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									MISG 2000

Equation Free Summaries

Centre for Industrial and Applicable Mathematics

University of South Australia

Table of contents



AD Instruments Biological Data Compression


Australian Rail Track Corporation Analysis of Train Wheel-rail Noise


Boral CMG SA Ltd Optimising Quarry Product Production and Delivery at Boral Adelaide


Email Ltd (Washing Products Division) A Washing Machine Balance Problem


Faulding Healthcare Ltd Warehouse Order Processing and Dispatching


National Rail Corporation Ltd AC versus DC Locomotives


Sola International R & D Australia Fracture Testing of Spectacle Lenses


Acknowledgement of Sponsorship



The Mathematics in Industry Study Group, MISG 2000, was held at the City East Campus of the University of South Australia from January 31st. to February 4th. Phil Howlett was the Director and David Panton the Project Coordinator. MISG is a special Interest Group of Australian and New Zealand Applied Mathematics, a division of The Australian Mathematical Society. The Study Group is an annual problem-solving workshop where university, government, and industrial mathematicians work with industry and business on specific industry projects. This year projects were presented by Boral Quarry Industries; Sola Optical; Email Washing Machine Products; The National Rail Corporation; The Australian Rail Track Corporation; AD Instruments; and Faulding Health Care Products. Around 120 delegates attended MISG 2000. MISG 2000 was opened by the Hon. Mr. Iain Evans (the SA Minister for Industry and Trade) who spoke of the need for industry to embrace new technology. His remarks were endorsed by the Chancellor of UniSA, Mr David Klingberg, when he responded on behalf of the University. Immediately after the opening speeches Ms. Joy Mettam from the Federal Department of Industry Science and Resources announced that the Centre for Industrial and Applicable Mathematics had won a $125,000 grant to run a Feasibility Study for a National MISG Technology Diffusion Program. This program could well be a significant development for Industrial Mathematics in Australia. There was another new development associated with MISG 2000 that may well become a permanent feature. The inaugural MISG Student Workshop was held on Sunday January 30th and was attended by more than 30 graduate and undergraduate students from all over Australia. They listened to lectures by international experts, Professor John King from the University of Nottingham and Dr Colin Please from the University of Southampton and then broke up into small groups to discuss problems from previous MISG study groups before engaging in preliminary discussions about MISG 2000. All students at the workshop went on to participate in MISG 2000. These equation free summaries were produced in conjunction with the project moderators and the industry representatives, by Tim Thwaites, a Melbourne based science journalist who attended MISG. Each summary briefly describes the work carried out during MISG and gives recommendations. Full technical reports will be available in the Proceedings which will appear later this year.

Associate Professor Phil Howlett Director, Centre for Industrial and Applicable Mathematics University of South Australia The Levels Campus Mawson Lakes Blvd Mawson Lakes 5095 South Australia Ph + 08 8302 3195


BIOLOGICAL DATA COMPRESSION AD Instruments Industry Representatives: John Gee, AD Instruments, Dunedin, NZ Boris Schlensky, AD Instruments, Sydney, NSW Project Managers: Lang White, University of Adelaide, Adelaide, SA Charles Pearce, University of Adelaide, Adelaide, SA

AD Instruments (ADI) develops and produces computer systems for collecting, recording and analysing scientific information. Using computers, measurements can be taken faster, more widely and over longer periods of time. With the resulting explosion in data, storage becomes a problem. Although ADI has always incorporated into its systems ways of compressing data so that information can be stored efficiently, the company felt it was time to assess and update the techniques it uses, particularly as it is now beginning to market systems which make it possible to record more complex data. For this review, ADI decided to draw upon the expertise of the MISG. The company presented the MISG team with a series of questions. In particular it was interested in finding out if better compression could be achieved using the newer techniques of data modelling, where a body of data is related to the form of a standard equation or system of equations. ADI also wanted to know if compression could be improved by allowing a controlled loss of data during storage. Using a series of different data sets, the MISG team showed using controlled data loss and data modelling could lead to better data compression than ADI’s present technique. The group then demonstrated the strengths and weaknesses of four different data models. The MISG team concluded that no one single method of data compression would suit all purposes. What constituted the best method depended on the form of the data—different models were useful for different sets of data. More efficient sampling could also help. * * * * * ADI’s data recording and analysis systems are used worldwide for research and teaching in the physical and biological sciences. The company’s head office is in Sydney, where all the hardware R&D is carried out. Software development occurs in Dunedin on New Zealand’s South Island. The conversion of the output of modern electronic measurement into a digital form is handled by the hardware, the analysis and management of storage by software. When ADI began making data acquisition systems, storage space on hard or floppy discs was at a premium, and so data compression was built in. Now, even though the increase in data storage capacity has been astronomical, the ability of computers to take measurements has grown even faster. For instance, ADI’s most widespread application, Chart, can record up to 16 channels of data simultaneously at rates of up to 400,000 samples a second. The system can be set to run for weeks at a time. Files as large as a gigabyte of data can be generated by one run of one experiment. So data compression is still crucial to the efficient management of information. The present data compression system used by ADI is based on tracking the difference between consecutive samples. Instead of storing absolute values, the system stores the difference between consecutive readings. This takes advantage of the fact that measurements are much more likely to cluster around an average than fill a whole range. Also, the difference between consecutive numbers is likely to be smaller than this average value. ADI’s data compression system performs reasonably well in comparison with other similar systems, but the company wanted to know if it could do better using one of the more modern data modelling techniques, where data sets are compressed by matching them to the equations for standard distributions.


