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 nonlinear equations; systems of linear and nonlinear equations and mathematical modelling; linear and integer programming; and nonlinear optimization for unconstrained and constrained minimisation problems. Knowledge from OU level 2 study of calculus and matrices is assumed.
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 see Future availability

This module is expected to start for the last time in October 2021. 
What you will study
The module is divided into three blocks of work: solutions of nonlinear equations, systems of linear and nonlinear equations and mathematical modelling; linear and integer programming; and nonlinear 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 nonlinear equations, systems of linear equations and systems of nonlinear equations; mathematical modelling; errors in numerical processes, convergence, illconditioning 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 nonlinear 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 nonlinear optimization methods
 the use of iterative methods in problem solving
 the use of Computer Algebra Packages for problem solving.
Entry
This is an OU level 3 module. OU 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; ability to differentiate 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 one of our level 2 mathematics modules Pure mathematics (M208), Mathematical methods, models and modelling (MST210), Mathematical methods (MST224) or the discontinued module Mathematical methods and models (MST209), or equivalent. You are more likely to successfully complete this module if you have acquired your prerequisite knowledge through passing at least one of these recommended modules.
You can try our selfassessment diagnostic quiz to help you determine if you are adequately prepared for this module.
If you have any doubt about the suitability of the module, please speak to an adviser.
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: 0495114758
 R. L. Burden, J. D. Faires (2011) Numerical Analysis, Brooks Cole, ISBN 10: 0538735635
For an introduction to linear algebra:
 H. Anton, C. Rorres (2010) Elementary Linear Algebra: With Supplemental Applications, John Wiley & Sons, ISBN 9780470561577
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.
Qualifications
M373 is an optional module in our:
It can also count towards most of our other degrees at bachelors level, where it is equally appropriate to a BA or BSc. We advise you to refer to the relevant qualification descriptions for information on the circumstances in which this module can count towards these qualifications because from time to time the structure and requirements may change.
Excluded combinations
Sometimes you will not be able to count a module towards a qualification if you have already taken another module with similar content. To check any excluded combinations relating to this module, visit our excluded combination finder or check with an adviser before registering.
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. Find out more about our services for disabled students..
Study materials
What's included
Module texts and website, including access to Maxima mathematical software which you need to download.
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..
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
A computing device with a browser and broadband internet access is required for this module. Any modern browser will be suitable for most computer activities. Functionality may be limited on mobile devices.
Any additional software will be provided, or is generally freely available. However, some activities may have more specific requirements. For this reason, you will need to be able to install and run additional software on a device that meets the requirements below.
A desktop or laptop computer with either:
 Windows 7 or higher
 macOS 10.7 or higher
The screen of the device must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.
To participate in our onlinediscussion area you will need both a microphone and speakers/headphones.
Our Skills for OU study website has further information including computing skills for study, computer security, acquiring a computer and Microsoft software offers for students.
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 us 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.
You can choose whether to submit your tutormarked assignments (TMAs) on paper or online through the eTMA system. You may want to use the eTMA system for some of your assignments but submit on paper for others. This is entirely your choice.
Professional recognition
This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.
Students also studied
Students who studied this module also studied at some time:
Future availability
Optimization starts once a year – in October. This page describes the module that will start in October 2018. We expect it to start for the last time in October 2021.
How to register
We regret that we are currently unable to accept registrations for this module. Where the module is to be presented again in the future, relevant registration information will be displayed on this page as soon as it becomes available.
Regulations
As a student of The Open University, you should be aware of the content of the academic regulations which are available on our
Essential Documents website.