Critical paths and floatEdexcel A-Level Further Maths: Subtopic test

1. 1
   A project has eight activities. Each activity, its duration in days and its immediate predecessors are: A 4 (none); B 6 (none); C 5 (A); D 3 (A, B); E 7 (B); F 4 (C, D); G 2 (D, E); H 3 (F, G). The earliest start time of an activity is the earliest time at which all of its predecessors are complete.

   (a)

   What is the earliest start time of activity G?

   [1 mark]

   • A9
   • B15
   • C13
   • D18

   (b)

   Which activities are critical?

   [1 mark]

   • AA, C, F, H
   • BB, E, G, H
   • CB, D, G, H
   • DA, D, F, H

   (c)

   Find the total float of activity C.

   [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 3 (none); B 5 (none); C 4 (A); D 2 (A); E 6 (B, C); F 3 (D); G 4 (E, F). Each activity needs one worker.

   (a)

   What is the minimum completion time of the project?

   [1 mark]

   • A13
   • B27
   • C15
   • D17

   (b)

   What is the lower bound for the number of workers needed to complete the project in the minimum time?

   [1 mark]

   • A2
   • B1
   • C3
   • D4

   (c)

   Find the total float of activity D.

   [2 marks]

   Total for question 2: 4 marks

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

   (a)

   Carry out a forward pass to find the earliest start time of each activity. State the minimum project completion time and the critical activities.

   [3 marks]

   (b)

   Carry out a backward pass and hence find the total float of each non-critical activity.

   [4 marks]

   Total for question 3: 7 marks

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

   (a)

   Use a forward and backward pass to find the minimum project completion time and the critical activities. Hence find a lower bound for the number of workers needed to complete the project in this time.

   [6 marks]

   (b)

   Unexpectedly, activity C takes 8 days instead of 5. Decide, with working, whether the project is delayed, and find the new critical activities.

   [6 marks]

   Total for question 4: 12 marks

End of questions