Another question ADI wished to settle was whether it could afford to use “lossy” as opposed to “lossless” techniques. In fact, these are just two ends of a spectrum. With a “lossless” method all data is included and stored. In “lossy” methods, data is cleaned up (smoothed) by ignoring unusual outlying points. ADI had always argued that researchers (particularly in the life sciences) needed to keep all data. In certain cases, it was the outliers which provided significant clues to trends, patterns or explanations. The MISG team showed that the improvement in compression which can be achieved using “lossy” techniques is significant, and that it was possible to control the level of data loss, while building in measures of how much loss had occurred. There was even a suggestion of developing an expert system to guide users of ADI’s software as to the most efficient ways of sampling, how much loss they could afford in their data and for what benefit. Although John Gee from ADI thought such a system would be useful, he argued that only a minority of those who bought the company’s products would be likely to take advantage of it. “Most of our customers do not want to know much about data acquisition, do not want to take the trouble to learn, and do not care about the details of data compression.” But most of the effort of the MISG team was devoted to studying the application of data modelling techniques to the kinds of data which the ADI systems are used to record. The group examined four models—Fourier domain, autoregressive, wavelet domain, and the Karhunen-Loeve transform—testing them on data sets provided by the company. It found that each model had its strengths and weaknesses. Fourier domain makes use of the Fourier transform equation so widely employed in signal processing. It provides major gains when data has been oversampled, and is well-suited to data which is periodic in nature. It can compress data to about one-quarter its original size with an error of less than one in 200. The autoregressive model also works well on periodic data, but can also be used on more regular data. The compression factors are not so great as for Fourier, and the error rate is greater. Wavelet domain, a new model which was demonstrated by Jerry Kautsky of Flinders University, can be adjusted to the data, and works well with irregular data. It can compress data to about 20 per cent at an error rate of less than one in 30. The Karhunen-Loeve transform is well suited to cases where there are large numbers of similar measurements. While the compression factor and error rate were still to be determined, both looked promising. The team also investigated how the number and value of residuals—data points which do not fit the models well—could be used to measure and show data loss. By the end of the week, the MISG team had concluded that there was no universal best method for compressing data. In fact, the best method depended on the form of the data to be compressed. However, the group did put forward some guidelines. It recommended that the Fourier domain method always be applied first. If the data is irregular, then wavelets is probably the method of choice; if periodic, the autoregressive model; and if repetitive, the Karhunen-Loeve transform. John Gee from ADI said he had learned a great deal from working with the MISG team over the week, particularly that “there is not going to be some wonder technique that works on everything”.


ANALYSIS OF TRAIN WHEEL-RAIL NOISE Australian Rail Track Corporation Industry Representatives: Peter Jaehne, Australian Rail Track Corporation, Adelaide, SA Glenn Edwards, Australian Rail Track Corporation, Adelaide, SA Project Managers: Neville Fowkes, University of Western Australia, Nedlands, WA Graeme Hocking, Murdoch University, Murdoch, WA The high-pitched squeal from cornering trains has been a problem for more than a century. The noise is particularly bad for the residents of the Adelaide Hills, where there are some of the tightest curves in the Australian railway network. It is not a trivial problem. Maximum noise levels have been measured at well over 100 decibels 7.5 metres from the track. This can be physically painful. In addition, production of the squeal consumes extra fuel, and wears wheels and rails. Since the privatisation of the Australian interstate rail system, the problem of railway squeal has become the preserve of the Australian Rail Track Corporation (ARTC), whose job it is to manage sections of the nation’s rail infrastructure and sell access to it. ARTC commissioned a study from consulting engineers, Vipac, who measured the noise, suggested what the source might be and made a series of recommendations ranging from further testing to physical changes in wheels and rails. ARTC asked the MISG team to identify the cause of wheel squeal via a frequency analysis, develop models of vibration of the wheel-rail system which could be used to validate the source of the noise, and develop strategies to mitigate the noise. After a solid look at the literature on the problem, and much discussion both amongst those at the conference, and with experts outside, the MISG team concluded that a cycle of sticking and slipping as wheels move along the rail would stimulate vibrations at the right frequency, and that these conditions are most likely to occur on sharp corners, and with badly maintained bogies or worn wheels. Among other measures, the team recommended more stringent maintenance of bogies, and modifying the padding under the sleepers so that the rails are more flexible. * * * * * Train noise, including the squeal which concerns residents in the Adelaide Hills, has been with us as long as railways, and has been studied for almost as long. The research literature goes back more than 100 years, and there are several journals devoted to the topic. For ARTC, railway squeal is a problem inherited when it took over management of sections of the interstate rail infrastructure in 1998. ARTC neither built the tracks it operates and maintains, nor does it have total control over the locomotives and rolling stock that use its network. So the MISG team was aware at the outset that certain solutions—such as re-laying entire stretches of rail or modifying the shape of rolling stock wheels—were impractical, either too costly or out of ARTC’s control. The team’s recommendations had to be inexpensive, effective and relatively easy to implement. Around the world, many cures have been proposed for train squeal. Those that have been effective, however, do not suit the ARTC network. One solution, for instance, is to use some sort of oil to modify the friction on tight curves. But while this lubrication of the rails works well for relatively short passenger trains, the effect does not last long enough to make much difference to the long, heavy freight trains which constitute the bulk of ARTC traffic. Another common measure, fitting damping devices on wheels, is far too expensive. Consulting engineers Vipac measured 17 freight trains traversing a curve in the Adelaide Hills. They recorded squeal at frequencies between two and eight kilohertz at noise levels of about 100 decibels, about 10 to 15 decibels louder than the maximum recommended by the New South Wales Environment Protection Authority.


