Activity
A
B
C
D
E
F
G
H
I
J
K
Expected Time
6
4
3
5
1
4
2
6
5
3
5
Variance
4/9
4/9
0
1/9
1/36
1/9
4/9
1/9
1
1/9
4/9
Example: ABC Associates
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Activity
ES EF LS LF Slack
A
0
6
0
6
0 *critical
B
0
4
5
9
5
C
6
9
6
9
0*
D
6 11
15 20
9
E
6
7
12 13
6
F
9
13
9 13
0*
G
9 11
16 18
7
H
13 19
14 20
1
I
13 18
13 18
0*
J
19 22
20 23
1
K
18 23
18 23
0*
The estimated project completion time is the Max EF at node 7 =
23. Critical Path (A-C-F-I-K)
Example: ABC Associates
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Probability the project will be completed within 24 hrs
2 = 2A +  2C + 2F + 2H + 2K
= 4/9 + 0 + 1/9 + 1 + 4/9
=2
 = 1.414.
z = (24 - 23)/(24-23)/1.414 = .71
From the Standard Normal Distribution table:
P(z < .71) = .5 + .2612 = .7612