Key Path computing

The longest path from the source point to the sink point path is the Key Path of the project, that is, the key path can ensure that all the paths can be completed.

OK, go to our assignment questions again:

For example, you can see the AOE network (the arc weight indicates the duration of the activity)

1) How many days does it take to complete the project?
2) What are key activities? The key paths are shown in the figure.

First, calculate the earliest occurrence time ve and the latest occurrence time Vl:

	1	2	3	4	5	6	7	8	9	10
Ve	0	5	6	12	15	16	16	19	21	23
Vl	0	9	6	12	15	21	16	19	21	23

First, numbers 1 and 10 are the source and sink points of the project respectively. We stipulate that the earliest occurrence time is ve (source point) = 0, and the latest occurrence time is VL (sink point) = ve (sink ). How can we calculate the two quantities? See the figure below:

For event I, ve (I) = max {ve (m) + Dut (<m, I>)}, VL (I) = min {VL (j) -Dut (<I, j>)}; among them, ve (m) indicates the earliest occurrence time of the previous event I, dut (<m, I>) indicates the duration from m to I. You can regard it as a recursive algorithm and call ve () until ve (source point) and then take its maximum value, because it must ensure that all activities on all paths can be completed, and the latest occurrence time is the same as the former, it always recursively calls VL () until VL (sink point, then, you can obtain the minimum value.

Return to the original question and compare the example graph to get the content displayed in the table very quickly.

Let's look at two concepts: E (I) and L (I). The former is the earliest possible start time of active AI, and the latter is the latest consent start time of active AI. Different from the preceding VE and VL, E and l indicate the time of an activity, and the former indicates the time of an event.

Assume that the activity from event m to event I is Af, then E (f) = ve (M), L (f) = VL (I)-Dut (<m, i>), that is, the earliest possible start time of the active AF is the earliest possible time of the arc tail event; the latest consent start time is the difference between the latest occurrence time of the arc tail event and the duration of the two events. Is it a bit awkward, but it is very easy to understand these keywords. E (K) = L (k) is a key activity. See the chart:

	A1	A2	A3	A4	A5	A6	A7	A8	A9	A10	A11	A12	A13
Permission	5	6	3	6	3	3	4	1	4	5	2	4	2
E	0	0	5	6	6	12	12	15	15	16	19	16	21
L	4	0	9	6	12	12	17	15	15	16	19	21	21

The VE and VL calculated by the graph and the front edge can obtain e and L of each activity. As shown in the preceding table, the key activities are A2, A4, A6, A8, A9, A10, a11 and A13, so the number of days required for the project is at least d = 6 + 6 + 3 + 1 (4) + 5 (2) + 2 = 23, there are two key paths for this project, 1-> 3-> 4-> 5-> 7-> 9-> 10 and 1-> 3-> 4-> 5-> 8-> 9-> 10.

Key Path computing