Upload
khawwam
View
156
Download
0
Embed Size (px)
Citation preview
Probabilistic Time Estimates
The average or expected time, te, and the variance, (Vi or ) for
each activity is of special interest.
The expected time of an activity te is a weighted average of the
three time estimates:
The standard deviation of each activity’s time is estimated as one-
sixth of the difference between the pessimistic and optimistic
time estimates.
The larger the variance, the greater the uncertainty associated with
an activity’s time
2
i
6
4 pmo
e
tttt
366
22
2 opop ttttV
1
Determining Path Probabilities
The probability that a given path will be completed in a
specific length of time can be determine using the
following formula:
The z-score indicates how many standard deviations of
the path distribution the specified time is beyond the
expected path duration.
deviation standardPath
meanPath - timeSpecified)scorez( z
Determining Path Probabilities
Determining Path Probabilities: Example
In a CPM network, the critical
path includes five activities. Their
durations are tabulated next.
Compute the following:
Activity Times (in days)
to tm tp
A 2 4 7
B 5 8 14
C 4 6 8
D 2 2 2
E 7 10 21
1.The probability that the project will finish within 32 days of its start.
2.The probability that the project will finish within 34 days of its start.
3.The probability that the project will finish no later than the35th day
Determining Path Probabilities: Example
Activity Times (in days) Expected
duration
te
Standard
deviation
σ
Variance
V=to tm tp
A 2 4 7 4.167 0.833 0.694
B 5 8 14 8.5 1.5 2.25
C 4 6 8 6 0.667 0.444
D 2 2 2 2 0 0
E 7 10 21 11.333 2.333 5.444
Path mean =32 V path =8.833
2
1.Path mean = 32 days
Path variance = 8.833
Path standard deviation = 2.972 days
%505.0}32Pr{
0972.2
3232
Dp
z
deviation standardPath
meanPath - timeSpecified)scorez( z
09-7
2. Path mean = 32 days
Path variance = 8.833
Path standard deviation = 2.972 days
%86.747486.0}34Pr{
67.0972.2
3234
Dp
z
2. Path mean = 32 days
Path variance = 8.833
Path standard deviation = 2.972 days
%38.848438.0}35Pr{
01.1972.2
3235
Dp
z
Example :Project Delta
09-10
Name: Project Delta
Durations are listed in weeks
Activity Description Optimistic Likely Pessimistic
A Contract signing 3 4 11
B Questionnaire design 2 5 8
C Target market ID 3 6 9
D Survey sample 8 12 20
E Develop presentation 3 5 12
F Analyze results 2 4 7
G Demographic analysis 6 9 14
H Presentation to client 1 2 4
09-11
Example: Project Delta
Name the Critical activities
Total project completion time
Overall project variance
Project standard deviation
Determine the probability that the project would finish no later than 32 weeks
09-12
Project Delta: expected duration for each activity
09-13
Project Variance
14
Activity Variance
Activity A: [(11 - 3)/6]2 = 64/36, or 1.78 weeks.
Project managers should:
Not just know likely times for activities
but also how much confidence we can place in these estimates.
Activity A, most likely that it will finish in 5 weeks, however, considerable amount of variance in that estimate (nearly 2 weeks).
09-15
Project Variance
Calculate the overall project variance:
Project variance is found by summing the variances of all critical activities.
09-16
Project Variance
Critical activities: A – C – D – F – H.
Total project completion time: 30 weeks
Overall project variance:
1.78 + 1.00 + 4.00 + .69 + .25 = 7.72
Project standard deviation:
Square root(project variance) = 2.78 weeks.
09-17
Probability of Completing Project
According to PERT:
Total project completion times use a normal probability distribution.
50% likelihood that completion time will be less than 30 weeks and a 50% chance that it will be greater than 30 weeks.
09-18
Probability of Completing Project
09-19
Probability of Completing Project
Determine the probability that the project would finish no later than 32 weeks:
Use the standard normal equation:
09-20
deviation standardPath
meanPath - timeSpecified)scorez( z
72.0)scorez(2.78
30 - 32 z
09-21
Probability of Completing Project
A Z value of 0.72 indicates a probability of 0.7642.
Thus, 76.42% chance that Project Delta will finish on or before the critical date of 32 weeks.
09-22
Probability of Completing Project
76% chance of success in meeting the deadline is probably unacceptable!
Suppose the organization requires a 95% likelihood of on-time delivery.
What would be the project completion time to ensure a 95% likelihood of on-time completion?
09-23
09-24
Probability of Completing Project
Z-score tables indicate that for 95% probability, a Z-score of 1.65 most closely represents this likelihood.
09-25
Probability of Completing Project
Path mean = specified time + z* path standard deviation
= 30 weeks + (1.65)( 2. 78)
= 34.95 weeks
This means that If the project team can negotiate for an additional 4.59 weeks, they have a very strong (95%) likelihood of ensuring that Project Delta will be completed on time.
26
Note In our Project Delta example, activity B has only 1 day
of slack and there is sufficient variance of 1.00.
In fact, the pessimistic time for activity B is 8 weeks, which would cause the project to miss its target deadline of 30 weeks, even though activity B is not on the critical path.
For this reason, it may be necessary to calculate the individual task variances not only for critical activities, but for all project activities, especially those with higher variances
27