A meat packing plant produces $480$ hams, $400$ pork bellies and $230$ picnic hams every day; each of these products can be either fresh or smoked. The total number of hams, bellies and picnics that can be smoked during a normal working day is $420$; in addition, up to $250$ products can be smoked on overtime at higher cost. The net profits are as follows:
\begin{array}{|c|c|c|} \hline & Fresh & \mbox{Smoked on regular time} & \mbox{Smoked on overtime} \\ \hline Hams & 8$ & 14$ & 11$\\ Bellies & 4$ & 12$ &7$ \\ Picnics & 4$ & 13$ & 9$\\ \hline \end{array}
For example the following schedule yields a total net profit of $9,965$ $.
\begin{array}{|c|c|c|} \hline & Fresh & \mbox{Smoked on regular time} & \mbox{Smoked on overtime} \\ \hline Hams & 165 & 280 & 35\\ Bellies & 295 & 70 & 35 \\ Picnics & 55 & 70 & 105\\ \hline \end{array}
The objective is to find what maximizes the total net profit. Formulate as a Linear Programing problem in the standard form.
My try:
$x_1, x_2,x_3$ represents hams, bellies and picnics \begin{equation*} \begin{cases} \max C^Tx \\ Ax \le b\\ x\ge 0 \end{cases} \end{equation*}
\begin{equation*} \begin{cases} \max x_1,x_2,x_3 \\ \begin{pmatrix} 8 & 14 & 11\\ 4 & 12 &7 \\ 4 & 13 & 9\\ \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \\ x_3\\ \end{pmatrix} \le\begin{pmatrix} 480\\ 400\\ 230\\ \end{pmatrix}\\ \forall i, x_i\ge 0 \end{cases} \end{equation*}
Which is very different from the answer given by the book...
The answer given by the book:
\begin{equation*} \begin{cases} \max 6x_1 +3x_2+8x_3+ 3x_4+9x_5+5x_6 \\ x1 +x2 \le 480\\ x_3+x_4\le 400\\ x_5+x_6\le 230\\ x_1 + x_3 + x_5\le 420\\ x_2 + x_4 + x_6 \le 250\\ \forall i,x_i\ge 0 \end{cases} \end{equation*}
