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Questions: Algebra BusinessCalculus
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Use the simplex method to maximize \(\displaystyle p = 20 x + 8 y\) subject to \(\displaystyle \begin{cases}80 x + 48 y \leq 3840 \\ 10 x + 53 y \leq 530 \\x \geq 0, y \geq 0 \end{cases}\)
Adding the slack variables \(\displaystyle s\) and \(\displaystyle t\) to the inequalities gives:
\begin{equation*}\begin{cases}80 x + 48 y +s = 3840 \\ 10 x + 53 y+t = 530 \\ - 20 x - 8 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
\(\displaystyle s\) & \(\displaystyle 80\) & \(\displaystyle 48\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 3840\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 10\) & \(\displaystyle 53\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 530\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -20\) & \(\displaystyle -8\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) \\ \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:
\(\displaystyle 8R_2-1R_1\)
\(\displaystyle 4R_3+1R_1\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle x\) & \(\displaystyle 80\) & \(\displaystyle 48\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 3840\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 0\) & \(\displaystyle 376\) & \(\displaystyle -1\) & \(\displaystyle 8\) & \(\displaystyle 0\) & \(\displaystyle 400\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 0\) & \(\displaystyle 16\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 4\) & \(\displaystyle 3840\) \\ \hline
\end{tabular}
There are no negative values in row \(\displaystyle p\) and this is the final tableau.The value of \(\displaystyle x\) is \(\displaystyle 48\). The value of \(\displaystyle t\) is \(\displaystyle 50\). The max value is \(\displaystyle p = 960\)
\begin{question}Use the simplex method to maximize $p = 20 x + 8 y$ subject to $\begin{cases}80 x + 48 y \leq 3840 \\ 10 x + 53 y \leq 530 \\x \geq 0, y \geq 0 \end{cases}$
\soln{9cm}{Adding the slack variables $s$ and $t$ to the inequalities gives:\newline\begin{equation*}\begin{cases}80 x + 48 y +s = 3840 \\ 10 x + 53 y+t = 530 \\ - 20 x - 8 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
$s$ & $80$ & $48$ & $1$ & $0$ & $0$ & $3840$ \\ \hline
$t$ & $10$ & $53$ & $0$ & $1$ & $0$ & $530$ \\ \hline
$p$ & $-20$ & $-8$ & $0$ & $0$ & $1$ & $0$ \\ \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
$8R_2-1R_1$\newline
$4R_3+1R_1$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$x$ & $80$ & $48$ & $1$ & $0$ & $0$ & $3840$ \\ \hline
$t$ & $0$ & $376$ & $-1$ & $8$ & $0$ & $400$ \\ \hline
$p$ & $0$ & $16$ & $1$ & $0$ & $4$ & $3840$ \\ \hline
\end{tabular}\newline
There are no negative values in row $p$ and this is the final tableau.The value of $x$ is $48$. The value of $t$ is $50$. The max value is $p = 960$ }
\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 = 20 x + 8 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%2020%20x%20%2B%208%20y%20" alt="LaTeX: \displaystyle p = 20 x + 8 y " data-equation-content=" \displaystyle p = 20 x + 8 y " /> subject to <img class="equation_image" title=" \displaystyle \begin{cases}80 x + 48 y \leq 3840 \\ 10 x + 53 y \leq 530 \\x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D80%20x%20%2B%2048%20y%20%5Cleq%203840%20%5C%5C%2010%20x%20%2B%2053%20y%20%5Cleq%20530%20%5C%5Cx%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX: \displaystyle \begin{cases}80 x + 48 y \leq 3840 \\ 10 x + 53 y \leq 530 \\x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}80 x + 48 y \leq 3840 \\ 10 x + 53 y \leq 530 \\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}80 x + 48 y +s = 3840 \\ 10 x + 53 y+t = 530 \\ - 20 x - 8 y+p =0 \end{cases} " src="/equation_images/%20%5Cbegin%7Bcases%7D80%20x%20%2B%2048%20y%20%2Bs%20%3D%203840%20%5C%5C%2010%20x%20%2B%2053%20y%2Bt%20%3D%20530%20%5C%5C%20-%2020%20x%20-%208%20y%2Bp%20%3D0%20%5Cend%7Bcases%7D%20" alt="LaTeX: \begin{cases}80 x + 48 y +s = 3840 \\ 10 x + 53 y+t = 530 \\ - 20 x - 8 y+p =0 \end{cases} " data-equation-content=" \begin{cases}80 x + 48 y +s = 3840 \\ 10 x + 53 y+t = 530 \\ - 20 x - 8 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 80 " src="/equation_images/%20%5Cdisplaystyle%2080%20" alt="LaTeX: \displaystyle 