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graph-optimisation.tex
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\documentclass{beamer}
\usetheme{Boadilla}
%\beamertemplatenavigationsymbolsempty
%gets rid of navigation symbols
\setbeamertemplate{navigation symbols}{}
\usepackage[latin1]{inputenc}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage{stmaryrd}
\usepackage{tikz}
\usepackage{verbatim}
\usepackage{xcolor}
\usepackage{algorithm}
\usepackage[noend]{algpseudocode}
\setbeamertemplate{caption}{\raggedright\insertcaption\par}
\newlength{\arrowsize}
\pgfarrowsdeclare{biggertip}{biggertip}{
\setlength{\arrowsize}{0.3pt}
\addtolength{\arrowsize}{.5\pgflinewidth}
\pgfarrowsrightextend{0}
\pgfarrowsleftextend{-5\arrowsize}
}{
\setlength{\arrowsize}{0.3pt}
\addtolength{\arrowsize}{.5\pgflinewidth}
\pgfpathmoveto{\pgfpoint{-5\arrowsize}{4\arrowsize}}
\pgfpathlineto{\pgfpointorigin}
\pgfpathlineto{\pgfpoint{-5\arrowsize}{-4\arrowsize}}
\pgfusepathqstroke
}
\newcommand{\redge}[1]{\overset{{}_{\shortleftarrow}}{#1}}
\newcommand{\resedges}[0]{\overset{{}_{\leftrightarrow}}{E}}
\title{Graph algorithmic approach}
\author{Sebastian Schenker}
\date{}
\begin{document}
\frame{\titlepage}
\begin{frame}[fragile]
\frametitle{Feasibility and optimality}
\begin{figure}
\begin{minipage}{0.45\textwidth}
\caption{Simple instance}
\begin{verbatim}
4 (time slots)
2 (types)
0 1 0 0 (demand)
0 0 1 1 (demand)
3 (costs)
0 4 (costs)
5 0 (costs)
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth}
\begin{itemize}
\item Feasible schedule:
\begin{itemize}
\item demand of each type is fulfilled
\item at each time slot at most one type is produced
\end{itemize}
\item[]
\item Optimal schedule:
\begin{itemize}
\item Lowest cost (inventory + changeover) among all
feasible schedules
\end{itemize}
\end{itemize}
\end{minipage}
\end{figure}
\end{frame}
\begin{frame}[fragile]
\frametitle{Feasible schedules}
\begin{figure}
\begin{minipage}{0.45\textwidth}
\begin{verbatim}
0 1 0 0 (demand type 0)
0 0 1 1 (demand type 1)
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth}
\begin{itemize}
\item 6 feasible schedules:
\item[] a)~~[ -1 0 1 1 ]
\item[] b)~~[ 0 -1 1 1 ]
\item[] c)~~[ 0 1 -1 1 ]
\item[] d)~~[ 1 0 -1 1 ]
\item[] e)~~[ 0 1 1 -1 ]
\item[] f)~~[ 1 0 1 -1 ]
\end{itemize}
\end{minipage}
\end{figure}
\end{frame}
\begin{frame}{Variables}
\begin{itemize}
\item consider $x^i_j \in \{0,1\}$ with
\item[] $x^i_j = 1~\Leftrightarrow$ type $i$ is produced at time
slot $j$
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Graph instance}
\begin{itemize}
\item[] \small{0 {\color{red}1} 0 0~~(demand type 0)}
\item[] \small{0 0 {\color{green}1} {\color{blue}1}~~(demand type 1)}
\end{itemize}
\begin{figure}
\begin{tikzpicture}[transform shape,scale=0.6]
\useasboundingbox(0,0) rectangle (14,8);
\tikzset{V/.style = {shape=circle, draw, text=black}}
\tikzset{E/.style = {draw,text=black,->}}
\node[V,text=red](d0) at (3,7) {$a$};
\node[V,text=green](d1) at (7,7) {$b$};
\node[V,text=blue](d2) at (11,7) {$c$};
\node[V](x00) at (0,5) {$x^0_0$};
