variability in the project completion...
TRANSCRIPT
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PERT/CPM
Variability in the Project Completion Date
• In doing Critical Path calculations, estimated activity times are not certain.
• What is the effect of these uncertainties on calculated project completion date?
Critical Path determines this date
Critical Path is a sequence of activities
Variation in Critical Path activities can cause variation in the completion date
Variation in non-Critical Path activities will not usually cause variation in the completion date
But… if variation along a non-Critical Path uses all of the slack time, this path may become Critical
But if… if variation along the Critical Path results in an earlier completion date, a non-Critical Path may become Critical.
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3 Estimates for each Activity
Optimistic time ( a )
Everything progresses in an ideal manner
Most probable time ( m )
Most likely under normal conditions
Pessimistic time ( b )
Encounter breakdowns and/or delays
Expected time
4
6
t
a m bt
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( a ) ( m ) ( b ) Expected
Most Time
Activity Optimistic Probable Pessimistic (Weeks)
A 4 5 12 6
B 1 1.5 5 2
C 2 3 4 3
D 3 4 11 5
E 2 3 4 3
F 1.5 2 2.5 2
G 1.5 3 4.5 3
H 2.5 3.5 7.5 4
I 1.5 2 2.5 2
J 1 2 3 2
Expected time
4
6
t
a m bt
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Case Study (1) (2) (3) (4)
Earliest Latest Earliest Latest
Activity Start Start Finish Finish
A 0 0 6 6
B 0 7 2 9
C 6 10 9 13
D 6 7 11 12
E 6 6 9 9
F 9 13 11 15
G 11 12 14 15
H 9 9 13 13
I 13 13 15 15
J 15 15 17 17
C F
A D G
J
E
B H I
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(1) (2) (3) (4) (2) - (1) (4) - (3)
Earliest Latest Earliest Latest SLACK SLACK
Activity Start Start Finish Finish (LS - ES) (LF - EF)
A 0 0 6 6 0 0
B 0 7 2 9 7 7
C 6 10 9 13 4 4
D 6 7 11 12 1 1
E 6 6 9 9 0 0
F 9 13 11 15 4 4
G 11 12 14 15 1 1
H 9 9 13 13 0 0
I 13 13 15 15 0 0
J 15 15 17 17 0 0
C F
A D G
J
E
B H I
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Project Duration
T stands for the project duration
The activities along the Critical Path are A – E – H – I – J
E H IA Jt t t t tT
Expected
Time
Activity (Weeks)
A 6
B 2
C 3
D 5
E 3
F 2
G 3
H 4
I 2
J 2
6 3 4 2 2 17T
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Variance in Project Activities 2
Variance of an activity time = 6
b a
( a ) ( m ) ( b ) Expected
Most Time Variance
Activity Optimistic Probable Pessimistic (Weeks) σ2
A 4 5 12 6 1.78
B 1 1.5 5 2 0.44
C 2 3 4 3 0.11
D 3 4 11 5 1.78
E 2 3 4 3 0.11
F 1.5 2 2.5 2 0.03
G 1.5 3 4.5 3 0.25
H 2.5 3.5 7.5 4 0.69
I 1.5 2 2.5 2 0.03
J 1 2 3 2 0.11
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Variance in Project Duration 2 2 2 2 22
E H IA J
2 1.78 0.11 0.69 0.03 0.11 2.72E H IA J
Variance
Activity σ2
A 1.78
B 0.44
C 0.11
D 1.78
E 0.11
F 0.03
G 0.25
H 0.69
I 0.03
J 0.11
Standard Deviation in Project Duration
2.72 1.65
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Final Assumption
The distribution of project completion time T is
Normally distributed.
The next step:
• Given the Expected Completion time (mean of the Normal)
• Given the Expected Standard Deviation in the Expected
Completion time (standard deviation of the Normal)
• Convert this Normal Curve to an equivalent Z-Normal Curve
17
1.65
xz
This “says” that any time you want to know the
“what if” probability that another completion
time (x) might actually occur, it represents a
random variable associated with this normal
distribution … and can be converted to an
equivalent z normal value…
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What if… example
What is the probability that the project would be
completed in 20 weeks or less?
This would be an x = 20 from this normal distribution.
Convert the x to a z
17
1.65
171.82
1.65
xz
xz
This “says” that a completion time
of 20 is 1.82 standard deviations
above the mean (17).
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Here is where we use the Table Handout
Z = 0 Z = 1.82 µ = 17 x = 20
Z table gives the area to the left, but only to the mean line. You need to add in the 0.50 to get the probability.
171.82
1.65
xz
xz
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Steps
1. Develop list of activities
2. Draw the network
3. Estimate expected activity time and variance for each
activity
4. Use expected activity time estimates to determine
earliest start & finish and latest finish and start for each
activity
5. Use the project completion time as the finishing time
6. Compute the slack for each activity & identify the
Critical Path
7. Use the variability in the project completion date to
compute probabilities of meeting a specified date