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Questions: Algebra BusinessCalculus
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Use the simplex method to maximize \(\displaystyle p = 24 x + 3 y\) subject to \(\displaystyle \begin{cases}76 x + 47 y \leq 3572 \\ 62 x + 59 y \leq 3658 \\x \geq 0, y \geq 0 \end{cases}\)
Adding the slack variables \(\displaystyle s\) and \(\displaystyle t\) to the inequalities gives:
\begin{equation*}\begin{cases}76 x + 47 y +s = 3572 \\ 62 x + 59 y+t = 3658 \\ - 24 x - 3 y+p =0 \end{cases}\end{equation*}This gives the first tableau:\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle s\) & \(\displaystyle 76\) & \(\displaystyle 47\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 3572\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 62\) & \(\displaystyle 59\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 3658\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -24\) & \(\displaystyle -3\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) \\ \hline
\end{tabular}
The pivot row is \(\displaystyle s\) and the pivot column is \(\displaystyle x\) . The departing variable is \(\displaystyle s\) and the incoming variable is \(\displaystyle x\). Pivoting using the row operations:
\(\displaystyle 38R_2-31R_1\)
\(\displaystyle 19R_3+6R_1\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle x\) & \(\displaystyle 76\) & \(\displaystyle 47\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 3572\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 0\) & \(\displaystyle 785\) & \(\displaystyle -31\) & \(\displaystyle 38\) & \(\displaystyle 0\) & \(\displaystyle 28272\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 0\) & \(\displaystyle 225\) & \(\displaystyle 6\) & \(\displaystyle 0\) & \(\displaystyle 19\) & \(\displaystyle 21432\) \\ \hline
\end{tabular}
There are no negative values in row \(\displaystyle p\) and this is the final tableau.The value of \(\displaystyle x\) is \(\displaystyle 47\). The value of \(\displaystyle t\) is \(\displaystyle 744\). The max value is \(\displaystyle p = 1128\)
\begin{question}Use the simplex method to maximize $p = 24 x + 3 y$ subject to $\begin{cases}76 x + 47 y \leq 3572 \\ 62 x + 59 y \leq 3658 \\x \geq 0, y \geq 0 \end{cases}$
\soln{9cm}{Adding the slack variables $s$ and $t$ to the inequalities gives:\newline\begin{equation*}\begin{cases}76 x + 47 y +s = 3572 \\ 62 x + 59 y+t = 3658 \\ - 24 x - 3 y+p =0 \end{cases}\end{equation*}This gives the first tableau:\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$s$ & $76$ & $47$ & $1$ & $0$ & $0$ & $3572$ \\ \hline
$t$ & $62$ & $59$ & $0$ & $1$ & $0$ & $3658$ \\ \hline
$p$ & $-24$ & $-3$ & $0$ & $0$ & $1$ & $0$ \\ \hline
\end{tabular}\newline
The pivot row is $s$ and the pivot column is $x$ . The departing variable is $s$ and the incoming variable is $x$. Pivoting using the row operations:\newline
$38R_2-31R_1$\newline
$19R_3+6R_1$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$x$ & $76$ & $47$ & $1$ & $0$ & $0$ & $3572$ \\ \hline
$t$ & $0$ & $785$ & $-31$ & $38$ & $0$ & $28272$ \\ \hline
$p$ & $0$ & $225$ & $6$ & $0$ & $19$ & $21432$ \\ \hline
\end{tabular}\newline
There are no negative values in row $p$ and this is the final tableau.The value of $x$ is $47$. The value of $t$ is $744$. The max value is $p = 1128$ }
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Use the simplex method to maximize <img class="equation_image" title=" \displaystyle p = 24 x + 3 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%2024%20x%20%2B%203%20y%20" alt="LaTeX: \displaystyle p = 24 x + 3 y " data-equation-content=" \displaystyle p = 24 x + 3 y " /> subject to <img class="equation_image" title=" \displaystyle \begin{cases}76 x + 47 y \leq 3572 \\ 62 x + 59 y \leq 3658 \\x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D76%20x%20%2B%2047%20y%20%5Cleq%203572%20%5C%5C%2062%20x%20%2B%2059%20y%20%5Cleq%203658%20%5C%5Cx%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX: \displaystyle \begin{cases}76 x + 47 y \leq 3572 \\ 62 x + 59 y \leq 3658 \\x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}76 x + 47 y \leq 3572 \\ 62 x + 59 y \leq 3658 \\x \geq 0, y \geq 0 \end{cases} " /> </p> </p><p> <p>Adding the slack variables <img class="equation_image" title=" \displaystyle s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> and <img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> to the inequalities gives:<br> <img class="equation_image" title=" \begin{cases}76 x + 47 y +s = 3572 \\ 62 x + 59 y+t = 3658 \\ - 24 x - 3 y+p =0 \end{cases} " src="/equation_images/%20%5Cbegin%7Bcases%7D76%20x%20%2B%2047%20y%20%2Bs%20%3D%203572%20%5C%5C%2062%20x%20%2B%2059%20y%2Bt%20%3D%203658%20%5C%5C%20-%2024%20x%20-%203%20y%2Bp%20%3D0%20%5Cend%7Bcases%7D%20" alt="LaTeX: \begin{cases}76 x + 47 y +s = 3572 \\ 62 x + 59 y+t = 3658 \\ - 24 x - 3 y+p =0 \end{cases} " data-equation-content=" \begin{cases}76 x + 47 y +s = 3572 \\ 62 x + 59 y+t = 3658 \\ - 24 x - 3 y+p =0 \end{cases} " /> This gives the first tableau:\begin{tabular}{|c|c|c|c|c|c|c|}\hline