\node[V](x01) at (2,5) {$x^0_1$};
\node[V](x02) at (4,5) {$x^0_2$};
\node[V](x03) at (6,5) {$x^0_3$};
\node[V](x10) at (8,5) {$x^1_0$};
\node[V](x11) at (10,5) {$x^1_1$};
\node[V](x12) at (12,5) {$x^1_2$};
\node[V](x13) at (14,5) {$x^1_3$};
\node[V](t0) at (3,3) {$t0$};
\node[V](t1) at (6,3) {$t1$};
\node[V](t2) at (9,3) {$t2$};
\node[V](t3) at (12,3) {$t3$};
\node[V](t) at (7,1) {\small{$sink$}};
%% \draw[E](d0) to (x00);
%% \draw[E](d0) to (x01);
%% \draw[E](d1) to (x10);
%% \draw[E](d1) to (x11);
%% \draw[E](d1) to (x12);
%% \draw[E](d2) to (x10);
%% \draw[E](d2) to (x11);
%% \draw[E](d2) to (x12);
%% \draw[E](d2) to (x13);
%% \draw[E](x00) to (t0);
%% \draw[E](x10) to (t0);
%% \draw[E](x01) to (t1);
%% \draw[E](x11) to (t1);
%% \draw[E](x02) to (t2);
%% \draw[E](x12) to (t2);
%% \draw[E](x03) to (t3);
%% \draw[E](x13) to (t3);
%% \draw[E](t0) to (t);
%% \draw[E](t1) to (t);
%% \draw[E](t2) to (t);
%% \draw[E](t3) to (t);
%% \draw[red,->,dashed](3,8) to (d0);
%% \draw[green,->,dashed](7,8) to (d1);
%% \draw[blue,->,dashed](11,8) to (d2);
%% \draw[red,->,dashed](t) to (6.5,0);
%% \draw[green,->,dashed](t) to (7,0);
%% \draw[blue,->,dashed](t) to (7.5,0);
\end{tikzpicture}
\end{figure}
\begin{itemize}
\item Number of vertices: $|\text{items}| + (|\text{time\_slots}| + 1) \cdot |\text{types}| + 1$
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Graph instance}
\begin{itemize}
\item[] \small{0 {\color{red}1} 0 0~~(demand type 0)}
\item[] \small{0 0 {\color{green}1} {\color{blue}1}~~(demand type 1)}
\end{itemize}
\begin{figure}
\begin{tikzpicture}[transform shape,scale=0.6]
\useasboundingbox(0,0) rectangle (14,8);
\tikzset{V/.style = {shape=circle, draw, text=black}}
\tikzset{E/.style = {draw,text=black,->}}
\node[V,text=red](d0) at (3,7) {$a$};
\node[V,text=green](d1) at (7,7) {$b$};
\node[V,text=blue](d2) at (11,7) {$c$};
\node[V](x00) at (0,5) {$x^0_0$};
\node[V](x01) at (2,5) {$x^0_1$};
\node[V](x02) at (4,5) {$x^0_2$};
\node[V](x03) at (6,5) {$x^0_3$};
\node[V](x10) at (8,5) {$x^1_0$};
\node[V](x11) at (10,5) {$x^1_1$};
\node[V](x12) at (12,5) {$x^1_2$};
\node[V](x13) at (14,5) {$x^1_3$};
\node[V](t0) at (3,3) {$t0$};
\node[V](t1) at (6,3) {$t1$};
\node[V](t2) at (9,3) {$t2$};
\node[V](t3) at (12,3) {$t3$};
\node[V](t) at (7,1) {\small{$sink$}};
\draw[E](d0) to (x00);
\draw[E](d0) to (x01);
\draw[E](d1) to (x10);
\draw[E](d1) to (x11);
\draw[E](d1) to (x12);
\draw[E](d2) to (x10);
\draw[E](d2) to (x11);
\draw[E](d2) to (x12);
\draw[E](d2) to (x13);
\draw[E](x00) to (t0);
\draw[E](x10) to (t0);
\draw[E](x01) to (t1);
\draw[E](x11) to (t1);
\draw[E](x02) to (t2);
\draw[E](x12) to (t2);
\draw[E](x03) to (t3);
\draw[E](x13) to (t3);
\draw[E](t0) to (t);
\draw[E](t1) to (t);
\draw[E](t2) to (t);
\draw[E](t3) to (t);
\draw[red,->,dashed](3,8) to (d0);
\draw[green,->,dashed](7,8) to (d1);
\draw[blue,->,dashed](11,8) to (d2);
\draw[red,->,dashed](t) to (6.5,0);
\draw[green,->,dashed](t) to (7,0);
\draw[blue,->,dashed](t) to (7.5,0);
\end{tikzpicture}
\end{figure}
\begin{itemize}
\item Unit capacity on each edge