The problem appears to be exacerbated by the use of the newer concrete sleepers, which are more rigid than wood. The squeal is centred on the bogies, and can keep sounding long after a train emerges from a curve. Squeal occurs in loaded or unloaded trains running in either direction independent of speed. The general consensus of the literature is that the squeal is caused by wheels sticking and slipping along the inside rail of the corner. It is the interaction of this process with the wheels which excites the noisy vibration. The MISG team came up with a simple model of the process to determine what factors influenced the frequency of oscillation of the rail and wheel during slip-stick. What could be modified to reduce their influence could then be determined. A standard set of rail wheels is a pair of truncated cones lying on their sides joined by a solid axle. (In fact, most sets come as bogies, in which there are at least two of these pairs of wheels.) On each wheel the smaller crosssection of the cone faces outwards, and a flange hooks over the inner side of the rail head. The cone shape is to ensure that the wheels are self-steering. When the train corners, each pair of wheels will be thrown to the outside of the curve, and the wheel on the inner side of the curve will trace a shorter path than its counterpart on the outside. If the curve is very tight, however, the flange of the outer wheel can ride up on its rail, and the inner wheel will not be able to turn fast enough to keep up—so it slips. This, together with some misalignment, is one possible cause of the squeal (which occurs along the inner rail), according the MISG team. But the persistence of the squeal into and along the next straight section of rail could well be due to misaligned or rogue bogies, the MISG team thought. It only takes the plane of a wheel to be more than half a degree out of alignment for the flange to scrape along the rail. This could be caused by worn wheels, or a sticky central turning bowl where the bogie attaches to the wagon. The team then set about developing equations to describe the movements of the wheels of a cornering bogie. The idea was to determine under what conditions the stick-slip mechanism which leads to the squeal is inevitable, and also to develop sensitive ways of determining rogue bogies. In the light of its analysis, the team put together a series of practical recommendations. Some focus on modifying the tracks and others on the trains. The group thought there may be a way of padding the sleepers in sensitive areas, such as tight corners, to help damp the vibration of the wheels. If the problem is precipitated by the riding of the flange on the outer rail of the curve, it may also help to narrow the gauge slightly on sharp corners. Cutting small gaps in the rails might prevent the sound from propagating. And to alleviate sticking of the bogies, it was thought that, after a tight turn, a shallow turn the other way may help. The most obvious solution for trains themselves is to identify and eliminate rogue bogies. And it was thought that oiling the central bowl of a problem bogie where it connects with its wagon may assist. Peter Jaehne from ARTC admitted that the problem of train squeal was a difficult one. “I spent some very interesting sessions with the group,” he said. “We will definitely be looking at damping the rails more closely. We can even do some experimental work. We already have some rubber blocks for use at level crossings, and we may be able to place them around curves as well.”


OPTIMISING QUARRY PRODUCT PRODUCTION AND DELIVERY AT BORAL ADELAIDE Boral CMG SA Ltd Industry Representatives: Grant Douglas, Boral, Adelaide, SA Danielle Hughes, Boral, Adelaide, SA Project Managers: David Sier, CSIRO Mathematical and Information Sciences, Clayton, VIC Patrick Tobin, Swinburne University of Technology, Hawthorn, VIC

Boral CMG SA Ltd operates seven quarries which supply about 160 different stone products to the Adelaide metropolitan area. The Boral quarries account for about half the volume of stone used in Adelaide. Until recently each Boral quarry was independent in terms of production and sales, and had its own staff and equipment. In the face of declining volumes, Boral embarked on a rationalisation program. The aim was to manage the seven quarries as a portfolio. It resulted in greater efficiency, and a reduction of staff and equipment. Company policy so far has precluded closing down or selling any of the quarries, as Boral feels that even its most inefficient quarry could provide unnecessary competition if it passed out of the company’s hands. With all quarries still in production, customers are generally supplied simply from the closest quarry, rather than on any more sophisticated basis. The company asked the MISG to produce a modelling tool to help it maximise profits from its quarry portfolio. Boral hoped the model would be able to assist in decision making on strategic matters, such as appropriate cost structures for production and delivery or whether it would be profitable to take particular quarries out of production for a specified period. The MISG team showed that such a model was feasible. Indeed, the team built a prototype for production and transport prices, tested it, and used it to generate preliminary results. The team concluded, for instance, that Boral should look more closely at downgrading production at certain quarries, and should also analyse the internal charging system it uses to allocate the costs of operating its transport fleet. * * * * * No Australian city is as well served by quarries as Adelaide, as is evident in its widespread use of stone. One company, Boral CMG SA Ltd, supplies about half the market from a network of seven quarries in or close to the metropolitan area. These quarries produce about 160 products in four groups—aggregates, roadbase materials, sands, and fills. There are about 40 basic products, and these are augmented by others where water or cement has been added. Most are produced at all seven quarries. Although the performance of individual products varies slightly with quarry, the most significant difference is between the harder, denser dolomite from four quarries and the lighter quartzite from the other three. Each of the Boral quarries was developed and operated independently, and had its own staff and equipment. This resulted in high fixed costs, particularly a large expenditure in repairs, maintenance and capital replacement. It also limited flexibility during downturns in the market as in the past two decades where the demand for stone products in Adelaide has diminished from 6 million tonnes a year in 1982 to about 4 million tonnes a year at present. In order to keep costs down, over the past couple of years Boral has restructured its quarries into a single portfolio, moving staff and mobile equipment from quarry to quarry when needed. Quarrying can be divided into seven functional operations—removing or stripping overburden; drilling and blasting the useful rock; loading and hauling the rock from the quarry face to the crushing plant; crushing and screening to process the rock to the required size; dumping of the product into a stockpile; treatment with water


