# OPERATIONS RESEARCH AND OPTIMIZATION - 2023/4

Module code: MAT2009

## Module Overview

This module introduces a variety of commonly used techniques from Operations Research.  The module leads to a deeper understanding of linear programming problems and the theory that underpins their solving.  Tools such as the Simplex Method are presented and an introduction to nonlinear optimisation methods is also provided.  This module supports and complements other modules where optimisation and constrained optimisation is considered.

### Module provider

Mathematics

ROBERTS James (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 73

Lecture Hours: 33

Seminar Hours: 11

Captured Content: 33

Semester 2

## Prerequisites / Co-requisites

Vector Calculus (MAT1005) or Linear Algebra and Vector Calculus (MAT1037) Linear Algebra (MAT1034) or Linear Algebra and Vector Calculus (MAT1037)

## Module content

• Indicative content includes:

• Problem formulation for linear programming problems;

• Simplex Method and sensitivity analysis;

• Duality and complementary slackness;

• Theory and applications of the Transportation Algorithm;

• Convex sets, convex functions, concave functions;

• Nonlinear optimization and conditions for local/global optima;

• Lagrange multipliers and Lagrange Multiplier Theory.

• Lagrangian duality.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 minutes) 20
Examination Exam (2 hours) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Ability to formulate linear programming problems and to use their decision-making skills to identify the most appropriate method of solution.

• Subject knowledge through explicit and implicit recall of key definitions and theorems as well as interpreting this theory.

• Understanding and application of subject knowledge to solve constrained optimisation problems.

Thus, the summative assessment for this module consists of:

• One final examination worth 80% of the module mark.

• One in-semester test; worth 20% of module mark.

Formative assessment and feedback

Students receive individual written feedback via a number of marked formative coursework assignments over an 11-week period.  The lecturer also provides verbal group feedback during lectures.  (Occasionally group feedback may be provided online when applicable.)

## Module aims

• Introduce students to linear programming, the Simplex Method and the Transportation Algorithm.
• Enable students to solve linear programming problems as primal problems or by using duality.
• Illustrate introductory theory for nonlinear programming problems that have equality constraints by demonstrating the application of Lagrange Multiplier Theory and to enable students to solve similar optimisation problems. Introduce students to Lagrangian duality.

## Learning outcomes

 Attributes Developed 001 Formulate simple Operations Research and Optimisation problems mathematically as well as quote and apply definitions and theorems relating to the Simplex Method to solve such linear programming problems. KCT 002 Identify when the Simplex Method is no longer suitable and to suggest and use more appropriate algorithms for solving optimisation problems. KCP 003 Demonstrate an understanding of convexity and concavity. KCT 004 Identify a nonlinear programming problem, understand the limits of Lagrange Multiplier Theory and solve nonlinear programming problems with equality constraints by analysing conditions to determine the optimal solution . KC 005 Construct and solve Lagrangian dual problems. KCP

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

• Give a detailed introduction to formulating linear and nonlinear programming problems as well as discussing an array of optimisation methods (and underpinning theory) used for solving such problems.

• Ensure experience is gained (through demonstration) of the methods typically used to solve constrained optimisation problems so that students can later apply their own decision-making to solve any viable programming problem that they encounter.

The learning and teaching methods include:

• lectures and tutorials.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MAT2009

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics MMath 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Music BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Economics and Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Financial Mathematics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2023/4 academic year.