With that introduction, in scheduling, the main objective and goal that PERT help us with is to answer the following question. What is the probability that the project can be completed in x number of days given the variable distributions of all the construction activities. So, let's move forward and I will give you an example with actual calculations to relate to what we just explained in the theory behind PERT. So let's say here on this example, as we can see we have what? This is the start and this is, let's test you here, this is an activity on node or activity on arrow? This is an activity on node as you can you can see the name of the activity inside the box not above the arrow. And we have one, two, three, four, five, six, seven, eight, and nine activities. The beginning of the project starts with activity A, B, and C and we finish the project with activity H, F and I. From the first example here, I will ask you to do the following. One, calculate the expected duration on each of the activities. We have them here on the project. And I will provide to you with all the three durations we just spoke about. So for each activity on the project here I will give you in the following slide the optimistic, the pessimistic and the most likely durations for each one of them. So as you can just apply the equation and find the expected duration on each activity you have in the project, this is one. Second, after you do that, I want you to find the critical path of the project. Third, I will ask you to calculate the mean of the normal distribution of the project, the expected value TE for the entire project. Number four, then to calculate the standard deviation of the project, sigma. Last, and after you calculate these four points, we will start by going through the PERT calculations and find some probabilities on finishing the project in x number of days. So let's start here, if I would ask you to write this down on a piece of paper before we move forward, to start to build on your project here. So, on the project I just gave, I highlighted for you here all the activities in the project. And I gave you the second column, the first column the name of the activity, the second column the most likely duration in your project for all these activities. The third column TA, we have the optimistic duration for each of the activities, and the last one, we have the pessimistic duration. And I gave you here as a reminder, the two equations that you would need to do. And I asked you to calculate the beginning here, what is the expected duration for each activity you find here? Of all these activities, using this equation and what is the variance f each single activity we have here? So, do the calculations and let's run them together in a minute. So let's go through the calculations quickly and answer. We found that The expected durations on and the values of the last two columns as you can see. Let's take just one or two quickly. Activity A for example, the expected duration for it would be 5 plus 4 times 3 plus 1. Or if you want to follow the same equation here you can start actually, of course, 1 plus 3 times 4 plus 5 and then you divide it by 6 to give you 3. The variance for A then with B, 5 minus 1 divided by 6, all to the power of 2 to give you the variance of that activity. For example activity H here, we have the same exact thing, expected duration of 3 plus 4 times 6 plus 9 all divided by 6 to give you 6. And the variance will be in this case 9 minus 36 divide by 61 to the power of 21 that the variance for activity h. Of course all same for in the table here. So now we found the first question that I mentioned which is own the expected values, TE, of all the activities in our project. And what we found also, all the variances for all the activities. The second question is to find the critical path. So that being said, let's go back and fill what you wrote on piece of paper the diagram that I just showed you on each activity to write the duration for that activity. The duration of that activity, what you want to use would be the expected value calculated from the three values provided to you. So that being said, then this is the duration from the table before. That activity A has three days to be completed B six days and so on. The question we are highlighting is, what is the critical path of the project? If you remember, we covered this in the previous module so let's take as an exercise the critical path without doing the forward or backward pass calculations here. I don't want you to do it. I want you to look at all possible combinations in this project from the path on to the view from the start of the project to the finish of the project. How many combinations we will have here? We would have around six combinations. Let's go one by one and find the total duration of the project, or what we are trying to find here, the expected value of the project. So let's move forward 1 would be ADH which would be 3 for A plus 8.5 plus 6 that will give you 17.5 days. And then we have BF, start B and then going to F here, 6 plus 10 equals 16. We have A, E and F 3 plus 3.5 plus 10, 16.5 days. And we do have B, G, and then I which will be 6 plus 7.5, 3.5 17. And we have another A E G and then I with a total duration of 17.5 and the last combination is C I with a total duration of 16. So, in this case what we see in our project here that we do have actually two critical path. Because path ADH and AEGI has the longest path in the project with 17.5 days as we can see 17.5 here and 17.5 here. So in that case, that mean of the normal distribution is therefore assumed to be 17.5, the total duration of the project. Now as we said, we have two critical path. In this case, for the PERT calculations, we need to find the path which we will choose for further PERT calculations. So we first find, to figure out which path to further the calculations in the PERT technique, we first find the variance on each path. As following. The first path A-D-H, A-D-H the duration of 17.5, this is the second path. The variance, if you remember the equation, we did it already in that table. From the table we took the variance for A, the variance for D, the variance for H, and it give us 3.694. Second, variance for the second critical path, AEGI. We have the variance for A, the variance for E, The variance for G and the variance for I to give a total variance for that path of 5.416. As we notice here V2, we have two variances. How to choose, V2 is greater than V1. That means that there are greater spread, if you remember the curve of the normal distribution. So the bigger the variance, the bigger the sigma, the bigger the standard deviation. So that means that there are a greater spread of the probable total project durations under the curve. So in this case, the V2 for the AEGI path, the second one here, is selected as the variance to be used for further PERT calculations on this project. So in case something like this happens, always take the bigger one because it takes the greater spread of the normal distribution. So the standard deviation then will be the square root of the 5.416 which is 2.33.