model_solutions.tex

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\title{Mathematical Modeling Solutions}

\maketitle

\section{Furniture Company}
Let $x, y, z$ be the number of tables, chairs, and desks that you decide to produce respectively.  

\begin{equation}
\label{eq:furniture_opt}
\begin{array}{rl}
\max~ & 60x + 40y + 100z \vspace{3pt}\\
\textup{s.t.}~ & 6x + 2y + 5z \leq 500 \vspace{3pt}\\
& 2x + 2y + 3z \leq 300 \vspace{3pt}\\
& 4x + 2y + 6z \leq 450 \vspace{3pt}\\
& x, y, z \geq 0 \vspace{3pt}\\
& x, y, z ~\textup{integer}\\
\end{array}
\end{equation}

\section{New England Power}
Let $AW, BW, CW, DW$ be the amounts of gigawatts supplied to Western Massachusetts from power plants A, B, C, D respectively, and let $AE, BE, CE, DE$ be the amounts of gigawatts supplied to Eastern Massachusetts from power plants A, B, C, D respectively.

\begin{equation}
\label{eq:new_england_power_opt}
\begin{array}{rl}
\min~ & 2200AE + 2200BE + 4000CE + 2300DE + 2800AW + 2200BW + 3000CW + 3100DW \vspace{3pt}\\
\textup{s.t.}~ & AE + BE + CE + DE \geq 900,000 \vspace{3pt}\\
& AW + BW + CW + DW \geq 700,000 \vspace{3pt}\\
& AW + AE \leq 800,000 \vspace{3pt}\\
& BW + BE \leq 400,000 \vspace{3pt}\\
& CW + CE \leq 600,000 \vspace{3pt}\\
& DW + DE \leq 350,000 \vspace{3pt}\\
& AW + AE \leq \frac{1}{4}(AE + BE + CE + DE + AW + BW + CW + DW) \vspace{3pt}\\
& AE, BE, CE, DE, AW, BW, CW, DW \geq 0 \\
\end{array}
\end{equation}

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