or cement if necessary; and loading and dispatching the product to the customer. Each quarry needs to be permanently equipped only to crush and screen, and to load and dispatch. All other functions can be serviced centrally, but this demands the development of intermediate stockpiles for storage of product between operations. Under the new approach, Boral was able to reduce its workforce, to replace ageing plant with bigger, more modern and more mobile equipment, and to use larger trucks and loaders. Maintenance costs, depreciation and capital replacement costs were reduced significantly, as was the delivered cost to the customer. At a time when the company revenue dropped by $6 million a year, Boral’s profits decreased only by about $900,000. There were limits to the new approach, however. The company decided it would keep all seven quarries in production. Given that planning and environmental considerations make it difficult to establish any new quarries in the Adelaide area, a working quarry is a valuable asset. And, if sold, even the most inefficient quarry in the Boral portfolio could provide unwelcome competition. Nearly all products are produced at all quarries so, in the absence of a sophisticated analysis of costs, the quarry closest to the customer generally delivers the ordered product. But this is unlikely to be the most efficient strategy overall. In order to reap maximum profits from the portfolio of quarries as a whole, Boral asked the MISG to produce an interactive model which could be used to determine the best margin available from selling each product, whatever the quarry. The company hoped that such a model could be used on a daily basis by production and salesforce staff to manage production and pricing, and by management as a planning and budgetary tool. The complexity of the model was in the significant numbers of variables—160 products from seven quarries, produced at between 40 and 500 tonnes an hour for a wide range of costs per tonne. In addition, there are costs of staff, equipment, delivery and asset management to be taken into account. The MISG team set about building several different equations. The first described the costs of production. It took into account the cost associated with each stage of production, as well as any royalties and taxes, and the costs of administering and filling orders. A second equation described the cost of delivery. (Boral uses its own delivery fleet where possible.) In fact, transport is a considerable portion of the cost of some products. Other equations were derived for revenue and for profits. Company data on filling orders were used to allocate the costs and prices involved in the equations. A prototype model of costs was constructed for individual products, which could be used to estimate the relative cost of filling orders from different quarries. Maps could be drawn for the delivered cost of a product from each quarry to a specified geographic area. It turned out that products from certain quarries were cost competitive even in the areas abutting a neighbouring quarry. This provided a strong argument for reducing production from the less competitive quarries for a specified period. The MISG team showed that it was feasible to construct a detailed model tracking profits, and that company data could be used to test it. A model of the cost of delivery would demonstrate the impact of the company’s artificial internal charging mechanisms used for managing its transport arm, and could even suggest the most efficient structure of the fleet of trucks Boral should use. The financial benefits or otherwise of using particular quarries for particular products, or of increasing or decreasing production at particular quarries, would also be easier to appreciate. Speaking for the company, Ms Danielle Hughes said Boral had been very happy with the MISG process and its outcomes, especially with the prototype models. The unblinkered view of the MISG team had been very useful, she said. “We learned a lot from it. Things we hadn’t considered before will now be taken into account. We will be very interested in the final report.”


A WASHING MACHINE BALANCE PROBLEM Email Ltd (Washing Products Division) Industry Representatives: Karkhee Lee, Email Washing Products, Adelaide, SA Wayne Burford, Email Washing Products, Adelaide, SA Project Managers: Bill Whiten, University of Queensland, St Lucia, QLD Phil Broadbridge, University of Wollongong, Wollongong, NSW Anyone who has ever used a washing machine is well aware of the problem posed by unbalanced loads during the spin cycle. The (inner) bowl of a standard machine spins at about 800 revolutions a minute. Uneven placement of wet clothes can lead to considerable sideways and tilting motion of the bowl, and dangerous and excessive vibration of the machine. Many models of washing machine alleviate this problem by installing a torus, partially filled with saline solution around the outside of the bowl. Within the torus, the liquid moves to the outside and tends to balance the uneven loading. But for each design change in height, weight, diameter and speed of the bowl, the dimensions of the balancing torus must change too. At present, torus design is a black art of experience, practical experimentation, trial and error, and intuition—which is time consuming and labour intensive. Email hoped that by developing a model of the balancing process, the MISG team may be able to provide a rational base for quicker and more efficient design of the torus. The team undertook an experimental program to gain a feeling for how the system vibrates under a range of conditions. Members of the group then developed a series of models of increasing complexity in one and two dimensions, along with a simplified three-dimensional picture. Where the models provided figures which could be tested by the experimental system, there was reasonable agreement. The models gave a better understanding of the system, and helped to identify future areas of study. The MISG group found that unbalanced loads lead to vibration because they move the centre of gravity of the bowl away from the centre of symmetry of the whole machine. Also, loads spread over different levels of the bowl tended to tilt the axis of rotation. The team reported that whereas one balance ring around the top of the bowl helps correct horizontal vibration, in view of the problem of tilt, two balance rings, top and bottom, give better control. * * * * * The market for washing machines is very competitive, which means that companies such as Email Ltd must continuously improve their products. One area of improvement readily noticed by consumers is in the vibration of the spin cycle. A well designed toroidal balance ring should allow a relatively smooth motion of the machine over a wide range of unbalanced loads. But the present design process tends to be labour intensive, and the sensitivity of its capacity to function to changes in the dimensions of design elements is unknown. If the torus is shaped wrongly or is too full or empty of liquid, it will not work properly. Email Washing Products Division approached MISG to devise a more efficient solution to designing the balance ring. The company asked the team to produce a model where measurable inputs such as spin speed, and diameter and depth of the bowl, could translate into outputs such as the amounts of solution (within the torus) necessary to compensate a particular weight of washing and the cross-section which would best match the job. The team began by measuring the scale of the problem. Email provided a special machine with a transparent side and lid, and members of the team used it to establish an experimental program. They were able to document typical vibration behaviour, as well as measuring the impact of specific changes. One result which surprised the team was that the same starting conditions produced different patterns of vibration. It was suggested that different modes of vibration could exist for the same configurations of mass.


