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SA Graduate Modelling Camp Problems

Problem 1: On-line sales and data analytics

Presenter: Professor Jeff Sanders, University of Stellenbosch and African Institute of Mathematical Sciences.

Problem Statement:
A typical industrial problem requires a solution which optimizes some quantity. When an optimal solution is unrealistic an acceptably efficient one is sought. For example the `target' might be a standard length of aluminium that minimises wastage over various uses; or treatment of the residue of sugar cane, for use in generating electricity all-year round, that minimizes its use of energy.

There is 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 we want. 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. So the mathematics is discrete and straightforward though perhaps unfamiliar.

With the widespread availability of the web, e-commerce has become hugely popular. Indeed the success of Amazon and e-books has lead to the closure of many bookshops. Apart from convenience to the shopper, on-line sales are able to offer features not readily available traditionally. Huge amounts of marketing data can be collected digitally and techniques of data analytics used to benefit further on-line sales. One example is the design of specials.

We'll think of an on-line shop as a 'data structure' on which certain operations are defined. We must decide what operations we want: like placing an order; validating payment; delivery; designing and advertising specials. Then we must decide what `state' the shop should have in order to be able to express those operations mathematically. Indeed the first step is to understand what exactly is a 'data structure' (states supporting operations) and how to describe it using discrete mathematics.

Next we must decide how our on-line shop should benefit from market research. This is exciting because it allows us to offer benefits not possible in the traditional setting. But can it also offer possibilities for insecurity (either personal or financial)? Our design must capture the desired features and avoid the undesirable ones. That is why we use discrete notation to express the design, and mathematical reasoning to show it behaves as we wish.

Finally we learn how to use 'big data', and the techniques of data analytics, to improve the profile of sales. An implementation (i.e. code) is not required; indeed our time will be spent on deciding features of a design which would be embodied in an implementation.

Problem 2: Gas venting

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

Problem Statement:
In the oil and gas industry toxic gases (unburned gases) are released during different phases in the treatment process. These gases accumulate and are stored in tanks to be eventually either burnt or released into the atmosphere. These gases are environmentally unfriendly and are a health hazard, so that concentration levels in the atmosphere should not exceed prescribed values (especially) in populated regions.

The companies can control the timing and release rate of gases. The concentration levels reached will depend on the particular location, the timing of the release (day or night etc.) and geographic and meteorological conditions. Our aim is to produce a series of models that may be used to devise a `best’ release strategy.

Problem 3: Instabilities in fluids

Presenter: Professor David P Mason, University of the Witwatersrand

Problem Statement:

Linear instabilities in fluids will be considered. The first instability analysed will be the Rayleigh-Taylor instability at the interface between two inviscid incompressible fluids. The task will be to show that the interface is unstable when the density of the upper fluid is greater than the density of the lower fluid. The effect of interfacial tension on the Rayleigh-Taylor instability will then be investigated. The Kelvin-Helmholtz instability will next be considered. An inviscid incompressible fluid is divided by a thin membrane which could model a flag or the sail of a yacht. The fluid is flowing with different velocities on each side of the membrane. The problem is to show that if the two velocities are different then the interface is unstable. The effect of interfacial tension on the Kelvin-Helmholtz instability will then be investigated. If there is time the Benard problem will be considered. This is concerned with the stability of a fluid at rest between two horizontal boundaries. There is a temperature difference between the two boundaries with the lower boundary being the hotter. The lower region of the fluid will have a slightly lower density due to thermal expansion. The task will be to show that if the temperature gradient between the two boundaries is too large then instability sets in.

Problem 4: Flaws in plate glass

Presenters: Professor Neville Fowkes and Dr Brendan Florio, University of Western Australia

Problem Statement

Glass is produced by melting a batch of sand in a large furnace. Excess melted glass in the chamber overflows into a second broad and shallow Pilkington floating chamber where it spreads out and is drawn off as sheet glass after solidifying. Bubbles are either carried into, or produced within, the furnace and these bubbles rise to the surface of the liquid glass in the furnace under buoyancy action and then burst. If the bubbles do not reach the surface before the glass leaves the chamber then they remain within the glass even when solidified and produce flaws. The plate glass is rejected if there are too many flaws. The objective is to accelerate the removal of bubbles from the furnace by either adding material that produces gases by chemical action, or by introducing large bubbles into the chamber; larger bubbles travel faster than smaller bubbles. Our first objective is to model the growth due to gas absorption and movement of an individual bubble in a chamber.

 

Supporting documents

The initiation of GuinnessThe initiation of Guinness

Saturday report-back presentation

Flaws in plate glass

Problem 5: Polynomial Time Approximation Scheme for Knapsack Problem

Presenter: Montaz Ali, University of Witwatersrand University, Johannesburg.

Problem Statement: 

Knapsack problems appear in real-world decision-making processes in a wide variety of fields, such as finding the least wasteful way to cut raw materials, selection of investments and portfolios. There are two types of Knapsack Problems: Linear Knapsack Problem (LKP) and Quadratic Knapsack Problem (QKP). In the LKP choice of any items is not related to the choice of any other items while in QKP choice of items is pairwise dependent.   Both problems are known to be NP-hard. The exact solutions techniques used are branch and bound, and branch and cut algorithm, both of which require valid inequalities and cutting planes. This project studies complexity issues of the solution times and solve the LKP using  the polynomial time approximation scheme.

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