Download Year 12 NETWORK SAC 2015

Year 12 Further SAC AT 4.1
NETWORKS
2015
Question 1:
Stuart and Lisa are wanting to promote St Leonards College to the nearby suburbs and wish to hold an
information session in the local town halls. They have agreed that they will visit 7 suburbs (vertices) and
will travel by car along main streets (edges) as shown below. They will visit the town halls in each suburb
to speak to potentially new parents, and at the end of their speech, they will get together to debrief, before
continuing into the next suburb. The distance in kilometres along each road is shown in the diagram.
1a) (i) What is the degree sum of the above graph?
[1 mark]
(ii) What percentage does the vertex degree of Sandringham make from the total degree sum of the
network vertices, correct to 2 decimal places?
[1 mark]
1b) In the network above, what is the length of the shortest route from Hampton to Bentleigh and
highlight this path in the diagram.
[2 marks]
1c) Stuart and Lisa want to start their marketing campaign from St Leonard’s and return to the school,
having travelled along each street only once. Stuart says that this is not possible, but Lisa insists that this
can be done.
(i) State with reason, whether, Stuart or Lisa is correct.
[2 marks]
(ii) Stuart proudly informs Lisa, that travelling every road exactly once is called __________________.
[1 mark]
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Year 12 Further SAC AT 4.1
NETWORKS
2015
1d) Find the route that minimises the estimated distance travelled if they start at St Leonards, visit all the
suburbs without travelling through all the roads, then return to St Leonards. What type of route is this
and state the distance?
[2 marks]
1e) Lisa realises that if she travels through a side street, she will be able to start at the school and return
to St Leonard’s having travelled on each road. Draw in the additional road/s to make this possible.
[1 mark]
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Year 12 Further SAC AT 4.1
NETWORKS
2015
1f) If however, Stuart wants to leave from his home in Brighton and finish up at the school, having
travelled every road once. Every time he arrives at the school he will provide an update on his marketing
progress to the School Council.
(i) What changes need to me made to the network to make this possible and show this on the
diagram above?
[1 mark]
(ii) What is the name given to such a path?
[1 mark]
(iii) state the minimum number of times he will need to debrief the Council.
[1 mark]
(ii) Verify Euler’s Formula, based on your diagram in 1c(iii).
[2 marks]
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Year 12 Further SAC AT 4.1
NETWORKS
2015
1g) Stuart realises that the traffic in Harefield and South roads in the mornings and afternoons can
become quite congested. As an enticement to parents, he has informed them that these roads will
become express laneways, called “St Leonard’s Way” which will be developed, exclusively for those St
Leonard’s students, who travel on the bus between 7am-9am and 3pm-5pm Monday to Friday. The town
halls in the suburbs will become the central pick up and drop off locations for the students.
(i) Highlight in the network below, the minimal spanning tree pathway that connects the town halls
to the school.
[1 mark]
(ii) What is the minimal distance that accomplishes this?
[1 mark]
1h)
A bus can pick up students from at most two suburbs at any given time on a single journey, before
arriving at St Leonards. The distance of the town halls is to be no more than 6.3km from the school.
(i)
Determine the minimum distance and highlight this on the network below.
[2 marks]
(ii)
Hence state how many buses should Stuart need to hire?
[1 mark]
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Year 12 Further SAC AT 4.1
NETWORKS
2015
Question 2:
Stuart is delighted by the canteen profits and the wonderful behaviour of the students at lunchtime. He is
intent on maintaining the jovial atmosphere and has formed a 5 piece musical band called “Big Stu and the
Heartbreakers”, consisting of himself, Lisa, Jules, Robbo and Deb. They will perform on the stage every
Friday lunchtime at the canteen, providing entertainment to students as they eat their lunch.
Each staff member can only fill one position in the band. The following bipartite graph illustrates the
positions that each is able to fill.
