\be
	\item Express each LP below in matrix inequality form. Then solve the LP using Maple provided it
is feasible and bounded.
	
		\be
        \item
        \begin{align*}
        \mbox{maximize } z &=6x_1+4x_2  \\
	\mbox{ subject to} \\
	2x_1 + 3x_2 & \leq 9 \\
	x_1 \phantom{ +20y} &\geq 4 \\
	\phantom{ +20x} x_2  &\leq 6  \\
	x_1,x_2 &\geq 0\notag
    % Unique solution
	\end{align*}
		\item \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 &= 10\notag \\
		x_1,x_2 &\geq 0\notag
		\end{align*}
        % Unique Solution	
		\item \begin{align*}
		 \mbox{maximize } z &=-x_1+4x_2  \\
		\mbox{ subject to} \notag\\
		-x_1 +x_2 &\leq 1 \notag\\
		x_1+\x &\leq 3  \notag\\
		x_1+x_2 &\geq 5 \notag \\
		x_1,x_2 &\geq 0\notag
		\end{align*}
        % Infeasible		
		\item \begin{align*}
		 \mbox{minimize } z &=-x_1+4x_2  \\
		\mbox{ subject to} \notag\\
		x_1 +3x_2 &\geq 5 \notag\\
		x_1+x_2 &\geq 4  \notag\\
		x_1-x_2 &\leq 2 \notag \\
		x_1,x_2 &\geq 0\notag
		\end{align*}
        (Hint: First change the LP so that the goal is ``maximize'' $\tilde{z}=-z=x_1-4x_2$.)
        % Minimization (Unique Solution)	
          \item \begin{align*}
		 \mbox{maximize } z &=2x_1-x_2  \\
		\mbox{ subject to} \notag\\
		x_1 +3x_2 &\geq 8 \notag\\
		x_1+x_2 &\geq 4  \notag\\
		x_1-x_2 &\leq 2 \notag \\
        x_1,x_2 &\geq 0\notag
        \end{align*}
        % Unbounded LP
        \item \begin{align*}
	       	 \mbox{minimize } z &=2x_1+3x_2  \\
		\mbox{ subject to} \notag\\
		3x_1 +x_2 &\geq 1 \notag\\
		x_1+x_2 &\leq 6  \notag\\
		x_2 &\geq 0\notag
        \end{align*}
        (Hint: The variable $x_1$ is unrestricted in sign. Define $x_1=x_{1,+}-x_{1,-}$, where
        $x_{1,+}$ and $x_{1,-}$ are nonnegative. Express the LP in terms of the three
        nonnegative decision variables, $x_2$, $x_{1,+}$ and $x_{1,-}$.)
        \ee
    \item In certain situations an optimization problem
    may not appear to be an LP, such as the case when constraints involve certain absolute value inequalities. For example,
    consider the following problem:
    \begin{align*}
	       	 \mbox{minimize } z &=x_1+4x_2  \\
		\mbox{ subject to} \notag\\
		x_1 +2x_2 &\leq 5 \notag\\
		|x_1-x_2| &\leq 2  \notag\\
		x_1, x_2 &\geq 0.\notag
        \end{align*}
    By restating the absolute value constraint as a combination of two linear constraints, show that
    this problem is in fact an LP. Write the LP in matrix inequality form, and determine its solution
    using Maple.
	\item Congratulations! Upon graduating from college, you've immediately been
	offered a high-paying position as president of the Lego
Furniture Company.\index{Lego Furniture Company}\footnote{Based upon Pendegraft, \cite{Pendegraft}, (1997). For this problem it helps to have some Legos, specifically 8 large rectangular
	blocks and 6 small square blocks.}
	Your company produces chairs (each requiring 2 square blocks
	and 1 rectangular block) as well as tables (each requiring 2 square blocks and 2 rectangular blocks) and has available resources consisting of 8 rectangular blocks and 6 square ones. Assume chairs and
tables each sell for \$5 and \$7, respectively, and that your company
 sells all of what it produces.
	\be
	\item Set up an LP whose objective is to maximize your company's revenue. Solve this LP through trial and error, using your Legos if you wish.
	\item Now solve the LP with the aid of Maple.
	\ee

\item (\emph{The Foraging Herbivore Model})\index{Foraging Herbivore Model} In certain parts of the U.S. the
	vole, or common field mouse, is an herbivore whose diet consists
	predominantly of grass and a type of broad-leafed herb known as forb.
	Research suggests that the vole searches for food in a manner that minimizes its
    total foraging time, subject to a set of two constraints.
	\footnote{Based upon Belovsky, \cite{Belovsky_herbivores}, (1984). }
	
    \newpar
	Table \ref{Ta:02x01x02} summarizes grass and forb data
	regarding digestive capacity and 	
	energy content. Food bulk records the extent to which the mass
    of a substance increases after it enters the digestive system and becomes
    liquid-saturated. For example,
    two grams of grass, when consumed, expands to $2 \times 1.64=3.28$ grams
    within the digestive system. To distinguish
    the mass of food prior to consumption from that in the digestive system,
    we use units of gm-dry and gm-wet, respectively.
    \begin{table}[h]
	\begin{center}\caption{Dietary data for vole}\label{Ta:02x01x02}
	\begin{tabular}{|l|c|c|}
	&
	{\scriptsize{Food bulk (gm-wet/gm-dry)}}
	& {\scriptsize{Energy content (kcal/gm)}} \\ \hline
	Grass & 1.64 &  2.11 \\ \hline
	Forb & 2.67 &  2.30 \\
	\end{tabular}
	\end{center}
	\end{table}
	
    The digestive capacity of the vole is 31.2 gm-wet per day, and the vole must consume enough food to meet an energy requirement of at least 13.9 kcal per day. Assume that the vole's foraging rate is 45.55 minutes per gram of grass and 21.87 minutes per gram of forb.
		\be
		\item Let $x_1$ and $x_2$ denote the number of grams of grass and number of grams of forb,
	respectively, consumed by the vole on a given day. Construct an LP that minimizes the vole's total daily foraging time given the digestive capacity and energy constraints. Verify that your units make sense for each constraint and for your
		objective function. Then write this LP in matrix inequality form.
		\item Solve the LP using Maple.
		\ee
	\ee
