Question 13
Based on this project network, what is the minimum number of days required to complete the project?
Not quite. 16 days is the length of B, D, F, G, the shortest path through the network. One way to get this is to read “minimum number of days” as “shortest path”. The project isn’t finished until every path is finished, so the longest path sets the minimum time. In 2021, 25.1% of Queensland students chose (A).
Not quite. 17 is the length of B, E, G. One way to get this is to start E as soon as B finishes and forget about A then C, which take longer to reach the same node. E can’t start until both B and C are done. In 2021, 9.6% of Queensland students chose (B).
That’s it. Step through the hints below to see why. In 2021, 61.4% of Queensland students chose (C).
Not quite. 31 is the sum of every duration, as if the activities had to happen one after another. Activities on different paths can run at the same time, so the project takes less than that. In 2021, 3.0% of Queensland students chose (D).
Pick a path to add up its days, or run a forward scan one node at a time.
Every activity has to be done. Activities on different paths can run at the same time, but the project can’t finish until the slowest path has finished.
So the minimum time for the project is the length of the longest path. That path is the critical path.
List every path from the first node to the last and add the durations.
$$\begin{aligned}\text{A, C, E, G}&:\ 4+3+8+4=19\\\text{A, C, D, F, G}&:\ 4+3+2+5+4=18\\\text{B, E, G}&:\ 5+8+4=17\\\text{B, D, F, G}&:\ 5+2+5+4=16\end{aligned}$$
Three of the four options are path lengths. Only one of them is the longest.
Check with a forward scan. Write the earliest start time in each node. Where two arrows arrive, take the larger total, because the next activity has to wait for both.
On a bigger network the forward scan is quicker than listing paths, and it’s the method the QCAA expects in short response.
$$\begin{aligned}\text{A, C, E, G}&=19\\\text{A, C, D, F, G}&=18\\\text{B, E, G}&=17\\\text{B, D, F, G}&=16\end{aligned}$$
The project can’t finish until its longest path has finished. The critical path is A, C, E, G, so the minimum completion time is 19 days.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2021 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2021, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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