<img class="equation_image" title=" \displaystyle " src="/equation_images/%20%5Cdisplaystyle%20%20" alt="LaTeX: \displaystyle " data-equation-content=" \displaystyle " /> & <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> & <img class="equation_image" title=" \displaystyle y " src="/equation_images/%20%5Cdisplaystyle%20y%20" alt="LaTeX: \displaystyle y " data-equation-content=" \displaystyle y " /> & <img class="equation_image" title=" \displaystyle s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> & <img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> & <img class="equation_image" title=" \displaystyle p " src="/equation_images/%20%5Cdisplaystyle%20p%20" alt="LaTeX: \displaystyle p " data-equation-content=" \displaystyle p " /> & <img class="equation_image" title=" \displaystyle " src="/equation_images/%20%5Cdisplaystyle%20%20" alt="LaTeX: \displaystyle " data-equation-content=" \displaystyle " /> \\ \hline
<img class="equation_image" title=" \displaystyle s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> & <img class="equation_image" title=" \displaystyle 76 " src="/equation_images/%20%5Cdisplaystyle%2076%20" alt="LaTeX: \displaystyle 76 " data-equation-content=" \displaystyle 76 " /> & <img class="equation_image" title=" \displaystyle 47 " src="/equation_images/%20%5Cdisplaystyle%2047%20" alt="LaTeX: \displaystyle 47 " data-equation-content=" \displaystyle 47 " /> & <img class="equation_image" title=" \displaystyle 1 " src="/equation_images/%20%5Cdisplaystyle%201%20" alt="LaTeX: \displaystyle 1 " data-equation-content=" \displaystyle 1 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 3572 " src="/equation_images/%20%5Cdisplaystyle%203572%20" alt="LaTeX: \displaystyle 3572 " data-equation-content=" \displaystyle 3572 " /> \\ \hline
<img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> & <img class="equation_image" title=" \displaystyle 62 " src="/equation_images/%20%5Cdisplaystyle%2062%20" alt="LaTeX: \displaystyle 62 " data-equation-content=" \displaystyle 62 " /> & <img class="equation_image" title=" \displaystyle 59 " src="/equation_images/%20%5Cdisplaystyle%2059%20" alt="LaTeX: \displaystyle 59 " data-equation-content=" \displaystyle 59 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 1 " src="/equation_images/%20%5Cdisplaystyle%201%20" alt="LaTeX: \displaystyle 1 " data-equation-content=" \displaystyle 1 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 3658 " src="/equation_images/%20%5Cdisplaystyle%203658%20" alt="LaTeX: \displaystyle 3658 " data-equation-content=" \displaystyle 3658 " /> \\ \hline
<img class="equation_image" title=" \displaystyle p " src="/equation_images/%20%5Cdisplaystyle%20p%20" alt="LaTeX: \displaystyle p " data-equation-content=" \displaystyle p " /> & <img class="equation_image" title=" \displaystyle -24 " src="/equation_images/%20%5Cdisplaystyle%20-24%20" alt="LaTeX: \displaystyle -24 " data-equation-content=" \displaystyle -24 " /> & <img class="equation_image" title=" \displaystyle -3 " src="/equation_images/%20%5Cdisplaystyle%20-3%20" alt="LaTeX: \displaystyle -3 " data-equation-content=" \displaystyle -3 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 1 " src="/equation_images/%20%5Cdisplaystyle%201%20" alt="LaTeX: \displaystyle 1 " data-equation-content=" \displaystyle 1 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> \\ \hline
\end{tabular}<br>
The pivot row is <img class="equation_image" title=" \displaystyle s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> and the pivot column is <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> . The departing variable is <img class="equation_image" title=" \displaystyle s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> and the incoming variable is <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> . Pivoting using the row operations:<br>
<img class="equation_image" title=" \displaystyle 38R_2-31R_1 " src="/equation_images/%20%5Cdisplaystyle%2038R_2-31R_1%20" alt="LaTeX: \displaystyle 38R_2-31R_1 " data-equation-content=" \displaystyle 38R_2-31R_1 " /> <br>
