[MUSIC] In this module, we will discuss program evaluation and review technique, or PERT as it is typical known as. PERT typically includes assimilation of the project. Historically, critical PERT method was developed by DuPont in the context of preventive maintenance which is carried out on a regular basis. The work breakdown structure resulted in the identification of the various activities constituting the preventive maintenance project. The precedence relationships among the activities leading to the network diagram and the activity durations which are unknown with a high degree of accuracy. The requirement was to complete the preventive maintenance as quickly as possible so as to reduce the downtime of the missionary. PERT, on the other hand, was developed in the context of an R & D project where there was considerable uncertainty in many of the activity durations. But assumes that an activity duration follows a beta distribution, which is different from a normal distribution in that the activity duration is always positive. And there are three parameters, namely optimistic estimate denoted by a, pessimistic estimate denoted as b, and the most likely estimate denoted as m. The concept of an optimistic estimate a, is that there is a chance of 1 in 100 of completing the activity in less than a. Similarly, the concept of a pessimistic estimate b, is that there's a chance of 1 in 100 for the activity exceeding the units of time. The assumptions of a better distribution for the activity duration implies that the expected or average duration is equal to a + 4 times the most likely estimate plus b. The entire thing divided by 6. As an administration, the better distribution for an activity, and the distribution of the project completion time, and shown in the diagram. The variance for the activity duration is given by somewhat complicated expression as shown in the diagram. As an illustration, the triangular distribution for an activity is also shown in the diagram. Knowing the distribution of the time for each activity, we would like to find the distribution of the project completion time, so that we can answer questions such as one, what is the probability of the project completion time exceeding a specified value? And two, what are the chances that an activity is on a critical path? One approach of finding the distribution of project completion time is to use the expected duration for each activity and find the critical path. Then assuming independence to the activity durations, the variance or the length on the duration of the critical path is obtained by adding the variances of the durations for the activities on the critical path. The standard deviation of the length of the critical path is, of course, the square root of the variance. Finally, assuming the distribution of the project completion time is a normal distribution, one can find the probability of the project completion time exceeding a specified value. The above approach is not correct, although it is simple to calculate. The approach has two flaws. First, because the uncertainty in activity durations, the same path my not be critical in different realizations in the activity durations. The same path is always the critical path only in very rare instances where the critical path is the dominant path in the sense that the length is always greater than the length of the other paths in all realizations of activity durations. Second, the assumption or normality of the distribution of project completion time would be valid only if the critical path has a large number of activities. Because the Central Limit Theorem which states that the sum of a large number of independent identically distributed random variables is approximately normal, is valid in most instances only if large is at least 25 to 30. Except possibly when the distribution of this activity duration is a uniform distribution, in which case the number may be as low as 12 to 15. The correct approach to finding a distribution of the completion time is to use simulation. It is a mathematical model that's an abstraction of the real world. And the model is run typically on the computer, and not in the real world, to find out the possible outcomes. And arrive at appropriate actions as necessary to be implemented in the real world. The user simulation in PERT is [INAUDIBLE] through a simple example. We consider a project where we have ten activities denoted as a, b, c, d, e, f, g, h, i, and j respectively. The president's relationship among the activities are shown in the table. The three estimates, minimum, mostly, and maximum estimates for the duration of each activity are also shown in the same table. The duration of each activity is seen to follow a triangular distribution with the associated three parameters for that activity. The associated network is shown in the diagram. Assuming a triangular distribution for each activity, the expected durations and the variances are shown in the next table. Each trial's simulation approach is to generate a random value for each activity duration from its assumed distribution. There are different approaches to generate random variants from an astute distribution. And these approaches do not cover, are available in most of the books on simulation. With the general directive activity durations, we find the critical part, and note the project completion time and the particular activities. Then, we repeat the drive safer about 10,000 times and find 10,000 project completion times which gives us a of all the completion times. In Azure, we also get the number of times an activity is on the critical part. The proportion of trials in which an activity is on the critical path is referred to as the criticality index for that activity. The summary statistics for 1,000 simulation runs using the software @RISK are shown in the table. The criticality index for each activity is shown in the next table. And this example if you use the expected duration of each activity the critical that is activity BG and I. The expected completion time for the project is 44 and the variance for completion time is 44.67. Currently the standard deviation is equal to square root 44.67 equal to 6.68. The table below gives the probability of the project completion time less than or equal to specified value as calculated by the incorrect approach of assuming of a normal distribution, the distribution of the length of the same critical path consisting of activities B, G, and I. The same table gives that correct values obtained by a simulation approach using the software @RISK. Note that the incorrect or the estimates required probabilities because in some realizations of the activity do, then the part B, G, I has a length less than or equal to specified value. Some other part may have lengths greater than the specified value. You briefly outline the simulation approach, when there are resources limitations. In this case, as the project progresses, decisions must be made as to which activities take priority. It should be noted the realizations or the duration of the activities to be done in the future are not known currently, and will be known only in the future, perhaps after the activity is completed. With resource limitations, each trial and simulation of the project must be done starting from the beginning. And incrementing the crop but unilateral time in the activity durations are measured. When one of the activity is completed, a decision has to be made as to which of the succeeding activities maybe started if the resources required exceed availability. A heuristic as discussed in the module and resource should be used. The duration for the activities that had to be started immediately must be generated from their respective distributions. The clock is then incremented by a day and the process is continued until the project is completed. One try the simulation is completed then the project is completed. The activity that constitute the critical change should be noted. The entire process is then repeated for the remaining times of simulation then by generating say 10,000 values for the project completion time. This would give the distribution of the project completion time and the criticality index for each activity can be calculated. This completes the module on PERT. [MUSIC]