Getting hooked on mathematics through its applications
Christine Mangelsdorf, Marcus Brazil and Antoinette Tordesillas
Abstract In this paper, we describe a one-day event which changed student perceptions about the relevance of mathematics in today’s world. In particular, we discuss in detail a competition held during this event in which senior level secondary school students were given an opportunity to experience consulting in the mining industry. Many of the students found the event to be an eye-opener, exposing them to a wealth of career opportunities that they were previously unaware of.
Engaging students through applications If mathematics is the language of science, then why is there a decline of student interest in mathematics at a time when science based industries and technologies are booming? Why do students who were good at mathematics in high school end up forgoing mathematics for other subjects at university? The answer to these questions is not all that hard to come by: many students are simply unaware of the wide range of career opportunities awaiting mathematics graduates. Somehow, we are still not getting through to our students just how extensively mathematics is used in everyday life.
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In an effort to raise student awareness of the power of mathematics in modern society, we ran a one-day fair for senior level secondary school students (i.e. Year 11 and 12) in Victoria [1]. The core event was called the Maths in Industry and Technology or MIT Challenge. It was designed to give the students a taste of real-world consulting in the format of a competition. Teams of four students, acting as consultants, were presented with a design and optimisation problem from the Western Mining Corporation.
A representative from the
Australian Mining Consultants, here acting as the client, gave the students a detailed background of the mining industry before presenting the actual MIT Challenge problem. The MIT Challenge was supported by a smorgasbord of activities, which showcased careers of mathematics graduates in a wide range of disciplines. There was “Mathematicians Exposed” which was a series of lectures given by mathematics graduates working in healthcare, telecommunications, finance, superannuation, information technology and the military. There were exhibits on various mathematical research and statistical consulting problems in transportation, food manufacturing, meteorology, operations research, geology, medicine and the military. The exhibits not only highlighted the extensive use of mathematics in the modern world but also exposed a myriad of unanswered questions that are fuelling developments in mathematics and stimulating the emergence of new areas (e.g. bioinformatics, nanotechnology and micromechanics). The survey of the fair showed the event to be an overwhelming success with 91% of respondents saying that it was both enjoyable and informative. In particular, a number of Victoria's top ranking students who participated in the fair have commented on how the event has influenced their choice of subject majors at University. The personal interactions with various staff in the Department, combined with the opportunity to meet graduates from such a
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wide range of areas and to learn about where their mathematical training has taken them, have served as a great inspiration to the students. The following were some of the students’ comments about the fair: "I saw areas I didn’t realise Maths was important to.” " Showed a wide range of applications for mathematics in the workforce, especially in areas which you might not have thought of or realised had maths as a basis.” "They had a wide range of companies that used maths in everyday problems which we didn’t know about." "It gave me a wider range of knowledge of how maths is used in industry. It's made me reconsider my career path.”
Design and presentation of the MIT Challenge A key aspect of organising the MIT Challenge was to find a real life problem that was accessible to senior level secondary mathematics students, specifically, one which they could make a reasonable attempt at solving in under three hours. The problem for the MIT Challenge was developed by University of Melbourne researchers who have been working on the design of underground mining tunnels, in collaboration with the Australian Mining Consultants for Western Mining Corporation. The MIT Challenge was presented to the students in five stages as follows.
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Stage 1: Kick-start MIT In preparation for the competition, the students were given a half hour “crash course” on the art of mathematical modelling for a simple real world problem and were provided with ideas and techniques they could use in the competition. Specifically, the problem that was discussed was that of minimising the fuel consumption for a train travelling on a flat track between two stations. Stage 2: Problem Presentation Mr. Brian Hall, a principal mining engineer from the Australian Mining Consultants, presented the MIT problem. To motivate the problem, Mr. Hall gave an overview of the mining sector, briefly discussing the importance of mining to the world economy, and how mathematics is used to solve a wide range of problems in management, design and operations in the mining industry. Stage 3: MIT Competition At this stage, the students tackle the MIT problem where they must learn to work together effectively and efficiently to meet their strict deadline. Within three hours they were required to develop a solution approach and come up with a proposal to the client, i.e. a written report of their work. The teams were then ranked according to the quality of their proposals. Stage 4: MIT Student Presentations The top six teams were required to give an oral presentation of their proposed solution to a panel of judges. In this case, the panel members included mathematicians and engineers who worked on related design problems for the mining industry. Stage 5: MIT Solution Presentation: In closing, one of the judges Dr Marcus Brazil, presented the solution to the MIT problem.
