Decision Mathematics 1: Critical path analysisEdexcel A-Level Further Maths: Topic test

1. 1
   A project is modelled by an activity network drawn with activities on arcs. The events are numbered 1 to 6. The activities are $A$ (event 1 to 2), $B$ (1 to 3), $C$ (2 to 5), $D$ (3 to 4), $E$ (5 to 6) and $F$ (4 to 6). There is also a dummy activity from event 2 to event 3 and another dummy from event 4 to event 5.

   (a)

   What are the immediate predecessors of activity $D$?

   [1 mark]

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

   (b)

   Which activity can start as soon as $D$ has finished, without waiting for any other activity to finish?

   [1 mark]

   • A$E$
   • B$E$ and $F$
   • CNeither, because of the dummy
   • D$F$

   (c)

   Complete the precedence table by giving the immediate predecessors of each of the activities $A$ to $F$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A project has seven activities. Each activity, its duration in days and its immediate predecessors are: $A$ 4 (none); $B$ 6 (none); $C$ 3 ($A$); $D$ 5 ($A$, $B$); $E$ 7 ($C$); $F$ 4 ($D$); $G$ 5 ($E$, $F$).

   (a)

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

   [1 mark]

   • A$20$ days
   • B$19$ days
   • C$34$ days
   • D$15$ days

   (b)

   Which activities are on the critical path?

   [1 mark]

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

   (c)

   Find the total float of activity $C$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A project has six activities. Each activity, its duration in days, the number of workers it needs and its immediate predecessors are: $A$ 2 days, 3 workers (none); $B$ 4 days, 2 workers (none); $C$ 3 days, 1 worker (none); $D$ 4 days, 1 worker ($C$); $E$ 4 days, 3 workers ($A$); $F$ 3 days, 2 workers ($D$). The minimum completion time of the project is 10 days, and the critical activities are $C$, $D$ and $F$.

   (a)

   Every activity starts at its earliest possible time. Find the number of workers needed on each of the ten days.

   [3 marks]

   (b)

   Only 4 workers are available at any time. Show that the project can still be completed in 10 days by delaying one activity, and give the start time of every activity.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A project has seven activities, each needing one worker. Each activity, its duration in days and its immediate predecessors are: $A$ 4 (none); $B$ 4 (none); $C$ 6 ($A$); $D$ 6 ($A$); $E$ 3 ($B$, $C$, $D$); $F$ 5 (none); $G$ 2 ($E$).

   (a)

   Carry out a forward pass and a backward pass to find the earliest and latest start times of every activity, the minimum completion time, the critical activities and the total float of $B$ and of $F$.

   [6 marks]

   (b)

   Find a lower bound for the number of workers needed to complete the project in the minimum time, and construct a schedule, stating which activities each worker does and when, that achieves this.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A project has four activities, $K$, $L$, $M$ and $N$. Their immediate predecessors are: $K$ none; $L$ none; $M$ is preceded by $K$ and $L$; $N$ is preceded by $L$ only. The project is to be modelled by an activity network with activities on arcs.

   (a)

   What is the least number of dummy activities needed in the network?

   [1 mark]

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

   (b)

   Which statement describes the dummy activity?

   [1 mark]

   • AIt runs from the end of $L$ to the start of $M$, so that $M$ follows both $K$ and $L$ while $N$ follows only $L$
   • BIt runs from the end of $K$ to the start of $N$, so that $N$ follows $K$
   • CIt runs from the end of $M$ to the start of $N$, so that $N$ follows $M$
   • DIt runs from the end of $N$ to the start of $M$, so that $M$ follows $N$

   (c)

   Two more activities are added: $O$ is preceded by $M$ and $N$, and $P$ is preceded by $N$ only. State, with a reason, how many further dummy activities are needed.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   In a project each activity needs one worker. The sum of all the activity durations is 70 days and the minimum project time is 30 days. Activity $X$ lasts 6 days. Its start event has earliest event time 5 and its finish event has latest event time 17.

   (a)

   What is the total float of activity $X$?

   [1 mark]

   • A$12$ days
   • B$11$ days
   • C$28$ days
   • D$6$ days

   (b)

   Activity $X$ is delayed by 4 days. What is the effect on the minimum project time?

   [1 mark]

   • AIt increases by 4 days
   • BThere is no effect
   • CIt increases by 2 days
   • DIt increases by 6 days

   (c)

   Find a lower bound for the number of workers needed to complete the project in 30 days.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A project has six activities. Each activity, its duration in days, the number of workers it needs and its immediate predecessors are: $A$ 5 days, 2 workers (none); $B$ 2 days, 1 worker (none); $C$ 5 days, 1 worker (none); $D$ 5 days, 1 worker ($B$); $E$ 2 days, 3 workers ($B$); $F$ 3 days, 2 workers ($A$, $D$, $E$). The minimum completion time is 10 days and the critical activities are $B$, $D$ and $F$.

   (a)

   Every activity starts at its earliest possible time. Find the number of workers needed on each of the ten days.

   [3 marks]

   (b)

   Only 4 workers are available at any time. Show that the project can still be completed in 10 days by delaying one activity, and give the start time of every activity.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A project has seven activities, each needing one worker. Each activity, its duration in days and its immediate predecessors are: $A$ 7 (none); $B$ 2 (none); $C$ 7 ($B$); $D$ 5 (none); $E$ 2 ($C$); $F$ 4 ($A$, $C$); $G$ 5 ($E$).

   (a)

   Carry out a forward pass and a backward pass to find the minimum completion time, the critical activities and the total float of $A$, $D$ and $F$.

   [6 marks]

   (b)

   Find a lower bound for the number of workers needed to complete the project in the minimum time, and construct a schedule, stating which activities each worker does and when, that achieves this.

   [6 marks]

   Total for question 8: 12 marks

End of questions