Year 12 Mathematics Standard 2

MS-N3 Critical path analysis Unit duration

Networks involve the graphical representation and modelling of situations as an approach to decision-making processes. Knowledge of networks enables 2 weeks
development of a logical sequence of tasks or a clear understanding of connections between people or items. Study of networks is important in developing
students’ ability to interpret a set of connections or sequence of tasks as a concise diagram in order to solve related problems.

Subtopic focus Outcomes

The principal focus of this subtopic is to use critical path analysis in the optimisation of A student:
real-life problems. Students develop awareness that critical path analysis is a useful tool  solves problems using networks to model decision-making in practical problems MS2-12-
in project planning, management and logistics. 8
 chooses and uses appropriate technology effectively in a range of contexts, and applies
critical thinking to recognise appropriate times and methods for such use MS2-12-9
 uses mathematical argument and reasoning to evaluate conclusions, communicating a
position clearly to others and justifying a response MS2-12-10
Related Life Skills outcomes: MALS6-11, MALS6-12, MALS6-13, MALS6-14

Prerequisite knowledge Assessment strategies

Students should have completed the Stage 6 topic MS-N2 Networks concepts. Staff could adopt formative assessment strategies and informal diagnostic testing to identify and
address students’ misconceptions.
All outcomes referred to in this unit come from Mathematics Standard 2019 Syllabus
© NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2017

© NSW Department of Education, March 2020 MS-N3 Critical path analysis 1

 Term Definition
Critical path The critical path is the sequence of network activities which combine to have the longest overall duration so as to determine the shortest
possible time needed to complete a project.
Earliest starting time (EST) The earliest starting time is the earliest time that any activity can be started after all prior activities have been completed.
Earliest finish time (EFT) The earliest time an activity can be finished after all the prerequisite activities have also been finished.
Float time Float time is the amount of time that a task in a project network can be delayed without causing a delay to subsequent tasks.
Flow capacity The flow capacity of a network can be found using the maximum-flow minimum-cut theorem and depends upon the capacity of each edge in
the network.
Latest finish time (LFT) The latest time an activity can be finished after all the prerequisite activities have also been finished, and with the project still running on
time.
Latest starting time (LST) The latest starting time is the latest time an activity may be started after all prior activities have been completed and without delaying the
project.
Maximum flow minimum cut The maximum-flow minimum-cut theorem states that the flow through a network cannot exceed the value of any cut in the network and that
the maximum flow equals the value of the minimum cut, i.e. it identifies the ‘bottle-neck’ in the system.

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 Lesson sequence Content Suggested teaching strategies and resources Date and Comments, feedback,
initial additional resources used
Considerations Technology note
The following technology can be used throughout this unit to
assist students in drawing networks:
 Graph creator
 Team Gantt
 Lucid chart
 Geogebra can also be used to create a network
diagram with weightings as captions.
Constructing  construct a network to represent the Introducing networks to model duration and
networks to model duration and interdependencies of interdependencies
duration and activities that must be completed during a  The teacher poses a question or scenario which can
interdependencies particular project, for example a student be explored to develop an understanding of the duration
(1 lesson) schedule, or preparing a meal AAM and interdependencies of tasks. Students need to consider
which activities are prerequisites of others and the impact
certain activities might have on the overall time. Sample
question:
o How long does it really take to get ready for school?
Resource: getting-ready-for-school.DOCX
When examining this activity, students can consider
which activities can be outsourced and therefore can
run at the same time. For example, if the student is
showering, they could outsource the making of their
breakfast to another family member.
Constructing networks to model duration and
interdependencies
 There are three different methods for developing and
labelling a network diagram to model duration and
interdependencies:

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 Lesson sequence Content Suggested teaching strategies and resources Date and Comments, feedback,
initial additional resources used

o Method 1: Attach the activities and the times (weights)

to vertices. This is referred to as ‘activity on node’.

o Method 2: Attach the activities and the times (weights)

to the edges and start and finish at a vertex. This is
referred to as ‘activity on arrow’.

o Method 3: Attach the activities to a vertex and the

times (weights) to the edges.