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The final winners were judged according to the quality of both their written and oral work. To give the judges sufficient time to carefully examine the students’ work and confer with each other, the winners were announced a week later. In the following sections, we present a summary of the MIT Challenge problem, a numerical solution to the problem, and various solution strategies for the problem, including the proposal submitted by the winning team.
Industrial background and significance of the MIT Challenge problem The mining industry provides significant economic and social benefits to Australia and the world at large, making this MIT problem especially relevant today and hence ideal for this competition [2].
“The mining industry has been key to the development of civilisation,
underpinning the iron and bronze ages, the industrial revolution and the infrastructure of today’s information age. In 2001, the mining industry produced over 6 billion tons of raw product valued at several trillion dollars” [3]. The USA, Canada, Australia, South Africa and Chile dominate the global mining scene and lead the world in mining and exploration technology. Australia, in particular, is well endowed with most minerals and fossil fuels (with oil being the notable exception). It holds the world's largest known resources of iron ore, uranium, lead, zinc, bauxite, silver, industrial diamonds and mineral sands. To give an idea of the economic and social significance of the mining industry in Australia:
more than 18,000 people worked in the area in 1996-1997 with mining services exports exceeding $1 billion [4]
the proportion of Australia’s wealth derived from minerals and fossil fuel is 2.5 times the corresponding proportion for the wealthiest 20 countries [5]
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in 1997-1998, the mining sector directly added $8.6 billion to Australia’s wealth [5]
20% of research and development expenditure in Australia is by mining related industries [5]
If Australia’s mineral resources increased by 10% it is estimated that Australia’s gross development product (GDP) would be 0.7% higher after five years than it would otherwise have been [5].
The MIT Challenge problem A mining company is planning to extract ore (say gold) from a deep underground ore body. The size, position and approximate volume of ore in the ore body are known from geological surveys. A method of extracting the ore has also been chosen. Extraction equipment will be sent into the ore body and ore will be removed via four access points at the base of the ore body. The positions of these access points have been determined and they are all at the same depth, exactly 300 metres below the surface. The ore will be taken to the surface via a system of tunnels (known as "drives") and a vertical shaft. The position on the surface of the top of the shaft is already known, and cannot be changed. You have been asked to design the network of drives connecting the four access points to the base of the shaft, so as to minimise the development and construction costs, that is, the costs of building the tunnels and shaft. Part (1): It has been decided by the mining company's engineers to sink the shaft to the same depth as the access points. This means that the drives will all be horizontal. The mining company wants you to optimise as far as you can the cost of building the drives. For this part of the problem, the cost of the shaft can be treated as a
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fixed amount and so does not need to be considered. How would you design the drives network so that the construction costs are minimum? Part (2): Assume that the base of the shaft does not need to be at the same depth as the access points. Although the position of the top of the shaft is fixed, you may be able to make further savings by minimising the construction costs of the shaft plus the drives. There is also an added constraint: the vehicles that haul ore from the access points to the shaft cannot travel at a gradient greater than 1/8, that is, they cannot move upwards by more than a metre for each 8 metres they travel horizontally. Hence, 1/8 is the maximum gradient for any drive, though drives are allowed to zigzag or spiral in 3 dimensions in order to achieve any desired vertical displacement between two points. Can further savings be achieved by changing the depth of the shaft? How would you design the network of shaft and drives so that the construction costs are minimized? Given information on locations and costs: Assume the x-axis points east and the y-axis points north and the z-axis points vertically upwards. Then in both parts of the problem the (x,y,z) coordinates in metres of the access points and the top of the shaft are: Access points:
A1 (4, 22, 0)
A2 (45, 52, 0)
Shaft Top:
ST (98, 6, 300)
A3 (92, 117, 0)
A4 (138, 75, 0)
Cost of building drives: $2,000 per metre Cost of building shaft:
$10,000 per metre
For Part (1), the position of the base of the shaft is (98, 6, 0). For Part (2), the position of the base of the shaft is (98, 6, k), where 0 ≤ k ≤ 300. The positions of the orebody, access points and shaft base are illustrated in Figure 1.
