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- 1. Maths A Yr 12 - Ch. 07 Page 355 Friday, September 13, 2002 9:33 AM 7 Networks syllabus reference eference Elective topic Operations research — networks and queuing In this chapter chapter 7A 7B 7C 7D Networks, nodes and arcs Minimal spanning trees Shortest paths Network ﬂow
- 2. Maths A Yr 12 - Ch. 07 Page 356 Wednesday, September 11, 2002 4:24 PM 356 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d Introduction to networks Mathematical models may be computer programs, drawings, a system of equations or a combination of these. Through models, people attempt to understand real situations. A postman plans the shortest delivery route or a builder schedules jobs on a large construction project so that the formwork is done as soon as the foundations are completed and the plasterers do not arrive before the walls have gone up. Models allow these people to think about and plan tasks before actually doing them. In particular operations research is the science of planning and executing an operation to make the most economical use of available resources. Networks, nodes and arcs Networks are maps that can represent an amazing variety of different things: simpliﬁed maps, relationships between people, sub-tasks in a building project, computer terminals or the ﬂow of trafﬁc through a city. In each case the network provides a means of studying real-life situations so that decisions can be made. When drawing a network, irrelevant information, such as bends in the roads of a map, is ignored. 1. A network is a collection of objects connected to each other in some way. 2. Networks are made up of nodes joined by arcs. If nodes are connected they are joined by an arc. 3. When the arcs have arrows they are called directed networks and travel is possible only in the direction of the arrows. There are many examples where networks can be used to model a situation. The ﬁrst worked example uses a network to plan a drive that takes the shortest possible path or distance. The network can be drawn and each node labelled. A path is a speciﬁc set of arcs connecting nodes and can be represented by the letters in the nodes, as we will see.
- 3. Maths A Yr 12 - Ch. 07 Page 357 Wednesday, September 11, 2002 4:24 PM 357 Chapter 7 Networks WORKED Example 1 George and Efﬁe want to drive from Airlie to Gillespie using the map at right. a Draw a network which represents the map. b Given that each road taken must bring them closer to Gillespie, list the number of ways from Airlie to Gillespie. How many ways are possible? c Identify the shortest path from the possible routes in b. 68 km Charles Friday Moon Mountain 66 km 39 km 48 km 50 km Davis Airlie 60 km 46 km Barnard THINK Represent towns with circles, called nodes, labelled with the ﬁrst letter of the town. Ignore the bends in the roads and use straight lines to represent roads connecting the towns. Check that towns not connected by roads on the map are not joined with an arc. a Each road taken from Airlie must go towards Gillespie. Indicate the direction on each arc with an arrow. b 54 km Ellis 83 km WRITE/DRAW a Lake Kawana 41 km Gillespie 1 2 3 b 1 66 39 A 46 60 B C 3 c Use the network to list the number of ways from A to G. Answer the question. 1 Add the lengths of the nodes to calculate the distances of the 6 routes in part b. 2 Answer the question. G D 41 54 83 68 E F 48 50 D 60 F 48 50 66 39 A 46 B 2 68 C 83 G 41 54 E A–B–D–E–G A–B–D–F–G A–B–E–G A–C–D–E–G A–C–D–F–G A–C–F–G There are 6 ways to go from Airlie to Gillespie. c A–B–D–E–G (60 + 46 + 41 + 54) 201 km A–B–D–F–G 204 km A–B–E–G 197 km A–C–D–E–G 200 km A–C–D–F–G 203 km A–C–F–G 184 km The shortest path is A–C–F–G: Airlie to Charles to Friday to Gillespie. If Efﬁe and George were more concerned with time, rather than distance, they might have consulted their travel adviser about the times for each of these stages and redrawn the network with the arcs representing average times for travelling on each connecting road. This network would help them to ﬁnd the shortest time.
- 4. Maths A Yr 12 - Ch. 07 Page 358 Wednesday, September 11, 2002 4:24 PM 358 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d WORKED Example 2 Efﬁe consults the local travel adviser about the travel times for the stages in the journey planned in worked example 1. She then redraws the network with the average time (in minutes) taken to drive between the towns as shown at right. Which path would take the least time and what is that time? THINK C 29 50 A 68 F 36 38 D 45 36 B 67 31 E 41 G WRITE 1 List all the possible paths and the times they will take. 2 The path of least time is ACDEG. A-B-D-E-G 153 min A-B-D-F-G 155 min A-B-E-G 153 min A-C-D-E-G 151 min A-C-D-F-G 153 min A-C-F-G 156 min The path ACDEG takes 151 min, the least time. So, Efﬁe and George would plan different routes depending on whether they were interested n shortest distance or shortest time. In addition to distances and times, arcs may also represent other relationships between nodes. In the following worked example we look at cost relationships between nodes. WORKED Example 3 The costs of connecting various locations on a university campus with computer cable are given in the table below. A blank space indicates no direct connection. A B A —— 4000 B —— —— C —— D —— C D E 5000 3000 1500 2200 4500 —— —— 2200 1500 —— —— —— 2500 Draw a network to represent this situation, showing the cost of connection along each arc. THINK 1 There are 5 nodes. Draw them as labelled circles. Because A and C have 3 connections, put them on the outside. WRITE/DRAW B A D E C
