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bunte Bildchen für NUME

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79
3. Semester/NUME/TeX_files/lineare_Optimierung_Vorspann.tex
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@ -42,16 +42,87 @@ definierte Funktion $f:\real^n\to\real$ als \begriff{Zielfunktion}. Allgemein ka
\end{itemize}
Zur graphischen Lösung kann man $x_3=1-x_1-x_2$ eliminieren. Dies ergibt $-10x_1+5x_2+20\to\min$, bei
\begin{align}
x_1-x_2+0.4 &\le 0 \notag \\
-x_1+x_2-1 &\le 0 \notag \\
3x_1-x_2+0.2 &\le 0 \notag \\
x_1+x_2-1 &\le 0\notag \\
x_1-x_2+0.4 &\le 0 \quad\textcolor{blue}{x_2\ge 0.4+x_1} \notag \\
-x_1+x_2-1 &\le 0 \quad\textcolor{yellow}{x_2 \le 1+x_1} \notag \\
3x_1-x_2+0.2 &\le 0 \quad\textcolor{red}{x_2\ge 3x_1+0.2} \notag \\
x_1+x_2-1 &\le 0 \quad\textcolor{green}{x_2 \le 1-x_1}\notag \\
x_1,x_2 &\ge 0 \notag
\end{align}
\begin{center}
\begin{tikzpicture}[scale=0.7]
\begin{scope}[transparency group]
\begin{scope}[blend mode=multiply]
\draw[->] (0,0) -- (10,0);
\draw[->] (0,0) -- (0,10);
\node at (10.3,0) (y) {$x$};
\node at (0,10.3) (x) {$y$};
\draw (-0.1,5) -- (0.1,5);
\node at (-0.3,5) (1y) {1};
\draw (-0.1,2) -- (0.1,2);
\node at (-0.4,2) (0.4y) {0.4};
\draw (-0.1,1) -- (0.1,1);
\node at (-0.4,1) (0.4y) {0.2};
\draw (5,-0.1) -- (5,0.1);
\node at (5,-0.3) (1x) {1};
\draw[yellow,thick] (0,5) -- (5,10);
\draw[green,thick] (0,5) -- (5,0);
\draw[blue,thick] (0,2) -- (8,10);
\draw[red,thick] (0,1) -- (3,10);
\draw[yellow, fill=yellow, opacity=0.2] (0,5) -- (5,10) -- (10,10) -- (10,0) -- (0,0) -- (0,5);
\draw[green, fill=green, opacity=0.2] (0,5) -- (5,0) -- (0,0) -- (0,5);
\draw[blue, fill=blue, opacity=0.2] (0,2) -- (8,10) -- (0,10) --(0,2);
\draw[red, fill=red, opacity=0.2] (0,1) -- (3,10) -- (0,10) -- (0,1);
\draw[dashed, thick, black] (0,2) -- (0.5,2.5) -- (1,4) -- (0,5) -- (0,2);
\node at (0.4,3.5) (GP) {$G_P$};
\end{scope}
\end{scope}
\end{tikzpicture}
\end{center}
Aus der graphischen Darstellung des zulässigen Bereiches und Parallelverschieben der Niveaulinie der Zielfunktion
\begin{align}
-2x_1+x_2 =c \notag
\end{align}
\begin{center}
\begin{tikzpicture}[scale=0.7]
\draw[->] (0,0) -- (10,0);
\draw[->] (0,0) -- (0,10);
\node at (10.3,0) (y) {$x$};
\node at (0,10.3) (x) {$y$};
\draw (-0.1,5) -- (0.1,5);
\node at (-0.3,5) (1y) {1};
\draw (-0.1,2) -- (0.1,2);
\node at (-0.4,2) (0.4y) {0.4};
\draw (-0.1,1) -- (0.1,1);
\node at (-0.4,1) (0.4y) {0.2};
\draw (5,-0.1) -- (5,0.1);
\node at (5,-0.3) (1x) {1};
\draw[gray,thick] (0,5) -- (5,10);
\draw[gray,thick] (0,5) -- (5,0);
\draw[gray,thick] (0,2) -- (8,10);
\draw[gray,thick] (0,1) -- (3,10);
\draw[dashed, thick, black, fill=gray] (0,2) -- (0.5,2.5) -- (1,4) -- (0,5) -- (0,2);
\node at (0.4,3.5) (GP) {$G_P$};
\draw[cyan] (0,5) -- (2.5,10);
\draw[cyan] (0,4.5) -- (2.5,9.5);
\draw[cyan] (0,4) -- (2.5,9);
\draw[cyan] (0,3.5) -- (2.5,8.5);
\draw[cyan] (0,3) -- (2.5,8);
\draw[cyan] (0,2.5) -- (2.5,7.5);
