# OPERATIONS RESEARCH AND OPTIMIZATION - 2024/5

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 are considered.

### Module provider

Mathematics & Physics

ROBERTS James (Maths & Phys)

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

Independent Learning Hours: 58

Lecture Hours: 33

Seminar Hours: 11

Guided Learning: 15

Captured Content: 33

Semester 1

None.

## 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 optimality;

• Lagrange multipliers and Lagrange Multiplier Theory.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-Semester Test (50 minutes) 20
Examination End-of-Semester Examination (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 in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1 and 2.

• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to Learning Outcomes 1 to 4.

Formative assessment

There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

Feedback

Students receive individual written feedback on the formative unassessed coursework and the in-semester test. The feedback is timed so that feedback from the first unassessed coursework assists students with preparation for the in-semester test. The feedback from both unassessed courseworks and the in-semester test assists students with preparation for the end-of-semester examination. This written feedback is complemented by verbal and written feedback given in tutorials. Students also receive verbal and written feedback in office hours.

## 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.

## Learning outcomes

 Attributes Developed 001 Students will be able to 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 For problems unsuitable for the Simplex Method, students will be able to identify and use more appropriate algorithms for solving optimisation problems. KCP 003 Students will demonstrate an understanding of convexity and concavity. KCT 004 Students will be able to 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

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:

• Three one-hour lectures per week for eleven weeks, with typeset notes to complement the lectures. The lectures provide a structured learning environment with opportunities for students to ask questions and to practice methods taught.

• There are two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these as guidance on their progress and understanding.

• The lecturer may demonstrate selected algorithms using software. Students will receive the necessary commands to independently execute these algorithms. This approach empowers students to practice and gain a more profound comprehension of the methods presented.

• Lectures may be recorded. Lecture recordings are intended to give students the opportunity to review parts of the session that they might not have understood fully and should not be seen as an alternative to attendance at lectures.

• Weekly one-hour tutorials, to discuss examples and feedback.

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

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourcefulness and

Resilience and Sustainability. This module is designed to allow students to develop knowledge,

skills, and capabilities in the following areas:

Digital Capabilities: Students have the opportunity to execute algorithms encountered during the module via software, thus enhancing their digital experience.

Employability: Students are equipped with the ability to optimize complex processes, allocate resources efficiently, and find innovative solutions to organizational challenges. These skills are highly sought after by employers in various industries, including logistics, finance, healthcare, and technology.

Global and Cultural Capabilities: Students enrolled in MAT2009 originate from various countries and possess a wide range of cultural backgrounds. During tutorials, student engagement in discussions naturally cultivates the sharing of different cultures.

Resourcefulness and Resilience: Students taking MAT2009 gain skills in problem solving techniques, becoming adept at optimizing processes and maximizing the efficient use of resources in various contexts. This enhances their ability to make informed decisions and promotes resourcefulness and resilience in adapting to changing circumstances.

Sustainability: Students learn how operations research and optimization techniques benefit sustainability. For example, in supply chain management, operations research optimises transportation routes and production schedules, minimizing energy consumption and emissions. This reduces operational costs and also lowers the environmental footprint. Furthermore, operations research aids in sustainable decision-making by quantifying trade-offs between economic, environmental, and social factors.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics MMath 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Music BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Financial Mathematics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 1 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 2024/5 academic year.