lec 9 pert 2

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

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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



2. Path mean = 32 days



%86.747486.0}34Pr{

67.0972.2

3234

Dp

z

2. Path mean = 32 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

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


09-19


Determine the probability that the project would finish no later than 32 weeks:

Use the standard normal equation:

09-20



72.0)scorez(2.78

30 - 32 z


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


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


Z-score tables indicate that for 95% probability, a Z-score of 1.65 most closely represents this likelihood.

09-25


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.

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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

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lec 9 pert 2

Engineering