\documentclass[11 pt]{article}

\setlength{\oddsidemargin}{-.45in}
%\setlength{\evensidemargin}{-.5in}
\setlength{\textwidth}{7.1in}
\setlength{\topmargin}{-0.95in}
\setlength{\textheight}{9.7in}

\usepackage{graphics}

%\usepackage{epsfig}
%\usepackage{wrapfig}
%\input epsf.sty 

\usepackage{maple2e}
 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Maple Output}
\DefineParaStyle{Warning}
\DefineCharStyle{2D Math}
\DefineCharStyle{2D Output}

%\usepackage{bbding}
%\usepackage{pstcol,pst-node,pstricks,pst-text,pst-grad,pst-3d}
\usepackage{pstcol,pstricks,pst-text,pst-grad,pst-node}
\usepackage{graphicx}
\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{amsfonts}
%\usepackage[intlimits]{amsmath}
\usepackage{wasysym}
\usepackage{amsthm}
\usepackage{multicol}
\usepackage{psfrag}
%\usepackage{pxfonts} % combine with \varointctrclockwise\limits_{C_2} for counterclockwise line integral
%comment out wasysym, amstheorem, and theoremstyle[plain]

\newtheorem*{weak_duality}{Weak Duality}
\newtheorem*{strong_duality}{Strong Duality}
\newtheorem*{duality_theorem}{The Duality Theorem}

%COMMANDS
\newcommand{\be}{\begin{enumerate}} 
\newcommand{\ee}{\end{enumerate}}

\newcommand{\bi}{\begin{itemize}} 
\newcommand{\ei}{\end{itemize}}

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\Nat}{\mathbb N}
\newcommand{\Q}{\mathbb Q}
\newcommand{\Z}{\mathbb Z}

\newcommand{\vecc}{\mathbf c}
\newcommand{\vecx}{\mathbf x}
\newcommand{\vecs}{\mathbf s}
\newcommand{\vece}{\mathbf e}
\newcommand{\veca}{\mathbf a}

\newcommand{\vecxs}{{\mathbf {x}}_s}
\newcommand{\vecxsh}{{\mathbf {x}}_{s,h}}
\newcommand{\vecxsp}{{\mathbf {x}}_{s,p}}
\newcommand{\vecb}{\mathbf b}
\newcommand{\vecu}{\mathbf u}
\newcommand{\vecv}{\mathbf v}
\newcommand{\vecw}{\mathbf w}
\newcommand{\vecn}{\mathbf n}
\newcommand{\vecy}{\mathbf y}
\newcommand{\vecz}{\mathbf z}
\newcommand{\onevec}{\ensuremath{\mathbf{1}}}
\newcommand{\zerovec}{\ensuremath{\mathbf{0}}}
\newcommand{\deltavec}{\boldsymbol{\delta}}

\newcommand{\bs}{\blacksquare}

\newcommand{\F}{\mathbf F}
\newcommand{\T}{\mathbf T}
\newcommand{\N}{\mathbf N}
\newcommand{\B}{\mathbf B}

\newcommand{\tor}{\boldsymbol{\tau}}


\newcommand{\dr}{d\mathbf r}
\newcommand{\dS}{d\mathbf S}

\newcommand{\la}{\langle} 
\newcommand{\ra}{\rangle}

\newcommand{\x}{\phantom{ +x_1}}

\newcommand{\bv}{\mbox{BV}}
\newcommand{\BV}{\mbox{BV }}
\newcommand{\NBV}{\mbox{NBV }}

\newcommand{\LPC}{\emph{LinPro Petroleum Company }}
\newcommand{\LP}{\emph{LinPro }}

\newcommand{\newpar}{\vspace{.15in}\noindent}
\newcommand{\noi}{\noindent}

\newcommand{\ds}{\displaystyle}
\newcommand{\cbox}{\framebox{\parbox[c][.05 in][c]{.05 in}{\makebox[.05in][l]{}} } \hspace{.025in}}