- 5. Maths A Yr 12 - Ch. 07 Page 359 Wednesday, September 11, 2002 4:24 PM 359 Chapter 7 Networks THINK 2 WRITE/DRAW From the table insert, in a systematic way, each arc and label each arc with its cost. A 3000 4000 2200 5000 D 4500 1500 2500 E 1500 B 2200 C remember remember 1. A network is a collection of objects connected to each other in some speciﬁc way. 2. A network consists of nodes which may be connected by arcs. 3. In a directed network, the arcs will have a direction indicated by arrows. 4. Networks can be used to model situations and calculate shortest paths. 7A Networks, nodes and arcs 1 Examine the network at right. (All the lengths are in metres.) a Which is the longest path? 1c b Which is the shortest path? A WORKED Example 6m C 4m 5m B 3m D E 9m 2 A traveller plans a journey from Ulawatu to Example Yallingup Bargara (shown on the road map at right). 160 km 1 a Draw a network to represent this situation. 120 km b Calculate the longest path if no road is 118 km travelled twice. Ulawatu Bargara c Calculate the shortest path. 45 km Angourie d The travelling times between each town are: 100 km Ulawatu–Yallingup 85 min 109 km Ulawatu–Black Rock 75 min Black Rock Yallingup–Angourie 80 min Black Rock–Angourie 82 min Yallingup–Bargara 120 min Angourie–Bargara 34 min. i Draw a network of this situation showing the time taken to travel between towns on each arc of the network. ii Calculate the longest time taken to travel from Ulawatu to Bargara. WORKED
- 6. Maths A Yr 12 - Ch. 07 Page 360 Wednesday, September 11, 2002 4:24 PM 360 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d iii Calculate the shortest time taken to travel from Ulawatu to Bargara. iv Complete the table showing the shortest distance between each of the towns. Black Rock Ulawatu Yallingup Angourie Ulawatu —— 120 100 Yallingup —— —— 220 Black Rock —— —— —— Angourie —— —— —— Bargara 209 —— v Produce a similar table showing the travelling times between each of the towns shown on the map. 3 A traveller plans a journey from Renoir to 85 Gauguin. The distances between various nearby Matisse 2 Pissarro towns are shown on the map at right. 62 Renoir a Calculate the shortest path. 46 60 b The travelling times between the following Van Gogh 58 41 65 towns are: 38 Monet Renoir–Pissarro 47 min 30 Renoir–Monet 44 min 46 Monet–Cezanne 40 min 75 Gauguin Pissarro–Cezanne 45 min Cezanne Pissarro–Van Gogh 34 min Pissarro–Matisse 75 min Pissarro–Monet 25 min Cezanne–Van Gogh 20 min Van Gogh–Matisse 38 min Cezanne–Gauguin 59 min Matisse–Gauguin 28 min i Draw a network of this situation showing the time taken to travel between towns on each arc of the network. ii Calculate the longest time to travel from Renoir to Gauguin. iii Calculate the shortest time to travel from Renoir to Gauguin. c Complete the table below showing the shortest distance between each of the towns. WORKED Example Renoir Pissarro Monet Cezanne Van Gogh Renoir —— Pissarro —— —— 41 Monet —— —— —— —— —— —— Van Gogh —— —— —— —— —— Matisse —— —— —— —— —— Gauguin —— Cezanne Matisse 179 123 —— d Produce a similar table showing the travelling times between each of the towns shown on the map.
- 7. Maths A Yr 12 - Ch. 07 Page 361 Wednesday, September 11, 2002 4:24 PM Chapter 7 Networks WORKED Example 361 4 The cost of trips on McFlaherty’s Bus service are given in the table below. 3 Port St Land St Port St —— 2.40 Land St —— —— Tork Rd —— —— Bell St —— —— Tork Rd Bell St Key St 1.80 2.40 1.50 —— 1.80 1.50 —— —— 2.00 a Draw a network representing this information. b What is the minimum cost of travelling from Port St to Tork Rd? c What is the minimum cost of travelling from Bell St to Port St? The distances, in kilometres, between towns in a region are given in the table below. Note: Where a blank appears no direct link between the towns exists. Grantha Tamwor Armida Beech Kianga 85 104 122 Grantha —— Tamwor —— —— 43 Armida —— —— —— 100 85 In a big storm the bridge on the Armida to Beech road was washed out. How far is the journey from Beech to Armida now? A 163 km B 128 km C 189 km D 154 km Minimal spanning trees The diagram at right represents a farm complex. 240 Sheds Workshop Each site needs to be connected directly or 250 200 indirectly to the transformer so that it can get 200 Garage 250 electrical power. For example, the garage can get 150 250 its power directly from the transformer or indirectly from the house, if the house is connected. The Pump House 350 numbers represent the distance between each site. How should the connections be arranged so 390 that the minimum length of cabling is used? 350 400 To answer this question in a systematic way Transformer we consider the following aspects of networks. A tree is a series of connections in a network that does not contain a loop. A spanning tree in a network is a tree that contains each node. HEET SkillS 5 multiple choice 7.1
- 8. Maths A Yr 12 - Ch. 07 Page 362 Wednesday, September 11, 2002 4:24 PM 362 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d To identify a minimal spanning tree, we use the minimal spanning algorithm which has the following steps. Step 1 Choose any node at random and connect it to its closest neighbour. Step 2 Choose an unconnected node which is the closest to any connected node. Connect this node to the nearest connected node. (If two or more nodes are nearest; that is have the same value, just select any one.) Step 3 Repeat step 2 until all the nodes are connected. The minimal spanning algorithm can be used to determine the least length of cable needed to connect each building of the farm complex considered above. WORKED Example 4 240 Sheds Find the minimal spanning tree to determine the minimum amount of cable needed to connect all the buildings in this farm complex to the transformer. Distances between locations are shown in this plan and are in metres. Workshop 250 200 Garage 250 150 200 250 Pump House 350 390 350 400 Transformer THINK 1 2 3 4 5 6 7 Draw a network with nodes using the ﬁrst letter of each building. Use dotted lines for the arcs and label each arc with distances between the nodes. Start with the transformer and ﬁnd the shortest arc. The unconnected node closest to T is P, so join T to P with an arc. Find the unconnected node closest to P or T. It is G. Connect P and G with an arc. Find the unconnected node closest to P, T, or G. It is W. Connect G and W with an arc. Find the unconnected node closest to P, T, G or W. It is H. Connect W and H. The sheds, S, are still not connected. Find the node closest to P, T, G, W or H which is closest to the unconnected node S. Connect W and S with an arc. WRITE/DRAW 240 S W 200 200 250 250 250 G 150 H 350 P 390 400 350 T 240 S W 200 200 150 G H P 350 T 8 9 Add up the lengths in the minimal spanning tree. Answer the question. 350 + 150 + 200 + 200 + 240 = 1140 The minimal length of cable to connect the buildings is 1140 m.
