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

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

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

Explain why activity B cannot be critical.

State which other activities are not critical.

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 = 11

Determine the missing early event times and late event times, and hence complete Diagram 1 in your answer book (opens in a new tab).

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.

Schedule the activities using Grid 1 in the answer book (opens in a new tab).

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.

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

State the critical path for the network.

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. (opens in a new tab)

Complete Diagram 1 in the answer book (opens in a new tab) to show the early event times and the late event times.

(i) State the minimum project completion time. 

(ii) List the critical activities. 

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.

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

Draw a cascade chart for this project on Grid 1 in the answer book (opens in a new tab).

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).

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.

Complete the table in the answer book (opens in a new tab) to show the immediately preceding activities for each activity.

(i) Complete Diagram 1 in the answer book (opens in a new tab) 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.

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 (opens in a new tab), draw a resource histogram to show the number of workers required at each time when each activity begins at its earliest possible start time.

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.

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, , of activity N is unknown, but it is given that 

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

Complete the precedence table in the answer book (opens in a new tab).

Complete Diagram 1 in the answer book (opens in a new tab) to show the early event times and the late event times.

List the critical activities.

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 . You must make the numbers used in your calculation clear.

Draw a cascade chart for this project on Grid 1 in the answer book (opens in a new tab).

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 (opens in a new tab) to show the early event times and the late event times.

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.

Schedule the activities using Grid 1 in the answer book (opens in a new tab).

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 (opens in a new tab) for the 8 activities in Figure 5.

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

Complete the activity network in the answer book (opens in a new tab) for the project. Your completed activity network must contain only the minimum number of dummies.

Given that all 14 activities have the same duration,

explain why activity D cannot be critical.

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 (opens in a new tab).

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

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

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.

The network in Figure 2 shows the activities that need to be completed for 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 are shown in Figure 2.

Complete Table 1 below to show the immediately preceding activities for each activity.

It is given that

State the largest possible integer value of .

(i) Complete Diagram 1 below to show the late event times.

(ii) State the activities that must be critical.

Calculate the total float for activity G.

The resource histogram in Figure 3 shows the number of workers required when each activity starts at its earliest possible time. The histogram also shows which activities happen at each time.

Complete Table 2 below to show the number of workers required for each activity of the project.

Draw a Gantt chart on Grid 1 below to represent the activity network.

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, in hours, of the corresponding activity is shown in brackets.

(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 of the project.

The table below lists the number of workers required for each activity in the project.

Each worker is able to do any of the activities. Once an activity is started it must be completed without interruption. It is given that each activity begins at its earliest possible start time.

(i) On Grid 1 in the answer book, draw a resource histogram to show the number of workers required at each time.

(ii) Hence state the time interval(s) when six workers are required.

The precedence table below shows the twelve activities required to complete a project.

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

Figure 6 shows a partially completed cascade chart for the project. The non‑critical activities F, J and K are not shown in Figure 6.

The time taken to complete each activity is given in hours and the project is to be completed in the minimum possible time.

State the critical activities.

Given that the total float of activity F is 2 hours,

State the duration of activity F.

The duration of activity J is hours, and the duration of activity K is hours, where and

(i) State, in terms of , the maximum possible total float for activity K.

(ii) State, in terms of and , the total float for activity J.

The precedence table below shows the twelve activities required to complete a project.

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

Figure 6 shows a partially completed cascade chart for the project. The non‑critical activities F, J and K are not shown in Figure 6.

The time taken to complete each activity is given in hours and the project is to be completed in the minimum possible time.

State the critical activities.

Given that the total float of activity F is 2 hours,

State the duration of activity F.

The duration of activity J is hours, and the duration of activity K is hours, where and

(i) State, in terms of , the maximum possible total float for activity K.

(ii) State, in terms of and , the total float for activity J.

The precedence table below shows the 12 activities required to complete a project.

Draw the activity network described in the precedence table, using activity on arc.

Your activity network must contain the minimum number of dummies only.

Each of the activities shown in the precedence table requires one worker. The project is to be completed in the minimum possible time.

Figure 3 shows a schedule for the project using three workers.

(i) State the critical path for the network.

(ii) State the minimum completion time for the project.

(iii) Calculate the total float on activity B.

(iv) Calculate the total float on activity G.

Immediately after the start of the project, it is found that the duration of activity I, as shown in Figure 3, is incorrect. In fact, activity I will take 8 hours. The durations of all the other activities remain as shown in Figure 3.

Determine whether the project can still be completed in the minimum completion time using only three workers when the duration of activity I is 8 hours. Your answer must make specific reference to workers, times and activities.