Note: In the third diagram, one dummy activity, X, was

introduced and the incoming edges have no weight.
Critical path analysis  given activity charts, prepare Prepare network diagrams from activity charts
(2 lessons) network diagrams and use critical path  The teacher sets the context in terms of project
analysis to determine the minimum time management:
for a project to be completed AAM

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 Lesson sequence Content Suggested teaching strategies and resources Date and Comments, feedback,
initial additional resources used
o use forward and backward scanning to o If you were managing any project you'd create a table
determine the earliest starting time that describes each of the activities, their duration and
(EST) and latest starting time (LST) for their dependencies or prerequisites.
each activity in a project (ACMGM105) o Try to establish authentic contexts to use with students
o understand why the EST for an activity Resource: activity-charts-to-network-diagram.DOCX
could be zero, and in what
circumstances it would be greater than Introducing critical paths
zero  The teacher introduces the lesson by posing the
o calculate float times of non-critical question: How can we minimise how long it takes to
activities (ACMGM108) complete a series of tasks, while maximising the time we
have available to do them?
o understand what is meant by critical
path o Students investigate the concept of critical paths using
the making a meal activity.
o use ESTs and LSTs to locate the critical
path(s) for the project (ACMGM106) Resource: Making a meal, introducing-critical-
paths.DOCX
o The teacher defines key terms or concepts for critical
path analysis
Resource: key-terms-and-concepts.DOCX
Forward and backward scanning
 The teacher models the identification of a critical
path using forward and backward scanning.
 Forward scanning consists of the following steps
where a network is constructed using :
o Draw the activity chart with an empty circle at each
vertex.
o In each circle write the EST of the activity that starts
from that vertex. Assign the start vertex an EST of zero.
This means that each activity with no prerequisites can
be started immediately.
o Proceed from left to right along each path from the
start, writing the EST into each circle as it is
encountered. When two or more paths join, select the
highest total.
o Continue until the finish vertex is reached. The EST of

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 Lesson sequence Content Suggested teaching strategies and resources Date and Comments, feedback,
initial additional resources used
the finish vertex is the critical time of the project.
 Backward scanning consists of the following steps:
o Begin at the finish vertex.
o In each circle write the LST of any activity that ends at
that vertex. Assign the finish vertex an LFT equal to the
critical time of the project. Meaning that the latest time
the project can finish is also the earliest time it can be
finished.
o Proceed from right to left, along each path from the
finish, writing the LFT into each circle as it is
encountered. When two or more paths join going
backwards, select the lowest total.
o Continue until the start vertex is reached. The LFT of
the start vertex must always be zero.
 Identify the critical path once forward and backwards
scanning have been completed by following the path with
zero float times.(i.e. EST =LST )
 Staff may like to use this Critical Path Analysis
Geogebra App to demonstrate forward and backward
scanning techniques to find the critical path.
Resource: critical-path-analysis.PPTX
Forward and backward scanning (alternate method)
 An ‘activity on node’ network diagram can be
modified to allow for forward and backward scanning by
replacing each vertex with an activity box:

 Consider the previous sample network diagram:

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 Lesson sequence Content Suggested teaching strategies and resources Date and Comments, feedback,
initial additional resources used

 Add the activity box:

 Forward scanning:
o Start by entering the EST on node A as 1, to
indicate that the process A starts at the start of day
1.

o Calculate the EFT as EFT =EST +duration−1,

for example the EFT for process A is 1. This means
that process A needs to be completed by the end
of day 1.
o The next process, B, will start the next day so enter
the EST on node B as 2 (the EFT of node A plus 1).

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 Lesson sequence Content Suggested teaching strategies and resources Date and Comments, feedback,
initial additional resources used

o Calculate the EFT for process B as above.

o Complete the EST and EFT on each node using
forward scanning techniques.

 Backward scanning:
o Start by entering the LFT on node G to be equal to
the EFT. In the case above this is 10.

o Calculate the LFT as LST =LFT−duration+1,

therefore in the example above the LST on node G
is 10−1+1=10, which means that process G
must start on day 10 (LST) and finish at the end of
day 10 (LFT).
o The previous process(es), E and F, will have to
finish the day before the LST of process G.
Therefore the LFT for nodes E and F will be 9 (the

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 Lesson sequence Content Suggested teaching strategies and resources Date and Comments, feedback,
initial additional resources used
LST of node G – 1).

o Complete the LFT and LST for each node using

backward scanning techniques.

