Module 5: Networks and Decision Mathematics

2010 VCE Further Mathematics SAC 4 Module 5: Networks and Decision Mathematics

Name……………………………………………………………… ……

Teacher’s Name: ………………………………………………………...

Time allowed: 65 plus 5 minutes reading time

Materials allowed: One bound reference and one CAS and/ or Scientific calculator

All answers correct to 2 decimal places unless otherwise specified. Diagrams are NOT to scale.

Section A - Multiple Choice = 20 marks Section B - Short Answer = 20 marks

Total Marks = 40

I understand I must not intentionally or unintentionally disclose any details of this SAC or imply what is or is not included; or in any way gain an unfair advantage over other students for myself or any other student. If I do, I understand that disciplinary action will occur and my result will be downgraded. In fairness to fellow students it is my responsibility to inform the VCE office if I am aware that information about the SAC is being passed on, or that a student has gained unfair advantage.

______Student Signature

Module 5: Networks and decision mathematics

SECTION A – MULTIPLE CHOICE

Question 1

The number of vertices with an even degree in the graph above is

A. 1 B. 2 C. 3 D. 4 E. 5

The information below relates to questions 2 and 3.

A graph with vertices M - S is shown below.

Question 2

A Hamiltonian path for the graph above is

A. P N O Q R S B. QRSOQPNOMN

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C. P Q R S O M N P D. P Q R S O N M E. S R Q P N M O Q Question 3

Which one of the following statements is true for the graph shown?

A. The graph is a tree. B. The graph is complete. C. The graph has at least one loop. D. The graph contains an Euler path. E. The graph contains an Euler circuit.

Question 4

The undirected graph with vertices A - D is shown below.

An adjacency matrix that could be used to represent this graph is

A. 0 1 1 1   1 0 1 1 1 1 0 1 B.   0 2 2 2 1 1 1 0   2 0 2 2 2 2 0 2   C. 0 1 1 2 2 2 2 0   1 0 1 1 1 1 0 1   2 1 1 0 D. 1 0 0 0   0 1 0 0 0 0 1 0 E. 0 1 1 1     0 0 0 1 1 0 1 1 1 1 0 2   2 1 1 1

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Question 5

3 7 5 3 5 2 6 start finish k 4

On the directed graph above the capacity of each edge is given. The capacity of the cut shown is 28. The value of k is

A. 4 B. 6 C. 7 D. 8 E. 10

Question 6

When drawn as a planar graph, the number of faces on the graph above is

A. 5 B. 7 C. 8 D. 10 E. 11

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Question 7

The graph shown above can be correctly described as

A. connected but not planar B. connected but not simple C. complete but not connected D. complete but not planar E. simple but not complete

Question 8

The network below shows the distances, in kilometres, of bike tracks that run between the west gate and the east gate of a large park.

1 2 5 3 2 west gate east gate 2

5 2 1 3 2

The length in kilometres, of the shortest route between the west gate and the east gate is

A. 7 B. 8 C. 9 D. 10 E. 11

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Question 9

A connected, planar graph has 7 vertices and 12 edges. The number of faces that this graph has is

A. 7 B. 9 C. 17 D. 19 E. 21

Question 10

2 5 2 4 1 4 4 3 3 2

For the network above, the total weight of the minimal spanning tree will be

A. 10 B. 12 C. 15 D. 20 E. 37

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Question 11 6

7 5 3 A B 5 2 2 5 7

For the graph above the maximum flow possible through the network from A to B is

A. 9 B. 10 C. 11 D. 12 E. 13

Question 12

Four manufacturers have been asked to provide the wholesale cost of producing 4 items for a large hardware distributor. Each manufacturer will be allocated one item so that the total cost to the distributor is minimised.

Item Manufacturer 1 2 3 4 A 12 8 8 12 B 11 10 9 10 C 8 12 9 7 D 10 9 11 7

That total minimum cost will be

A. $31 B. $32 C. $34 D. $37 E. $38

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Question 13

Which one of the following graphs does Euler’s formula involving faces, edges and vertices not apply to?

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The following information below relates to Questions 14 and 15

The activities together with the time, in weeks, needed to complete them for a particular project are shown on the network below.

C,3 H,2

A,2 D,5 I,1 M,4 G,4 L,5 start finish E,6 J,4 B,3 N,5

F,5 K,4

Question 14

The critical path for this project is

A. A,C,H,M B. A,D,G,L C. B,E,G,L D. B,F,J,L E. B,F,K,N

Question 15

The event that will cause an extra four activities to become critical to the completion of the project is

A. Activity D is decreased by 1 day B. Activity F is increased by 1 day C. Activity J is increased by 1 day D. Activity K is increased by 1 day E. Activity N is decreased by 1 day

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Question 16

Activity Prerequisites A  B  C B D C,F E A,D F E G C,F H E

Which one of the following could be the graph for the above table?

A. E B. E A H A H D F D F B G B G C C

C. E D. E A H A H D F D F B G B C C G

E. E H A F G B C D

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Question 17

Annie, Ben Carrie, David, Ellie and Frank all play a game of tennis against every other person. The results are shown in the directed graph below, where X Y indicates X beat Y.

The person who has a two step dominance over David, D, is

A. Annie

B. Ben

C. Carrie

D. Ellie

E. Frank

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Question 18

Activity Duration (days) Earliest Start Latest Start Crash Time Time (day) Time (day) (days) A 2 0 1 1 B 3 0 0 0 C 4 3 3 2 D 2 7 7 0 E 1 3 5 2 F 3 4 6 1 G 2 9 9 1 H 4 4 7 1

Given the data in the table above, which one of the following statements is true?

