Applied Mathematics with Open-Source Software

Operational Research Problems with Python and R

Vincent Knight, Geraint Palmer

• Publisher: Chapman and Hall/CRC
• Year: 2022
• eBook ISBN: 9781000582109
• Edition: 1
• Topic: Business
• Subtopic: Operations
• Index: Business

Applied Mathematics with Open-source Software: Operational Research Problems with Python and R is aimed at a broad segment of readers who wish to learn how to use open-source software to solve problems in applied mathematics. The book has an innovative structure with 4 sections of two chapters covering a large range of applied mathematical techniques: probabilistic modelling, dynamical systems, emergent behaviour and optimisation. The pairs of chapters in each section demonstrate different families of solution approaches. Each chapter starts with a problem, gives an overview of the relevant theory, shows a solution approach in R and in Python, and finally gives wider context by including a number of published references. This structure will allow for maximum accessibility, with minimal prerequisites in mathematics or programming as well as giving the right opportunities for a reader wanting to delve deeper into a particular topic.

Features

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• An excellent resource for scholars of applied mathematics and operational research, and indeed any academics who want to learn how to use open-source software.

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• Offers more general and accessible treatment of the subject than other texts, both in terms of programming language but also in terms of the subjects considered.

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• The R and Python sections purposefully mirror each other so that a reader can read only the section that interests them.

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• An accompanying open-source repository with source files and further examples is posted online at https://bit.ly/3kpoKSd.

V Optimisation

CHAPTER 8 Linear Programming

DOI: 10.1201/9780429328534-8

Finding the best configuration of some system can be challenging, especially when there is a seemingly endless amount of possible solutions. Optimisation techniques are a way to mathematically derive solutions that maximise or minimise some objective function, subject to a number of feasibility constraints. When all components of the problem can be written in a linear way, then linear programming is one technique that can be used to find the solution.

8.1Problem

A university runs 14 modules over three subjects: Art, Biology and Chemistry. Each subject runs core modules and optional modules. Table 8.1 gives the module numbers for each of these.

Table 8.1. List of modules on offer at the university.

Art Core	Biology Core	Chemistry Core
M00	M05	M09
M01	M06	M10
Art Optional	Biology Optional	Chemistry Optional
M02	M07	M11
M03	M08	M12
M04		M13

The university is required to schedule examinations for each of these modules. The university would like the exams to be scheduled using the least amount of time slots. However not all modules can be scheduled at the same time as they share some students:

• All art modules share students;
• all biology modules share students;
• all chemistry modules share students;
• biology students have the option of taking optional modules from chemistry, so all biology modules may share students with optional chemistry modules;
• chemistry students have the option of taking optional modules from biology, so all chemistry modules may share students with optional biology modules;
• biology students have the option of taking core art modules, and so all biology modules may share students with core art modules.

How can every exam be scheduled with no clashes, using the least amount of time slots?

8.2Theory

Linear programming is a method that solves a type of optimisation problem of a number of variables by making use of some concepts of higher dimensional geometry [10]. Optimisation here re...

Table of contents

• Cover Page
• Half-Title Page
• Series Page
• Title Page
• Copyright Page
• Contents
• Authors
• Section I Getting Started
• Section II Probabilistic Modelling
• Section III Dynamical Systems
• Section IV Emergent Behaviour
• Section V Optimisation
• Bibliography
• Index