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Graduate Student Workshop Problems

 

Problem 1:  Traffic lights or traffic circles?

Presenter:  Professor Neville Fowkes, School of Mathematics and  Statistics, University of Western Australia, Perth, Australia

Problem statement

The usual choice of  handling two intersecting streams of traffic is to use a set of traffic lights to regulate the flow in each stream.  The alternative is to use a traffic circle so that the streams initially merge before separating again.  Which is best?  There are two primary issues (safety and traffic flux), and of course there will be associated costs.  As with all modelling problems one needs to look at simple situations to identify the important issues.  A continuum traffic flow model has been developed that describes the flow of a single stream of cars so it is a matter of adapting this model.  This model will be described at the workshop.  This work may prove to be useful for the MISG air blast problem.

Presentation of problem solution:
Traffic lights and traffic circles.pdf

Problem 2: Predicting survival on the Titanic

Presenter: Charles Fodya, School of Computational and Applied Mathematics, University of the Witwatersrand

Problem statement

The sinking of the RMS Titanic is one of the most infamous shipwrecks in history.  On April 15, 1912, during her maiden voyage, the Titanic sank after colliding with an iceberg, killing 1502 out of 2224 passengers and crew.  This sensational tragedy shocked the international community and led to better safety regulations for ships. One of the reasons that the shipwreck led to such loss of life was that there were not enough lifeboats for the passengers and crew.  Although there was some element of luck involved in surviving the sinking, some groups of people were more likely to survive than others, such as women, children, and the upper-class. In this contest, we ask you to complete the analysis of what sorts of people were likely to survive.  In particular, we ask you to apply the tools of mathematical modelling and machine learning, which are discussed in the workshop, to predict which passengers survived the tragedy. The model that predicts most out of the survivors is deemed the best.

Presentation of problem solution:
Titanic.pdf 

Problem 3:  Application of Euler-Bernoulli beam theory to fracturing in rock layers and surface buckling

Presenter:  Ashleigh Hutchinson, School of Computational and Applied Mathematics, University of the Witwatersrand

Problem statement

Euler-Bernoulli beam theory has many applications in mining because rock masses generally consist of layers of rock. If a beam cracks at the supporting pillars there will be a change in the boundary conditions at the pillars which will determine the progression of the fractures. The development of fractures in a layer of rock supported by pillars at each end will be investigated. Rock bursts are sudden failures of rock near the surface of an excavation. A model of a rock burst as surface buckling of an Euler-Bernoulli beam will be considered. The growth of a fracture just below the surface can also give rise to a rock burst. This will be modelled by using an Euler-Bernoulli beam to describe the layer of rock between the fracture and the surface of the excavation.  

Presentation of problem solution:
Application of the EulerBernoulli Beam Equation.pdf 

Problem 4:  Warehouse stock layout

Presenter:  Dr Alan Watson

Problem Statement

Given

  • A warehouse comprising bins (shelf locations) on a floor grid, and stacked N levels high.
  • A list of products (a few 1000s) with movement profiles, e.g. average and modal units demand per day, plus variance or 90%-iles; and average, modal and 90%-ile “pick” sizes (the number of units in an order).

Constraints

  • If stock is available anywhere in the warehouse, it must be picked before the next delivery for an order placed in the delivery period.  There are 3 shifts per day, 6 a.m., 11 a.m. and 3 p.m..
  • Each bin holds only 1 product (even if there is very little of it in stock).
  • There is limited floor space: only one stock picker can access a bin at a time, or the floor space in front of a bin.
  • There is a central collation point to which all picked stock must be delivered for packing and invoicing.
  • All of the stock for a single pick must be delivered at once (the pickers have trolleys for large picks),

The picker may pick multiple picks together, but (per above point) each pick must be 100% complete on delivery to the collation and invoicing station.

Variant 1

Question: Where do we locate product in the warehouse?

Problem

  • Do we scatter high-demand product across the warehouse (to avoid congestion), or do we locate it in a concentrated area near to the collation station?  How near?
  • What does “high demand” mean?  By “high demand” we mean number of orders (“picks”), not number of units.  We start by assuming that a large pick and a small pick take the same time.  Then we need to extend the model to incorporate pick size, because large picks are definitely much slower than small ones, in practice.
  • How do we model the congestion problem?

Presentation of problem solution:
Warehouse.pdf

 

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