<img class="equation_image" title=" \displaystyle 19R_3+6R_1 " src="/equation_images/%20%5Cdisplaystyle%2019R_3%2B6R_1%20" alt="LaTeX: \displaystyle 19R_3+6R_1 " data-equation-content=" \displaystyle 19R_3+6R_1 " /> <br>
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
<img class="equation_image" title=" \displaystyle " src="/equation_images/%20%5Cdisplaystyle%20%20" alt="LaTeX: \displaystyle " data-equation-content=" \displaystyle " /> & <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> & <img class="equation_image" title=" \displaystyle y " src="/equation_images/%20%5Cdisplaystyle%20y%20" alt="LaTeX: \displaystyle y " data-equation-content=" \displaystyle y " /> & <img class="equation_image" title=" \displaystyle s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> & <img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> & <img class="equation_image" title=" \displaystyle p " src="/equation_images/%20%5Cdisplaystyle%20p%20" alt="LaTeX: \displaystyle p " data-equation-content=" \displaystyle p " /> & <img class="equation_image" title=" \displaystyle " src="/equation_images/%20%5Cdisplaystyle%20%20" alt="LaTeX: \displaystyle " data-equation-content=" \displaystyle " /> \\ \hline
<img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> & <img class="equation_image" title=" \displaystyle 76 " src="/equation_images/%20%5Cdisplaystyle%2076%20" alt="LaTeX: \displaystyle 76 " data-equation-content=" \displaystyle 76 " /> & <img class="equation_image" title=" \displaystyle 47 " src="/equation_images/%20%5Cdisplaystyle%2047%20" alt="LaTeX: \displaystyle 47 " data-equation-content=" \displaystyle 47 " /> & <img class="equation_image" title=" \displaystyle 1 " src="/equation_images/%20%5Cdisplaystyle%201%20" alt="LaTeX: \displaystyle 1 " data-equation-content=" \displaystyle 1 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 3572 " src="/equation_images/%20%5Cdisplaystyle%203572%20" alt="LaTeX: \displaystyle 3572 " data-equation-content=" \displaystyle 3572 " /> \\ \hline
<img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 785 " src="/equation_images/%20%5Cdisplaystyle%20785%20" alt="LaTeX: \displaystyle 785 " data-equation-content=" \displaystyle 785 " /> & <img class="equation_image" title=" \displaystyle -31 " src="/equation_images/%20%5Cdisplaystyle%20-31%20" alt="LaTeX: \displaystyle -31 " data-equation-content=" \displaystyle -31 " /> & <img class="equation_image" title=" \displaystyle 38 " src="/equation_images/%20%5Cdisplaystyle%2038%20" alt="LaTeX: \displaystyle 38 " data-equation-content=" \displaystyle 38 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 28272 " src="/equation_images/%20%5Cdisplaystyle%2028272%20" alt="LaTeX: \displaystyle 28272 " data-equation-content=" \displaystyle 28272 " /> \\ \hline
<img class="equation_image" title=" \displaystyle p " src="/equation_images/%20%5Cdisplaystyle%20p%20" alt="LaTeX: \displaystyle p " data-equation-content=" \displaystyle p " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 225 " src="/equation_images/%20%5Cdisplaystyle%20225%20" alt="LaTeX: \displaystyle 225 " data-equation-content=" \displaystyle 225 " /> & <img class="equation_image" title=" \displaystyle 6 " src="/equation_images/%20%5Cdisplaystyle%206%20" alt="LaTeX: \displaystyle 6 " data-equation-content=" \displaystyle 6 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 19 " src="/equation_images/%20%5Cdisplaystyle%2019%20" alt="LaTeX: \displaystyle 19 " data-equation-content=" \displaystyle 19 " /> & <img class="equation_image" title=" \displaystyle 21432 " src="/equation_images/%20%5Cdisplaystyle%2021432%20" alt="LaTeX: \displaystyle 21432 " data-equation-content=" \displaystyle 21432 " /> \\ \hline
\end{tabular}<br>
There are no negative values in row <img class="equation_image" title=" \displaystyle p " src="/equation_images/%20%5Cdisplaystyle%20p%20" alt="LaTeX: \displaystyle p " data-equation-content=" \displaystyle p " /> and this is the final tableau.The value of <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> is <img class="equation_image" title=" \displaystyle 47 " src="/equation_images/%20%5Cdisplaystyle%2047%20" alt="LaTeX: \displaystyle 47 " data-equation-content=" \displaystyle 47 " /> . The value of <img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> is <img class="equation_image" title=" \displaystyle 744 " src="/equation_images/%20%5Cdisplaystyle%20744%20" alt="LaTeX: \displaystyle 744 " data-equation-content=" \displaystyle 744 " /> . The max value is <img class="equation_image" title=" \displaystyle p = 1128 " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%201128%20" alt="LaTeX: \displaystyle p = 1128 " data-equation-content=" \displaystyle p = 1128 " /> </p> </p>