- 9. Maths A Yr 12 - Ch. 07 Page 363 Wednesday, September 11, 2002 4:24 PM Chapter 7 Networks 363 For the minimal spanning tree in the previous worked example it does not matter which node was used as the starting point. The same spanning tree would have resulted. However, suppose that the distance between the sheds and the pump had been 240 — the same distance from the sheds to the workshop. Then we could have chosen the ﬁnal arc as either SW or SP but not both. However, the total length of the minimal spanning tree would have been the same. History of mathematics JOHN FORBES NASH (1928–) When the movie A Beautiful Mind won an Oscar for best ﬁlm in 2002, John Nash was in the audience. The movie, based on a book by the same name, is his story. John Nash was born in Blueﬁeld, West Virginia in the United States. His schoolteachers did not recognise his brilliance and they focussed on his lack of social skills. His mother was a schoolteacher who encouraged his love of books and experiments. One of his chemistry experiments with explosives caused the death of a school friend. He enjoyed Compton’s Pictured Encyclopedia, and the book, Men of Mathematics by E T Bell, ﬁrst excited him about mathematics. He succeeded in proving difﬁcult mathematical problems such as Fermat’s Theorem for himself. He entered Carnegie Technical College in Pittsburgh to follow his father’s footsteps in engineering. He moved to chemistry to avoid the rigidity of mechanical drawing. Then, encouraged by the mathematics faculty, he moved from chemistry to major in mathematics, realising that it was possible to make a good career in America as a mathematician. He excelled in mathematics and graduated with an MS as well as a BS because of his advanced mathematical knowledge. On graduation from Carnegie, where an elective course in international economics inﬂuenced his mathematical ideas, he was offered fellowships at both Harvard and Princeton. In 1948, he chose Princeton where he was closer to his family in Blueﬁeld. He avoided lectures and studied on his own, and was full of mathematical ideas. His interest in game theory grew and he developed the mathematics of equilibrium strategies to predict behaviour. In two papers Equilibrium Points in n-person Games and Non-cooperative Games, Nash proved the existence of a strategic equilibrium for non-cooperative games, the Nash equilibrium, and suggested approaching the study of cooperative games by their reduction to non-cooperative form. In his two papers on bargaining theory, he proved the existence of the Nash bargaining solution and provided the ﬁrst execution of the Nash program. He was awarded the Nobel Prize in Economic Science in 1994, for this work on game theory 45 years earlier. In the movie, A Beautiful Mind, we see a version of how his ideas were stimulated by thinking about non-predictable strategies in a bar scene. In another scene we see him
- 10. Maths A Yr 12 - Ch. 07 Page 364 Wednesday, September 11, 2002 4:24 PM 364 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d mapping the interactions between pigeons and saying that he is developing an algorithm to predict their behaviour. An algorithm is a step-by-step procedure for a particular mathematical problem and is the idea that lies at the heart of all the computer programming and the code which drives digital computers. After obtaining his degree in 1950 he worked as an instructor at Princeton but moved to the mathematics faculty of Massachusetts Institute of Technology (MIT) where he met his wife, Alicia, a physics graduate. In 1958 he was described as the most promising mathematician in the world. He became mentally disturbed in 1959 when Alicia was pregnant. Nash attributes his recovery from mental illness to a determined effort to think rationally, aided by light mathematical work. He rejected his delusions and in his acceptance speech for the Nobel Prize in 1994 said, ‘I am still making the effort and it is conceivable that with the gap period of about 25 years of partially deluded thinking providing a sort of vacation, my situation may be atypical. Thus I have hopes of being able to achieve something of value through my current studies or with any new ideas that come in the future.’ In 1999 John Nash was also awarded the Leroy P Steele Prize by the American Mathematical Society for contributions to research. Questions 1. Which book ﬁrst stimulated John Nash’s interest in mathematics? 2. Which two prizes did John Nash receive? 3. What is an algorithm? Research 1. Find out about game theory. 2. What opportunities are there to study mathematics after ﬁnishing school? WORKED Example 5 The cost, in dollars, of connecting 7 ofﬁces with a computer network is given in the table. A —— —— —— —— —— A B C D E B 45 —— —— —— —— C 70 150 —— —— —— D 100 50 100 —— —— E 65 90 85 40 —— F 140 95 50 55 70 Use the minimal spanning algorithm to calculate the minimum cost of connecting the ofﬁces. THINK 1 2 Draw a network to represent the information given in the table. Select any starting point, say C. WRITE/DRAW A 140 45 70 B 100 65 F 95 90 55 70 150 50 E 50 100 C 85 D 40
- 11. Maths A Yr 12 - Ch. 07 Page 365 Wednesday, September 11, 2002 4:24 PM Chapter 7 Networks THINK 3 365 WRITE/DRAW Identify the shortest arc connected to C. This is arc CF. A 140 45 70 B 100 65 F 95 90 55 70 150 50 E 50 100 C D 40 85 4 Identify the shortest arc connected to C or F to an unconnected node. This is arc FD. A 140 45 70 B 100 65 F 95 90 55 70 150 50 E 50 100 C D 40 85 5 Continue, using the minimal spanning algorithm to get the ﬁgure opposite. A 140 45 70 B 100 65 F 95 90 55 70 150 50 E 50 100 C D 40 85 6 Use the minimal spanning tree to answer the question. The minimum cost of linking the ofﬁces is $45 + $50 + $50 + $55 + $40 = $240.
