[MUSIC] This module is about scheduling activities. In this module we'll see how to calculate the earliest completion of the project. And are able to schedule for the various activities. As in the previous module, on work breakdown structure, and a truck diagram, you would not worry about resources required to perform each activity. We continue to assume that the necessary resources will be available as and when required. In a later module, we'll consider the case where the available resources are limited. We'll use the same software example that we considered in the previous module. Recall that in that software project, we identified eight activities as shown in the table. We'll use the following notation in arriving at the schedule. Activity starts at time t implies that is start at the end of the day t, are actually at the beginning of the day t+1. And activity ends at time k implies that activity is completed at the end of day k. With this notation in the example, we have only activity A starting at time 0, and ending at two weeks. Activity B, C, D started at the end of week 2 or actually at the start of week 3 and end at week 6, 5 and 4 respectively Now activity E can start only at the both activity B and C are completed. Activity B is completed at week 6 while activity C is completed at week five. So the earliest start for activity E is the end of week 6, or actually the beginning of week 7. So the earliest finish of activity E is 6 plus the duration of activity E, which is 6 plus 2 is equal to the eighth week. The earliest start of activity F is week eight and the earliest finish of activity F is week nine since the duration is one week. Proceeding in this way we find that the earliest start of activity G is week 9 and the earliest finish of G is week 10. Finally, we find that the earliest start of activity H is week 10 and the earliest finish of activity H is week 11. So the earliest completion of the project is 11 weeks from the start of the project. The process of finding the earliest start and the earliest finish of each activity and consequently, the earliest completion time of the project as we have done just now is often referred to as, forward pass, through the network. In the forward pass, the earliest start of an activity is the maximum of the earliest finish of all its predecessor activities. So, the earliest start of an activity we can calculated only after the earliest finish of all these process have been calculated. The earliest finish of an activity simply the earliest start of that activity plus the duration of that activity. In part of the network, is a sequence of activities from the start of the project to the completion of the project. The length of the path is defined as the sum of the durations of the activities in the path. In this simple example, we have three paths only, as given below, which is path one, activities A, B, E, F, G, and H with a length of total duration of 2 + 4 + 2 + 1 + 1 + 1 = 11 weeks. Part two has activities A, C, E, F, G, and H, with a total length, a total duration, of 2 + 3 + 2 + 1 + 1 + 1 = 10 weeks. Part 3 has activities A, D, G and H only, with a length of total duration 2 + 2 + 1 + 1 = 6 weeks. The earliest completion time of the project is the length of then longest part which in the simple is 11 weeks as we had calculated using the forward paths. In general, there may be several longest paths, ie there may be several paths with the same maximum length. Conceptually, one can enumerate all the paths and their respective lengths to identify the longest path which represents the earliest completion time of the project. In practice the number of notes would be large and consequently the number of parts would be too large to enumerate all of them. The forward parts that we have talked about average the numeration of all the parts to find out the earliest completion time of the project. The longest part is referred to as a critical part. The activities in the longest part in the network consist of of a set of critical activities. The notion of a critical part and critical activities comes from the fact that is any of the activities in the critical part is delayed, that is it takes more time than the estimated duration for that activity, the project completion time will be delayed from its earliest completion time. This also implies that you have to pay more attention to particular activities. And ensure that none of them is delayed in order for the project to be completed as early as possible. Of course, it does not mean that you can ignore the other activities. Because substantial delay in one or more activities may delay the completion time of the project. As we have seen earlier, the forward pass through the network views the earliest start earliest finish of each activity and the earliest completion time of the project. We don't enumerating all the pass, the critical path and the critical activities. Can be identified by using the forward pass that we have done and another procedure, which is often referred to as the backward pass through the network. The backward pass will give for each activity the latest start and latest finish for each activity, assuming the project duration is the earliest completion time of the project as I identified in the forward pass through the network. We will next do a backward pass through the same simple network for which we had earlier done the forward pass through the network. Since the project is to be completed by the earliest completion time, which, in this example, is 11 weeks, we have that the latest finish of activity H is 11. Now the latest start of activity age is obtained by subtracting its duration from the latest finish. And the latest start of activity age is 10. Next we consider the predecessor activity age, the activity why. For which the latest start of all its success are activities have already been collected. Then the latest finish of activity, why is the minimum latest starts of its success are activities. In this example we have only one predecessor activity of activity H, which is activity G. Latest finish of activity G is hence equal to 10 which is the later start of activity H. Then the later start of activity G is nine, which is given by the latest finish of G minus it's duration. Activity G has two predecessors, namely activities D and F. Latest finish of both of activities D and F are equal to to the later start of activity G. Which is 9, the latest start of activity E is 8 since is duration is 1, while the latest start of activity D is 7 since is duration is 2. Proceeding in this manner, we get the latest start and latest finish of each activity as shown in the table. Note, that the latest finish of activity is the minimum of the latest start of activates B, C, and D Which are 2, 3, and 7 respectively. And as the Latest finish of activity is two and the latest start of activity A is 0. The difference between the latest start and the earliest start are equivalently when there are no logs than a drug as in this case. The difference between the latest finish and the earliest finish of an activity gives what is referred to as total slack. The critical activities are those activities for which total slack is zero. In this example, all activities, except activity c and d are critical. Activity c and d have a total slack of one and five, respectively. This means that if only activity C is delayed by one week, or if only activity D is delayed by five weeks, but not both, the project will still be completed by the end of week 11. We have done the backward pass through the network, assuming the project completion time is equal to the earliest completion time as found in the forward pass through the network. But if there's a project deadline which is later than the earliest completion time of the project, the critical activities have positive slack, but among all the activities in the network, the critical activities have the least positive slack. But if there's a project deadline which is earlier than the earliest completion time of the project, the critical activities are negative slack but among all the activities in the network, critical activities have the least in an algebraic sense, least slack negative. Thus in all cases it would be appropriate to say that the critical activities are the least like Among all activities that consider the project. For each activity we have the earlier start and the later start for that activity. Thus we have two extreme possible schedules, one corresponding to the earlier start times, and the other corresponding to later start times. The earliest start schedule may be appropriate if you feel that some activities may take more time than the duration you have estimated. But at the deli structure your work is sometimes done earlier than required, i.e funds are spent earlier than required and it's costly here the time value of money is considered. On the other hand, lay structured delays as finding a funds but may delay the completion time of the project if some activities take more time than estimated. So in effect you must balance between finding funds early as in the early start schedule, but increase possibility of delaying the project completion time with a late start schedule. In addition to defining total slack there's another one called the free slack. This concept essentially assumes that an early start schedule is being adopted. The free slack for an activity is the maximum time that the activity may be delayed without affecting the early start of all its immediate successors. For an activity say activity b has only one predecessor say activity a then it follows that activity a has zero free slack irrespective of whether it has positive total slack or not. It can easily verify that free slack for an activity is less than or equal to total slack. This completes the module. [MUSIC]