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Questions: Algebra BusinessCalculus
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Use the simplex method to maximize \(\displaystyle p = 7 x + 18 y\) subject to \(\displaystyle \begin{cases}86 x + 29 y \leq 2494 \\ 31 x + 62 y \leq 1922 \\x \geq 0, y \geq 0 \end{cases}\)
Adding the slack variables \(\displaystyle s\) and \(\displaystyle t\) to the inequalities gives:
\begin{equation*}\begin{cases}86 x + 29 y +s = 2494 \\ 31 x + 62 y+t = 1922 \\ - 7 x - 18 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
\(\displaystyle s\) & \(\displaystyle 86\) & \(\displaystyle 29\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 2494\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 31\) & \(\displaystyle 62\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 1922\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -7\) & \(\displaystyle -18\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) \\ \hline
\end{tabular}
The pivot row is \(\displaystyle t\) and the pivot column is \(\displaystyle y\) . The departing variable is \(\displaystyle t\) and the incoming variable is \(\displaystyle y\). Pivoting using the row operations:
\(\displaystyle 62R_1-29R_2\)
\(\displaystyle 31R_3+9R_2\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle s\) & \(\displaystyle 4433\) & \(\displaystyle 0\) & \(\displaystyle 62\) & \(\displaystyle -29\) & \(\displaystyle 0\) & \(\displaystyle 98890\) \\ \hline
\(\displaystyle y\) & \(\displaystyle 31\) & \(\displaystyle 62\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 1922\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 62\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 9\) & \(\displaystyle 31\) & \(\displaystyle 17298\) \\ \hline
\end{tabular}
There are no negative values in row \(\displaystyle p\) and this is the final tableau.The value of \(\displaystyle s\) is \(\displaystyle 1595\). The value of \(\displaystyle y\) is \(\displaystyle 31\). The max value is \(\displaystyle p = 558\)
\begin{question}Use the simplex method to maximize $p = 7 x + 18 y$ subject to $\begin{cases}86 x + 29 y \leq 2494 \\ 31 x + 62 y \leq 1922 \\x \geq 0, y \geq 0 \end{cases}$
\soln{9cm}{Adding the slack variables $s$ and $t$ to the inequalities gives:\newline\begin{equation*}\begin{cases}86 x + 29 y +s = 2494 \\ 31 x + 62 y+t = 1922 \\ - 7 x - 18 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
$s$ & $86$ & $29$ & $1$ & $0$ & $0$ & $2494$ \\ \hline
$t$ & $31$ & $62$ & $0$ & $1$ & $0$ & $1922$ \\ \hline
$p$ & $-7$ & $-18$ & $0$ & $0$ & $1$ & $0$ \\ \hline
\end{tabular}\newline
The pivot row is $t$ and the pivot column is $y$ . The departing variable is $t$ and the incoming variable is $y$. Pivoting using the row operations:\newline
$62R_1-29R_2$\newline
$31R_3+9R_2$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$s$ & $4433$ & $0$ & $62$ & $-29$ & $0$ & $98890$ \\ \hline
$y$ & $31$ & $62$ & $0$ & $1$ & $0$ & $1922$ \\ \hline
$p$ & $62$ & $0$ & $0$ & $9$ & $31$ & $17298$ \\ \hline
\end{tabular}\newline
There are no negative values in row $p$ and this is the final tableau.The value of $s$ is $1595$. The value of $y$ is $31$. The max value is $p = 558$ }
\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 = 7 x + 18 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%207%20x%20%2B%2018%20y%20" alt="LaTeX: \displaystyle p = 7 x + 18 y " data-equation-content=" \displaystyle p = 7 x + 18 y " /> subject to <img class="equation_image" title=" \displaystyle \begin{cases}86 x + 29 y \leq 2494 \\ 31 x + 62 y \leq 1922 \\x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D86%20x%20%2B%2029%20y%20%5Cleq%202494%20%5C%5C%2031%20x%20%2B%2062%20y%20%5Cleq%201922%20%5C%5Cx%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX: \displaystyle \begin{cases}86 x + 29 y \leq 2494 \\ 31 x + 62 y \leq 1922 \\x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}86 x + 29 y \leq 2494 \\ 31 x + 62 y \leq 1922 \\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}86 x + 29 y +s = 2494 \\ 31 x + 62 y+t = 1922 \\ - 7 x - 18 y+p =0 \end{cases} " src="/equation_images/%20%5Cbegin%7Bcases%7D86%20x%20%2B%2029%20y%20%2Bs%20%3D%202494%20%5C%5C%2031%20x%20%2B%2062%20y%2Bt%20%3D%201922%20%5C%5C%20-%207%20x%20-%2018%20y%2Bp%20%3D0%20%5Cend%7Bcases%7D%20" alt="LaTeX: \begin{cases}86 x + 29 y +s = 2494 \\ 31 x + 62 y+t = 1922 \\ - 7 x - 18 y+p =0 \end{cases} " data-equation-content=" \begin{cases}86 x + 29 y +s = 2494 \\ 31 x + 62 y+t = 1922 \\ - 7 x - 18 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 86 " src="/equation_images/%20%5Cdisplaystyle%2086%20" alt="LaTeX: \displaystyle 86 " data-equation-content=" \displaystyle 86 " /> & <img class="equation_image" title=" \displaystyle 29 " src="/equation_images/%20%5Cdisplaystyle%2029%20" alt="LaTeX: \displaystyle 29 " data-equation-content=" \displaystyle 