\be
\item Each of the following LP's has been assigned in a previous section. Now solve each using the revised simplex algorithm.
	\be
	\item Section \ref{S:Simplex}, Exercise 1
		\begin{align*}
		 \mbox{maximize } z &=3x_1+2x_2  \\
		\mbox{ subject to} \notag\\ 
		x_1 \phantom{ +20x_2} &\leq 4 \notag\\
		x_1+3x_2 &\leq 15  \notag\\
		2x_1+x_2 &\leq 10 \notag \\
		x_1,x_2 &\geq 0\notag
		\end{align*}	
	
	\item Section \ref{S:Simplex}, Exercise 1
		\begin{align}
		 \mbox{maximize } z &=4x_1 + x_2 +5x_3  \\
		\mbox{ subject to} \notag\\ 
		2x_1+x_2+3x_3 &\leq 14 \notag\\
		6x_1+3x_2 +3x_3 &\leq 22  \notag\\
		2x_1+3x_2  \x &\leq 14 \notag \\
		x_1,x_2,x_3 &\geq 0\notag
		\end{align}	
	\item Section \ref{S:Big_M}, Exercise 2
		\begin{align*}
		\mbox{minimize } z &=-x_1+4x_2   \\
		\mbox{ subject to} \\ 
		x_1 +3x_2 &\geq 8 \\
		x_1+x_2 &\geq 4  \\
		x_1-x_2 &\leq 2  \\
		x_1,x_2 &\geq 0
		\end{align*}
	\ee
\item Use the dual simplex algorithm to solve the following LP.
	\begin{align}
		 \mbox{maximize } z &=-x_1 -2 x_2  \\
		\mbox{ subject to} \notag\\ 
		-2x_1+2x_2+x_3 &\leq 8 \notag\\
		8x_1+2x_2 -2x_3 &\leq 5  \notag\\
		x_1,x_2,x_3 &\geq 0\notag
		\end{align}	
\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 current value
of $(x_1,x_2)=(4,10)$. Answer this question both graphically and using the dual simplex algorithm.
\item Consider the Herbivore Foraging Model LP from Sections \ref{S:LP_Graphical} and \ref{S:Big_M}. 
	\be
	\item Suppose that the vole begins a diet that requires it to limit its total grass and forb
	intake to 15 grams per day. Add this new constraint to the original model, and find the optimal solution 
	that maximizes
	the vole's daily consumption.
	\item Repeat the preceding question under the assumption that the vole limits its intake to 12 grams per
	day.
	\ee

	
	
	
	
\ee
