CASE STUDY - Probability in PERT
This is in continuation of PERT in my previous hub. As stated therein, PERT is probabilistic in nature. It is used in those projects which are being developed for the first time. Hence, the developer has no previous experience and seeks guidance from the knowledgeable persons. On the other hand, CPM is deterministic in nature as all estimates are based on own experience or track-records.
But it must be remembered that many internal and external events may frustrate the predictions. This happens frequently and is taken as norm rather than exception especially in unstable economic and political environments.
In order to make realistic estimates, PERT obtains three estimates of different scenarios as shown below:
- Optimistic Time Estimates if everything turns favorable.
- Pessimistic Time Estimates if all goes bad.
- Mostly likely time which would be experienced in normal conditions.
Time Estimation Formula
Since Optimistic and Pessimistic conditions would be far less than normal conditions, a weight of one each is assigned to Optimistic and Pessimistic Times. In case of mostly likely time, a weight of 4 is assigned. This is a standard practice. The result is divided by total weight of 6, to find out weighted average which would serve as Time Estimate or Te as in the formula shown on the right-hand side:
Time Expected (Te)
A Construction Project
Let us start with construction program of a yatch. Being our first venture, we would prepare a PERT and obtain necessary estimates from designers, yatch builders and other knowledgeable persons like carpenters, welders and electricians in their respective fields.
The three estimates, optimistic, most likely and pessimistic, are given in table titled Basic Question. In the next table Time Expected (Te) has been calculated based on the formula given previously.
As stated before, PERT uses a "Weighted Average" of three time-estimates to calculate Time Expected (Te) for a particular task. These estimates are not wild guesses but have come from reliable sources. When it comes to masonry work, who can better estimate time required for making a brick wall than an experienced mason.
Various researchers have criticized use of “weighted averages” in time estimates. They argue that in this way, time would often be underestimated. But whenever one tries to predict future, one is confronted with many problems. To be realistic, one should make meticulous efforts and double check every figure.
Once we know the Te for each task, the rest is like CPM i.e. (i) the boxes representing variou activities would be placed keeping in view the predecesssor and successor activities, (ii) clear cut linkages shown between the activities, (iii) forward passes made to workout project duration, (iv) all possible paths identified and (iv) the longest path, being the Critical Path highlighted with red-line. This has been shown in the net work given below:
PERT Network - Activity On Node (AON)
PERT FORMULAE: Variance & Standard Deviation
Since PERT recognizes uncertainty in estimates of durations, it gives rough estimates about final completion. Now what-if analysis can be conducted like what is the probability of completion if project is delayed by certain period of time. Please note that probability of being completed by critical time is 50%. If more days are added the probability would increased and can be quantified by using normal curve method. For this we need a z-table and a standard deviation. PERT has special formula for calculating Standard Deviation. First, it would identify the activities on the critical path. Second, it would calculate variance for each activity on the path. Finally, square-root would be worked of sum total of variances of activities on the critical path. Necessary working is shown in the right hand side table.
Normal Curve with properties
Probability of completion
Normal distribution is natural distribution. I teach Project Management to a class of 40 students. Since all the students are reading from the same books, are being taught by the same teacher in the same environments, their marks in any test would be normally distributed. About 68% of the students would gain around the average, a few would well above it and a few well below. In a recently conducted test, the average score was 80 with a standard deviation of 6. It means that 68% of them got marks between 74 & 86. There were few exceptions. A few students were well above 86 while a few were much below 74.
With this back ground, we can find probability of a student getting 95 marks. It gives a z-value of 2.5 which means there is hardly 1% chance of scoring 95%. What about chances of 87%? The probability would increase to 12%
In our estimates for construction of the yatch, average time was 28 days with a probability of 50%. If we increase the project duration, the probability will increase and vice versa. When a sponsor asks for say 99% probability, the duration can be extended accordingly. In this case, if the construction team is given 32 days, there is hardly 1% chance that they would disappoint the sponsors. It does not mean that they would necessarily take that much time; maybe they complete it in 28 days or even less. But one should not expect a miracle.