Several subgroups began working on algebraic models of the vibrating system at different levels of complexity in one, two and three dimensions. The two-dimensional model provided good understanding of how the torus worked. The outer cabinet is centred on an axis at the geometric centre of the machine. But an unbalanced load in the inner bowl pushes its centre of rotation to one side. The centre of mass of the water in the torus moves to the other side partially compensating for this displacement and thus reducing the amount of vibration. Using the two-dimensional model, the group could calculate the size of the torus and the mass of water needed to reduce the vibration to a specified maximum. And from the equation derived for the two-dimensional model, members of the team were able to predict the maximum amplitude of vibration for different masses of washing at different speeds. The numbers were in pretty good agreement with what the experimental team were measuring. The three-dimensional model was more complex, and only a simplification could be constructed. The equations linked the angular momentum of rotation, the eccentricity of rotation (or deviation from a circular path), and the angle of tilt of the bowl from vertical. No regular precession of the bowl around the vertical was detected experimentally, which agrees with predictions of the effect of damping on the vibrations. Members of the subgroup working on the theory derived a system of 16 differential equations describing the motion they observed. These equations fell into two categories. One set describes the oscillation of the inner bowl around the centre of gravity, which arises from the uneven distribution of washing around the circumference of the bowl. The other set describes the tilting of the axis of rotation, which arises mainly from uneven vertical distribution of washing over the depth of the bowl. This subgroup came to the conclusion that for stable operation of the machine, the movement of the centre of gravity from the centre of symmetry of the machine needed to be minimised, together with the movement of the rotation axis from the vertical. Given these two elements of vibration, the subgroup recommended that two balance rings at the top and bottom of the bowl be used instead of one, to give better control of the tilting of the rotation axis in particular. “Over the week, what was a complex problem, became a whole lot more complex for me, but is now described in a much better way,” said the project engineer for R&D department of Email washing products, Mr Lee Kharkhee. “We achieved a lot in a short period of time. It was most inspiring to work with people who show a real passion for what they are doing. And we were able to relate the theory to real practical outcomes. Already, Email can use the two-dimensional model to set design criteria.”


WAREHOUSE ORDER PROCESSING AND DISPATCHING Faulding Healthcare Ltd Industry Representatives: John Eleftheriou, Faulding Healthcare, Adelaide, SA Manios Dolmas, Faulding Healthcare, Adelaide, SA Kevin Burt, Faulding Healthcare, Adelaide, SA Project Managers: Mohan Krishnamoorthy, CSIRO Maths and Information Sciences, Clayton, VIC Lou Caccetta, Curtin University of Technology, Perth, WA Faulding Healthcare Ltd operates an automated warehouse which distributes pharmaceutical supplies to about 550 customers in the Sydney metropolitan area together with others in the country. Two deliveries are made each day to city customers and one to those in the country. The warehouse is arranged in aisles so that operators can use 27 large computer-controlled trolleys, known as datamobils, to “pick” products from the shelves. Each operator gathers orders together into a suite of eight plastic tubs, or totes, carried by his or her datamobil. The datamobils follow a preset cyclic path through the warehouse aisles, stopping at the positions of the products required to fill individual orders. But at present the datamobils are making only about 270 cycles out of a potential 405 during each 10-hour shift. Also, the system has difficulty coping with late orders. So the company asked the MISG team if it could help to reschedule cycles to ensure optimal use of the datamobils by minimising the stops they make, and by ordering and grouping products more efficiently to reduce overall cycle times. Faulding also requested the MISG team to build a model robust enough to cope with disruptions and late orders. The MISG team incorporated these requests in a mathematical model which maximised usage of the datamobils given within physical and logical constraints. In trying to construct mathematical equations to describe the process of filling orders, the team recognised the large numbers of interacting factors and constraints involved. Although the question of how best to distribute products within the warehouse became significant, including the issue of clumping or spreading popular (fast moving) products throughout the warehouse, this was not considered as part of the study. By the end of the week, the MISG had split the overall problem into a series of solvable components. The team had developed formulations and methods to solve those components, and had identified several operational issues and strategies, such as ways of deciding how to generate a more even flow of datamobils through the warehouse. * * * * * In the past 150 years, Fauldings has grown from a pharmacy in colonial South Australia to one of Australia’s top 100 companies with several divisions. Faulding Healthcare distributes products and healthcare services to hospitals and pharmacies. The company’s Sydney warehouse processes orders from about 550 metropolitan customers and more from surrounding country areas. In the metropolitan area there are two deliveries a day, morning and afternoon. Most pharmacies place their afternoon orders only after receiving the morning delivery. To fill the day’s orders typically involves 40 delivery runs in the morning and 37 in the afternoon, each servicing about a dozen pharmacies. Additional deliveries are made to country areas. The warehouse is highly automated, and makes use of computer-controlled electronic trolleys, datamobils, to pick out and put together orders from about 16,000 available products. After each order is checked for price and availability of stock, the details are fed into a computer program which divides up the specified products into a series of totes or plastic tubs into which they will be collected. Each datamobil carries eight totes.


