D1: Critical path analysisEdexcel International A Level Maths: Topic test

1. 1
   A conference hall is prepared in five activities. Activities $A$ and $B$ have no predecessors. $C$ has immediate predecessor $A$ only. $D$ has immediate predecessors $A$ and $B$. $E$ has immediate predecessors $C$ and $D$. The project is modelled by an activity network, using activity on arc.

   (a)

   How many dummy activities are needed in the network?

   [1 mark]

   • A$0$
   • B$1$
   • C$2$
   • D$3$

   (b)

   Where should the dummy be drawn?

   [1 mark]

   • AFrom the end of $B$ to the start of $C$
   • BFrom the end of $C$ to the start of $E$
   • CFrom the end of $A$ to the start of $D$
   • DFrom the end of $D$ to the start of $E$

   (c)

   Explain why a dummy activity is needed in the network.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A house extension is modelled by an activity network with events numbered $1$ to $5$. The activities, with the events they join and their durations in days, are $A$ (from event $1$ to $2$, $4$ days), $B$ ($1$ to $3$, $6$ days), $C$ ($2$ to $4$, $5$ days), $D$ ($3$ to $4$, $3$ days), $E$ ($3$ to $5$, $7$ days) and $F$ ($4$ to $5$, $2$ days).

   (a)

   What is the minimum time in which the project can be completed?

   [1 mark]

   • A$11$ days
   • B$27$ days
   • C$9$ days
   • D$13$ days

   (b)

   Which activities are critical?

   [1 mark]

   • A$B$ and $E$
   • B$A$, $C$ and $F$
   • C$B$, $D$ and $F$
   • D$E$ and $F$

   (c)

   Find the total float of activity $C$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A stage set is built by an activity network with events numbered $1$ to $7$. The activities, with the events they join and their durations in days, are $A$ ($1$ to $2$, $5$), $B$ ($1$ to $3$, $4$), $C$ ($2$ to $4$, $6$), $D$ ($2$ to $5$, $3$), $E$ ($3$ to $5$, $7$), $F$ ($3$ to $6$, $2$), $G$ ($4$ to $7$, $4$), $H$ ($5$ to $7$, $5$) and $I$ ($6$ to $7$, $6$). There is also a dummy activity from event $4$ to event $5$.

   (a)

   Carry out a forward pass to find the earliest time of every event, and state the minimum completion time of the project.

   [3 marks]

   (b)

   Carry out a backward pass, hence list the critical activities, and find the total float of activity $G$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A film crew prepares a set using nine activities. Their durations in hours and immediate predecessors are: $A$ ($5$, none); $B$ ($4$, none); $C$ ($3$, none); $D$ ($6$, $A$); $E$ ($4$, $A$ and $B$); $F$ ($5$, $C$); $G$ ($3$, $D$ and $E$); $H$ ($4$, $F$); $I$ ($2$, $G$ and $H$). Each activity needs exactly one worker for its whole duration and cannot be split.

   (a)

   Find the minimum completion time of the project and the critical activities, and find the total float of activity $H$.

   [6 marks]

   (b)

   Show that at least $3$ workers are needed to complete the project in the minimum time, and construct a schedule that does it with $3$ workers.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A product launch is modelled by an activity network with events numbered $1$ to $6$, using activity on arc. The activities, with the events they join, are $A$ ($1$ to $2$), $B$ ($1$ to $3$), $C$ ($2$ to $4$), $D$ ($3$ to $4$), $E$ ($3$ to $5$), $F$ ($4$ to $6$) and $G$ ($5$ to $6$). There is a dummy activity from event $2$ to event $3$.

   (a)

   What are the immediate predecessors of activity $E$?

   [1 mark]

   • A$A$ and $B$
   • B$A$ only
   • C$B$ only
   • D$C$ and $D$

   (b)

   Activity $B$ is delayed. Which activities can still start as soon as $A$ is complete?

   [1 mark]

   • A$C$, $D$ and $E$
   • B$C$ and $D$
   • C$D$ and $E$
   • D$C$ only

   (c)

   Write down the immediate predecessors of activity $F$ and of activity $G$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   In an activity network, activity $X$ runs from event $3$ to event $5$ and has duration $6$ days. The earliest time of event $3$ is $8$ days and the latest time of event $5$ is $22$ days.

   (a)

   What is the total float of activity $X$?

   [1 mark]

   • A$14$ days
   • B$16$ days
   • C$8$ days
   • D$36$ days

   (b)

   Activity $X$ takes $10$ days longer than planned. By how much is the project completion time delayed?

   [1 mark]

   • AIt is not delayed
   • B$2$ days
   • C$10$ days
   • D$8$ days

   (c)

   Explain why activity $X$ is not a critical activity.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A kitchen refit has seven activities. Durations in hours and immediate predecessors are: $A$ ($4$, none); $B$ ($6$, none); $C$ ($3$, $A$); $D$ ($5$, $A$); $E$ ($4$, $B$ and $C$); $F$ ($2$, $D$ and $E$); $G$ ($3$, $D$). Each activity needs exactly one worker for its whole duration and cannot be split.

   (a)

   Find the earliest start time of every activity and the minimum completion time of the project.

   [3 marks]

   (b)

   Only two workers are available. Show that the project cannot be completed in $13$ hours, and construct a schedule that completes it in $14$ hours.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A railway repair is modelled by an activity network with events numbered $1$ to $6$. The activities, with the events they join and their durations in days, are $A$ ($1$ to $2$, $6$), $B$ ($1$ to $3$, $3$), $C$ ($2$ to $4$, $5$), $D$ ($3$ to $4$, $4$), $E$ ($3$ to $5$, $7$), $F$ ($4$ to $6$, $3$) and $G$ ($5$ to $6$, $2$). There is a dummy activity from event $2$ to event $3$.

   (a)

   Carry out a forward pass and a backward pass. Hence state the minimum completion time, the critical activities, and the total float of activity $C$.

   [6 marks]

   (b)

   Activity $C$ is found to take $8$ days instead of $5$. Find the new minimum completion time and the new critical activities. Use the total float of $C$ to explain why the project is delayed by less than $3$ days.

   [6 marks]

   Total for question 8: 12 marks

End of questions