NESA exemplar question

1. The following table gives details of a set of six tasks
which have to be completed to finish a project. The
immediate predecessors are those tasks which must be
completed before a task may be started.

a. Draw an activity network

b. Perform a forward scan and backward scan to find the
critical path
c. What is the critical time for the project?

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 Lesson sequence Content Suggested teaching strategies and resources Date and Comments, feedback,
initial additional resources used
d. List the critical activities
e. Identify the float time of activity D
f. A time lag of one day is needed between activities B
and E due to issues of supply of resources. Describe
the effect it will have on the critical path.
Resource: ms-n3-nesa-examples-solutions.DOCX
Applying critical path analysis
 Student activity: Students to use forward and
backward scanning to analyse organising a family meal and
a get together with friends at the movies. Students are
asked to consider how long each activity takes, the earliest
time dinner will be ready and what time they will make it to
a movie after dinner as well as what if scenarios.
Resource: meal-and-a-movie.DOCX
 Student activity: Students can use this website from
brighthubpm.com to learn how to use Microsoft Excel to
determine the critical path and plan a project by developing
a Gantt chart.
Solve flow problems  solve small-scale network flow Constructing a network diagram representing flow
using network problems, including the use of the  Teacher introduces the concept of a network
diagrams. ‘maximum-flow minimum-cut’ theorem, representing flow by showing the shimap.org video.
(1 lesson) for example determining the maximum
volume of oil that can flow through a  Driving question: Australia has a trade agreement
network of pipes from an oil storage tank with China. There are many travel routes between the two
(the source) to a terminal (the sink) countries with different capacities of flow. How do we
(ACMGM109) AAM represent this as a network diagram?

o convert information presented in a o Student activity: Students draw a network diagram to

table into a network diagram represent their answers and consider the driving
question. Teacher can introduce network terminology
o determine the flow capacity of a (source and sink) by using each student's diagram.
network and whether the flow is
sufficient to meet the demand in o The graphs should have direction and weightings.
various contexts Resource: australias-trade-with-asia.DOCX
Maximum-flow  solve small-scale network flow Introducing flows and cuts
minimum-cut problems, including the use of the

© NSW Department of Education, March 2020 MS-N3 Critical path analysis 10

 Lesson sequence Content Suggested teaching strategies and resources Date and Comments, feedback,
initial additional resources used
(2 lessons) ‘maximum-flow minimum-cut’ theorem,  The teacher introduces key terms and concepts in
for example determining the maximum network flow diagrams which are networks where the
volume of oil that can flow through a weights on the edges are often referred to as capacities and
network of pipes from an oil storage tank is the maximum rate of flow through that edge. Terms and
(the source) to a terminal (the sink) concepts include:
(ACMGM109) AAM o Source
o determine the flow capacity of a o Sink
network and whether the flow is
sufficient to meet the demand in o Flow rules
various contexts o Excess flow
o Saturated edges
o Cut
Resource: key-terms-and-concepts.DOCX
The maximum-flow minimum-cut theorem
 The teacher introduces the concepts of maximum
flow and minimum cut.
o The maximum flow from source to sink is called the
network’s flow capacity.
o The minimum cut is the cut that has the least capacity.
o The maximum-flow minimum-cut theorem states that
the maximum flow through a network is equal to the
value of the minimum cut.
o If you have a flow of a certain capacity, and a cut of the
same capacity then this theorem says that the flow is
the maximum flow, and the cut is the minimum cut.
This is useful to show that a cut or flow found by trial
and error is actually the best possible.
o A minimum cut will only include edges which are
saturated. However, there may be saturated edges
which are not part of a minimum cut.
 The teacher models the methods of finding the
maximum flow and minimum cut.
Resources: maximum-flow-minimum-cut-methods.DOCX,

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 Lesson sequence Content Suggested teaching strategies and resources Date and Comments, feedback,
initial additional resources used
maximum-flow-minimum-cut-methods.PPTX, maximum
flow (youtube)
 Student activity: Students to practise finding the
minimum cut and maximum flow (flow capacity) for a range
of networks.
Resource: maximum-flow-minimum-cut-sample-
questions.DOCX

Reflection and evaluation

Please include feedback about the engagement of the students and the difficulty of the content included in this section. You may also refer to the
sequencing of the lessons and the placement of the topic within the scope and sequence. All ICT, literacy, numeracy and group activities should
be recorded in the ‘Comments, feedback, additional resources used’ section.

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