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Ore Deposit A3 North
A4
A2
A1 Shaft
Figure 1:
Plan view of the ore body showing the access points to the ore body relative to the base of the shaft.
Solution to the MIT Challenge problem The following is a summary of the numerical solution to the MIT Challenge problem and some of the mathematical theory involved in rigorously solving the underlying geometrical optimisation problem. The students were not expected to be familiar with any of this mathematical theory. In the next section we discuss some of the possible solution methods, including the solution given by the winning team. Part (1):
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The network required for Part (1) is an example of a Steiner minimum network - a network connecting a given set of points in a plane such that the total length of edges in the network is as small as possible. Such networks can contain extra junctions, which are referred to as Steiner Points. Steiner minimum networks have been studied extensively by mathematicians and engineers interested in network optimisation [6]. The main geometric properties of a Steiner minimum network, resulting from its minimality, are: 1) All edges are straight lines. 2) The network contains no circuits (or cycles). 3) No two edges meet at an angle of less than 120 degrees. 4) Exactly 3 edges meet at any Steiner point and the angle between each pair of them is exactly 120 degrees. 5) There are at most n-2 Steiner points, where n is the number of fixed nodes given in the original problem (n=5 in this problem). 6) There are many possible topologies (or patterns of connections) to choose between. Using these properties, the applied mathematician Z.A. Melzak developed a simple geometric algorithm [7] for computing the Steiner minimum network for a given topology. By applying this algorithm to all feasible topologies and choosing the shortest length network, one can obtain the following solution:
Junctions: s0 = (81.5, 46.8, 0), s1 = (107.1, 79.4, 0)
Length of Drives network: 244.9 m
Cost of Drives: $489,725
Total cost (including shaft $3M): $3,489,725
A plan view of the optimal drives network for Part (1) is shown in Figure 2.
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Ore Deposit A3 North
S1
A2
A4
S0
A1 Shaft
Figure 2: A plan view of the optimal drives network for Part (1)
Part (2): The second part is much more difficult. In particular, the geometric properties involving 120 degree angles no longer apply. It is useful to note that: to go up 1 metre by shaft costs $10,000, but to go up 1 metre by a (zigzag) drive costs $16,000. This suggests that we should never use zigzag drives in the optimal solution. The shaft is sufficiently expensive, however, that some savings can be made by raising the base of the shaft. In essence, this part involves lifting the shaft base as much as possible without creating any zigzag drives. The optimal solution was obtained using a computer program [8] employing iterative techniques. The optimal solution is:
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Junctions: s0 = (92.5, 73.6, 6.5), s1 = (105.9, 86, 4.25)
Shaft base: (98, 6, 15)
Length of Drives network: 58.6 m
Cost of Drives: $517,121
Total cost (including shaft): $3,367,121
A plan view of the optimal solution for Part (2) is shown in Figure 3. The drives with arrows pointing in the upward direction have a slope of exactly 1/8. The drive between A1 and A2 is horizontal.
Ore Deposit A3 North
S1 S0
A4
A2
A1 Shaft
Figure 3: A plan view of the optimal solution for Part (2)
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Solution methods and the winning team’s proposal The judges for the MIT competition did not expect any students to find the optimal solutions outlined in the previous section. In the presentation of the MIT Challenge problem, it was emphasised that the students should be concentrating on developing a good solution method, rather than necessarily finding a numerical solution for the given data. Of course, most students found that the most effective way of developing good methods was to experiment with the given numerical data. For Part (1), it was hoped that the students would uncover some of the geometric properties of minimum networks listed in the previous section, in particular properties 1), 2), 6) and perhaps a weaker version of property 3), for example, showing that no two edges meet at an angle of less than 90 degrees. The key question is then how to construct such a minimum network, for a given set of nodes. Some of the many possible approaches to this aspect of the problem are as follows: Exact Methods
Set up a system of equations whose variables are the coordinates of the Steiner points, and minimise the lengths of the edges, by attempting to minimise a suitable objective function.
