The activities of a PERT network and their corresponding activity time estimates (in weeks), namely optimistic (\(t_o\)), most likely (\(t_m\)), and pessimistic (\(t_p\)), are given in the table below.
Activity
\(t_o\)
\(t_m\)
\(t_p\)
1-2
2
4
12
1-3
2
4
6
1-4
3
4
11
2-5
3
6
9
3-4
2
5
14
3-5
3
3
3
4-5
3
6
15
5-6
2
5
8
The expected project length is ________ weeks (in integer).
Show Hint
Convert each activity's three time estimates into one expected time, then add along every path to find the longest one.
Step 1: Turn the three time estimates into one expected time per activity, using $t_e = (t_o + 4t_m + t_p)/6$.
This gives: 1-2 = 5, 1-3 = 4, 1-4 = 5, 2-5 = 6, 3-4 = 6, 3-5 = 3, 4-5 = 7, 5-6 = 5 weeks.
Step 2: Draw out the network and trace every route from node 1 to node 6.
There are four such routes once the expected times replace the raw estimates.
Step 3: Add up the expected times along each route.
1-2-5-6 sums to 16, 1-3-5-6 sums to 12, 1-4-5-6 sums to 17, and 1-3-4-5-6 sums to 22 weeks.
Step 4: The project cannot finish before its longest route is done, so find the maximum.
The route 1-3-4-5-6 is longest at 22 weeks and forms the critical path.
Final Answer:
The expected length of the project is 22 weeks.
\[ \boxed{22 \text{ weeks}} \]