After the software determines what products are to go into each tote, it allocates totes to datamobils. Each datamobil is then programmed to stop at the location of the products required to fill its totes. When a product is picked off the shelf by a datamobil operator, its identity is verified by means of a barcode, and the correct quantity by weighing on scales. The datamobil will not move onto the next location unless these two attributes match the order. “Picking” takes between 10 and 18 seconds, depending on the skill and experience of the operator. The warehouse is laid out so that all datamobils follow a pre-set course which cycles up and down the aisles, stopping where necessary. Passing is not possible, and considerable vehicle interference occurs. The company thought that there may be ways of reducing datamobil cycle times. Specifically, if efficiency could be increased to the point where fewer datamobils were needed, not only would this reduce interference, but it would also lower capital and maintenance costs. The MISG team began by reformulating the problem to make it easier to attack in mathematical terms. Essentially it decided on two objectives—to allocate products to totes and cycles so as to minimise the overall time taken to pick them off the shelves and to minimise the time wasted in all cycles. The team then began the task of developing a model of the picking system in the warehouse. First, it defined a large number of terms and symbols to be used in developing equations. Some of these incorporated the constraints of the system, for instance, no more than 27 datamobils and 70 operators. It soon became clear that this was a problem of allocation—allocation of products to totes, allocation of totes and drivers to datamobils, and in particular, allocation of products to aisle locations. In fact the problem was split up into several components along these lines. The issue of just how products should be arranged in the warehouse became a major point of discussion. Products are classed according to how popular (fast moving) they are. Debate centred around whether a more efficient traffic flow could be maintained if the faster moving products (where the datamobils are likely to spend the most time) were spread throughout the warehouse or clumped together in one spot. In the end it was decided that both these strategies should be modelled. For instance, in approaching the question of how best to allocate totes to datamobils, the team decided it could be done according to three different criteria. One way to allocate the totes would be so as to ensure minimal variation per cycle in the time it takes to pick products off the shelves. This would provide a means for operator experience to be built into the model, as the pick time depends on the operator. Another way to allocate totes would be to maximise the spread of locations to be picked, in the hope of minimising datamobil interference. Or you could do the reverse, and maximise the clumping of locations to be picked, in the hope of increasing the efficiency and predictability of pick time. Scheduling when each datamobil should set off on its round (cycle) could also be done in different ways. At present, datamobils which are likely to complete their cycle fastest are sent off first, but the allocation could also be made in terms of average distance to where the products are located, those datamobils with longer to go before picking products being sent off first. The MISG team partitioned the problem into a series of tractable components, and provided methods and guidance as to how to solve those components. The process helped Faulding Healthcare to identify the issues it needed to confront in order to increase efficiency of datamobil use. “Looking at this problem from outside has produced a fresh approach,” said company representative, Mr Manios Dolmas. “I’m sure we’ll be back next year.”


AC VERSUS DC LOCOMOTIVES National Rail Corporation Ltd Industry Representatives: Con Alexandrides, National Rail Corporation, Adelaide, SA Gary Alexander, National Rail Corporation, Adelaide, SA Project Managers: Basil Benjamin, University of South Australia, The Levels, SA Philip Laird, University of Wollongong, Wollongong, NSW National Rail Corporation Ltd operates about 300 freight and passenger trains a week across the interstate standard gauge network connecting Brisbane, Sydney, Melbourne, Adelaide, Broken Hill, Alice Springs and Perth. Its business priority is long-distance freight trains. In 1996 and 1997, the company invested more than $350 million in 120 new NR-class DC diesel electric locomotives. Since then AC locomotive technology has developed to the stage where it is competitive with DC. National Rail wanted an MISG team to provide an external perspective on the strengths and weaknesses of DC and AC locomotives, and to suggest the opportunities AC technology might provide. To make the AC-DC comparison practical and relevant, the company proposed doing it for the Melbourne to Brisbane rail corridor. Given that context, National Rail wanted to know was if it were worth converting at least some of its existing DC locomotive stock to AC, at a nominal cost of about $1 million for each locomotive. Any meaningful comparison involves many factors including: the performance of the locomotives and the load each can haul; the gradients along the track; crew and maintenance costs; fuel capacity and consumption; depreciation; track access fees; the cost of conversion; the axle weight of each engine; and the amount of wear locomotives cause to the track. After pulling together the relevant data, the use of a simulation model to calculate fuel consumption and transit times, and much discussion, the Study Group looked at the options for locomotives to haul freight between Melbourne and Brisbane. Calculations were made for both a 3500-tonne train and a 3000-tonne train being hauled by two locomotives, sometimes with a third locomotive assisting at the steepest sections of track. Under the assumptions made—including for depreciation and the cost of conversion—AC locomotives were cost effective.

