This module will interest you if you need to create mathematical models or if you use numerical software in industry, science, commerce or research. It’s concerned with the skills needed to represent real optimization problems as mathematical models, and with techniques used in numerical analysis and operational research for solving these models by computer. Explaining how and when modelling and numerical techniques can be applied, the module covers 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. Knowledge from OU level 2 study of calculus and matrices is assumed.
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.
Read the full content list
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.
This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.
There is no formal pre-requisite study, but you must have the required mathematical skills.
You can check you’re ready for M373 and see the topics it covers here.
Talk to an advisor if you’re not sure you’re ready.
You should aim to be confident and fluent with the concepts covered in the Are you ready? quiz here, and follow the advice in the quiz.
The key topics to revise include:
- Taylor’s theorem
- partial derivatives
- continuity and convergence
- matrices and vectors
- Gaussian elimination
- eigenvalues and eigenvectors
- linear dependence and independence.
Mathematical methods, models and modelling (MST210) or Mathematical methods (MST224) is ideal preparation.
You'll have access to a module website, which includes:
- a week-by-week study planner
- course-specific module materials
- relevant computer software and associated guidance
- audio and video content
- assessment details, instructions and guidance
- online tutorial access
- access to student and tutor group forums.
You'll be provided with printed books covering the content of the module, including explanations, examples and activities to aid your understanding of the concepts and associated skills and techniques. You'll also receive a printed module handbook.
You will need
Scientific calculator, but not one that is designed or adapted to offer any of the following facilities: Algebraic manipulation, differentiation or integration, language translation or can communicate with other devices or the internet. It also cannot have retrievable information stored in it such as databanks, dictionaries, mathematical formulae or text.
You'll need a desktop or laptop computer with an up-to-date version of 64-bit Windows 10 (note that Windows 7 is no longer supported) or macOS and broadband internet access.
To join in spoken conversations in tutorials we recommend a wired headset (headphones/earphones with a built-in microphone).
Our module websites comply with web standards and any modern browser is suitable for most activities.
Our OU Study mobile App will operate on all current, supported, versions of Android and iOS. It's not available on Kindle.
It's also possible to access some module materials on a mobile phone, tablet device or Chromebook, however, as you may be asked to install additional software or use certain applications, you'll also require a desktop or laptop as described above.