Use the geometric properties of the network to solve exactly, along the lines of the work of Melzak [7].
Iterative Search Methods
Start with an approximate solution; look for Steiner points with edges meeting at small angles (say, < 90 degrees, or, better still, < 120 degrees); move each such Steiner point
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a small distance into the small angle to make it larger. Repeat until all angles are suitably large.
Superimpose a grid upon the region. For a given topology, restrict the Steiner points to grid points and exhaustively search for the grid points which minimise the length of the network. Now construct a tighter grid on a neighbourhood around the current Steiner points, and repeat the search. There are many other possible approaches, including some nice physical models based on
soap bubbles or weighted strings [6]. (Such approaches would have been infeasible in the MIT competition, due to the time restriction, but could work well in a classroom situation.) Part (2), which was included to challenge the better students, is most suited to iterative methods. However, an added difficulty in devising a good iterative algorithm for this part is that it is not easy to decide what direction to move the Steiner points to improve a given approximate solution. The teams were given access to basic computing facilities during the MIT challenge so some used spreadsheets to help with calculations. In a couple of hours, competing teams of students came up with practical solutions that gave huge savings on tunnelling costs and devised effective strategies for determining optimal tunnel layouts. Some of the student oral presentations were of an extremely high standard as was the mathematical insight displayed in the written reports. Several of the teams achieved solutions that were close to the optimal solution for Part (1). As expected, few teams made any real progress in Part (2). Below, we summarise the written proposal submitted by the winning team from Caulfield Grammar School.
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Part 1: Consider an arbitrary network. If the network ever extends outside of the convex hull of the points we are interested in (A1, A2, A3, A4, ST) then it won’t be the solution. A better network may be obtained by constraining the network within the hull.
constrained outside hull
Better network If any two points are connected by a curved line, a better network is obtained by making it straight.
Better network
Hence, the minimal network lies within the convex hull and has straight lines only. What is not known is how many extra points need to be added (where an extra point is an intersection of three or more drives) for an optimal solution.
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Solution with no extra points
Solution with one extra point
Solution with two extra points
Summary of method for finding a solution: 1) Define an algorithm linking points together, to produce all possible ways of doing so. 2) Define an algorithm to determine the minimum network for each possible way of connecting. 3) Run algorithms on a computer.
Proof that there cannot be more than 3 extra points: The minimal network will be a “tree” (a graph with no cycles) and hence has one less edge than it does vertices.
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Each extra point must have at least three edges connecting to it. Join n fixed points with n-2 extra points like so:
If one more extra point were to be added, then one and only one edge must be added. But the extra point needs to connect to at least three other points and each other extra point is already connected to three things, so more edges are required. But no more can be added. So the most amount of extra points that may be in a tree with n fixed points is n-2. Algorithm 1: Start with the points you wish to connect. Add in either 0, 1, 2 or 3 variable points (cannot be more than 3 since there are 5 fixed points). Pick two points and join them.
B
A
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Out of the other points, they must connect (directly or not) to either one of A or B but not both. If both, then a cycle would exist and a tree wouldn’t exist. So there will be a set of points connected to A and a set connected to B.
B
A
The other points must be partitioned so that they go into one of these two sets. Also, each set must have at most 2 less variable points as fixed points. For each partition, we now have a similar, but reduced problem. We need to find all the ways of making a tree given some points. So the whole algorithm can be applied recursively to each partition. This gives all the trees with A and B connected. Then you connect A with another point C and so on.
B
B D
A
C
A
C
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Do the whole process again, but every time A needs to be connected to B, then don’t consider that case. Do this for all points A can connect to. We now have all the possible ways of connecting the network. Algorithm 2: For each tree produced by algorithm 1, the minimum placement of points is found like so: 1) The fixed points are fixed. The variable points can vary within the convex hull. 2) Use a computer to vary the points within the domain to find an approximate minimum of all distances (systematically, at regular spaced points). Use fine intervals near the approximate minimum.
3) Each tree will give a resultant minimum placement of the variable points this way. 4) Out of all the minimum placements, select the least one. This is the solution because the shortest distance is the least cost.
Part 2 Same as Part 1, but: 1) Shaft point becomes variable with obvious restrictions.