\newcommand{\marginlabel}[1]{\mbox{}\marginpar{\raggedleft\hspace{0pt}#1}}

\newcommand{\mf}[1]{ {\ttfamily{\upshape{#1}}  }}

%LP
%\begin{align}
% \mbox{maximize } z &=10x_1+8x_2  \\
%	\mbox{ subject to} \notag\\ 
%	x_1 \phantom{ +20x_2} &\leq 10 \notag\\ 
%	x_1+2x_2 &\leq 20  \notag\\
%	10x_1+2x_2 &\leq 120 \notag \\
%	x_1,x_2 &\geq 0.\notag
%	\end{align}	

%TABLEAU

%\begin{table}[h]
%\centering
%\begin{tabular}{|c||c|c|c|c|c||c|}
%\hline
%$z$ & $x_1$ & $x_2$ & $s_1$ & $s_2$ & $s_3$ & RHS \\
%\hline \hline 
%1 & 0 & -8 & 10 & 0 & 0 & 100 \\ \hline 
%0 &  1 & 0 & 1 & 0 & 0 & 10 \\ \hline 
%0 &  0 & 2 & -1 & 1 & 0 & 10 \\ \hline 
%0 &  0 & 7 & -10 & 0 & 1 & 20 \\ \hline 
%\end{tabular}
%\caption{Tableau after first iteration}\label{Ta:petroleum_second_tableau}
%\end{table}

%LABELS
%Section: \label{S:Name}
%Theorem/Explore: \label{Th:Name}
%Table: \label{Ta:Name}
%Reference \cite{Burch}

%CUSTOMIZED ITEMIZED LIST SYMBOL.
%\begin{list}{default_label}{\setlength{\leftmargin}{.5in} \setlength{\rightmargin}{.5in}}

%CHANGE TO DOUBLE AND 1.5 SPACING
%\renewcommand{\baselinestretch}{1.5}

%****************************************
%LABELED EQUATION. TO REFER TO THE EQUATION LATER IN TEXT TYPE "(\ref{E:quadratic})".
%begin{equation} \label{E:quadratic}
% f(x)=x^2 \text{hi}
%\end{equation}

%USE OF SPLIT
%begin{equation} \label{E:quadratic}
% \begin{split} &f(x)=long formula \\
% & broken over two lines \end{split} 
%\end{equation}

%ALIGNED FORMULA
%\begin{align*}
%f(x)&= x^2+2x+1 \\ \intertext{because..}
%   &=(x+1)^2 
%\end{align*}

%TWO ALIGNED FORMULA COLUMNS
%\begin{alignat}{2}
%P_0(x)&=1, & \qquad P_1(x)&=x \notag \\
%P_2(x)&=\frac{1}{2}(3x^2-1), & \qquad P_3(x)&=\frac{1}{2}(5x^3-3x) \notag 
 % \end{alignat}
 
%GATHERING FORMULAS: TWO EQUATIONS, WITH NO ALIGNMENT
% \begin{gather*}
% x^2+2x+1
%  (x+1)^2
% \end{gather*}

%SPLITTING A LONG FORMULA WITH MULTLINE
%\begin{multline*}
% x^5+x^4+x^3 \\
% +x^2+x+1  
%\end{multline*}

%BOX
%The second argument in the parbox (1in} controls the box width; first argument controls height
%\framebox{\parbox[c][.2in][c]{1in}{\makebox[.5in][l]{}} }

%DOTFILL TO CHECKBOX
%\dotfill \cbox

%CONTOUR AND DOUBLE INTEGRALS WITH CONTOUR/REGION BELOW INTEGRAL SIGN
% \oint\limits_C
% \iint\limits_D

%SAMPLE TRUE-FALSE QUESTION
%	\item \cbox 
%	If $A$ and $B$ are matrices and $AB=0$, then $A=0$ or $B=0$
%	\vspace{2.3in}

