Coursework template CO343
Author
Simon
Last Updated
5년 전
License
Creative Commons CC BY 4.0
Abstract
Template for LaTeX beginners, designed with students taking the Imperial College London CO343: Operations Research course in mind.
Template for LaTeX beginners, designed with students taking the Imperial College London CO343: Operations Research course in mind.
\documentclass{article}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb}
\usepackage{parskip}
\usepackage{graphicx}
% Margins
\usepackage[top=2.5cm, left=3cm, right=3cm, bottom=4.0cm]{geometry}
% Colour table cells
\usepackage[table]{xcolor}
% Get larger line spacing in table
\newcommand{\tablespace}{\\[1.25mm]}
\newcommand\Tstrut{\rule{0pt}{2.6ex}} % = `top' strut
\newcommand\tstrut{\rule{0pt}{2.0ex}} % = `top' strut
\newcommand\Bstrut{\rule[-0.9ex]{0pt}{0pt}} % = `bottom' strut
%%%%%%%%%%%%%%%%%
% Title %
%%%%%%%%%%%%%%%%%
\title{Coursework template CO343}
\author{Firstname Lastname \\ CID 01234567}
\date{\today}
\begin{document}
\maketitle
%%%%%%%%%%%%%%%%%
% Problem 1 %
%%%%%%%%%%%%%%%%%
\section{Problem 1}
The problem states that we should find $x$ that solves the following equation
\begin{align}
\label{eq:example_equation} % Equation label; can be used for referencing
2 x^2 + 4 x - 6 = 0 \,.
\end{align}
We take the standard algorithm for solving equations of the form $a x^2 + bx + c$ and apply it to Equation~\ref{eq:example_equation}. This gives us
\begin{align}
x
&= \frac{2}{2 \cdot 2} \pm \sqrt{\left( \frac{2}{2 \cdot 2} \right)^2 + \frac{6}{2} } \\
&= 1 \pm 2
\end{align}
So the solutions are $x=3$ and $x = -1$.
% Example of how to add figure (can be used for jpeg, png, pdf, eps etc)
In Figure~\ref{fig:universe}, we can see an example of a galaxy.
\begin{figure}[h!]
\centering
\includegraphics[scale=2]{universe.jpg}
\caption{Example figure}
\label{fig:universe}
\end{figure}
%%%%%%%%%%%%%%%%%
% Problem 2 %
%%%%%%%%%%%%%%%%%
\pagebreak
\section{Problem 2}
Example of Simplex tableau:
\begin{align}
\begin{array}{c | cccccc | c}
BV & z & x_1 & x_2 & x_3 & x_4 & x_5 & RHS \\
\hline % horizontal line
z & 1 & 0 & 0 & -\tfrac{2}{5} & -\tfrac{1}{5} & 0 & -8 \\
x_2 & 0 & 0 & 1 & -\tfrac{1}{5} & \tfrac{2}{5} & 0 & 5 \\
x_5 & 0 & 0 & 0 & -\tfrac{3}{5} & \tfrac{1}{5} & 1 & 1 \\
x_1 & 0 & 1 & 0 & \tfrac{3}{5} & -\tfrac{1}{5} & 0 & 3
\end{array}
\end{align}
We can define the \LaTeX{} commands \texttt{Tstrut} and \texttt{Bstrut} to get more spacing between rows in the tableau and make it look nicer:
\begin{align}
\begin{array}{c | cccccc | c}
BV & z & x_1 & x_2 & x_3 & x_4 & x_5 & RHS \Tstrut\Bstrut \\
\hline
z & 1 & 0 & 0 & -\tfrac{2}{5} & -\tfrac{1}{5} & 0 & -8 \Tstrut\Bstrut \\
x_2 & 0 & 0 & 1 & -\tfrac{1}{5} & \tfrac{2}{5} & 0 & 5 \Tstrut\Bstrut \\
x_5 & 0 & 0 & 0 & -\tfrac{3}{5} & \tfrac{1}{5} & 1 & 1 \Tstrut\Bstrut \\
x_1 & 0 & 1 & 0 & \tfrac{3}{5} & -\tfrac{1}{5} & 0 & 3 \Tstrut\Bstrut
\end{array}
\end{align}
We can colour text and highlight cells in tableau, or just leave them empty:
\begin{align}
\begin{array}{c | cccccc | c}
BV & z & x_1 & x_2 & x_3 & x_4 & x_5 & RHS \Tstrut\Bstrut \\
\hline
z & 1 & & & -\tfrac{2}{5} & -\tfrac{1}{5} & & -8 \Tstrut\Bstrut \\
x_2 & & & \cellcolor{gray!50}1 & -\tfrac{1}{5} & \tfrac{2}{5} & & 5 \Tstrut\Bstrut \\
x_5 & & & & -\tfrac{3}{5} & \tfrac{1}{5} & 1 & 1 \Tstrut\Bstrut \\
x_1 & & \textcolor{red}{1} & & \tfrac{3}{5} & -\tfrac{1}{5} & & 3 \Tstrut\Bstrut
\end{array}
\end{align}
Here is how you make vectors and matrices:
\begin{align}
\mathbf x = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}^\top \\
\mathbf A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}^{-1}
\end{align}
Here is a formulation of a linear program:
\begin{align*}
\min_{x} \quad & c^\top x \\
\mathrm{s.t.} \quad
& A x \leq b \\
&-1 \leq x_n \leq 1 \,, \quad n = 1, \dots, N
\end{align*}
There is an ocean of Latex questions and answers online. If you have a question, most likely someone else will have asked the same question before.
\end{document}