mirror of https://github.com/2martens/uni.git
217 lines
6.9 KiB
TeX
217 lines
6.9 KiB
TeX
\documentclass[10pt,a4paper,oneside,ngerman,numbers=noenddot]{scrartcl}
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\usepackage[T1]{fontenc}
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\usepackage[utf8]{inputenc}
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\usepackage[ngerman]{babel}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage{amssymb}
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\usepackage{paralist}
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\usepackage{gauss}
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\usepackage{pgfplots}
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\usepackage[locale=DE,exponent-product=\cdot,detect-all]{siunitx}
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\usepackage{tikz}
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\usetikzlibrary{matrix,fadings,calc,positioning,decorations.pathreplacing,arrows,decorations.markings}
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\usepackage{polynom}
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\usepackage{multirow}
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\polyset{style=C, div=:,vars=x}
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\pgfplotsset{compat=1.8}
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\pagenumbering{arabic}
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% ensures that paragraphs are separated by empty lines
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\parskip 12pt plus 1pt minus 1pt
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\parindent 0pt
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% define how the sections are rendered
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\def\thesection{\arabic{section})}
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\def\thesubsection{\alph{subsection})}
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\def\thesubsubsection{(\roman{subsubsection})}
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% some matrix magic
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\makeatletter
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\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
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\hskip -\arraycolsep
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\let\@ifnextchar\new@ifnextchar
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\array{#1}}
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\makeatother
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\begin{document}
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\author{Jan Branitz (6326955), Jim Martens (6420323),\\
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Stephan Niendorf (6242417)}
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\title{Hausaufgaben zum 6. Januar}
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\maketitle
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\section{} %1
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\begin{tikzpicture}[node distance=2cm]
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\node (s) {s};
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\node (a) [above right=of s] {a};
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\node (b) [right=of s] {b};
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\node (c) [below right=of s] {c};
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\node (d) [right=of c] {d};
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\node (e) [right=of b] {e};
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\node (t) [right=of e] {t};
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\node (sCap) [below=0.1 of s] {(-, $\infty$)};
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\node (aCap) [above=0.1 of a] {(s, +, 3)};
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\node (bCap) [below=0.1 of b] {(s, +, 5)};
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\node (cCap) [below=0.1 of c] {(s, +, 6)};
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\node (tCap) [above=0.1 of t] {(a, +, 3)};
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\path[->] (s) edge[line width=2pt] node [above left] {0(3)} (a)
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(s) edge node [below] {0(5)} (b)
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(s) edge node [below left] {0(6)} (c)
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(a) edge[line width=2pt] node [above right] {0(5)} (t)
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(a) edge node [left] {0(4)} (b)
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(b) edge node [below] {0(2)} (e)
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(b) edge node [below left] {0(6)} (d)
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(c) edge node [below] {0(4)} (d)
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(d) edge node [below right] {0(5)} (t)
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(e) edge node [below] {0(3)} (t);
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\end{tikzpicture}
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\begin{tikzpicture}[node distance=2cm]
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\node (s) {s};
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\node (a) [above right=of s] {a};
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\node (b) [right=of s] {b};
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\node (c) [below right=of s] {c};
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\node (d) [right=of c] {d};
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\node (e) [right=of b] {e};
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\node (t) [right=of e] {t};
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\node (sCap) [below=0.1 of s] {(-, $\infty$)};
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\node (bCap) [below=0.1 of b] {(s, +, 5)};
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\node (cCap) [below=0.1 of c] {(s, +, 6)};
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\node (dCap) [below=0.1 of d] {(b, +, 5)};
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\node (eCap) [below=0.1 of e] {(b, +, 2)};
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\node (tCap) [above=0.1 of t] {(d, +, 5)};
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\path[->] (s) edge node [above left] {3(3)} (a)
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(s) edge[line width=2pt] node [below] {0(5)} (b)
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(s) edge node [below left] {0(6)} (c)
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(a) edge node [above right] {3(5)} (t)
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(a) edge node [left] {0(4)} (b)
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(b) edge node [below] {0(2)} (e)
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(b) edge[line width=2pt] node [below left] {0(6)} (d)
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(c) edge node [below] {0(4)} (d)
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(d) edge[line width=2pt] node [below right] {0(5)} (t)
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(e) edge node [below] {0(3)} (t);
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\end{tikzpicture}
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\begin{tikzpicture}[node distance=2cm]
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\node (s) {s};
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\node (a) [above right=of s] {a};
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\node (b) [right=of s] {b};
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\node (c) [below right=of s] {c};
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\node (d) [right=of c] {d};
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\node (e) [right=of b] {e};
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\node (t) [right=of e] {t};
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\node (sCap) [below=0.1 of s] {(-, $\infty$)};
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\node (eCap) [below=0.1 of e] {(b, +, 2)};
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\node (cCap) [below=0.1 of c] {(s, +, 6)};
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\node (dCap) [below=0.1 of d] {(c, +, 4)};
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\node (tCap) [above=0.1 of t] {(e, +, 2)};
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\node (bCap) [below=0.1 of b] {(d, -, 4)};
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\path[->] (s) edge node [above left] {3(3)} (a)
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(s) edge node [below] {5(5)} (b)