\draw[cyan] (0,2) -- (2.5,7);
\draw[cyan, ultra thick] (0,1.5) -- (2.5,6.5);
\draw[cyan] (0,1) -- (2.5,6);
\draw[cyan] (0,0.5) -- (2.5,5.5);
\draw[cyan] (0,0) -- (2.5,5);
\end{tikzpicture}
\end{center}
erhält man das minimale $c$, bei dem die Niveaulinie und der zulässige Bereich noch den gemeinsamen Punkt $(x_1^\ast,x_2^\ast)=(0.1,0.5)$ haben und $c=0.3$. Zur Überführung der Optimierungsaufgabe in Standardform fügt man nicht-negative Schlupfvariablen $y_1,...,y_4$ hinzu:
\begin{align}
x_1-x_2+y_1+0.4 &= 0 \notag \\

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3. Semester/NUME/Vorlesung NUME.pdf
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Material/lineare_optimierung.pdf Normal file
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73
Material/lineare_optimierung.tex Normal file
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@ -0,0 +1,73 @@
\documentclass{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}{scale=0.5}
\begin{scope}[transparency group]
\begin{scope}[blend mode=multiply]
\draw[->] (0,0) -- (10,0);
\draw[->] (0,0) -- (0,10);
\node at (10.3,0) (y) {$x$};
\node at (0,10.3) (x) {$y$};
\draw (-0.1,5) -- (0.1,5);
\node at (-0.3,5) (1y) {1};
\draw (-0.1,2) -- (0.1,2);
\node at (-0.4,2) (0.4y) {0.4};
\draw (-0.1,1) -- (0.1,1);
\node at (-0.4,1) (0.4y) {0.2};
\draw (5,-0.1) -- (5,0.1);
\node at (5,-0.3) (1x) {1};
\draw[yellow,thick] (0,5) -- (5,10);
\draw[green,thick] (0,5) -- (5,0);
\draw[blue,thick] (0,2) -- (8,10);
\draw[red,thick] (0,1) -- (3,10);
\draw[yellow, fill=yellow, opacity=0.2] (0,5) -- (5,10) -- (10,10) -- (10,0) -- (0,0) -- (0,5);
\draw[green, fill=green, opacity=0.2] (0,5) -- (5,0) -- (0,0) -- (0,5);
\draw[blue, fill=blue, opacity=0.2] (0,2) -- (8,10) -- (0,10) --(0,2);
\draw[red, fill=red, opacity=0.2] (0,1) -- (3,10) -- (0,10) -- (0,1);
\draw[dashed, thick, black] (0,2) -- (0.5,2.5) -- (1,4) -- (0,5) -- (0,2);
\node at (0.4,3.5) (GP) {$G_P$};
\end{scope}
\end{scope}
\end{tikzpicture}\\
\begin{tikzpicture}
\draw[->] (0,0) -- (10,0);
\draw[->] (0,0) -- (0,10);
\node at (10.3,0) (y) {$x$};
\node at (0,10.3) (x) {$y$};
\draw (-0.1,5) -- (0.1,5);
\node at (-0.3,5) (1y) {1};
\draw (-0.1,2) -- (0.1,2);
\node at (-0.4,2) (0.4y) {0.4};
\draw (-0.1,1) -- (0.1,1);
\node at (-0.4,1) (0.4y) {0.2};
\draw (5,-0.1) -- (5,0.1);
\node at (5,-0.3) (1x) {1};
\draw[gray,thick] (0,5) -- (5,10);
\draw[gray,thick] (0,5) -- (5,0);
\draw[gray,thick] (0,2) -- (8,10);
\draw[gray,thick] (0,1) -- (3,10);
\draw[dashed, thick, black] (0,2) -- (0.5,2.5) -- (1,4) -- (0,5) -- (0,2);
\node at (0.4,3.5) (GP) {$G_P$};
\draw[cyan] (0,5) -- (2.5,10);
\draw[cyan] (0,4.5) -- (2.5,9.5);
\draw[cyan] (0,4) -- (2.5,9);
\draw[cyan] (0,3.5) -- (2.5,8.5);
\draw[cyan] (0,3) -- (2.5,8);
\draw[cyan] (0,2.5) -- (2.5,7.5);
\draw[cyan] (0,2) -- (2.5,7);
\draw[cyan, ultra thick] (0,1.5) -- (2.5,6.5);
\draw[cyan] (0,1) -- (2.5,6);
\draw[cyan] (0,0.5) -- (2.5,5.5);
\draw[cyan] (0,0) -- (2.5,5);
\end{tikzpicture}
\end{document}