A. A and B lie on the critical path.

B. A crash time of 2 hours for E will not reduce the overall time for the project.

C. The overall reduction in time for the project is 8 days if all the crash times are implemented.

D. If only two activities can be crashed, they should be C and E

E. The total time for the project is 21 days.

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Module 5: Networks and decision mathematics

SECTION B – SHORT ANSWER

Question 1

A group of five friends, Tom, Leigh, Pete, Sanjo and Jack set up a landscape gardening business. Each of the five has an expertise in at least one of the five main areas of the business. Each of the men is to be put in charge of one of the areas that they have expertise in so that all areas have someone in charge. The bipartite graph below shows the names of the five men and the areas in which they have expertise.

Tom paving

Leigh planting

Pete fencing

Sanjo concreting

Jack maintenance

a. Which area must Tom be put in charge of?

1 mark b. How many areas have only one person who can feasibly be put in charge?

1 mark

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Question 2

The landscape gardening business has 7 jobs currently running. The network below shows the roads that are used to travel between the jobs which are labelled A – G.

D G E a. Find the sum of the degrees of the vertices of the network shown above.

1 mark b. Write down a route that starts at A and represents a Hamiltonian path.

1 mark

Whilst travelling between jobs, Leigh loses something from the back of his ute. He plans a route so that he travels just once on each of the roads shown on the network in order to look for the missing item. c. Which vertex or vertices could he start from to achieve this?

1 mark

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Question 3

The five men are keen golfers and play each other regularly. The directed graph below shows the most recent results when the men have played in pairs.

Tom Pete Key Tom Leigh indicates Tom defeated Leigh

Jack Sanjo

The dominance matrix, showing one-step dominance for the men in their recent golf results is shown below. loser Tom Leigh Pete Sanjo Jack Tom  0 1 0 0 0    Leigh  0 0 0  1  winner Pete  1 1 0 0 1    Sanjo  1 0 1 0 1    Jack  1 0 0 0 0  a. Write down the missing entry in the matrix above.

1 mark b. Which two men have equal highest totals of one-step dominance?

1 mark

______17 c. By adding together the total of one-step and two-step dominances, find the most dominant golfer and the least dominant golfer amongst the five men.

most dominant golfer

least dominant golfer 2 marks

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Question 4

Jack has identified ten activities that must be performed in order to complete a job that the business has running. Those activities together with the time that they take to complete in days are shown on the directed network below.

A,2 F,5

J,3 start D,6 finish G,4 B,3 I,2

E,3 H,3

a. What is the earliest start time for activity F?

1 mark b. What is the latest start time for activity H?

1 mark c. What is the minimum time that the job can be completed in?

1 mark

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Just prior to the job beginning, the property owner changes the design so that activity D is removed from the job. The new directed network reflecting this change is shown below. C,5

A,2 F,5

J,3 start finish G,4 B,3 I,2

E,3 H,3

d. By how many days does this reduce the completion time?

1 mark

The property owner also wants the job to be completed in 10 days. Jack determines that only activities F, G, H and J can be reduced and that the minimum time each can be reduced to is 1 day. The cost to the owner is $1000 per day for each activity that is reduced. e. Complete the table below showing the number of days each of these four activities should be reduced so that the job is reduced to 10 days at the least cost.

Activity Number of days to be reduced F G H J

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2 marks Total 15 marks Question 5

An upgrade is being carried out on the wharf on island E. There are 9 activities that need to be completed. The directed network below shows these 9 activities, A – I, and the time, in days, required to complete each one.

E,4 F,5

A,2 I,3 start finish D,3 G,4 B,4 H,3

a. What is the shortest time in which these improvements can be completed?

1 mark b. Which two activities have the greatest slack time?

1 mark c. What is the latest start time for activity E?

1 mark

Activity C is to be reduced by 3 days. d. Which 3 activities have now become critical to the upgrade being completed in the shortest possible time?

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1 mark e. Explain whether it was worth reducing activity C by 3 days given that it cost money each day to do so. Use completion times for the project to justify your opinion.

1 mark Total 15 marks

Question 6

Activities A, B, C, D, E, F, G, H and I are necessary to establish a garden.

Activity Immediate Expected Extra Workers Cost of Predecessor Duration reduce time by reduction per (days) (days) day ($) A none 15 3 2000 B none 23 3 1800 C none 17 1 1200 D C 19 1 900 E A 31 3 1300 F B,D 18 2 400 G B,D 29 1 800 H C 25 2 1700 I E,F 22 4 1000 J H 22 7 1000 a. Complete the network below to show all the information given in the first three columns of the table.

E, 31 I, 22 A, 15 F, 18 B, 23 G, 29 C, 17 J, 22 ______H, 25 22

1 mark b. What is the critical path for this project?

______1 mark c. What is the shortest time possible to complete this project?

______1 mark d. What is the maximum number of days late that activity B can start and still not delay the finishing time?

______1 mark e. What is the maximum number of days late that activity F can start and still not delay the finishing time?

______1 mark f. If each day the total project takes costs the company $5000, use the information in the table given above to find the maximum amount of money that can be saved if extra workers are brought in?

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______2 marks g. If activity C could be completed in just three days, how long would the whole project take if no other activity had its time reduced?

______1 mark

Total = 15 marks

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