School of Mathematics and Physics

SOR Level 3 Modules

  • SOR3001 Linear and Dynamic Programming (1st semester)

Pre-requisite:

Introduction
Operational Research is the application of quantitative analysis to problems outside of the physical sciences and in particular to the problems of business, industry and administration. However, all the techniques can be used outside of Operational Research and emphasis is given to formulation of problems and recognition of the appropriate technique. Blackboard examples are given for each technique and the homework problems are drawn from a wide field. Most techniques are in the form of algorithms and for hand calculations the problems have to be fairly simple, but indications are given how computers could be applied to bigger problems and computer outputs are examined.

The aim of this course is to develop competence in two of the most important mathematical techniques used in Operational Research, in formulating problems and expressing answers clearly.

This course covers the two main general-purpose mathematical techniques used in Operational Research together with some specialized applications. These are all optimisation techniques, although they also give insight into the problems and the interpretation of answers is stressed. Calculus cannot be used because either (i) the objective function and the constraints cannot be expressed simply, or (ii) the constraints dominate the problem and the optimal solution will be at the edge of the feasible region, or (iii) the variables are not continuous (e.g., they are integer).

Some students may have met some of the techniques in courses at other levels, but no previous knowledge is assumed and more mathematical rigour and understanding is required than at those levels. However, the mathematical knowledge assumed does not extend beyond that in Level 1. The main knowledge assumed is elementary linear algebra (bases, linear dependence of vectors, matrix notation, partitioning of matrices). No knowledge of economics is required but the economic interpretation of some results is explored. Apart from a passing reference to stochastic Dynamic Programming this course is purely deterministic and no knowledge of statistics is required.

Contents
The scope of Operational Research, formulating a problem from a verbal description.
Dynamic programming: formulation, principle of optimality, value iteration, applications including equipment replacement, allocation, production planning and optimal routes. Special algorithms for production planning and optimal routes.
Linear programming: formulation, theory, Primal Simplex Method, interpretation of the final tableau, Revised Simplex Method, duality theory including economic interpretation, Dual Simplex Method, Post-optimal Analysis, Transportation and Assignment problems. Network flows: maximal flow problem, minimal cost flows, the out-of-kilter algorithm. A wide variety of practical problems and applications is discussed.

Assessment

Exam 70%   Group Project 20%   Presentation 10%

  • SOR3008 Statistical Data Mining (2nd semester)

Pre-requisite: SOR2004

Introduction
In the 1990's there was an explosive growth in both the generation and collection of data due mainly to the advancement of computing technology in processing and storage of data and the ease of scientific data collection. As a result, overwhelming mountains of data are being generated and stored. For example, in the business world large supermarket chains such as Wal-Mart and Sainsbury's collect data amounting to millions of transactions per day. In the US all health-care transactions are stored on computers yielding terabyte databases which are constantly being analysed by insurance companies. There are huge scientific databases as well. Examples include the human genome database project and NASA's Earth Observatory System. This has brought about a need for vital techniques for the modelling and analysis of these large quantities of data: data mining.

Data Mining is the process of selection, exploration, and modelling of large quantities of data to discover previously unknown regularities or relations with the aim of obtaining clear and useful results for the owner of the database. The application of data mining includes many different areas, such as market research (customer preferences), medicine, epidemiology, risk analysis, fraud detection and more recently within bioinformatics for modelling DNA.

This module will focus on data mining techniques which have evolved from and are strongly based on statistical theory.

Contents
Introduction to Data Mining.
Exploratory data analysis: Principal Component Analysis; handling big data - reformatting data; Multiple Imputation.
Cluster analysis: Hierarchical clustering; Partitioning algorithms.
Classification: Nearest neighbour algorithms; Classification trees; Naïve Bayes Classifier; Bayesian networks; Ordinal Regression; Multinomial Logit; Conditional Logit; Nested Logit;Techniques for comparing classifiers - including bagging and boosting in ensemble methods.
Prediction (continuous targets): Regression Trees; Random Forests; Neural Networks; Support Vector Machines; Regression splines.
Association Rule mining.

Assessment

Exam 60%    2x Computer Practical 40%

  • SOR3012 Stochastic Processes and Risk (1st semester)

Pre-requisite: SOR1001

Introduction
Uncertainty or risk is a natural part of life. Continually faced with choices, we are required to make decisions based on the possible consequences or outcomes. We often use the phrase ‘a calculated risk’ to describe how we analyse a problem and come up with our decision. Now the question is: how do we calculate risk - mathematically.

Statistical analysis of a random process helps determine whether there are any underlying patterns. If there are, we can use this to our advantage in predicting the future - or at least make an educated guess! Not surprisingly, this topic is primarily recognised for its career opportunities where ‘risk’ assessment and planning matter: for example the provision of power stations, telecoms, transport planning, the spread of infectious diseases (swine flu, for example). A major application of this kind of mathematics is the financial services sector: insurance and investment, credit risk, capital market trading etc. Many students that have taken this course are currently employed as highly-paid actuaries.

Aside from the educational and commercial value of this subject, it leads into very interesting and fun topics such as measure theory, martingales and potential theory, noise, fractals, etc.

Contents
Probability theory, random variables and simulating random events
Dealing with multiple random variables.  The conditional expectation theorem.
Discrete time Markov chains including gamblers ruin.  Simulating discrete time Markov chains
Continuous time Markov chains including the poisson process and the M/M/1 Queue.  Simulating continuous time Markov chains.
Simulating non-Markovian processes.

Assessment

Exam 60%   Logbook 20%   2x Report 20%