What you will study
The module is divided into three blocks of work: solutions of non-linear equations, systems of linear and non-linear equations and mathematical modelling; linear and integer programming; and non-linear optimization for unconstrained and constrained minimization problems. You will be expected to run given computer programs as part of your study, but you will not be required to write any computer programs.
In the broad area of operational research, the module will enable you to formulate a real problem in mathematical terms; to recognise whether the problem can be solved numerically; to choose a suitable method; to understand the conditions required for the method to work; to evaluate the results and to estimate their accuracy and their sensitivity to changes in the data.
Optimization is a practical subject, although it is supported by a growing body of mathematical theory. Problems that require the creation of mathematical models and their numerical solutions arise in science, technology, business and economics as well as in many other fields. Creating and solving a mathematical model usually involves the following main stages:
- formulation of the problem in mathematical terms: this is the creation of a mathematical model
- devising a method of obtaining a numerical solution from the mathematical model
- making observations of the numerical quantities relevant to the solution of the problem
- calculating the solution, usually with a computer or at least with a scientific calculator
- interpreting the solution in relation to the real problem
- evaluating the success or failure of the mathematical model.
Many of the problems discussed in the module arise in operational research and optimization: for example, how to get the most revenue from mining china clay when there is a choice of several mines. In this example the mathematical model consists of a set of linear inequalities defining the output from each mine, the number of mines that can be worked, the correct blend of clay and the total amount of clay mined each year. The method of solving the problem uses mixed linear and integer programming; the numerical data that need to be observed include the financial implications of opening a mine, the number of mines that can be worked with the labour force, and the quality of clay from potential mines. These data will be fed into a computer, which will combine them with the chosen method of solving the equations to produce solutions consisting of outputs from each mine in each year of operation.
This module examines all the stages but concentrates on: the first stage, creating the mathematical model; the second stage, devising a method; the fourth stage, calculating numerical solutions; and the fifth stage, interpreting the solution. Each of the three blocks of work takes about ten weeks of study:
- Block I – Direct and iterative methods of solving single non-linear equations, systems of linear equations and systems of non-linear equations; mathematical modelling; errors in numerical processes, convergence, ill-conditioning and induced instability.
- Block II – Formulation and numerical solution of linear programming problems using the two-phase simplex method; formulation of integer programming problems and the branch and bound method of solution; sensitivity analysis.
- Block III – Formulation and numerical solution of unconstrained and constrained non-linear optimization problems using, among others, the DFP and BFGS methods with line searches; illustrative applications.
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You will learn
Successful study of this module should enhance your skills in:
- mathematical modelling
- operational research
- linear programming and non-linear optimization methods
- the use of iterative methods in problem solving
- the use of Computer Algebra Packages for problem solving.
Mastering the material in this module will enable you to mathematically model and solve real-world problems in operational research and optimization. Such problems frequently occur in many fields including science, technology, business and economics.
This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.