How do you find a latest event time?

Backward pass: take the smallest of l(j)-d over all activities leaving the event; l(n)=e(n).

What is the minimum project duration?

The earliest event time at the finish event.

What is l(1) in a correct backward pass?

0 (a useful check on your arithmetic).

What does a total float of 0 mean?

The activity is critical: any delay delays the whole project.

What is the critical path?

A path of critical activities from start to finish; its length is the minimum project duration.

Can a network have more than one critical path?

Yes: if two or more routes have the same longest length.

Why can a forward pass use a dummy of duration 0?

The dummy still forces its end event to wait: e(j)\ge e(i)+0.

What does float of 4 days on an activity mean?

It can be delayed by up to 4 days (starting at its earliest time) without delaying the project.

Critical paths and floatEdexcel International A Level Maths: Flashcards

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How do you find an earliest event time?

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How do you find an earliest event time?: Forward pass: take the largest of $e(i)+d$ over all activities arriving at the event; $e(1)=0$ .
How do you find a latest event time?: Backward pass: take the smallest of $l(j)-d$ over all activities leaving the event; $l(n)=e(n)$ .
What is the minimum project duration?: The earliest event time at the finish event.
What is $l(1)$ in a correct backward pass?: $0$ (a useful check on your arithmetic).
Earliest start time of an activity from event $i$ to $j$ ?: $e(i)$
Earliest finish time of an activity from event $i$ to $j$ with duration $d$ ?: $e(i)+d$
Latest finish time of an activity from event $i$ to $j$ ?: $l(j)$
Latest start time of an activity from event $i$ to $j$ with duration $d$ ?: $l(j)-d$
State the total float formula.: $F(i,j)=l(j)-e(i)-d(i,j)$
What does a total float of 0 mean?: The activity is critical: any delay delays the whole project.
What is the critical path?: A path of critical activities from start to finish; its length is the minimum project duration.
Can a network have more than one critical path?: Yes: if two or more routes have the same longest length.
Why can a forward pass use a dummy of duration 0?: The dummy still forces its end event to wait: $e(j)\ge e(i)+0$ .
What does float of 4 days on an activity mean?: It can be delayed by up to 4 days (starting at its earliest time) without delaying the project.

Exam questions on Critical paths and float

A project is modelled by an activity network with start event 1 and finish event 6. The activities are: A from event 1 to 2 (3 days); B from 1 to 3 (5 days); C from 2 to 4 (6 days); D from 3 to 4 (2 days); E from 3 to 5 (4 days); F from 4 to 6 (3 days); G from 5 to 6 (5 days).
Calculate the total float on activity D.2 marks
A project is modelled by an activity network with start event 1 and finish event 5. The activities are: A from event 1 to 2 (6 days); B from 1 to 3 (4 days); C from 2 to 4 (3 days); a dummy from 2 to 3; D from 3 to 4 (7 days); E from 4 to 5 (5 days); F from 3 to 5 (9 days).
Find the earliest finish time and the latest finish time of activity B, and hence its total float.2 marks
A project is modelled by an activity network with start event 1 and finish event 5. The activities are: A from event 1 to 2 (5 days); B from 1 to 3 (7 days); C from 2 to 4 (6 days); D from 3 to 4 (5 days); E from 3 to 5 (8 days); F from 4 to 5 (3 days).
Carry out a forward pass and state the minimum project duration.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).