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Start exercice 3

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optimisation-complexite/exercices.tex
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@ -18,14 +18,16 @@
On considère le programme linéaire suivant~:
\begin{equation*}
Z = 6x_1 + 7x_2 + 8x_3
\end{equation*}
\begin{align*}
x_1 + 2x_2 + x_3 &\leq 100 \\
3x_1 + 4x_2 + 2x_3 &\leq 120 \\
2x_1 + 6x_2 + 4x_3 &\leq 20 \\
x_1, x_2, x_3 &\geq 0
\text{Max}Z = 6x_1 + 7x_2 + 8x_3 \\
\left\{
\begin{array}{l}
x_1 + 2x_2 + x_3 \leq 100 \\
3x_1 + 4x_2 + 2x_3 \leq 120 \\
2x_1 + 6x_2 + 4x_3 \leq 20 \\
x_1, x_2, x_3 \geq 0 \\
\end{array}
\right.
\end{align*}
\begin{enumerate}
@ -61,7 +63,7 @@
\end{align*}
\begin{tabularx}{\linewidth}{|Y|Y|Y|Y|Y|Y|Y|Y|Y|}
\hline
Max & \multicolumn{2}{c}{$C_i$} & 6 & 7 & 8 & 0 & 0 & 0 \\
Max & \multicolumn{2}{|c|}{$C_i$} & 6 & 7 & 8 & 0 & 0 & 0 \\
\hline
$C_B$ & B & b & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ \\
\hline
@ -76,7 +78,7 @@
\end{tabularx}
\begin{tabularx}{\linewidth}{|Y|Y|Y|Y|Y|Y|Y|Y|Y|}
\hline
Max & \multicolumn{2}{c}{$C_i$} & 6 & 7 & 8 & 0 & 0 & 0 \\
Max & \multicolumn{2}{|c|}{$C_i$} & 6 & 7 & 8 & 0 & 0 & 0 \\
\hline
$C_B$ & B & b & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ \\
\hline
@ -91,7 +93,7 @@
\end{tabularx}
\begin{tabularx}{\linewidth}{|Y|Y|Y|Y|Y|Y|Y|Y|Y|}
\hline
Max & \multicolumn{2}{c}{$C_i$} & 6 & 7 & 8 & 0 & 0 & 0 \\
Max & \multicolumn{2}{|c|}{$C_i$} & 6 & 7 & 8 & 0 & 0 & 0 \\
\hline
$C_B$ & B & b & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ \\
\hline
@ -107,4 +109,51 @@
\end{enumerate}
\section{Exercice 3}
On considère le programme linéaire suivant~:
\begin{align*}
\text{Max}Z = x_1 - x_2 \\
\left\{
\begin{array}{l}
2x_1 + x_2 \leq -4 \\
x_1 - x_2 \leq 4 \\
x_1 + x_2 \leq 10 \\
x_1, x_2 \geq 0 \\
\end{array}
\right.
\end{align*}
\begin{itemize}
\item Résoudre le programme par la méthode du simplexe, et déduire la solution optimale $S_1$.
\begin{align*}
\left\{
\begin{array}{l}
-2x_1 + x_2 + x_3 = 4 \\
x_1 - x_2 + x_4 = 4 \\
x_1 + x_2 + x_5 = 10 \\
x_1, x_2, x_3, x_4, x_5 \geq 0 \\
\end{array}
\right.
\end{align*}
\begin{tabularx}{\linewidth}{|Y|Y|Y|Y|Y|Y|Y|Y|Y|}
\hline
Max & \multicolumn{2}{|c|}{$C_i$} & 1 & -1 & 0 & 0 & 0 \\
\hline
$C_B$ & B & b & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ \\
\hline
0 & $x_4$ & 4 & -2 & 1 & 1 & 0 & 0 \\
0 & $x_5$ & 4 & 1 & -1 & 0 & 1 & 0 \\
0 & $x_6$ & 10 & 1 & 1 & 0 & 0 & 1 \\
\hline
\multicolumn{2}{|c|}{$Z_i$} & 0 & 0 & 0 & 0 & 0 & 0 \\
\hline
\multicolumn{3}{|c|}{$C_i - Z_i$} & 1 & -1 & 0 & 0 & 0 \\
\hline
\end{tabularx}
\end{itemize}
\end{document}