SA Graduate Modelling Camp Problems

Presenter: Dr Thama Duba, Durban University of Technology

Problem Statement:

It was on New Year¡¯s Day in 1995 when a monster wave hit the Statoil Draupner platform, which monitors the pressure and gas flows in Norway¡¯s offshore pipelines. A wave of height more than 23 meters struck the site.

Rogue waves, also known as giant waves, freak waves or monster waves, appear from nowhere and are much steeper than normal surrounding waves. They are highly destructive and can no longer be reserved for maritime folklore. They have become a focus for much research now especially because they can break a mainliner to pieces and destroy the surrounding offshore structures.

Satellite measurements confirm the existence of such waves. Studies of these waves involve several mathematical models but their behaviour rendered linear models to a level of redundancy. Modulation instability as an approach to studying these waves in several research activities failed, as there are no boundaries in the ocean. A new mathematical model published in Scientific Reports successfully simulated the true height of the Draupner wave as well as two other rogue waves using wind speed estimates based on satellite data collected during storms. Neither approach has succeeded in predicting when and where the monster waves might occur.

So what is the mechanism behind the occurrence of such extreme waves? Much of research has now come to the suggestion that rogue waves form due to nonlinear interactions between surface waves. The remaining question is how to model these nonlinear properties.

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Modelling Rogue Waves - Wake of Destruction

Presenter: Matthews M. Sejeso, University of the Witwatersrand, Johannesburg

Problem Statement

A brewing company in North America has established itself in the market with the concept of beer blending. This concept consists of mixing different types of ready-to-drink beers in order to create uniquely new flavours of beer. One famous example of beer blending is the ¡°Black and Tan¡±, which is obtained by layering a pale ale and dark beer. In order to achieve a perfect blending, the company considers a large set of beers as raw materials. Each raw material has multiple attributes such as brightness, colour, thickness, coarseness, and flavour. Each of these attributes is graded on a numeric scale. Thus, a raw material is ¡°finger printed¡± for different attributes, that is, raw material is given a numeric score. On the other hand, each final blend of beer is also ¡°figure printed¡± against the same attributes.

The objective is then to find a combination of raw materials to produce the blends to match their attributes score requirements, within the availability of the given raw materials and based on the weekly demand. In the event that the attributes score requirements cannot be matched, it is important to find the combination of raw materials that achieves the closest match to the attributes score requirements. Usually, some of the attributes of a blend may be more critical in achieving the blend¡¯s quality than others. Therefore, in order to achieve the closest match of a blend, the decision maker would prefer the scores of the critical attributes not to be violated even if that implies some violations to the scores of the less critical attributes.

The aim of this project is to develop a decision support tool that is able to recommend lowest cost option for recipes for beer blends based on the price, the availability and the attributes of raw materials. The solution should ensure the attributes score requirements are matched whenever possible. When it is not possible to match these scores, the closest match solution should be found.

Some data are attached for testing the solution. The ¡°Raw_Materials.xls¡± file is a set of raw materials with numeric scores of the attributes, their availability as well as their costs. The file ¡°Some_Testing_Scenarios.xlsx¡± presents three test scenarios which can be used to assess the decision support tool on a small scale. Finally the file ¡°Blends.xls¡± is a set of beer blends with the score requirement of each of their attributes as well as the amount of demand that needs to be achieved using the raw materials in the file ¡°Raw_Materials.xls¡±. This will be used to assess the efficiency of the decision support tool.

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Decision Support Tool for Optimal Beer Blending

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Decision Support Tool for Optimal Beer Blending - Report

Presenter: Professor Neville Fowkes, University of Western Australia, Perth, Australia

Problem Statement:

Contact free and fast determinations of the thermo-physical properties of thin films are required in the semiconductor industry and elsewhere. Such thin films may be deposited on the surface of a substrate or obtained by altering the surface of a substrate or obtained by altering the surface properties of a material by annealing or by chemical treatment. The conductivity of the surface is usually much less than that of the substrate. One technique used is to shine a laser beam on the surface and measure the emitted radiation from the surface photographically. This temperature rise will be dependent on the thermal properties of both the film and its substrate and heat loss from the surface. How can one use the temperature rise results to determine the conductivity of the film?

