Critical Path Analysis (Edexcel A Level Further Maths: Decision Maths 1)

Topic Questions

1a
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3 marks

Activity Time taken (days) Immediately preceding activities
A 5 -
B 8 -
C 4 -
D 14 A
E 10 A
F 3 B, C, E
G 7 C
H 5 D, F, G
I 7 H
J 9 H


The table above shows the activities required for the completion of a building project. For each activity, the table shows the time it takes, in days, and the immediately preceding activities. Each activity requires one worker. The project is to be completed in the shortest possible time.

q3-8fmo-27-june-2018

Figure 2

Figure 2 shows a partially completed activity network used to model the project. The activities are represented by the arcs and the number in brackets on each arc is the time taken, in days, to complete the corresponding activity.

Add the missing activities and necessary dummies to Diagram 1 in the answer book.

1b
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3 marks

Complete Diagram 1 in the answer book to show the early event times and the late event times.

1c
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1 mark

State the critical activities.

1d
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3 marks

At the beginning of the project it is decided that activity G is no longer required.

Explain what effect, if any, this will have on

(i)
the shortest completion time of the project if activity G is no longer required, 

(ii)
the timing of the remaining activities.

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2a
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5 marks
Activity Immediately preceding activities
A -
B -
C A
D A
E A
F B, C
G B, C
H D
I D, E, F, G
J D, E, F, G
K G

Draw the activity network described in the precedence table above, using activity on arc.
Your activity network must contain the minimum number of dummies.

2b
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1 mark

Every activity shown in the precedence table has the same duration.

Explain why activity B cannot be critical.

2c
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1 mark

State which other activities are not critical.

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3a
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2 marks

q4-fig-3-june-2019-9fm0-3d

Figure 3

The network in Figure 3 shows the activities that need to be undertaken to complete a project. Each activity is represented by an arc and the duration of the activity, in days, is shown in brackets. The early event times and late event times are to be shown at each vertex and one late event time has been completed for you.

The total float of activity H is 7 days.

Explain, with detailed reasoning, why x = 11

3b
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3 marks

Determine the missing early event times and late event times, and hence complete Diagram 1 in your answer book.

3c
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1 mark

Each activity requires one worker and the project must be completed in the shortest possible time using as few workers as possible.

Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time.

3d
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3 marks

Schedule the activities using Grid 1 in the answer book.

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4a
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5 marks
Activity Immediately preceding activities
A
B
C
D A
E C
F B, C, D
G A
H B, C, D
I B, C, D, G
J B, C, D, G
K E, H

Draw the activity network described in the precedence table above, using activity on arc. Your activity network must contain only the minimum number of dummies.

4b
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1 mark

Given that all the activities shown in the precedence table have the same duration,

State the critical path for the network.

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5a
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2 marks

q2-fig1-oct-2020-8fm0-27

Figure 1

A project is modelled by the activity network shown in Figure 1. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, to complete the corresponding activity. Each activity requires one worker. The project is to be completed in the shortest possible time.

Complete the precedence table in the answer book.

5b
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3 marks

Complete Diagram 1 in the answer book to show the early event times and the late event times.

5c
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2 marks
(i)
State the minimum project completion time. 

(ii)
List the critical activities. 
5d
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1 mark

Calculate the maximum number of hours by which activity H could be delayed without affecting the shortest possible completion time of the project. You must make the numbers used in your calculation clear.

5e
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2 marks

Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.

5f
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3 marks

Draw a cascade chart for this project on Grid 1 in the answer book.

5g
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1 mark

Using the answer to (f), explain why it is not possible to complete the project in the shortest possible time using the number of workers found in (e).

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6a
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2 marks

fig1-november-2020-9fm0-3d

Figure 1

The network in Figure 1 shows the activities that need to be undertaken to complete a project. Each activity is represented by an arc and the duration, in hours, of the corresponding activity is shown in brackets.

Explain why each of the dummy activities is required.

6b
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2 marks

Complete the table in the answer book to show the immediately preceding activities for each activity.

6c
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6 marks
(i)
Complete Diagram 1 in the answer book to show the early event times and the late event times.
(ii)
State the minimum completion time for the project.
(iii)
State the critical activities.
6d
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3 marks

Each activity requires one worker. Each worker is able to do any of the activities. Once an activity is started it must be completed without interruption.

On Grid 1 in the answer book, draw a resource histogram to show the number of workers required at each time when each activity begins at its earliest possible start time.

6e
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2 marks

Determine whether or not the project can be completed in the minimum possible time using fewer workers than the number indicated by the resource histogram in (d). You must justify your answer with reference to the resource histogram and the completed Diagram 1.

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7a
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2 marks

fig-1-oct-2021-8fm0-27

Figure 1

A project is modelled by the activity network shown in Figure 1. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, to complete the corresponding activity. The exact duration, x, of activity N is unknown, but it is given that 5 space less than space x space less than space 10.

Each activity requires one worker. The project is to be completed in the shortest possible time.

Complete the precedence table in the answer book.

7b
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4 marks

Complete Diagram 1 in the answer book to show the early event times and the late event times.

7c
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1 mark

List the critical activities.

7d
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1 mark

It is given that activity J can be delayed by up to 4 hours without affecting the shortest possible completion time of the project.

Determine the value of x. You must make the numbers used in your calculation clear.

7e
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4 marks

Draw a cascade chart for this project on Grid 1 in the answer book.

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8a
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4 marks

fig-2-november-2021-9fm0-3d

Figure 2

A project is modelled by the activity network shown in Figure 2. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, to complete the corresponding activity.

Complete Diagram 1 in the answer book to show the early event times and the late event times.

8b
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2 marks

Each activity requires one worker and the project must be completed in the shortest possible time using as few workers as possible.

Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time. You must show your working.

8c
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3 marks

Schedule the activities using Grid 1 in the answer book.

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9a
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2 marks

fig-5-november-2021-9fm0-3d

Figure 5

Figure 5 shows a partially completed activity network for a project that consists of 14 activities.

Complete the precedence table in the answer book for the 8 activities in Figure 5.

9b
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4 marks

The precedence table for the remaining 6 activities is given below.

Activity Immediately preceding activities
I D, E, G, H
J D, E, G, H
K E, G, H
L I, J, K
M J, K
N J, K

Complete the activity network in the answer book for the project. Your completed activity network must contain only the minimum number of dummies.

9c
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2 marks

Given that all 14 activities have the same duration,

explain why activity D cannot be critical.

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10a
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3 marks

fig-4-specimen-2017-9fm0-3d

Figure 4

A project is modelled by the activity network shown in Figure 4. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete that activity. Each activity requires one worker. The project is to be completed in the shortest possible time.

Calculate the early time and the late time for each event, using Diagram 1 in the answer book.

10b
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3 marks

On Grid 1 in the answer book, complete the cascade (Gantt) chart for this project.

10c
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3 marks

On Grid 2 in the answer book, draw a resource histogram to show the number of workers required each day when each activity begins at its earliest time.

10d
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3 marks

The supervisor of the project states that only three workers are required to complete the project in the minimum time.

Use Grid 2 to determine if the project can be completed in the minimum time by only three workers. Give reasons for your answer.

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