80 " data-equation-content=" \displaystyle 80 " /> & <img class="equation_image" title=" \displaystyle 48 " src="/equation_images/%20%5Cdisplaystyle%2048%20" alt="LaTeX: \displaystyle 48 " data-equation-content=" \displaystyle 48 " /> & <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 3840 " src="/equation_images/%20%5Cdisplaystyle%203840%20" alt="LaTeX: \displaystyle 3840 " data-equation-content=" \displaystyle 3840 " /> \\ \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 10 " src="/equation_images/%20%5Cdisplaystyle%2010%20" alt="LaTeX: \displaystyle 10 " data-equation-content=" \displaystyle 10 " /> & <img class="equation_image" title=" \displaystyle 53 " src="/equation_images/%20%5Cdisplaystyle%2053%20" alt="LaTeX: \displaystyle 53 " data-equation-content=" \displaystyle 53 " /> & <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 530 " src="/equation_images/%20%5Cdisplaystyle%20530%20" alt="LaTeX: \displaystyle 530 " data-equation-content=" \displaystyle 530 " /> \\ \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 -20 " src="/equation_images/%20%5Cdisplaystyle%20-20%20" alt="LaTeX: \displaystyle -20 " data-equation-content=" \displaystyle -20 " /> & <img class="equation_image" title=" \displaystyle -8 " src="/equation_images/%20%5Cdisplaystyle%20-8%20" alt="LaTeX: \displaystyle -8 " data-equation-content=" \displaystyle -8 " /> & <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 8R_2-1R_1 " src="/equation_images/%20%5Cdisplaystyle%208R_2-1R_1%20" alt="LaTeX: \displaystyle 8R_2-1R_1 " data-equation-content=" \displaystyle 8R_2-1R_1 " /> <br>
<img class="equation_image" title=" \displaystyle 4R_3+1R_1 " src="/equation_images/%20%5Cdisplaystyle%204R_3%2B1R_1%20" alt="LaTeX: \displaystyle 4R_3+1R_1 " data-equation-content=" \displaystyle 4R_3+1R_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 80 " src="/equation_images/%20%5Cdisplaystyle%2080%20" alt="LaTeX: \displaystyle 80 " data-equation-content=" \displaystyle 80 " /> & <img class="equation_image" title=" \displaystyle 48 " src="/equation_images/%20%5Cdisplaystyle%2048%20" alt="LaTeX: \displaystyle 48 " data-equation-content=" \displaystyle 48 " /> & <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 3840 " src="/equation_images/%20%5Cdisplaystyle%203840%20" alt="LaTeX: \displaystyle 3840 " data-equation-content=" \displaystyle 3840 " /> \\ \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 376 " src="/equation_images/%20%5Cdisplaystyle%20376%20" alt="LaTeX: \displaystyle 376 " data-equation-content=" \displaystyle 376 " /> & <img class="equation_image" title=" \displaystyle -1 " src="/equation_images/%20%5Cdisplaystyle%20-1%20" alt="LaTeX: \displaystyle -1 " data-equation-content=" \displaystyle -1 " /> & <img class="equation_image" title=" \displaystyle 8 " src="/equation_images/%20%5Cdisplaystyle%208%20" alt="LaTeX: \displaystyle 8 " data-equation-content=" \displaystyle 8 " /> & <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 400 " src="/equation_images/%20%5Cdisplaystyle%20400%20" alt="LaTeX: \displaystyle 400 " data-equation-content=" \displaystyle 400 " /> \\ \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 16 " src="/equation_images/%20%5Cdisplaystyle%2016%20" alt="LaTeX: \displaystyle 16 " data-equation-content=" \displaystyle 16 " /> & <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 4 " src="/equation_images/%20%5Cdisplaystyle%204%20" alt="LaTeX: \displaystyle 4 " data-equation-content=" \displaystyle 4 " /> & <img class="equation_image" title=" \displaystyle 3840 " src="/equation_images/%20%5Cdisplaystyle%203840%20" alt="LaTeX: \displaystyle 3840 " data-equation-content=" \displaystyle 3840 " /> \\ \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 48 " src="/equation_images/%20%5Cdisplaystyle%2048%20" alt="LaTeX: \displaystyle 48 " data-equation-content=" \displaystyle 48 " /> . 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 50 " src="/equation_images/%20%5Cdisplaystyle%2050%20" alt="LaTeX: \displaystyle 50 " data-equation-content=" \displaystyle 50 " /> . The max value is <img class="equation_image" title=" \displaystyle p = 960 " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%20960%20" alt="LaTeX: \displaystyle p = 960 " data-equation-content=" \displaystyle p = 960 " /> </p> </p>