9.5 is the total of the path M2(2.5)→M3(2.0)→M4(5.0), which runs only through the brand M side. The node just before M3 also has S3 and M1 (via the dummy activity) entering it, and M3 cannot start until those finish. This value comes from picking a short path instead of the longest one, so it is wrong.
×Incorrectイ
11.0 is the total of the path M1(4.0)→(dummy activity)→M3(2.0)→M4(5.0). The dummy activity has a duration of 0 and expresses only an ordering constraint, so this path does exist, but S1→S2→S3, which enters the same node, takes a longer 7.0 hours. The error comes from computing with the M1 path instead of taking the maximum.
×Incorrectウ
12.0 is the total of the path S1(2.0)→S2(3.0)→S4(2.0)→M4(5.0). However, M3 also enters the node just before M4, and its earliest time is 9.0 hours, later than the 7.0 hours of the S4 path. Since M4 can start only at 9.0 hours, the work cannot end at 12.0.
○エ
正解In an arrow diagram, the shortest time in which the whole project can finish equals the length of the path from start to end whose total duration is the longest (the critical path). At a junction node where arrows merge, the next activity cannot start until every activity entering that node has finished, so the earliest start time of each node is the largest of the total times of the paths coming into it. Calculating in order for Figure 1: S1(2.0) followed by S2(3.0) finishes 5.0 hours after the start, and from there S3(2.0) and S4(2.0) branch off. The node just before M3 has three arrows entering it, S3, M2, and M1 (via the dummy activity), so M3(2.0) can start only at max(5.0+2.0=7.0, M2 at 2.5, M1 at 4.0)=7.0 hours, and M3 finishes 9.0 hours after the start. The node just before M4 has S4 and M3 entering it, so M4(5.0) starts at max(5.0+2.0=7.0, 9.0)=9.0 hours, and the work ends at 9.0+5.0=14.0 hours. In other words, the critical path is S1→S2→S3→M3→M4, totaling 2.0+3.0+2.0+2.0+5.0=14.0 hours. Therefore 14.0 in choice エ is correct.
ここがポイント
To find the shortest overall duration with an arrow diagram (PERT chart), total the durations of every path from start to end and find the critical path, which is the largest of those totals. The earliest event time of a merging node is the largest of the total times of the paths entering it.
A dummy activity, drawn with a dotted line, has a duration of 0 and expresses only the order relationship between activities (an ordering constraint). It adds no time, but it must always be taken into account as a dependency.