mirror of
https://github.com/vale981/TUD_MATH_BA
synced 2025-03-05 09:31:39 -05:00
bunte Bildchen für NUME
This commit is contained in:
henrydatei
parent
10b12ad060
commit
7db95609e0
4 changed files with 148 additions and 4 deletions
|
|
@ -42,16 +42,87 @@ definierte Funktion $f:\real^n\to\real$ als \begriff{Zielfunktion}. Allgemein ka
|
||||||
\end{itemize}
|
\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
|
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}
|
\begin{align}
|
||||||
x_1-x_2+0.4 &\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 \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 \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\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
|
x_1,x_2 &\ge 0 \notag
|
||||||
\end{align}
|
\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
|
Aus der graphischen Darstellung des zulässigen Bereiches und Parallelverschieben der Niveaulinie der Zielfunktion
|
||||||
\begin{align}
|
\begin{align}
|
||||||
-2x_1+x_2 =c \notag
|
-2x_1+x_2 =c \notag
|
||||||
\end{align}
|
\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:
|
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}
|
\begin{align}
|
||||||
x_1-x_2+y_1+0.4 &= 0 \notag \\
|
x_1-x_2+y_1+0.4 &= 0 \notag \\
|
||||||
|
|
|
||||||
Binary file not shown.
BIN
Material/lineare_optimierung.pdf
Normal file
BIN
Material/lineare_optimierung.pdf
Normal file
Binary file not shown.
73
Material/lineare_optimierung.tex
Normal file
73
Material/lineare_optimierung.tex
Normal file
|
|
@ -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}
|
||||||
Loading…
Add table
Reference in a new issue