%FIGURE WITH GRAPHICS PACKAGE
%\begin{figure}[h]
%\hfil {  \includegraphics{c:/images/OR/blending_feasible_region.eps} }  \hfil
%\caption{Feasible region for Fishback Petroleum LP}\label{Fi:blending_feasible_region} 
%\end{figure}

%scale with \scalebox{.5}{ }

%FIGURE WITH EPSFIG PACKAGE: height SCALES PROPORTIONALY (SEE LATEX COMPANION, P. 317)
% \epsfig{file=c:/images/Lanczos/roof_function.eps,height=1cm.}

%TWO FIGURES SIDE BY SIDE, SCALED, AND LABELLED SEPARATELY
%\begin{figure}[h]
%\centering
%\raisebox{1.4in}{I)}
%\scalebox{.6}{\includegraphics{c:/mth300/Exams/exam2_gradientfield3.eps} } 
%\raisebox{1.4in}{II)}  
%\scalebox{.6}{\includegraphics{c:/mth300/Exams/exam2_gradientfield1.eps} }
%\end{figure}

%TEXT AND FIGURE SIDE BY SIDE
%
%\begin{tabular}{lr}
%
%\raisebox{1.1in}{\parbox[b]{3in}{Suppose that $f$ is the function whose graph is shown to the right. Assume
%that this function fails to have partial derivatives at every point along the $x$- and $y$-axes.}}
%
%&
%
%\epsfig{file=c:/images/Lanczos/roof_function.eps,height=2in.}
%
%\end{tabular}
%
%\vspace{.2in}
%******************************************

%MATRIX COMMANDS
%\begin{bmatrix}a & b \\ c & d \end{bmatrix}

%TABLE COMMANDS
%\begin{tabular}{c|c} $x$ & $y$ \\ \hline a & b \\ c & d \\ e & f \ \end{tabular}

%VARIABLE SPACED COLUMNS IN A TABLE
%\begin{tabular}{p{2.5in} p{2.5in}}
%	$\det(BA)$ &   $\det(A^{-1})$ \vspace{2in} \\
%	$\det(3B)$ &   $\displaystyle \det \left( \begin{bmatrix} 1 & 2 & 4 \\ 2 & 0 & -2 \\ 1 & 2 & 2 \end{bmatrix} \right)$ 
%	\end{tabular}

%TABLE/GRAPH ON ONE LINE COMMANDS
%\begin{figure}[h]
%\centering
%\begin{tabular}{cc}
%	\raisebox{.75in}{\begin{tabular}[b]{c|c} $x$ & $y$ \\ \hline a & b \\ c & d \\ e & f  \end{tabular} } &	
%	\hspace{1in} \includegraphics[2in,2in]{c:/mth310/project/Hamilton.bmp} 
%\end{tabular}
%\caption{{\scriptsize Table and graph for $f$.}}
%\end{figure}

%***********************************************
%BOX ENVIRONMENTS

%TEXT IN A BOX VIA \framebox[width][position]{text} WHERE position = l,r,c
%\framebox[2in][l]{Name}

%TEXT IN A BOX VIA \fbox
%\setlength{\fboxrule}{2pt}\setlength{\fboxsep}{2mm}
%\fbox{Text in a box}

%RAISEBOX / ROTATEBOX 
%\raisebox{lift}{contents}  \rotatebox{-90}{contents} 

%SCALEBOX
%\scalebox{fraction}{contents}
%\scalebox{horz fraction }[vert fraction]{contents} 

%PARBOX: POSITIONED RELATIVE TO BASELINE c, t, b
%\parbox[pos]{width}{text}

%MINIPAGE: CREATES A MINIVERSION OF THE PAGE WITHIN THE PAGE; CAN BE USED INSIDE FIGURE AND TABLE ENVS
%\begin{minipage}[pos]{width} text
%\end{minipage}