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Feasible flow corresponds to schedule}
\begin{figure}
\begin{tikzpicture}[transform shape,scale=0.6]
\useasboundingbox(0,0) rectangle (14,8);
\tikzset{V/.style = {shape=circle, draw, text=black}}
\tikzset{E/.style = {draw,text=black,->}}
\node[V,text=red](d0) at (3,7) {$a$};
\node[V,text=green](d1) at (7,7) {$b$};
\node[V,text=blue](d2) at (11,7) {$c$};
\node[V](x00) at (0,5) {$x^0_0$};
\node[V](x01) at (2,5) {$x^0_1$};
\node[V](x02) at (4,5) {$x^0_2$};
\node[V](x03) at (6,5) {$x^0_3$};
\node[V](x10) at (8,5) {$x^1_0$};
\node[V](x11) at (10,5) {$x^1_1$};
\node[V](x12) at (12,5) {$x^1_2$};
\node[V](x13) at (14,5) {$x^1_3$};
\node[V](t0) at (3,3) {$t0$};
\node[V](t1) at (6,3) {$t1$};
\node[V](t2) at (9,3) {$t2$};
\node[V](t3) at (12,3) {$t3$};
\node[V](t) at (7,1) {\small{$sink$}};
\draw[E](d0) to (x00);
\draw[E,red](d0) to (x01);
\draw[E,green](d1) to (x10);
\draw[E](d1) to (x11);
\draw[E](d1) to (x12);
\draw[E](d2) to (x10);
\draw[E](d2) to (x11);
\draw[E](d2) to (x12);
\draw[E,blue](d2) to (x13);
\draw[E](x00) to (t0);
\draw[E,green](x10) to (t0);
\draw[E,red](x01) to (t1);
\draw[E](x11) to (t1);
\draw[E](x02) to (t2);
\draw[E](x12) to (t2);
\draw[E](x03) to (t3);
\draw[E,blue](x13) to (t3);
\draw[E,green](t0) to (t);
\draw[E,red](t1) to (t);
\draw[E](t2) to (t);
\draw[E,blue](t3) to (t);
\draw[red,->,dashed](3,8) to (d0);
\draw[green,->,dashed](7,8) to (d1);
\draw[blue,->,dashed](11,8) to (d2);
\draw[red,->,dashed](t) to (6.5,0);
\draw[green,->,dashed](t) to (7,0);
\draw[blue,->,dashed](t) to (7.5,0);
\end{tikzpicture}
\end{figure}
\begin{itemize}
\item corresponding schedule: [ 1 0 -1 1 ]
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Consider directed cycles in residual graph}
\begin{figure}
\begin{tikzpicture}[transform shape,scale=0.6]
\useasboundingbox(0,0) rectangle (14,8);
\tikzset{V/.style = {shape=circle, draw, text=black}}
\tikzset{E/.style = {draw,text=black,->}}
\node[V,text=red](d0) at (3,7) {$a$};
\node[V,text=green](d1) at (7,7) {$b$};
\node[V,text=blue](d2) at (11,7) {$c$};
\node[V](x00) at (0,5) {$x^0_0$};
\node[V](x01) at (2,5) {$x^0_1$};
\node[V](x02) at (4,5) {$x^0_2$};
\node[V](x03) at (6,5) {$x^0_3$};
\node[V](x10) at (8,5) {$x^1_0$};
\node[V](x11) at (10,5) {$x^1_1$};
\node[V](x12) at (12,5) {$x^1_2$};
\node[V](x13) at (14,5) {$x^1_3$};
\node[V](t0) at (3,3) {$t0$};
\node[V](t1) at (6,3) {$t1$};
\node[V](t2) at (9,3) {$t2$};
\node[V](t3) at (12,3) {$t3$};
\node[V](t) at (7,1) {\small{$sink$}};
\draw[E](d0) to (x00);
\draw[E] (x01) to (d0);
\draw[E] (x10) to (d1);
\draw[E](d1) to (x11);
\draw[E](d1) to (x12);
\draw[E](d2) to (x10);
\draw[E](d2) to (x11);
\draw[E](d2) to (x12);
\draw[E] (x13) to (d2);
\draw[E](x00) to (t0);
\draw[E] (t0) to (x10);
\draw[E] (t1) to (x01);
\draw[E](x11) to (t1);
\draw[E](x02) to (t2);
\draw[E](x12) to (t2);
\draw[E](x03) to (t3);
\draw[E] (t3) to (x13);
\draw[E] (t) to (t0);
\draw[E] (t) to (t1);
\draw[E] (t2) to (t);
\draw[E] (t) to (t3);
% \draw[red,->,dashed](3,8) to (d0);
% \draw[green,->,dashed](7,8) to (d1);
% \draw[blue,->,dashed](11,8) to (d2);
