\be
\item Use the dual simplex algorithm to solve each of the following LPs.
\be
\item
	\begin{align*}
		 \mbox{minimize } z &=x_1 + 4x_2  \\
		\mbox{ subject to} \\
		x_1+3x_2 &\geq 1 \\
        4x_1+18x_2 &\geq 5 \\
		x_1,x_2 &\geq 0
		\end{align*}
\item
    \begin{align*}
		 \mbox{minimize } z &=x_1 + x_2  \\
		\mbox{ subject to} \\
        x_1 \phantom{+x_2} \leq 8 \\
		-x_1+x_2 &\leq 4 \\
        -x_1+2x_2 &\geq 6 \\
         2x_1+x_2 &\leq 25 \\
          3x_1+x_2 &\geq 18 \\
           -x_1+2x_2 &\geq 6 \\
		x_1,x_2 &\geq 0
		\end{align*}
\ee
	
\item Consider again the scenario when a third stock, stock C, is added to each of the fuel
types in the \LP LP. Assume that each gallon of premium requires 5 units
of stock C, each unit of regular unleaded requires 6 units of stock C. What is the minimum
number of available units of stock C that will change the optimal solution from its value
of $(x_1,x_2)=(4,10)$ in the original LP?
	
\ee