- 12. Maths A Yr 12 - Ch. 07 Page 366 Wednesday, September 11, 2002 4:24 PM 366 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d remember remember 1. A spanning tree connects all nodes in the network and does not contain any loops. 2. A minimal spanning tree is the smallest spanning tree. 3. To ﬁnd the minimal spanning tree use the minimal spanning tree algorithm. Step 1 Choose any node at random and connect it to its closest neighbour. Step 2 Choose an unconnected node which is the closest to any connected node. Connect this node to the nearest connected node. Step 3 Repeat Step 2 until all nodes are connected. 7B WORKED Example 4 Minimal spanning trees 1 Find the minimal spanning tree for each of the following networks. a b B B 4 8 4 A D 5 A 5 9 6 7 4 C c C 12 A D d 20 A D 40 21 30 18 C 17 B D 20 40 30 15 15 C E 2 The rail authority plans to connect the country centres shown with a rail network (distances are in kilometres). What is the minimum length of track required to achieve this? Use a minimal spanning tree algorithm as follows. a Begin at Pallas and connect it to its nearest neighbour. Which town is this? b Which unconnected town is closest to Pallas or to the town selected in a? c Connect this town to the existing link in the shortest way possible. d Continue by connecting the closest unconnected nodes to any connected ones, one at a time, until all nodes are connected. 30 B E 15 Yule 65 View 42 Zenith 50 80 70 Pallas 52 88 82 65 79 Xavier 50 55 Walga 88 67 Rockdale 55 52 Urchin 62 Sturt
- 13. Maths A Yr 12 - Ch. 07 Page 367 Wednesday, September 11, 2002 4:24 PM 367 Chapter 7 Networks 3 The paths between the various cages at the Monkeys Nolonger Park Zoo are dirt and when it rains 70 55 65 they become muddy. The ﬁgure at right shows Crocodiles 65 Lions 50 all paths, with distances in metres. Management has decided to put in concrete paths. Kiosk a What total length of path would be required if 30 60 60 50 each dotted line was to become a concrete path? b Use the minimal spanning tree algorithm to ﬁnd the minimum length of concrete path Entrance Birds 80 that is required so that patrons could see each exhibit and visit the kiosk without walking on a dirt path. c Repeat the minimal spanning tree algorithm using a different starting point and show that it does not matter where you start. 4 Use the minimal spanning tree algorithm to ﬁnd the minimal spanning tree for the following networks. 30 a b 54 54 C E B E A 23 18 20 45 D 18 B 31 23 C 40 A C D 20 d B 30 30 23 A F 24 20 50 6 B 7 23 6 C E D 9 4 8 8 G 40 F 6 E 5 Find the minimal spanning tree for each of the following networks. a b c A D 12 14 D 18 17 15 A F 20 22 G E 15 I 5 C 15 B 18 22 6 A number of small, private mines have opened up in Waller Flats and the local shire council wants to link them by bitumen roads as shown in the ﬁgure at right. What is the minimum length of road that is needed? (Assume the only connections that can be made are those marked on the map of Waller Flats at right.) F E G D 5 5 J 12 5 5 8 8 H 5 12 C F 5 8 8 22 17 10 12 13 10 18 15 B E 12 14 C B A 10 18 F 48 7 7 50 F D 48 45 60 55 A c 45 8 8 G K 5 Mine 1 Mine 4 5 km 15 km Mine 5 Mine 2 6 km 14 km Mine 6 12 km 10 km Mine 3 7 km 15 km 11 km Mine 7
- 14. Maths A Yr 12 - Ch. 07 Page 368 Wednesday, September 11, 2002 4:24 PM 368 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d 7 In question 6 the dotted lines connecting the mines represent dirt roads. If an inspector wants to visit all the mines and is willing to travel on dirt roads, what is the shortest distance he or she needs to travel to visit each of them, starting from Mine 1? WORKED Example 5 8 A gas pipeline is to be connected between 5 towns so that each town has at least one connection to the system. The gas pipeline costs $25 000 per kilometre. The distance (in km) between the towns is given in this table. A B C D E A —— 16 23 10 43 B —— —— 32 17 19 C —— —— —— 35 43 D —— —— —— —— 38 a Find the length of the network connecting these towns in the shortest way. b What is the cost of this connection? 9 An ofﬁce computer system requires the linking of 8 terminals. Each terminal has to have at least one connection with the system. The cost (in dollars) of connecting each terminal with another is given in the table. A B C D E F G H A —— 35 50 75 50 100 65 105 B —— —— 100 40 65 70 90 105 C —— —— —— 70 60 40 55 15 D —— —— —— —— 30 40 105 100 E —— —— —— —— —— 55 40 30 F —— —— —— —— —— —— 25 50 G —— —— —— —— —— —— —— 75 a What is the smallest possible cost for linking the computer terminals if each terminal has at least one connection with the system? b If each terminal is connected to every other terminal, what is the cost of the linking? Use the network at right to answer questions 10 and 11. The dimensions are in km. 10 multiple choice Which of the following arcs are not in the spanning tree? A AB B AC C BC D BG G A B 12 27 C 15 18 20 D 25 E 11 multiple choice What is the length of the minimal spanning tree? A 120 km B 105 km C 98 km F 40 18 22 18 22 D 103 km
