Introducing Probability with PERT

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Introducing Probability with PERT

Many factors may cause time taken to complete a task to vary. Parkinson’s Law says that work always expands to fill the time allowed. That means that tasks may take longer than the estimated time, but seldom take less. One reason is that when people find themselves with some time left, they tend to refine what they have done. Another is that people fear that if they turn work in early, they may be expected to do the task faster the next time or that they may be given more work to do.Image result for Introducing Probability with PERT in Production Management diagram
In addition, we also have to understand variation. If the same person types a page on a word processor, the typing times will vary. Sometimes it will take ten minutes, while other times it will take fifteen. The average may be twelve, but we may expect that half the time it will take twelve minutes or less and half the time it will take twelve minutes or more.
In recent years, a new method of estimating time duration for/by ‘knowledge work’ has been developed. Rather than have individuals estimate task durations, at least three people are asked to estimate each activity in the project. They do this without discussing their ideas with one another. They then meet to find out what they have put on paper.
Time estimates often vary because objectivity has not been properly applied. Assumptions and tolerances are not properly codified. There are some guidelines for documenting estimates:
Show the percent tolerance that is likely to apply.
Tell how the estimate was made and what assumptions were used.
Specify any factors that might affect the validity of the estimate (such as time–will the estimate still be valid in six months).
To deal with possible variations, the Program Evaluation and Review Technique (PERT) uses the ‘beta probability distribution’. The PERT charts remain exactly the same as for single-time estimate. The variability associated with ‘activity performance’ times is considered for computing completion time probabilities only.
The beta distribution, unlike the normal probability distribution, allows the most likely time estimate to be close to the pessimistic time, close to the optimistic time, or anywhere in between. It is not symmetrical like the normal distribution.
By obtaining three time-estimates for each activity, it is possible to calculate the expected duration of each activity and the standard deviation of that duration. Those values can then be used to determine an expected completion time for the project, as well as the probability of completing the project within a given time period. The three time-estimates used to calculate expected activity time are:
Optimistic time(a): the shortest time the activity will reasonably take.
Most likely time(m): the time this activity would take most of the time.
Pessimistic time(b): the longest time the activity would be expected to take.
Using these three values, it is possible to calculate the expected duration of an activity. The formula is given as:
te = (a + 4m + b) / 6
The variance of activity duration is given by the formula:
2 = [(b–a) / 6]2
Where, ’te’ is the expected time
‘2’ is the variance
‘a’ is the optimistic time estimate
‘m’ is the most likely time estimate
‘b’ is the pessimistic time estimate
For determining the probable dates of completion, the network is drawn and variance for all the events falling on the critical path is computed. The variance of a particular event is calculated, assuming zero variance for initial event and then adding the variance of the activity (or of activities) on the critical path up to the event.
Similarly, expected duration up to a particular event can be calculated by adding up all the mean times for the activities on the path, from the start to this event. The variance and statistical-mean-time for a particular event can be calculated using the Control Limit Theorem, which states:
Event Expected Duration = TE = (te )1-2 + (te )2-3+…………………….
Variance of time for the event = VTE = Vt1-2 + Vt2-3+…………
It is assumed that standard normal curve is applicable to the distribution of actual occurrence time of the event. For knowing the probability of accomplishing an event at a particular time, the normal curve area tables are used. These tables give probability values for all the possible different units (a measure in terms of standard deviation ‘z’). Therefore, it becomes necessary to calculate ‘z’ before referring to the normal curve area tables. In case of PERT network charts, ‘z’ can be calculated with the formula:
z = (TS – TE ) / (VTE )½
Where z = units in terms of standard deviation,
TS = the scheduled time for the completion of the event, and
TE = the event meantime computed using Control Limit Theorem like above.
Finally, the probability values for any particular completion time of any of the events can directly be read for the normal curve tables against the calculated values of ‘z’.