%**************************************************

%\footnote{text}

%\marginpar[left-text]{right-text} USE SCRIPTSIZE AND CHANGE TEXTWIDTH AT START OF FILE

%PIECEWISE DEFINED FUNCTION
%p(t)=\begin{cases}
%		\frac{1}{2},	&\text{if $t \in [-1,1]$};\\
%		0,	&\text{otherwise,}
%	\end{cases}

%WEB URL
%\texttt{http://www.phy.syr.edu/courses/java-suite/crosspro.html} 

%MAPLE FILE NAME
%\textbf{extrema.mws}

% POSITIONING
% TO CREATE ENVIRONOMENT: \begin{  center, flushleft, flushright,}
% WITHIN ANOTHER ENVIRONMENT \centering, \raggedright, \raggedleft
\begin{document}
\begin{center}
{\Large {{\textbf {Eating for Cheap at McDonalds}} }} \\
\end{center}

\newpar	
A classic application of linear programming is what is known as the \emph{diet problem}. There is no single
such problem, but the central theme is the same, namely to optimize some function, i.e. minimize cost, maximize protein, etc., subject to a set of constraints, e.g. nutritional needs. The \emph{Third World Diet Model} from the text exercises is a simple two-variable example of such a problem. This activity involves many more variables.

\newpar
The following table summarizes cost and nutrional information for
various McDonald's menu
items.


\begin{table}[h]
\begin{center}
	\begin{tabular}{|l|c|c|c|c|c|c|c|c|} 
	\hline
	& & \multicolumn{7}{c|}{Nutritional Information}\\ \hline
	Item & Price (\$) & Cal & \begin{tabular}{l} Carb \\ (gm) \end{tabular} & 
	\begin{tabular}{l} Protein \\ (gm) \end{tabular} & Vit A  & Vit C  & Calc  & Iron \\ \hline

	\begin{tabular}{l} Quarter Pounder \\ with Cheese \end{tabular} & 1.84 & 510 & 34 & 28 & 15 & 6 & 30 & 20 \\ \hline 
	\begin{tabular}{l} McLean Deluxe \\ with Cheese \end{tabular} & 2.19 & 370 & 35 & 24 & 15 & 10 & 20 & 20 \\ \hline
	Big Mac & 1.84 & 500 & 42 & 25 & 6 & 2 & 25 & 20 \\ \hline
	Filet-O-Fish & 1.44 & 370 & 38 & 14 & 2 & 0 & 15 & 10 \\ \hline
	McGrilled Chicken & 2.29 & 400 & 42 & 31 & 8 & 15 & 15 & 8 \\ \hline
	Small Fries & .77 & 220 & 26 & 3 & 0 & 15 & 0 & 2 \\ \hline
	Sausage McMuffin & 1.29 & 345 & 27 & 15 & 4 & 0 & 20 & 15 \\ \hline
	Lowfat Milk & .60 & 110 & 12 & 9 & 10 & 4 & 30 & 0 \\ \hline 
	Orange Juice & .72 &  80 & 20 & 1 & 2 & 120 & 2 & 2 \\ \hline
	\end{tabular}
\end{center}	
	\end{table}


\newpar
Here is a table summarizing the average adult's daily nutritional guidelines.
\begin{table}[h]
	\begin{center}
	\begin{tabular}{|c|c|c|} 
	\hline
	Category & Minimum & Maximum \\ \hline 
	Cal & 2000 & none  \\ \hline
	Carb & 350 & 375 \\ \hline
	Protein & 55 & none \\ \hline
	Vit A & 100 & none  \\ \hline 
Vit C & 100 & none  \\ \hline 
Calc & 100 & none  \\ \hline 
Iron & 100 & none  \\ \hline 
	\end{tabular}
	\end{center}
	\end{table}

\newpar
Set up and solve an LP that determines the daily diet consisting only of McDonald's menu items that meets these nutritional 
guidelines and does so at a minimum cost. Solve this LP using either Maple and/or Excel.


\end{document}