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(s) edge[line width=2pt] node [below left] {0(6)} (c)
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(a) edge node [above right] {3(5)} (t)
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(a) edge node [left] {0(4)} (b)
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(b) edge[line width=2pt] node [below] {0(2)} (e)
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(b) edge[line width=2pt] node [below left] {5(6)} (d)
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(c) edge[line width=2pt] node [below] {0(4)} (d)
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(d) edge node [below right] {5(5)} (t)
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(e) edge[line width=2pt] node [below] {0(3)} (t);
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\end{tikzpicture}
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\begin{tikzpicture}[node distance=2cm]
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\node (s) {s};
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\node (a) [above right=of s] {a};
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\node (b) [right=of s] {b};
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\node (c) [below right=of s] {c};
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\node (d) [right=of c] {d};
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\node (e) [right=of b] {e};
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\node (t) [right=of e] {t};
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\node (sCap) [below=0.1 of s] {(-, $\infty$)};
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\node (cCap) [below=0.1 of c] {(s, +, 4)};
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\node (dCap) [below=0.1 of d] {(c, +, 2)};
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\node (bCap) [below=0.1 of b] {(d, -, 2)};
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\path[->] (s) edge node [above left] {3(3)} (a)
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(s) edge node [below] {5(5)} (b)
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(s) edge node [below left] {2(6)} (c)
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(a) edge node [above right] {3(5)} (t)
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(a) edge node [left] {0(4)} (b)
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(b) edge node [below] {2(2)} (e)
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(b) edge node [below left] {3(6)} (d)
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(c) edge node [below] {2(4)} (d)
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(d) edge node [below right] {5(5)} (t)
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(e) edge node [below] {2(3)} (t);
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\end{tikzpicture}
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\section{} %2
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\subsection{} %a
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\begin{alignat*}{9}
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\text{minimiere}\; & 67w &+& 120k &+& 100h &+& 60f &+& 97b &+& 124n &+& 22s &+& 62m && \\
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\multicolumn{18}{l}{\text{unter den Nebenbedingungen}} && \\
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& 8w &+& 25k &+& 30h &+& 22f &+& 3b &+& 8n &+& 6s && &\geq &\, 75 \\
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& w &+& 35k &+& 8h &+& f && &+& 33n &+& 13s &+& 98m &\geq &\, 90 \\
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& 54w && && && &+& 42b &+& 4n &+& 63s && &\geq &\, 300 \\
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\multicolumn{16}{r}{$w, k, h, f, b, n, s, m$} \,&\geq &\, 0
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\end{alignat*}
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\subsection{} %b
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In das verlinkte Tool wird folgendes eingegeben:
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\begin{verbatim}
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Minimize p = 67w + 120k + 100h + 60f + 97b + 124n + 22s + 62m subject to
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8w + 25k + 30h + 22f + 3b + 8n + 6s + 0m >= 75
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w + 35k + 8h + f + 0b + 33n + 13s + 98m >= 90
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54w + 0k + 0h + 0f + 42b + 4n + 63s + 0m >= 300
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w >= 0
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k >= 0
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h >= 0
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f >= 0
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b >= 0
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n >= 0
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s >= 0
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m >= 0
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\end{verbatim}
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Die Ausgabe ist:
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\begin{verbatim}
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Optimal Solution: p = 6801/28; w = 0, k = 0, h = 0, f = 87/56,
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b = 0, n = 0, s = 381/56, m = 0
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\end{verbatim}
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\subsection{} %c
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In das verlinkte Tool wird folgendes eingegeben:
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\begin{verbatim}
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Minimize p = 3x1 + 24x2 + 13x3 + 9x4 + 20x5 + 19x6 subject to
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110x1 + 205x2 + 160x3 + 160x4 + 420x5 + 260x6 >= 2000
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4x1 + 32x2 + 13x3 + 8x4 + 4x5 + 14x6 >= 55
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2x1 + 12x2 + 54x3 + 285x4 + 22x5 + 80x6 >= 800
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x1 <= 4
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x2 <= 3
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x3 <= 2
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x4 <= 8
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x5 <= 2
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x6 <= 2
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x1 >= 0
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x2 >= 0
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x3 >= 0
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x4 >= 0
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x5 >= 0
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x6 >= 0
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\end{verbatim}
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Die Ausgabe ist:
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\begin{verbatim}
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Optimal Solution: p = 185/2; x1 = 4, x2 = 0, x3 = 0, x4 = 9/2, x5 = 2, x6 = 0
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\end{verbatim}
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\subsection{} %d
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In das verlinkte Tool wird folgendes eingegeben:
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\begin{verbatim}
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Maximize p = 8a1 + 3a2 + 6b1 + 3b2 + 9c1 + 5c2 subject to
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a1 + a2 <= 400
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b1 + b2 <= 480
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c1 + c2 <= 230
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a1 + b1 + c1 <= 420
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a2 + b2 + c2 <= 250
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a1 >= 0
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a2 >= 0
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b1 >= 0
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b2 >= 0
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c1 >= 0
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c2 >= 0
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\end{verbatim}
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Die Ausgabe ist:
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\begin{verbatim}
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Optimal Solution: p = 4550; a1 = 400, a2 = 0, b1 = 0, b2 = 40,
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c1 = 20, c2 = 210
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\end{verbatim}
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\end{document}
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