- 15. Maths A Yr 12 - Ch. 07 Page 369 Wednesday, September 11, 2002 4:24 PM 369 Chapter 7 Networks Shortest paths Given a network representing the distance between towns, consider the question, ‘How far is it from town A to town X?’. In earlier sections we have approached such a question using a trial and error method. However, when networks become more complex, a systematic method is required. The method used is called the shortest path algorithm. Shortest path algorithm To ﬁnd the shortest path between A and X in a network, follow these steps. Step 1 For all nodes that are one step away from A, write the shortest distance from A inside the circle representing the closest node. Step 2 For all nodes which are two steps away from A, write the shortest distance from A inside the circle representing the closest node two steps away. Step 3 Continue in this way until X is reached. Step 4 The shortest path can be identiﬁed by starting at X and moving back to the node from which the minimum value at X was obtained, then continuing this process until A is reached. This will be explored in the next worked example. WORKED Example 6 A 2 3E 3 5I 2 3 M 3 Find the shortest path from A to P in the network at right. The units are in minutes and represent time taken. Note: We have placed the labels outside the nodes so that the times can be placed inside the circles. THINK 1 2 3 4 5 6 Beginning at A write inside the nodes at B and E the shortest time taken to get to them. Then write in the shortest time for all nodes which are two steps away from A. That is, C = 4, F = 5 and I = 8. Continue in this way until P is reached. For example, at node J, the time from I would be 10, so the shorter time, 9, from F is put in the node. Now back-track from P moving from node to node along the arcs which produced the minimum values. Check to see if this is the shortest path. This is the shortest path. Put arrows on this path. Write the answer. WRITE/DRAW A 2 3 E 3 3 5 I 8 2 3 M 11 3 2 B 2 3 4 C 2 3 5 F 3 4 9 1 7 10 2 D 6 H 3 12 9 5 K 11 2 3 N C 2 3G 3 5K 2 3 O 3 3 G 5 J 2 B 2 3F 3 4J 2 1 N 2 13 L 1 O 3 14 P The shortest path from A to P is A–B–F–J–K–L–P and is 14 minutes long. D 3H 5L 1 P
- 16. Maths A Yr 12 - Ch. 07 Page 370 Wednesday, September 11, 2002 4:24 PM 370 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d remember remember To ﬁnd the shortest path from A to X in a network: 1. For all nodes one step away from A, write the shortest distance. 2. For all nodes two steps away from A, write the shortest distance. 3. Continue until X is reached. The shortest path is located by starting at X and working backwards to A. 7C WORKED Example 6 Shortest paths 1 Find the length of the shortest path from A to B in each of the following networks. a b 7 10 5 5 4 3 8 A 20 B 4 10 20 13 10 A 16 12 12 25 15 25 25 2 B 14 c d A 23 23 30 24 A 16 45 2 2 3 B 4 4 6 4 7 27 34 3 34 18 4 5 4 12 5 30 8 10 4 10 7 20 6 7 6 5 7 6 B e A 5 6 6 7 5 25 50 18 3 4 4 45 35 7 5 4 A 35 5 5 5 f 4 5 14 19 16 23 15 3 12 29 25 40 25 15 16 26 30 16 32 22 24 B 5 B
- 17. Maths A Yr 12 - Ch. 07 Page 371 Wednesday, September 11, 2002 4:24 PM 371 Chapter 7 Networks 64 2 From the map at right, where the units are km, answer the following questions. a What is the shortest distance from Fourier to Rolle? b What is the shortest distance from Fourier to Stokes? c What is the shortest distance from Fourier to Stokes travelling through Reynolds? Aiken Stokes 32 24 60 25 Feynman 56 95 44 Reynolds Rolle Hardy 36 45 32 34 Gauss Ahmes 51 45 27 Fourier 26 Lebesgue 3 For each of the following networks, ﬁnd the shortest path from A to B. a b 25 10 7 10 7 A 25 15 35 45 25 B 28 25 10 5 6 50 31 20 40 12 15 d 15 5 30 7 20 13 14 8 8 7 8 A 12 7 9 12 35 25 B 8 37 15 30 15 26 12 13 8 7 6 8 10 12 11 10 10 7 8 8 6 8 6 6 26 6 8 12 11 6 A f 12 7 8 B 35 8 6 7 40 8 14 e 34 26 8 6 8 15 25 13 11 23 12 15 16 A 14 15 9 7 50 30 8 10 B 30 35 8 c 35 50 35 32 10 25 40 A 15 5 40 6 11 14 13 6 13 7 A 9 12 8 10 6 11 12 12 8 B 8 12 5 7 11 8 7 8 15 14 B 13 11 13 12 14 9
- 18. Maths A Yr 12 - Ch. 07 Page 372 Wednesday, September 11, 2002 4:24 PM 372 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d 4 This table shows the travelling times in minutes between towns which are connected directly to each other. Note: The line indicates that towns are not connected directly to each other. Addisba Eric 0 50 20 25 — 50 0 25 30 30 Callop 20 25 0 — 60 Dilger Work Dilger Bundong 7.1 Callop Addisba ET SHE Bundong 25 30 — 0 70 Eric — 30 60 70 0 a Draw a network to show the connection of the towns by these roads. b Find the shortest travelling time between Addisba and Eric. 1 This network at right represents the potential cost of a covered walkway between various locations on a campus. 1 How many nodes are there in this network? 3600 A 6400 2 How many arcs are there in this network? B 4000 3000 E 8000 D C 2000 6000 4000 6400 F 3 Which node/s have more than 4 arcs meeting? The cost of the walkway is to be kept to a minimum but it should be possible to go from any location to any other via a covered walkway. 4 Find the minimal spanning tree. 5 What arcs are not included in the minimal spanning tree? 6 What is the minimum cost of such an arrangement of walkways? 7 If one is to travel from D to F under cover, what path should be taken? It is found that there was an error in the estimate for the walkway connecting A to C. The correct value should be $3600. 8 Find the new minimal spanning tree 9 What is the new minimum cost for a suitable arrangement of walkways? 10 If one were to travel from B to C under cover, what path should be taken?