The question of upgrading the track was briefly considered particularly for the operation of heavier trains.
* * * * * In the past three or four years, National Rail has become aware of an upsurge of interest in AC locomotive technology. The major US locomotive manufacturers now market a range of AC engines, and about one locomotive in 20 operating on US Class 1 railways is an AC locomotive. A significant advantage of AC locomotives is that they perform better than DC locomotives at low speeds. They have better adhesion and can operate for longer at slower speeds, so their ability to haul heavy loads slowly up sustained steep gradients is superior to that of DC locomotives. They also have better braking. AC locomotives are more fuel-efficient and their maintenance costs are lower. But the initial cost of DC locomotives is less and they are a little lighter. The Melbourne to Brisbane standard gauge line is about 1912 kilometres of mainly single track with occasional passing loops to accommodate trains up to 1500 m long. The steepest gradients of 1 in 40 or 2.5 per cent are in the middle section between Junee and Taree (via Sydney). Clearly, any locomotives operating along the corridor would have to be capable of managing the 1 in 40 gradients. Apart from the performance of the locomotives, other considerations include the cost of track access (a flagfall plus a set rate per gross tonne per kilometre), the cost of the crews to operate the engines, and the weight of the


locomotives. In fact, conversion from DC to AC would add about two or three tonnes to the weight of an engine, taking it about 0.5 tonnes over the maximum axle load currently allowed on the track. Either this limit will have to be changed, or each converted locomotive will have to carry less fuel. Given all the considerations, the team decided that any comparisons would eventually have to be made on the basis of a standard figure of cost per tonne of goods carried between Melbourne and Brisbane. This initial discussion also raised several other issues. For instance, would easing the gradients at the steepest parts of the track allow less expensive locomotive combinations to be used? The sections with gradients as steep as 1 in 40, only amount to about 17 km of track northbound from Melbourne—the direction in which most of the freight flows—and about double that southbound from Brisbane. Another issue—never fully resolved—was whether different combinations of locomotives led to significant differences in track wear and maintenance. The team decided that in order to provide a practical comparison, it would analyse several different combinations of locomotives. It then calculated a bottom-line figure of cost per tonne of goods hauled by adding together the cost of the following components: crew; fuel; track access; locomotive maintenance; and locomotive depreciation. The comparison is not simple. For instance, it is likely that a two-locomotive train will use less fuel than a three-locomotive train, but the latter may take less time for the trip and thus reduce crew costs. The team gained access to track file data via the University of South Australia, and was able to use the simulation program MTRAIN to calculate transit times and fuel use for the various locomotive combinations it studied. Initially the team compared what it thought were the two most likely locomotive combinations. The first was to haul a 3500-tonne load using two DC locomotives between Melbourne and Junee, three between Junee and Taree, and two from there into Brisbane. This strategy was compared with hauling 3000 tonnes the whole way using two AC locomotives. But then the group decided to look at other options. For instance, hauling 3500 tonnes using two AC locomotives supplemented by a third just from Sydney to Broadmeadow (near Newcastle). Calculations showed that two AC locomotives could, in fact, haul 3500 tonnes the whole way from Melbourne to Brisbane in 31.3 hours, and that this was the best option at a cost of $12.46 per tonne. However this journey time was appreciably longer than the 28.2 hours needed for 2 DC locomotives to haul 2600 tonnes at a cost of $13.52 per tonne. The best DC locomotive options involved two DC locomotives supplemented by a third for part of the way, and the lowest cost for these combinations was $13.02 per tonne. The question of upgrading the track was briefly considered. Reduction of the steepest gradients would save time and fuel. There is also advantage in reducing the need for assisting locomotives for the operation of heavier trains. The current level of NRC traffic, however, would hardly justify the cost, and the total traffic at present probably would not justify the payback period. A representative of the National Rail Corporation, Mr Con Alexandrides said the calculations and conclusions of the Study Group had shown his company that it needed to consider its options seriously. “The dialogue was important and raised issues that would not normally come out. There was a lot of passion, and that is a really powerful tool in industry.”

The project managers acknowledge the contribution of all members of this project team and particularly the contributions of the NRC representatives.