2a)
Musician
Position
Lisa
drums
Jules
guitar
Robbo
keyboards
Deb
saxophone
Stuart
vocal
Which musician must play the guitar?
[1 mark]
2b)
Complete the table showing the positions that the following musicians must fill in the band.
Person
Position
Robbo
Deb
Stuart
[3 marks]
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Year 12 Further SAC AT 4.1
NETWORKS
2015
The five musicians will compete in a music trivia game. Each musician competes once against every other
musician and there can only be one winner and one loser. The results are represented in the dominance
matrix , Matrix 2.1 and also in the incomplete directed graph below.
L J R DS
L  0 0 1 0 1
Matrix 2.1
2c)




R 0 0 0 1 0 


D 1 0 0 0 0 
S  0 1 1 1 0 
J 1 0 1 1 0
Explain why the leading diagonal in Matrix 1 are all zero.
[1 mark]
On the directed graph below, an arrow from Jules to Lisa shows that Jules won against Lisa.
Directed Graph
One of the edges on the directed graph is missing.
2d)
Using the information in Matrix 2.1, draw in the missing edge on the directed graph above and
clearly show its direction.
[2 marks]
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Year 12 Further SAC AT 4.1
2e)
NETWORKS
2015
Determine the one-step dominance for each of the musicians and their position.
Person
Dominance
Position
Lisa
Jules
Robbo
Deb
Stuart
[5 marks]
2f)
Based on the information above, no musician can be declared the winner. Explain why this is the
case.
[1 mark]
2g)
By calculating the two stage dominance, state the order in which the musicians have won the
contest by filling in the table below, and hence the overall winner of the competition.
[3 marks]
Person
Dominance
Position
Lisa
Jules
Robbo
Deb
Stuart
[5 marks]
Overall winner:_________________
[1 mark]
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Year 12 Further SAC AT 4.1
NETWORKS
2015
Question 3:
After releasing a number of one-hit wonders, “Big Stu and the Heartbreakers” have decided to split up.
Before doing so, they will perform one last time at the Plenary. There are four main tasks which need to
be completed in preparation for the final concert. Table 3.1 shows the number of hours that 4 members of
the band would take to complete these tasks. Deb has been assigned as the designated chauffeur and will
not be performing on stage.
3a)
Deb will also allocate tacks so that the total time for completing the four tasks is kept to a
minimum, using the Hungarian algorithm.
Table 3.1
Task
Props
Tickets
Costumes
Advertising
(i)
Lisa
11
10
12
9
Jules
10
11
8
12
Robbo
14
9
14
13
Stuart
8
10
7
9
Complete step 1 of the Hungarian algorithm and fill in the cells below.
Task
Lisa
Jules
Robbo
Stuart
Props
Tickets
Costumes
Advertising
[2 marks]
(ii)
After completing step 1, Deb decided that an allocation of tasks to minimise the total time taken
was not yet possible. Explain why Deb made this decision.
[1 mark]
(iii)
Complete the steps of the Hungarian algorithm to produce a table from which the optimal
allocation of tasks can be made.
Task
Lisa
Jules
Robbo
Stuart
Props
Tickets
Costumes
Advertising
[4 marks]
3b)
Write the name of the task that each person should do for the optimal allocation of the tasks and
the time they will take to complete their allocated task.
Task
Time to complete
Lisa
Jules
Robbo
Stuart
[4 marks]
3c)
How many hours in total will it take to prepare for their concert?
[1 mark]
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Year 12 Further SAC AT 4.1
NETWORKS
2015
Question 4:
Due to the popularity of the band, the performers are giving away some free tickets to their final concert. Winners
will come from six towns and fly in using St. Leonard’s private Cessna aircraft, called “The Groove”. They will fly from
Mildura and terminate in Melbourne. On this network is listed the available spaces for passengers flying out of
various locations for the concert.