- 19. Maths A Yr 12 - Ch. 07 Page 373 Wednesday, September 11, 2002 4:24 PM Chapter 7 Networks 373 Network ﬂow An application of networks used to analyse ﬂow of trafﬁc or water is network ﬂow. These usually involve directed networks where arrows show the direction of ﬂow. An example is described below. A driver starts for work in the city at 7.30 am each morning. He lives in an outer suburb and as he travels from his driveway through a few streets in his local neighbourhood, there is not much trafﬁc on the roads. As he joins the road that connects his suburb to the next suburb, he notices an increase in the volume of the trafﬁc. As this two-lane road joins the four-lane freeway into the city, the ﬂow of trafﬁc becomes immense. Cars are following bumper to bumper, with drivers changing lanes to drive in the fastest lane. The costs involved, ﬁnancial and otherwise, for those who participate in the morning rush are signiﬁcant. It is in everyone’s best interest that the trafﬁc ﬂow smoothly and that trafﬁc jams be avoided at all costs. Engineers use mathematical models of network ﬂow to ensure smooth ﬂow of trafﬁc. Flow capacities and maximum ﬂow The network’s starting node(s) is called the source. This is where all ﬂows commence. The ﬂow goes through the network to the end node(s) which is called the sink. The ﬂow capacity (capacity) of an arc is the amount of ﬂow that an arc can allow through if it is not connected to any other arcs. The inﬂow of a node is the total of the ﬂows of all arcs leading into the node. The outﬂow of a node is the minimum value obtained when one compares the inﬂow to the sum of the capacities of all the arcs leaving the node. Consider the following ﬁgures. D A B C Source F Sink E All ﬂow commences at A. It is therefore the source. All ﬂow converges on F indicating it is the sink. 100 B 30 20 10 B has an inﬂow of 100. The ﬂow capacity of the arcs leaving B is 30 + 20 + 10 = 60. The outﬂow is the minimum of 100 and 60, which is 60. 30 100 B 20 80 B still has an inﬂow of 100 but now the capacity of the arcs leaving B is (80 + 20 + 30) = 130. The outﬂow from B is now 100. The ﬂow capacity of the network is the total ﬂow possible through the entire network.
- 20. Maths A Yr 12 - Ch. 07 Page 374 Wednesday, September 11, 2002 4:24 PM 374 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d WORKED Example 7 Consider the information presented in the following table. From Quantity (litres per minute) Demand (E) 1000 — 200 To 200 Rockybank Reservoir (R) Marginal Dam Marginal Dam (M) Freerange Marginal Dam (M) Waterlogged (W) 200 200 Marginal Dam (M) Dervishville (D) 300 300 (F) a Convert the information to a network diagram, clearly indicating the direction and quantity of the ﬂow. b Determine the ﬂow capacity of the network. c Determine whether the ﬂow through the network is sufﬁcient to meet the demand of all the towns. THINK WRITE a Construct and label the required number of nodes. The nodes are labelled with the names of the source of the ﬂow and the corresponding quantities are recorded on the arcs. a F 200 R 1000 M 200 200 W 300 200 E 300 D b 1 Examine the ﬂow into and out of the Marginal Dam node. Record the smaller of the two at the node. This is the maximum ﬂow through this point in the network. b F 200 R 1000 M 200 W 300 D Even though it is possible for the reservoir to send 1000 L/min (in theory), the maximum ﬂow that the dam can pass on is 700 L/min (the minimum of the inﬂow and the sum of the capacities of the arcs leaving the dam). 2 In this case the maximum ﬂow through Marginal Dam is also the maximum ﬂow of the entire network. Maximum ﬂow is 700 L/min.
