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Optimization

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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 minimisation problems. Knowledge from Level 2 study of calculus and matrices is assumed.

Modules at Level 3 assume that you are suitably prepared for study at this level. If you want to take a single module to satisfy your career development needs or pursue particular interests, you don’t need to start at Level 1 but you do need to have adequately prepared yourself for OU study in some other way. Check with our Student Registration & Enquiry Service to make sure that you are sufficiently prepared.

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No current presentation - see Future availability

This course is expected to start for the last time in October 2017.

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. About a quarter of your study time will be devoted to practical work. Computer programming is not part of the module.

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 revised 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.

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.

Entry

This is a Level 3 module. Level 3 modules build on study skills and subject knowledge acquired from studies at Levels 1 and 2. They are intended only for students who have recent experience of higher education in a related subject, preferably with The Open University. You are expected to bring to the module some knowledge of:

  • Calculus Definition of differentiation and integration; ability to differentiate and integrate a variety of functions; Taylor’s theorem with remainder; partial derivatives; understanding of continuity and convergence
  • Matrices Ability to manipulate equations with matrices and vectors; Gaussian elimination; eigenvalues and eigenvectors; linear dependence and independence.

You could get the necessary background from our Level 2 mathematics modules Pure mathematics (M208), or Mathematical methods, models and modelling (MST210) (or its predecessor MST209), or equivalent. Students are more likely to successfully complete this module if they have acquired their prerequisite knowledge through passing at least one of these recommended OU modules.

If you have any doubt about the suitability of the module, please contact our Student Registration & Enquiry Service.

Preparatory work

If you would like to do some preparatory reading, you could choose from:

  • E. W. Cheney, D. R. Kincaid (2008) Numerical Mathematics and Computing, Brooks Cole, ISBN 10: 0-495-11475-8
  • R. L. Burden, J. D. Faires (2011) Numerical Analysis, Brooks Cole, ISBN 10: 0-538-73563-5

For an introduction to linear algebra:

  • H. Anton, C. Rorres (2010) Elementary Linear Algebra: With Supplemental Applications, John Wiley & Sons, ISBN 978-0-470-56157-7

The following material from Pure mathematics (M208) would be very useful:

  • Linear Algebra Block: Unit 2 Linear Equations and Matrices; Unit 3 Vector Spaces; Unit 5 Eigenvectors.
  • Analysis Block A: Unit 2 Sequences; Unit 4 Continuity.
  • Analysis Block B: Unit 1 Limits, Unit 2 Differentiation.

Your regional or national centre will be able to tell you where you can see reference copies, or you can buy selected materials from Open University Worldwide Ltd.

Regulations

As a student of The Open University, you should be aware of the content of the Module Regulations and the Student Regulations which are available on our Essential documents website.

If you have a disability

You will need to spend considerable amounts of time using a personal computer.

If you have particular study requirements please tell us as soon as possible, as some of our support services may take several weeks to arrange. Visit our Services for disabled students website for more information, including:

  • help to determine your study requirements and how to request the support that you need  
  • Disabled Students' Allowances (DSAs)
  • using a computer for OU study
  • equipment and other support services that we offer
  • examination arrangements
  • how to contact us for advice and support both before you register and while you are studying.

Study materials

What's included

Main study texts, module guide, Mathcad 14 and multimedia packages supplied on CD-ROM, website.

You will need

Scientific calculator.

We recommend you access the internet at least once a week during the module to download module resources and assignments, and to keep up to date with module news.

Computing requirements

You will need a computer with internet access to study this course. It includes online activities – you can access using a web browser – and some course software provided on disk.

  • If you have purchased a new desktop or laptop computer running Windows since 2007 you should have no problems completing the computer-based activities.
  • A netbook, tablet or other mobile computing device is not suitable for this course – check our Technical requirements section.
  • If you have an Apple Mac or Linux computer – please note that you can only use it for this course by running Windows on it using Boot Camp or a similar dual-boot system.

You can also visit the Technical requirements section for further computing information including the details of the support we provide.

Teaching and assessment

Support from your tutor

You will have a tutor who will help you with the study material and mark and comment on your written work, and whom you can ask for advice and guidance. We may also be able to offer group tutorials or day schools that you are encouraged, but not obliged, to attend. Where your tutorials are held will depend on the distribution of students taking the module. 

Contact our Student Registration & Enquiry Service if you want to know more about study with The Open University before you register.

Assessment

The assessment details for this module can be found in the facts box above.

Please note that TMAs for all undergraduate mathematics and statistics modules must be submitted on paper as – due to technical reasons – we are unable to accept TMAs via our eTMA system.

Future availability

The details given here are for the module that starts in October 2015. We then expect it to be available once a year, in October.

Students also studied

Students who studied this course also studied at some time:

How to register

We regret that we are currently unable to accept registrations for this course. Where the course is to be presented again in the future, relevant registration information will be displayed on this page as soon as it becomes available.

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Distance learning

The Open University is the world’s leading provider of flexible, high quality distance learning. Unlike other universities we are not campus based. You will study in a flexible way that works for you whether you’re at home, at work or on the move. As an OU student you’ll be supported throughout your studies – your tutor or study adviser will guide and advise you, offer detailed feedback on your assignments, and help with any study issues. Tuition might be in face-to-face groups, via online tutorials, or by phone.

For more information read Distance learning explained.

Are you already an OU student ? Go to StudentHome for information on choosing your next module.
Course facts
About this course:
Course code M373
Credits 30
OU Level 3
SCQF level 10
FHEQ level 6
Course work includes:
4 Tutor-marked assignments (TMAs)
Examination
No residential school

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