Applied mathematics problems with Open Source Software: Operational Research with Python and R.
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 in to a particular topic.
Features
This chapter aims to help a reader understand if this book is for them and what background they need to use it. It starts by answering the following questions:
The chapter structure of the book is then discussed, apart from the 1st chapter all chapters have a similar structure and then some discussion is given to how the code is written and presented.
This chapter can be skipped if a reader is confident in their use of programming and wants to skip ahead to a specific problem.
This chapter models a barber shop as a queueing system using a continuous time Markov chain. The goal is to identify which particular reconfiguration of the barber shop would be most beneficial to customers.
A theoretic overview of concepts related to stationary distributions of continuous Markov chains is given before solutions approaches are given in both R and Python. This is done using Numpy in Python and expm in R for all the underlying linear algebra.
This chapter models a bicycle repair shop as a network of two queueing processes. The goal is to identify the effect of two particular improvements to the service configuration.
A theoretic overview of two types of discrete event simulation frameworks is given: event scheduling and process based simulation. As well as this the topics of random processes and pseudorandom number generation are also covered.
The two types of simulation frameworks are addressed in the solution approaches: in Python Ciw is used which is an implementation of event scheduling. In R simmer is used which is an implementation of process based simulation. As a result, perhaps this chapter more than any other is worthwhile reading both the R and Python section (even if the reader is only a user of one language) as this gives a good contrast to the two frameworks.
This chapter models the spread of a cold in a population as a single dimensional problem. The goal is to identify whether or not an investment in cold medicine should be made to offset a loss of productivity.
A theoretic overview of what a differential equation is and what a solution of a differential equation represents is given.
The solution offered in Python is done symbolically using the Sympy library whereas the solution offered in R is done through numerical integration using deSolve.
This chapter models the spread of an infectious disease within a population as a compartmental model. The goal is to understand what interventions could be made to help minimise the long term effects of the disease.
A theoretic overview of systems of ordinary differential equations is presented in terms of stocks and flows. The specific model used here: an SIR model is discussed in detail. This is all complemented with a brief discussion of some numerical techniques used to integrate systems of ordinary differential equations.
The solution offered in Python makes use of Numpy and Scipy which is equivalent to the solution offered in R which makes use of deSolve.
This chapters models a policy decision that has implications on the actions of two companies in a duopoly modelled as a normal form game. The goal is to understand what financial incentives can be put in place to ensure competition between the companies that benefits the consumer.
A theoretic overview of the definition of normal form games and Nash equilibria is given.
The solution offered in Python makes use of Nashpy which is a specific library for the computation of Nash equilibria. The solution offered in R makes use of Recon which is a general purpose library with functionality for a few topics in economics.
This chapters models a population of association football fans as a classic agent based model of segregation. The goal is to understand how the individual preferences of the population members affects the overall observed behaviour at the population level.
A theoretic overview of agent based modelling is given specifically concentrating on the roles of agents and the environment.
In Python the fully object oriented nature of the language is used to model agents and the environment as classes. The R solution makes use of the R6 library which allows for a similar implementation of classes with required functionality.
This chapter finds an exam schedule for a scenario where there are students that have a number of possible clashes. This is done by formulating and solving a linear programming problem.
A theoretic description of linear programming is given. Intuitive ideas that lead to solutions approaches are described. Direct and linear algebraic formulations are explained.
The Python solution makes use of the Pulp library which corresponds to the direct formulation of the problem. The R solution uses the ROI library which requires the linear algebraic formulation.
This chapter identifies the best delivery route for a delivery company. This is a traditional travelling salesman problem which aims to find a route that minimises the total distance travelled.
A theoretic description of neighbourhood search and a specific consideration of the "2-opt" algorithm are given. These are types of heuristic algorithms which may not guarantee optimality but can often perform better in practice than exact approaches.
The standard Python library and base R are used to build these algorithms.
Geraint Palmer is a Welsh Medium Lecturer at Cardiff University in the School of Mathematics. He is a member of the operational research group where his research interests are in simulation and probabilistic modelling, in particular applying these to model public services such as healthcare systems. He uses open source software in all aspects of his research: he is a maintainer of Ciw, an open source Python library for discrete event simulation, and won the OR Society's Doctoral Award in 2018. Geraint is also a fellow of the Software Sustainability Institute and has presented at a number of international conferences on the subject of best practice of scientific computing, and regularly teaches programming and runs coding workshops for people of all ages.
Vince Knight is a Senior Lecturer at Cardiff University in the School of Mathematics. His research interests are in emergent behaviour, probabilistic modelling, applications in healthcare and pedagogy. He maintains a number of open source research software projects, is a trustee of the UK Python association, is an editor for the Journal of Open Source Software, was awarded the 2017 John Pinner award for contribution to the Python community and is a fellow of the Sustainable Software Institute. He regularly wins awards for his teaching in the School of Mathematics. He does not only speak at conferences around the world but continues to organise conferences to bring the power of open source software to as many people as possible.
Assuming you have anaconda on your machine, run:
$ conda env create -f environment.ymlThere are some further dependencies that are needed.
Activate the environment:
$ conda activate ampwoss
$ inv envTo delete the environment:
$ inv delenvRun:
$ inv compileThis creates a build/ directory which contains slightly modified source files
(some annotations removed) as well as a main.pdf. This modifications are
prescribed by a substitions variables in tasks.py.
An intermediate step is to run:
$ inv buildWhich creates the modified source files but does not compile the document.
Note that as documented in
#72 the
texlive-langgreek latex package might be needed.
Run:
$ inv doctestTo doctest a specific chapter:
$ inv doctest --path="src/chapters/06/main.tex"To also check the style with black on the python code and lintr on
the R code:
$ inv doctest --styleRun:
$ inv analyseThis analyses the tex documents with style and diction:
style: analyses surface characteristics of a document, including sentence
length and other readability measures.diction: identifies wordy and commonly misused phrases.See http://wiki.christophchamp.com/index.php?title=Style_and_Diction#Lix for more information.
To install these packages:
Debian:
apt install diction
apt install style
OS X (with the brew package manager):
brew install diction
brew install style