- 21. Maths A Yr 12 - Ch. 07 Page 375 Wednesday, September 11, 2002 4:24 PM Chapter 7 Networks THINK WRITE c c 375 1 Determine that the maximum ﬂow through Marginal Dam meets the total ﬂow demanded by the towns. F 200 200 M 200 W 200 300 E 300 D 2 If the requirements of step 1 are able to be met, then determine that the ﬂow into each town is equal to the ﬂow demanded by them. Flow through Marginal Dam = 700 L/min Flow demanded = 200 + 300 + 200 = 700 L/min By inspection of the table, all town inﬂows equal town demands (capacity of arcs leaving the town nodes). Consider what would happen to the system if Rockybank Reservoir continually discharged 1000 L/min into Marginal Dam while its output remained at 700 L/min. Such ﬂow networks enable future planning. Future demand may change, the population may grow or a new industry that requires more water may come to one of the towns. The next worked example will examine such a case. Excess ﬂow capacity is the surplus of the capacity of an arc less the ﬂow into the arc. es in ion v t i gat n inv io es The seven bridges of Königsberg On the River Pregel in the European town of Königsberg, there were 7 bridges arranged as below. Island Land People wondered if it was possible to cross all 7 bridges without crossing any bridge more than once. Can you see if it can be done? t i gat
- 22. Maths A Yr 12 - Ch. 07 Page 376 Wednesday, September 11, 2002 4:24 PM 376 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d WORKED Example 8 A new dairy factory, Creamydale (C), is to be set up on the outskirts of Dervishville. The factory will require 250 L/min of water. a Determine whether the original ﬂow to Dervishville is sufﬁcient. b If the answer to part a is no, is there sufﬁcient ﬂow capacity into Marginal Dam to allow for a new pipeline to be constructed directly to the factory to meet their demand? c Determine the maximum ﬂow through the network if the new pipeline was constructed. THINK a 1 Add the demand of the new factory to Dervishville’s original ﬂow requirements. If this value exceeds the ﬂow into Dervishville then the new demand cannot be met. WRITE a F 200 R 1000 M 200 200 W 300 200 E 300 + 250 D 2 The new requirements exceed the ﬂow. The present network is not capable of meeting the new demands. M E 550 300 D b 1 Reconstruct the network including a new arc for the factory after Marginal Dam. b F 200 R 1000 M 200 200 W E 300 300 250 200 D 250 C 2 Repeat step 1 from worked example 7 to ﬁnd the outﬂow of node M. Marginal Dam inﬂow = 1000 Marginal Dam outﬂow = 200 + 200 + 300 + 250 R = 950 F 200 1000 M 200 W 300 250 D C 3 Determine if the ﬂow is sufﬁcient for a new pipeline to be constructed. c This answer can be gained from part b step 2 above. There is excess ﬂow capacity of 300 into Marginal Dam which is greater than the 250 demanded by the new factory. The existing ﬂow capacity to Marginal Dam is sufﬁcient. c The maximum ﬂow through the new network is 950 L/min.
- 23. Maths A Yr 12 - Ch. 07 Page 377 Wednesday, September 11, 2002 4:24 PM 377 Chapter 7 Networks C D- extension extension — Paths and circuits: Eulerian and Hamiltonian remember remember C 7D WORKED Example 7a Network ﬂow 1 Convert the following ﬂow tables into network diagrams, clearly indicating the direction and quantity of the ﬂow. a To Flow capacity A A B C D c From B C C D E 100 200 50 250 300 From To Flow capacity M M N N Q O R N Q O R R E E 20 20 15 5 10 12 12 b From To Flow capacity R S T T U d AC ER T D- 1. In a network ﬂow diagram, the arcs have quantities that indicate rates of ﬂow; for example, litres per minute, cars per second, people per hour and so on. 2. The starting node(s) from which all ﬂows commence is called the source. 3. The ﬂow goes through the network to the end node(s) which is called the sink. 4. The ﬂow capacity (or capacity) of an arc is the amount of ﬂow that an arc would allow if it were not connected to any other arcs. 5. The ﬂow capacity of the network is the total ﬂow possible through the network. 6. The inﬂow of a node is the total of the ﬂows of all arcs leading into the node. 7. The outﬂow of a node is the minimum of either the inﬂow or the sum of the capacities of all the arcs leaving the node. 8. Excess ﬂow capacity of an arc equals the ﬂow capacity of an arc minus the ﬂow into the arc. S T U E E 250 200 100 100 50 From To Flow capacity D D G G F F J H F G H J H J E E 8 8 5 3 2 6 8 8 RO IVE INT For more information on paths and circuits, click here when using the CD-ROM. M extension — Minimum cut–Maximum flow M extension AC ER T IVE INT The maximum ﬂow through most simple networks can be determined using these methods, but more complex networks require different methods to be used. RO
- 24. Maths A Yr 12 - Ch. 07 Page 378 Wednesday, September 11, 2002 4:24 PM 378 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d 2 For node B in the network at right, state: a the inﬂow at B B 23 b the arc capacities ﬂowing out of B c 16 A the outﬂow from B. D 27 34 C 3 Repeat question 2 for the network at right. A B 2 C 2 D 4 5 3 WORKED Example 3 4 E 6 4 For each of the networks in question 1, determine: i the ﬂow capacity 7b, c ii whether the ﬂow through the network is sufﬁcient to meet the demand. 5 Convert the following ﬂow diagrams to tables as in question 1. a b B B 4 5 A 3 c 4 5 A 3 3 4 2 C 2 D A E 3 6 B 2 D 4 5 d 3 E 4 5 A 6 12 7 8 7 C 2 C 2 D B 2 6 D 4 F 7 C 3 4 E 6 3 E 8 6 Calculate the capacity of each of the networks in question 5. WORKED Example 8 7 i Introduce new arcs, from the information which follows, to each of the network diagrams produced in question 1 ii calculate the new network ﬂow capacities. a From To Flow capacity A A B C D B B C C D E E 100 200 50 250 300 100 b From To Flow capacity R S T T U S S T U E E T 250 200 100 100 50 100
