From 996d99d39377d173892b6502e614b9190d9f5394 Mon Sep 17 00:00:00 2001 From: Jim Martens Date: Thu, 24 Oct 2013 15:27:08 +0200 Subject: [PATCH] MATH2-Inf-2: 2 geloest. --- optimierung/Uebungsblatt2.tex | 119 ++++++++++++++++++++++++++++++++-- 1 file changed, 115 insertions(+), 4 deletions(-) diff --git a/optimierung/Uebungsblatt2.tex b/optimierung/Uebungsblatt2.tex index 5d46eb2..4d2fe48 100644 --- a/optimierung/Uebungsblatt2.tex +++ b/optimierung/Uebungsblatt2.tex @@ -144,7 +144,7 @@ Stephan Niendorf (6242417)} x_{5} \,&=&\, 6 \,&-&\, x_{1} \,&-&\, 5x_{2} \,&-&\, 3x_{3} \\ x_{6} \,&=&\, 4 \,&+&\, x_{1} \,&-&\, 3x_{2} \,&-&\, 3x_{3} \\ x_{7} \,&=&\, 2 \,&-&\, x_{1} \,&+&\, x_{2} \,&-&\, 3x_{3} \\ \cline{1 - 9} - z &=& &-& 5x_{1} \,&+&\, 11x_{2} \,&-&\, 5x_{3} + z \,&=&\, &-& 5x_{1} \,&+&\, 11x_{2} \,&-&\, 5x_{3} \end{alignat*} \underline{1. Iteration}: @@ -175,7 +175,7 @@ Stephan Niendorf (6242417)} x_{5} \,&=&\, \frac{8}{3} \,&-&\, \frac{8}{3}x_{1} \,&-&\, \frac{29}{3}x_{3} \,&+&\, \frac{5}{3}x_{4} \\ x_{6} \,&=&\, 2 && \,&-&\, 7x_{3} \,&+&\, x_{4} \\ x_{7} \,&=&\, \frac{8}{3} \,&-&\, \frac{2}{3}x_{1} \,&-&\, \frac{5}{3}x_{3} \,&-&\, \frac{1}{3}x_{4} \\ \cline{1 - 9} - z &=& \frac{22}{3} \,&-&\, \frac{4}{3}x_{1} \,&+&\, \frac{29}{3}x_{3} \,&-&\, \frac{11}{3}x_{4} + z \,&=&\, \frac{22}{3} \,&-&\, \frac{4}{3}x_{1} \,&+&\, \frac{29}{3}x_{3} \,&-&\, \frac{11}{3}x_{4} \end{alignat*} \underline{2. Iteration}: @@ -207,7 +207,7 @@ Stephan Niendorf (6242417)} x_{2} \,&=&\, \frac{30}{29} \,&-&\, \frac{1}{29}x_{1} \,&-&\, \frac{1}{29}x_{4} \,&-&\, \frac{4}{29}x_{5} \\ x_{6} \,&=&\, \frac{2}{29} \,&+&\, \frac{56}{29}x_{1} \,&-&\, \frac{6}{29}x_{4} \,&+&\, \frac{21}{29}x_{5} \\ x_{7} \,&=&\, \frac{64}{29} \,&-&\, \frac{6}{29}x_{1} \,&-&\, \frac{18}{29}x_{4} \,&+&\, \frac{5}{29}x_{5} \\ \cline{1 - 9} - z &=& 10 \,&+&\, \frac{4}{3}x_{1} \,&-&\, 2x_{4} \,&-&\, x_{5} + z \,&=&\, 10 \,&+&\, \frac{4}{3}x_{1} \,&-&\, 2x_{4} \,&-&\, x_{5} \end{alignat*} \underline{3. Iteration}: @@ -239,7 +239,7 @@ Stephan Niendorf (6242417)} x_{2} \,&=&\, 1 \,&-&\, \frac{13}{232}x_{4} \,&-&\, \frac{1}{8}x_{5} \,&+&\, \frac{1}{8}x_{3} \\ x_{6} \,&=&\, 2 \,&+&\, x_{4} && \,&-&\, 7x_{3} \\ x_{7} \,&=&\, 2 \,&-&\, \frac{3}{4}x_{4} \,&+&\, \frac{1}{4}x_{5} \,&+&\, \frac{3}{4}x_{3} \\ \cline{1 - 9} - z &=& \frac{34}{3} \,&-&\, \frac{7}{6}x_{4} \,&-&\, \frac{3}{2}x_{5} \,&-&\, \frac{29}{6}x_{3} + z \,&=&\, \frac{34}{3} \,&-&\, \frac{7}{6}x_{4} \,&-&\, \frac{3}{2}x_{5} \,&-&\, \frac{29}{6}x_{3} \end{alignat*} Dieses Tableau liefert die optimale Lösung $x_{1} = 1, x_{2} = 1, x_{3} = 0$ mit $z = \frac{34}{3}$. @@ -260,5 +260,116 @@ Stephan Niendorf (6242417)} \[ x_{1} = 1, x_{2} = 1, x_{3} = 0, x_{4} = 0, x_{5} = 0, x_{6} = 2, x_{7} = 2 \text{ mit } z = \frac{34}{3} = 11\frac{1}{3} \] + % + % 2 startet hier + % + % \section{} %2 + \textbf{Aufgabe:} Lösen Sie das folgende LP-Problem mit dem Simplexverfahren: + \begin{alignat*}{5} + \text{maximiere}\; & x_{1} &-& 9x_{2} &-& 11x_{3} &+& 3x_{4} && \\ + \multicolumn{10}{l}{\text{unter den Nebenbedingungen}} && \\ + \;& x_{1} &+& x_{2} &+& 3x_{3} &+& x_{4} &\leq & 3 \\ + \;-& x_{1} &-& 3x_{2} &-& 7x_{3} &+& x_{4} &\leq & 1 \\ + \multicolumn{8}{r}{$x_{1}, x_{2}, x_{3}, x_{4}$} \,&\geq &\, 0 + \end{alignat*} + + \textbf{Lösung.} + + \underline{Starttableau}: + \begin{alignat*}{6} + x_{5} \,&=&\, 3 \,&-&\, x_{1} \,&-&\, x_{2} \,&-&\, 3x_{3} \,&-&\, x_{4} \\ + x_{6} \,&=&\, 1 \,&+&\, x_{1} \,&+&\, 3x_{2} \,&+&\, 7x_{3} \,&-&\, x_{4} \\ \cline{1 - 11} + z \,&=&\, && x_{1} \,&-&\, 9x_{2} \,&-&\, 11x_{3} \,&+&\, 3x_{4} + \end{alignat*} + + \underline{1. Iteration}: + + Eingangsvariable: $x_{4}$\\ + Ausgangsvariable: $x_{6}$ + + Es folgt + \begin{alignat*}{2} + x_{4} &=&& 1 + x_{1} + 3x_{2} + 7x_{3} - x_{6} \\ + x_{5} &=&& 3 - x_{1} - x_{2} - 3x_{3} - \left(1 + x_{1} + 3x_{2} + 7x_{3} - x_{6}\right) \\ + &=&& 3 - x_{1} - x_{2} - 3x_{3} - 1 - x_{1} - 3x_{2} - 7x_{3} + x_{6} \\ + &=&& 2 - 2x_{1} - 4x_{2} - 10x_{3} + x_{6} \\ + z &=&& x_{1} - 9x_{2} - 11x_{3} + 3\left(1 + x_{1} + 3x_{2} + 7x_{3} - x_{6}\right) \\ + &=&& x_{1} - 9x_{2} - 11x_{3} + 3 + 3x_{1} + 9x_{2} + 21x_{3} - 3x_{6} \\ + &=&& 3 + 4x_{1} + 10x_{3} - 3x_{6} + \end{alignat*} + + \underline{Ergebnis der 1. Iteration}: + \begin{alignat*}{6} + x_{4} \,&=&\, 1 \,&+&\, x_{1} \,&+&\, 3x_{2} \,&+&\, 7x_{3} \,&-&\, x_{6} \\ + x_{5} \,&=&\, 2 \,&-&\, 2x_{1} \,&-&\, 4x_{2} \,&-&\, 10x_{3} \,&+&\, x_{6} \\ \cline{1 - 11} + z \,&=&\, 3 \,&+&\, 4x_{1} && \,&+&\, 10x_{3} \,&-&\, 3x_{6} + \end{alignat*} + + \underline{2. Iteration}: + + Eingangsvariable: $x_{3}$ \\ + Ausgangsvariable: $x_{5}$ + + Es folgt + \begin{alignat*}{2} + 10x_{3} &=&& 2 - 2x_{1} - 4x_{2} + x_{6} - x_{5} \\ + x_{3} &=&& \frac{1}{5} - \frac{1}{5}x_{1} - \frac{2}{5}x_{2} + \frac{1}{10}x_{6} - \frac{1}{10}x_{5} \\ + x_{4} &=&& 1 + x_{1} + 3x_{2} + 7\left(\frac{1}{5} - \frac{1}{5}x_{1} - \frac{2}{5}x_{2} + \frac{1}{10}x_{6} - \frac{1}{10}x_{5}\right) - x_{6} \\ + &=&& 1 + x_{1} + 3x_{2} + \frac{7}{5} - \frac{7}{5}x_{1} - \frac{14}{5}x_{2} + \frac{7}{10}x_{6} - \frac{7}{10}x_{5} - x_{6} \\ + &=&& \frac{12}{5} - \frac{2}{5}x_{1} + \frac{1}{5}x_{2} - \frac{3}{10}x_{6} - \frac{7}{10}x_{5} \\ + z &=&& 3 + 4x_{1} + 10\left(\frac{1}{5} - \frac{1}{5}x_{1} - \frac{2}{5}x_{2} + \frac{1}{10}x_{6} - \frac{1}{10}x_{5}\right) - 3x_{6} \\ + &=&& 3 + 4x_{1} + 2 - 2x_{1} - 4x_{2} + x_{6} - x_{5} - 3x_{6} \\ + &=&& 5 + 2x_{1} - 4x_{2} - 2x_{6} - x_{5} + \end{alignat*} + + \underline{Ergebnis der 2. Iteration}: + \begin{alignat*}{6} + x_{3} \,&=&\, \frac{1}{5} \,&-&\, \frac{1}{5}x_{1} \,&-&\, \frac{2}{5}x_{2} \,&+&\, \frac{1}{10}x_{6} \,&-&\, \frac{1}{10}x_{5} \\ + x_{4} \,&=&\, \frac{12}{5} \,&-&\, \frac{2}{5}x_{1} \,&+&\, \frac{1}{5}x_{2} \,&-&\, \frac{3}{10}x_{6} \,&-&\, \frac{7}{10}x_{5} \\ \cline{1 - 11} + z \,&=&\, 5 \,&+&\, 2x_{1} \,&-&\, 4x_{2} \,&-&\, 2x_{6} \,&-&\, x_{5} + \end{alignat*} + + \underline{3. Iteration}: + + Eingangsvariable: $x_{1}$\\ + Ausgangsvariable: $x_{3}$ + + Es folgt + \begin{alignat*}{2} + \frac{1}{5}x_{1} &=&& \frac{1}{5} - \frac{2}{5}x_{2} + \frac{1}{10}x_{6} - \frac{1}{10}x_{5} - x_{3} \\ + x_{1} &=&& 1 - 2x_{2} + \frac{1}{2}x_{6} - \frac{1}{2}x_{5} - 5x_{3} \\ + x_{4} &=&& \frac{12}{5} - \frac{2}{5}\left(1 - 2x_{2} + \frac{1}{2}x_{6} - \frac{1}{2}x_{5} - 5x_{3}\right) + \frac{1}{5}x_{2} - \frac{3}{10}x_{6} - \frac{7}{10}x_{5} \\ + &=&& \frac{12}{5} - \frac{2}{5} + \frac{4}{5}x_{2} - \frac{1}{5}x_{6} + \frac{1}{5}x_{5} + 2x_{3} + \frac{1}{5}x_{2} - \frac{3}{10}x_{6} - \frac{7}{10}x_{5} \\ + &=&& 2 + x_{2} - \frac{1}{2}x_{6} - \frac{1}{2}x_{5} + 2x_{3} \\ + z &=&& 5 + 2\left(1 - 2x_{2} + \frac{1}{2}x_{6} - \frac{1}{2}x_{5} - 5x_{3}\right) - 4x_{2} - 2x_{6} - x_{5} \\ + &=&& 5 + 2 - 4x_{2} + x_{6} - x_{5} - 10x_{3} - 4x_{2} - 2x_{6} - x_{5} \\ + &=&& 7 - 8x_{2} - x_{6} - 2x_{5} - 10x_{3} + \end{alignat*} + + \underline{Ergebnis der 3. Iteration}: + \begin{alignat*}{6} + x_{1} \,&=&\, 1 \,&-&\, 2x_{2} \,&+&\, \frac{1}{2}x_{6} \,&-&\, \frac{1}{2}x_{5} \,&-&\, 5x_{3} \\ + x_{4} \,&=&\, 2 \,&+&\, x_{2} \,&-&\, \frac{1}{2}x_{6} \,&-&\, \frac{1}{2}x_{5} \,&+&\, 2x_{3} \\ \cline{1 - 11} + z \,&=&\, 7 \,&-&\, 8x_{2} \,&-&\, x_{6} \,&-&\, 2x_{5} \,&-&\, 10x_{3} + \end{alignat*} + + Dieses Tableau liefert die optimale Lösung $x_{1} = 1, x_{2} = 0, x_{3} = 0$ mit $z = 7$. + + \underline{Startlösung ("`zulässige Basislösung am Anfang"')}: + \[ + x_{1} = 0, x_{2} = 0, x_{3} = 0, x_{4} = 2, x_{5} = 3, x_{6} = 1 \text{ mit } z = 0 + \] + \underline{Zulässige Basislösung nach der 1. Iteration}: + \[ + x_{1} = 0, x_{2} = \frac{2}{3}, x_{3} = 0, x_{4} = 1, x_{5} = 2, x_{6} = 0 \text{ mit } z = 3 + \] + \underline{Zulässige Basislösung nach der 2. Iteration}: + \[ + x_{1} = 0, x_{2} = 0, x_{3} = \frac{1}{5}, x_{4} = \frac{12}{5}, x_{5} = 0, x_{6} = 2 \text{ mit } z = 5 + \] + \underline{Zulässige Basislösung nach der 3. Iteration}: + \[ + x_{1} = 1, x_{2} = 0, x_{3} = 0, x_{4} = 2, x_{5} = 0, x_{6} = 0 \text{ mit } z = 7 + \] \end{document}