\end{tikzpicture}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Simple directed cycle yielding different schedule}
\begin{figure}
\centering
\begin{tikzpicture}[transform shape,scale=0.6]
\useasboundingbox(0,0) rectangle (14,8);
\tikzset{V/.style = {shape=circle, draw, text=black}}
\tikzset{E/.style = {draw,text=black,->}}
\node[V,text=red](d0) at (3,7) {$a$};
\node[V,text=green](d1) at (7,7) {$b$};
\node[V,text=blue](d2) at (11,7) {$c$};
\node[V,thick](x00) at (0,5) {$x^0_0$};
\node[V,thick](x01) at (2,5) {$x^0_1$};
\node[V](x02) at (4,5) {$x^0_2$};
\node[V](x03) at (6,5) {$x^0_3$};
\node[V,thick](x10) at (8,5) {$x^1_0$};
\node[V,thick](x11) at (10,5) {$x^1_1$};
\node[V](x12) at (12,5) {$x^1_2$};
\node[V](x13) at (14,5) {$x^1_3$};
\node[V](t0) at (3,3) {$t0$};
\node[V](t1) at (6,3) {$t1$};
\node[V](t2) at (9,3) {$t2$};
\node[V](t3) at (12,3) {$t3$};
\node[V](t) at (7,1) {\small{$sink$}};
\draw[E,thick](d0) to (x00);
\draw[E,thick](x01) to (d0);
\draw[E,thick] (x10) to (d1);
\draw[E,thick](d1) to (x11);
\draw[E](d1) to (x12);
\draw[E](d2) to (x10);
\draw[E](d2) to (x11);
\draw[E](d2) to (x12);
\draw[E](x13) to (d2);
\draw[E,thick](x00) to (t0);
\draw[E,thick](t0) to (x10);
\draw[E,thick](t1) to (x01);
\draw[E,thick](x11) to (t1);
\draw[E](x02) to (t2);
\draw[E](x12) to (t2);
\draw[E](x03) to (t3);
\draw[E](t3) to (x13);
\draw[E](t) to (t0);
\draw[E](t) to (t1);
\draw[E](t2) to (t);
\draw[E](t) to (t3);
\end{tikzpicture}
\end{figure}
\begin{itemize}
\item schedule: [ 1 0 -1 1 ] $\Rightarrow$ [ 0 1 -1 1 ]
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Another simple directed cycle yielding another different schedule}
\begin{figure}
\begin{tikzpicture}[transform shape,scale=0.6]
\useasboundingbox(0,0) rectangle (14,8);
\tikzset{V/.style = {shape=circle, draw, text=black}}
\tikzset{E/.style = {draw,text=black,->}}
\node[V,text=red](d0) at (3,7) {$a$};
\node[V,text=green](d1) at (7,7) {$b$};
\node[V,text=blue](d2) at (11,7) {$c$};
\node[V](x00) at (0,5) {$x^0_0$};
\node[V](x01) at (2,5) {$x^0_1$};
\node[V](x02) at (4,5) {$x^0_2$};
\node[V](x03) at (6,5) {$x^0_3$};
\node[V](x10) at (8,5) {$x^1_0$};
\node[V](x11) at (10,5) {$x^1_1$};
\node[V](x12) at (12,5) {$x^1_2$};
\node[V](x13) at (14,5) {$x^1_3$};
\node[V](t0) at (3,3) {$t0$};
\node[V](t1) at (6,3) {$t1$};
\node[V](t2) at (9,3) {$t2$};
\node[V](t3) at (12,3) {$t3$};
\node[V](t) at (7,1) {\small{$sink$}};
\draw[E](d0) to (x00);
\draw[E](x01) to (d0);
\draw[E](x10) to (d1);
\draw[E](d1) to (x11);
\draw[E](d1) to (x12);
\draw[E](d2) to (x10);
\draw[E](d2) to (x11);
\draw[E,thick](d2) to (x12);
\draw[E,thick](x13) to (d2);
\draw[E](x00) to (t0);
\draw[E](t0) to (x10);
\draw[E](t1) to (x01);
\draw[E](x11) to (t1);
\draw[E](x02) to (t2);
\draw[E,thick](x12) to (t2);
\draw[E](x03) to (t3);
\draw[E,thick] (t3) to (x13);
\draw[E] (t) to (t0);
\draw[E] (t) to (t1);
\draw[E,thick](t2) to (t);
\draw[E,thick](t) to (t3);
\end{tikzpicture}
\end{figure}
\begin{itemize}
\item schedule: [ 1 0 -1 1 ] $\Rightarrow$ [ 1 0 1 -1 ]
\end{itemize}
\end{frame}
\end{document}