Cut 1
Cut 2
4a)
The network is cut as shown by the dotted lines. Determine the capacity of:
(i)
Cut 1:
(ii)
4b)
4c)
Cut 2:
[2 marks]
What is the maximum number of passengers who could depart from Mildura to Melbourne for the
final concert using “The Groove”?
[1 mark]
What does the cut tell us about the maximum number of passengers who would depart on “The
Groove” from Mildura to Melbourne for the final concert?
[1 mark]
4d)
Show the cut that indicates the maximum flow.
[1 mark]
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Year 12 Further SAC AT 4.1
NETWORKS
2015
Question 5:
An additional selling point to encourage enrolments at the school, is the “Sweat, Learn and Earn” program
that Stuart has introduced. This allows students to operate the newly designed tuck shop. Parents are
delighted that their child will learn practical skills and in turn get paid for their efforts.
The tasks involved are shown in the table 5.1 below:
Table 5.1: Activity durations
Task
A
B
C
D
E
F
G
H
I
J
5a)
Description
Close the tuck shop
Empty bins
Clean student tables
Heat hot dogs and pies
Type the menu on the ccTV
Stock the fridge with drinks
Wipe down kitchen surfaces
Count money
Sweep the floors
Open the tuck shop
Duration
(minutes)
4
5
7
5
2
4
4
10
6
3
Immediate
Predecessor/s
-------------A
B
B
B
C
E, F
E, F
D, G
H, I
An incomplete network for this project is shown below. Complete the network, labelling the
activity and its duration.
E, 2
A, 4
B,5
D,5
I,6
[5 marks]
5b)
The earliest and latest start times for project activities are shown in table 5.2 below. Use the
information from Table 1 to complete the table below by filling in the shaded cells.
Table 5.2: Earliest and latest start times for project activities
Task
Earliest Start Time
Latest Start Time
(minutes)
(minutes)
A
0
0
B
4
4
C
9
D
9
E
9
18
F
16
16
G
20
H
20
20
[3 marks]
I
24
24
J
30
30
5c)
State the critical path/s for this network?
[2 marks]
5d)
What is the minimum time needed to complete these student activities?
[1 mark]
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Year 12 Further SAC AT 4.1
NETWORKS
2015
By using more students it is possible to speed up some activities. However, this will increase costs.
Activities which can be reduced in time and the associated increased costs and maximum reduction are
shown in Table 5.3 below. The information in Table 5.1 still applies to all other activities.
Table 5.3:
5e)
Activity time reductions and associated increased costs
Activity
Cost
Maximum reduction
($/minute)
(minutes)
C
1.60
3
D
1.00
2
H
1.40
2
I
1.20
2
What is the shortest time in which the activities can now be finished? (Re-drawing the changes on the
diagram may be helpful, though is not essential).
E, 2
A, 4
5f)
B,5
D,___
_
I, __
[1 mark]
Stuart receives a bonus of $2.50 for each minute saved on the chores. Complete table 5.4 below,
which outlines the benefits of reducing the time of the project completion. (Re-drawing the changes to
the network diagram sheet on page 13 may be helpful, though is not essential).
Table 5.4:
Benefits from reducing the time of project completion
Project time Total cheapest cost
Activities
Total bonus paid
(minutes)
of reduction
reduced in time
($)
($0.00)
33
0.00
-----0
32
2.60
H,I
31
30
C, H, I
29
5g)
Total Savings
($)
0
[5 marks]
What is the shortest project completion time which benefits Stuart? Give a reason for your
answer?
[2 marks]
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Year 12 Further SAC AT 4.1
NETWORKS
Question 1
Question 2
Question 3
Question 4
Question 5
TOTAL MARKS
12
2015
Year 12 Further SAC AT 4.1
NETWORKS
2015
E, 2
A, 4
B,5
D,___
_
I, __
E, 2
A, 4
B,5
D,___
_
I, __
E, 2
A, 4
B,5
D,___
_
I, __
E, 2
A, 4
B,5
D,___
_
13
I, __