- 25. Maths A Yr 12 - Ch. 07 Page 379 Wednesday, September 11, 2002 4:24 PM 379 Chapter 7 Networks c From To Flow capacity M M N N Q O R N N Q O R R E E E d From 20 20 15 5 10 12 12 5 To Flow capacity D D G G F F J H D F G H J H J E E E 8 8 5 3 2 6 8 8 10 ET SHE In question 7c the outﬂow from N is: A 5 B 20 Work 8 multiple choice C 15 D 25 2 Questions 1 to 4 refer to the network at right. The network represents the distance between towns in kilometres. B 20 A 1 What is the shortest path from A to F? 20 15 C 40 32 30 20 D E 10 F 32 2 Give the length of the shortest path from A to F. 3 Give the shortest path from B to F. 4 What is the length of the shortest path from B to F? 5 In the network at right, what is the inﬂow at the node? 20 40 6 In the same network as question 5, what is the outﬂow at the node? 7 What is the excess ﬂow capacity of arc BC in the network at right? A 20 10 B 30 C Questions 8 to 10 also refer to the ﬁrst network above. This network shows the capacity of irrigation pipes in kilolitres per hour. 8 What is the inﬂow at C? 9 What is the outﬂow at C? 10 What is the maximum ﬂow in the network? 7.2
- 26. Maths A Yr 12 - Ch. 07 Page 380 Friday, September 13, 2002 10:36 AM 380 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d summary Networks, nodes and arcs • A network consists of a number of nodes connected by arcs. • When the arcs have arrows the network is called a directed network and travel is possible only in the direction of the arrows. Minimal spanning tree • A tree is a series of connections in a network that does not contain a loop. • A spanning tree in a network is a tree that contains each node of the network. • A minimal spanning tree is the arrangement of arcs in which every node is connected to at least one other node in such a way as to minimise the total length of these arcs. • To ﬁnd the minimal spanning tree use the minimal spanning tree algorithm: Step 1 Choose any node at random and connect it to its closest neighbour. Step 2 Choose an unconnected node which is the closest to any connected node. Connect this node to the nearest connected node. Step 3 Repeat Step 2 until all the nodes are connected. • A path is a series of nodes connected by arcs. • The shortest path is the shortest distance from a given starting point to a given end point. Shortest path • The shortest path is the shortest distance from a given starting point to a given end point. • To ﬁnd the shortest path between A and X: 1. For all nodes that are one step away from A, write the shortest distance from A inside the circle representing the node. 2. For all nodes which are two steps away from A, write the shortest distance from A inside the circle representing the node. 3. Continue in this way until X is reached. 4. The shortest path can be identiﬁed by starting at X and moving back to the node from which the minimum value at X was obtained, then continuing this process until A is reached. Network ﬂow • A network can be used to represent the network ﬂow of quantities such as water, trafﬁc or telephone calls. • Arcs indicate rates of ﬂow. The inﬂow at a node is the sum of the capacities of the arcs leading into the node. The outﬂow at a node is the minimum of either the inﬂow or the sum of the capacities of the arcs leaving the node. • In a network ﬂow diagram, the arcs have quantities that indicate rates of ﬂow, for example, litres per minute, cars per second people per hour and so on. • The starting node(s) is called the source, from which all ﬂows commence. • The ﬂow goes through the network to the end node(s) which is called the sink. • The ﬂow capacity (or capacity) of an arc is the amount of ﬂow that an arc would allow if it were not connected to any other arcs. • The ﬂow capacity of the network is the total ﬂow possible through the entire network. • The inﬂow of a node is the total of the ﬂows of all arcs leading into the node. • The outﬂow of a node is the minimum of either the inﬂow or the sum of the capacities of all the arcs leaving the node. • Excess ﬂow capacity equals the ﬂow capacity of an arc minus the ﬂow into the arc.
- 27. Maths A Yr 12 - Ch. 07 Page 381 Wednesday, September 11, 2002 4:24 PM 381 Chapter 7 Networks CHAPTER review 10 B 1 For the network at right, write down: a the number of nodes b the number of arcs. E 20 10 A 10 C 10 G 10 20 D 7A 20 5 F 10 2 The following table represents the cost, in tens of thousands of dollars, of resurfacing roads connecting various locations in a district. Draw a network representing this situation. A B C D A —— 5 E 11 B —— —— 4 C —— —— —— 8 D —— —— —— —— E —— —— —— —— 7A 12 7 —— 3 Describe an algorithm used to identify the minimal spanning tree. 4 Give the minimal spanning tree for the network in question 1. 5 Determine the minimal spanning tree for the ﬁgure at right. 30 B 15 15 20 E H 15 40 30 7B 7B 7B 30 I 15 15 F K 15 35 35 25 30 30 25 A 25 C 30 30 D 35 G 25 J 6 It is planned to join the towns shown on the map Caerleon 16 km at right by a rail link. Use a minimal spanning 15 km Freshwater Brownsville algorithm to ﬁnd the shortest length of track 15 km needed to connect each town by rail. 18 km Miriam 34 km Amesbury 19 km Manto 18 km 61 km 41 km Gaine 7B
- 28. Maths A Yr 12 - Ch. 07 Page 382 Wednesday, September 11, 2002 4:24 PM 382 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d 7C 7C 7D 7 Identify the shortest path from A to G in question 1. What is the length of this path? 7D 10 If the arcs in question 5 represent capacity for ﬂow, calculate each of the following: a inﬂow at C b outﬂow at C c maximum ﬂow from A to K. 7D 11 From the table at right produce a network ﬂow diagram. 8 Identify the shortest path from A to K in question 5. What is the length of this path? 9 If the arcs in the network in question 1 represent capacity for ﬂow, calculate the following: a inﬂow at C b outﬂow at C c the maximum ﬂow. 13 C 6 C 10 D 4 D 3 C E 14 D F 10 E F 15 From To Flow quantity A B 13 A C 6 A G 16 B C 10 B D 4 B G 2 C D 3 C E 14 D F 10 E F 15 G CHAPTER B C 7 A B test yourself Flow quantity B 12 Draw the network ﬂow diagram for the table at right. To A 7D From D 3 G H 10 H F 13

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