Scheduling Techniques

PERT

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Duration Estimation MethodsRegardless of the technique you use, the

tendency in project estimation is to provide one number for each estimate of activity duration

In other words, if you have 100 activities on your schedule, each activity would have one estimate associated with it. This is generally viewed as the "most likely" estimate.

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Duration Estimation Methods under certaintyPast experience:

use history as a guide and rely on past examples of similar projects.

assumes what worked in the past will continue to work today.

Expert opinion:contact a past project manager or expert

to get accurate information on activity estimates.

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Duration Estimation Methods under uncertaintyMathematical derivation method (know

also as beta distribution or three-point estimation or PERT :Program Evaluation and Review Technique model) develops duration probability based on a reasoned analysis of:best-casemost likely caseand worst-case scenarios

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Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

The PERT: Background In the 1950s, the US Navy had a budget overrun and a

schedule delay of as much as 50% in the Polaris missile system.

Main Problem: Lack of any relevant historical data. The project team launched a joint research effort to develop a

tool to assist in the planning of the Polaris project. The objective was to devise a method that predicts the

completion date of a project with a certain likelihood using the theory of probability.

In 1958, this tool was developed under the name program evaluation and review technique.

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Mathematical DerivationWe need to identify three values:

activity’s most likely duration: time expected assuming the development proceeds normally.

activity’s most pessimistic duration: time needed under the assumption that everything will go badly.

activity’s most optimistic duration: duration is under the assumption that development will proceed extremely well.

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Probability DistributionFor representing these time estimates,

we can use probability distributions that are either:Symmetrical: normal distribution.Asymmetrical: beta distribution.

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Normal DistributionA normal distribution implies:

probability of an event taking the most likely time is one that is centered on the mean of the distribution.

Pessimistic and optimistic values will cancel each other out.

Thus leaving the mean value as the expected duration time for the activity.

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Symmetrical (Normal) Distribution for Activity Duration Estimation

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Beta DistributionIn real life:

Rare to find examples in which optimistic and pessimistic durations are symmetrical.

In project management, we use beta distributions.

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Asymmetrical (Beta) Distribution for Activity Duration Estimation

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Probabilistic Time Estimates

• The beta distribution is generally used to describe the inherent variability in time estimates.

• The distribution may be skewed to either right or left according to the nature of a particular activity.

The optimistic and pessimistic duration values essentially serve as upper and lower bounds for the distribution range

• The mean and variance of the distribution can be readily obtained from the three time estimates.

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PERT Activity Times

3 time estimates Optimistic times (to) Most-likely time (tm) Pessimistic time (tp)

Expected time: t = (to + 4 tm + tp)/6

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Probabilistic Time EstimatesThe 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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Probabilistic Time EstimatesThe expected duration of a path (i.e. the path mean) is equal to the

sum of the expected times of the activities on the path:

Path mean = ∑ of expected times of activities on the path

The standard deviation of the expected time for each path is determined by summing the variances of the activities on a path and then taking the square root of that number:

pathon activities of variancespath

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Probabilistic Time Estimates : Example

1. Compute the expected time for each activity and expected duration for each path

2. Identify the critical path3. Compute the variance of each activity and standard deviation

of each path.

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Probabilistic Time Estimates : ExampleGiven the following :Activity Immediate predecessors Times (in weeks)

to tm tpa --- 1 3 4b a 2 4 6c b 2 3 5d --- 3 4 5e d 3 5 7f e 5 7 9g --- 2 3 6h g 4 6 8i h 3 4 6

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Example Network

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A B C

D E F

G H I

Probabilistic Time Estimates : Example1.Path Activity Times (in weeks) Path

totalto tm tp

a-b-c a 1 3 4 2.83 10.00b 2 4 6 4.00c 2 3 5 3.17

d-e-f d 3 4 5 4.00 16.00e 3 5 7 5.00f 5 7 9 7.00

g-h-i g 2 3 6 3.33 13.5h 4 6 8 6.00i 3 4 6 4.17

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Probabilistic Time Estimates : Example3.

Path Activity

Timesto tm tp

a-b-c a 1 3 4 (4 -1)2/36=9/3634/36=0.944 0.971b 2 4 6 (6 -2)2/36=16/36

c 2 3 5 (5 -2)2/36=9/36d-e-f d 3 4 5 (5 -3)2/36=4/36

36/36=1.00 1.00e 3 5 7 (7 -3)2/36=16/36f 5 7 9 (9 -5)2/36=16/36

g-h-i g 2 3 6 (6 -2)2/36=16/3641/36=1.139 1.067h 4 6 8 (8 -4)2/36=16/36

i 3 4 6 (6 -3)2/36=9/36

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path2 path

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Probabilistic Time Estimates

Knowledge of the expected path times and their standard deviations enables a manager to compute probabilistic estimates of the project completion time, such as these:

The probability that the project will be completed by a specific time

The probability that the project will take longer than its schedule completion time.

These estimates can be derived from the probability distribution that various paths will be completed by the specified time.

This involves the use of the normal distribution. The path distribution is represented by a normal distribution.

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Determining Path Probabilities

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