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Thin Film Conductivity Measurements - January 3, 2019

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Thin Film Conductivity Measurements - January 12, 2019

Presenter: Professor Jeff Sanders, University of Stellenbosch and AIMS South Africa

Problem Statement:

Can we design an on-line auction system which benefits from being distributed and digital?

With the widespread availability of the web, e-commerce has become hugely popular. Apart from convenience to the buyer, on-line sales and auctions are able to offer features not readily available traditionally. Marketing data can be collected digitally and techniques of data analytics used to benefit further sales.

This project concerns the design of an on-line (or virtual) auction system (you may know e-bay or gum tree).

(a) It begins with a brief survey of types of auction, in order that we appreciate the range of potential behaviour/protocols (e.g. English, Dutch, Sealed-bid, Vickrey).

(b) It then considers the features and functionality desired of an on-line auction, including those not possible in a standard auction. Important is avoidance of negative features where possible. (What about privacy? Accountability? Collusion between bidders? Can Data Science and the techniques of data analytics, be used to improve the profile of sales?)

(c) A system is designed which incorporates the desired features.

(d) Finally the chosen auction system is shown to behave as desired.

For (c) we will think of an on-line auction as a `data structure¡¯ i.e. as consisting of state on which certain operations are defined, like SubmitItem, Bid, Result. Then we must decide what state the system has in order to be able to express those operations mathematically. Indeed an important step is to understand what exactly is a `data structure¡¯ (states supporting operations) and how to describe it using discrete mathematics.

A typical industrial problem requires a solution which optimises some quantity. When an optimal solution is unrealistic an acceptably efficient one is sought. It might be: what standard length of aluminium minimises wastage over various uses; or what treatment of the residue of sugar cane, for use in generating electricity all-year round, minimises use of energy. The solution is typically described by a differential equation and then constraints imposed to optimise the solution.

This project is an example of an important family of contemporary problems which fit that paradigm, but whose solution is instead a design (of optimal or acceptable efficiency). Ingenuity is required to invent the design and mathematics is needed to show that it functions as desired. The design might enable several robots to coordinate in exploring a mine; it might be for a secure electronic voting system; or it might be for on-line sales featuring specials that benefit from market data. The mathematics is discrete and straightforward though perhaps unfamiliar.

An implementation (i.e. code) is not sought. Indeed out time will be spent on expressing features and deciding how to capture them in an implementation.

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Presentation - Online Auctions

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Report back presentation - Online Auctions

Presenter: Dr Ashleigh Jane Hutchinson, University of the Witwatersrand

Problem Statement:

You have purchased a piece of wild land that is suitable for animals. To make money, you decide to start developing a game farm. In order to ensure the success of your game farm, you have to investigate the topology of your land and the climate. You are given a loan from a bank. Using this money, you will need to purchase your animals. The goal of this problem is two-fold:

1. Firstly, you will need to carefully select the type of animals and number of animals that will be purchased given some budget. The goal is to create a sustainable environment with as little human interference as possible. You will need to think very carefully about choosing your animals.
2. In order to make a profit, you also need to think about what attracts tourists. For example, having rhinos will attract tourists but it may also attract poachers.

To begin this problem, you will need to consider a simple example, say purchasing five different types of animals where only one is part of a predator species. You will need to build more complicated models and decide on the most important factors that must be included in these models.

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A start-up game farm - presentation

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A start-up game farm - report back presentation

Presenter: Professor D P Mason, University of the Witwatersrand

Problem Statement:

Double diffusion convection has applications in many areas of science such as in oceanography and astrophysics and in chemical and other branches of engineering. It occurs in fluids when there are two components with different diffusion coefficients which have opposing effects on the vertical density gradient. The linear stability of a two-dimensional fluid between two horizontal planes due to the diffusion of heat, which is destabilizing, and the diffusion of salt, which is stabilizing, will be analysed. The aim of the problem is to provide an introduction to double diffusion convection in Lake Kivu and the formation of temperature steps in the lake.

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Double diffusive convection - presentation

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Linear stability of double diffusion convection