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2) All cases with gradients more than 1/8 are rejected when doing the calculation. There will be a corresponding domain restriction on the variable points (like the convex hull in Part 1). 3) All edges will be straight still, since any curve can be broken up into straight line segments which will have shorter length. 4) When calculating distances, the shaft distances should be multiplied by 5 to give an “effective drive distance” since the shaft costs 5 times as much the drives.
Mathematics in science and technology - making a connection There can be no question that there is widespread use of mathematics in science, engineering and commerce. Mathematics is the language for expressing ideas across these disciplines.
The problem is that this fact has remained somewhat hidden to the general
community, thereby contributing to the prevalent view that there are limited career opportunities for mathematics and statistics graduates. The help of industry and government is crucial in engaging students to the applications of mathematics, and for communicating that mathematics offers the necessary tools and ways of thinking to unite the concepts common to the many areas of science, engineering and commerce. Students must be given experiences which require them to make connections between mathematics and other disciplines in a way that is real and relevant. Only then will they truly see the power that mathematics brings to understanding and solving problems in diverse settings beyond the classroom.
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Acknowledgment We thank the Department of Mathematics and Statistics of the University of Melbourne, National Science Week 2000, The Australian Mathematical Society, Apple and Analytica for their financial support of the Maths Fair. We also thank Mr Brian Hall of Australian Mining Consultants for presenting the MIT problem.
References [1]
Mangelsdorf, C. and Tordesillas, A. (2001) “Real world maths in action! I’d like to see that!” The Australian Mathematical Society Gazette 28 No. 3, 135-142.
[2]
Cassimatis, V. (2000) “The importance of the mining industry to the economy” Presented at the Annual Conference of Economists, Gold Coast Queensland on July 4, 2000. URL: http://www.qmc.com.au/docs/general/sept2000_submission_economicbenefits.pdf
[3]
World Mining Outlook 2002, URL: http://www.mbendi.co.za/indy/ming/p0025.htm
[4]
Australian Bureau of Statistics, unpublished employment data.
[5]
Stoeckel, A. (1999) “Minerals - our wealth down under”, Centre for International Economics, Canberra.
[6]
Hwang, F.K., Richards, D.S. and Winter, P. (1992) The Steiner Tree Problem, Annals of Discrete Mathematics 53, Amsterdam: Elsvier Science Publishers.
[7]
Melzak, Z.A. (1961) “On the problem of Steiner” Canad. Math. Bull. 4, 143-148.
[8]
Brazil, M., Lee, D.H., Rubinstein, J.H., Thomas, D.A., Weng, J.F. and Wormald, N.C. (2000) “Network Optimisation of Underground Mine Design” Proc. Australasian Institute for Mining and Metallurgy 305 No.1, 57-65.
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Christine Mangelsdorf is a lecturer in the Department of Mathematics and Statistics at the University of Melbourne. Her research has focused on developing mathematical models for the behaviour of colloidal particles in both static and oscillating electric fields. She also does the scheduling of draws and match programs for various major sporting organizations in Australia. She currently serves as Treasurer of the Victoria branch of the Australia and New Zealand Industrial and Applied Mathematics (ANZIAM) society.
Marcus Brazil is a senior lecturer in the Department of Electrical and Electronic Engineering at The University of Melbourne. He is also a member of the ARC Special Research Centre for Ultra-Broadband Information Networks. His main research interest is in Optimal Network Design with applications to Telecommunications, VLSI Physical Design, and Underground Mining. He also does occasional consultancy work for mining companies such as Newmont Australia Limited.
Antoinette Tordesillas is a senior lecturer in the Department of Mathematics and Statistics at The University of Melbourne. She has worked on a number of modelling projects in collaboration with industry and government laboratories from Australia and the USA. Her research interests span various topics in mechanics: particulate (granular) systems, contact mechanics, solids, and soil dynamics. She currently serves as Chair of the Victoria branch of the Australia and New Zealand Industrial and Applied Mathematics (ANZIAM) society.
Address for correspondence: Christine Mangelsdorf, Department of Mathematics and Statistics, University of Melbourne, 3010, Victoria, Australia.
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Email:
[email protected]
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