FRACTURE TESTING OF SPECTACLE LENSES SOLA International R&D Australia Industry Representatives: Philip Stephenson, SOLA, Lonsdale, SA Matthew Cuthbertson, SOLA, Lonsdale, SA David Lewis, SOLA, Lonsdale, SA Chong Kok, SOLA, Lonsdale, SA Project Managers: Jim Hill, University of Wollongong, Wollongong, NSW Stephen Lucas, University of South Australia, The Levels, SA SOLA International, founded in Adelaide in 1960, is now the world’s largest manufacturer and supplier of plastic spectacle lenses. About 100 million people worldwide wear SOLA lenses. Although the company headquarters has moved to California, SOLA still maintains an R&D arm in Adelaide. Before new lenses are allowed to be sold, US and European authorities demand they meet certain safety standards, in particular that they are resistant to fracturing. In America, the lenses are tested by dropping a steel ball of specified size from a predetermined height onto the centre of the lens; in Europe a similar ball is pushed against the lens at a prescribed pressure. These test methods are laborious and can lead to unreproducible results. The drive towards lenses which are more efficient, lighter and thinner, makes them more susceptible to breakage, and the multilayered coatings applied to reduce scratches and glare are brittle and can crack. So SOLA asked the MISG to look at ways of modelling how plastic lenses fracture. The company felt that accurate numerical models of lens fracture and what happens at the point of impact may well save time, effort and money in laboratory testing. At the very least, the process of building such models could provide greater insight into the lens fracture process. The MISG team tried several different approaches, and came up with preliminary equations and assumptions that should make it possible to develop the necessary models. * * * * * SOLA International makes more than 20,000 different types of plastic lenses. Ever since the early days in Adelaide, the company’s business success has depended on a steady flow of innovation. More than 60 per cent of SOLA’s current sales derive from new products, which means that safety testing is a significant activity for the company. But the present test methods are labour intensive and the results are often unable to be reproduced. According to Dr Matthew Cuthbertson of SOLA International the fracture tests make creating a new product range “a nightmare”, and qualifying a new material for lenses “an even bigger nightmare”. So the company came to the MISG in the hopes that it could help develop a model to predict from the geometry of a lens, and the strength of the impact, whether the lens would fracture or not. The idea was to reduce the amount of laboratory testing by focusing attention on those types of lenses which were likely to be most at risk of breakage. It was also hoped that very process of construction of the model might well provide insight into those factors which affect fracture. The trend of modern lens manufacture is towards thinner, lighter and more durable lenses. Unfortunately, this makes them more liable to fracture. The curvature of lenses bends light and provides the desired properties. The radii of curvature of the front and back surfaces of the lens are different, and the difference between the inverse of these radii defines the power of the lens. For example, in minus power lenses (for the short-sighted) the back is more tightly curved than the front, which makes the centre of lens the thinnest part. Making lenses lighter demands flatter curves. But this leads to even thinner centres, thus lenses which are easier to fracture.


The material out of which lenses are made determines how they deform under pressure. In addition, most lenses nowadays are coated back and front with layers of material to make them scratch resistant and cut down glare. These coatings are often more brittle than the lens material itself. While such coatings are only a few microns thick and do not have a significant effect on the structural properties and strength of the lens, they are likely to be places where cracks start. As well as a model to predict the degree of deformation of the lens on impact of the steel ball, the company hoped the MISG team might provide an ability to predict stresses in the back surface coating, and a capacity to predict crack initiation and propagation. This problem involves the deformation of plastic, which is coated with materials with different physical characteristics, under conditions where cracking can take place. Some of the issues include the events surrounding impact; whether the material contains flaws and microcracks and how these affect the outcome; the difference between impact and static testing; and how cracks propagate through more than one material. The members of the MISG team came up with at least six different approaches. They included looking at a lens as represented by an elastic spherical shell and also at how cracks propagate across the interface between two different materials. Several approaches involved assumptions and approximations to make the mathematics more tractable. In the end, the team split into two major groups each coordinated by one of the project managers. One group, led by Dr Stephen Lucas, studied the deformation of a lens as represented by a flat elastic plate; the other, led by Prof Jim Hill, investigated the bending of a multi-layered material shaped as a cylinder (which was mathematically more convenient than a sphere). With assistance from Dr Dave Clements of the University of Adelaide and Mr Grant Cox of the University of Wollongong, the Hill group was able to construct and solve an equation to calculate the stress at the back of the cylindrical lens for a given force on the front. This is the stress that causes lens fracture. The values calculated for it were comparable with empirical figures obtained from the company. The Lucas group began by calculating the characteristics of the vibrations set up in the flat plastic plate when hit by a steel ball. The period of transverse and compressive waves was about a millionth of a second, bending and flexing of the plate took less than a thousandth of a second, whereas the duration of impact was about two thousandths of a second. The group also showed that the forces imparted by the US (drop ball) test and the European (static) test were almost equal, yet lenses tend to fracture more easily in the American test. The group derived an equation for predicting the force due to the ball’s impact, assuming the plastic plate is perfectly flat, thin, elastic and that any deformations are vertical and small. They found that the plate could not take the shape of a part of the ball, and that the contact region should in fact be a ring. But practically the ring’s radius is so small that it should really be considered as a point. Extensions to the equation to accommodate larger deformations, plates that are not flat, and vary in thickness were also proposed. Enough was achieved by the two groups to show that development of the kinds of models requested by SOLA International is possible with further work. “We’ve had a fascinating week,” said Dr Philip Stephenson from SOLA. “It is a difficult problem, and we were not sure how far people would get, but we are very encouraged by the many different approaches. The team has come up with interesting and useful ideas.”


Acknowledgment of Sponsorship
Financial Backing
University of South Australia     Centre for Industrial and Applicable Mathematics; School of Mathematics; Research Office; Division of ITEE.

ANZIAM (Australian and New Zealand Industrial and Applied Mathematics). CSIRO helped support student travel grants. DSTO generously supported the release of Summer vacation undergraduate students for the MISG week. CEANET Pty Ltd who are the sole distributors of MathWorks and Maple in Australia helped sponsor the Reception and provided the latest releases of both Maple and MATLAB for the participants to use during MISG. Southcorp helped provide wine for the Reception and lunches during MISG.

We would also like to acknowledge the dedicated and enthusiastic work of the moderators throughout MISG. This acknowledgment should also extend to all participating mathematicians and their associated institutions who gave their time without charge during the MISG week.


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