29 " /> & <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 2494 " src="/equation_images/%20%5Cdisplaystyle%202494%20" alt="LaTeX: \displaystyle 2494 " data-equation-content=" \displaystyle 2494 " /> \\ \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 31 " src="/equation_images/%20%5Cdisplaystyle%2031%20" alt="LaTeX: \displaystyle 31 " data-equation-content=" \displaystyle 31 " /> & <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 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 1922 " src="/equation_images/%20%5Cdisplaystyle%201922%20" alt="LaTeX: \displaystyle 1922 " data-equation-content=" \displaystyle 1922 " /> \\ \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 -7 " src="/equation_images/%20%5Cdisplaystyle%20-7%20" alt="LaTeX: \displaystyle -7 " data-equation-content=" \displaystyle -7 " /> & <img class="equation_image" title=" \displaystyle -18 " src="/equation_images/%20%5Cdisplaystyle%20-18%20" alt="LaTeX: \displaystyle -18 " data-equation-content=" \displaystyle -18 " /> & <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 t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> and the pivot column is <img class="equation_image" title=" \displaystyle y " src="/equation_images/%20%5Cdisplaystyle%20y%20" alt="LaTeX: \displaystyle y " data-equation-content=" \displaystyle y " /> . The departing variable is <img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> and the incoming variable is <img class="equation_image" title=" \displaystyle y " src="/equation_images/%20%5Cdisplaystyle%20y%20" alt="LaTeX: \displaystyle y " data-equation-content=" \displaystyle y " /> . Pivoting using the row operations:<br>
<img class="equation_image" title=" \displaystyle 62R_1-29R_2 " src="/equation_images/%20%5Cdisplaystyle%2062R_1-29R_2%20" alt="LaTeX: \displaystyle 62R_1-29R_2 " data-equation-content=" \displaystyle 62R_1-29R_2 " /> <br>
<img class="equation_image" title=" \displaystyle 31R_3+9R_2 " src="/equation_images/%20%5Cdisplaystyle%2031R_3%2B9R_2%20" alt="LaTeX: \displaystyle 31R_3+9R_2 " data-equation-content=" \displaystyle 31R_3+9R_2 " /> <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 s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> & <img class="equation_image" title=" \displaystyle 4433 " src="/equation_images/%20%5Cdisplaystyle%204433%20" alt="LaTeX: \displaystyle 4433 " data-equation-content=" \displaystyle 4433 " /> & <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 62 " src="/equation_images/%20%5Cdisplaystyle%2062%20" alt="LaTeX: \displaystyle 62 " data-equation-content=" \displaystyle 62 " /> & <img class="equation_image" title=" \displaystyle -29 " src="/equation_images/%20%5Cdisplaystyle%20-29%20" alt="LaTeX: \displaystyle -29 " data-equation-content=" \displaystyle -29 " /> & <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 98890 " src="/equation_images/%20%5Cdisplaystyle%2098890%20" alt="LaTeX: \displaystyle 98890 " data-equation-content=" \displaystyle 98890 " /> \\ \hline
<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 31 " src="/equation_images/%20%5Cdisplaystyle%2031%20" alt="LaTeX: \displaystyle 31 " data-equation-content=" \displaystyle 31 " /> & <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 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 1922 " src="/equation_images/%20%5Cdisplaystyle%201922%20" alt="LaTeX: \displaystyle 1922 " data-equation-content=" \displaystyle 1922 " /> \\ \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 62 " src="/equation_images/%20%5Cdisplaystyle%2062%20" alt="LaTeX: \displaystyle 62 " data-equation-content=" \displaystyle 62 " /> & <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 9 " src="/equation_images/%20%5Cdisplaystyle%209%20" alt="LaTeX: \displaystyle 9 " data-equation-content=" \displaystyle 9 " /> & <img class="equation_image" title=" \displaystyle 31 " src="/equation_images/%20%5Cdisplaystyle%2031%20" alt="LaTeX: \displaystyle 31 " data-equation-content=" \displaystyle 31 " /> & <img class="equation_image" title=" \displaystyle 17298 " src="/equation_images/%20%5Cdisplaystyle%2017298%20" alt="LaTeX: \displaystyle 17298 " data-equation-content=" \displaystyle 17298 " /> \\ \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 s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> is <img class="equation_image" title=" \displaystyle 1595 " src="/equation_images/%20%5Cdisplaystyle%201595%20" alt="LaTeX: \displaystyle 1595 " data-equation-content=" \displaystyle 1595 " /> . The value of <img class="equation_image" title=" \displaystyle y " src="/equation_images/%20%5Cdisplaystyle%20y%20" alt="LaTeX: \displaystyle y " data-equation-content=" \displaystyle y " /> is <img class="equation_image" title=" \displaystyle 31 " src="/equation_images/%20%5Cdisplaystyle%2031%20" alt="LaTeX: \displaystyle 31 " data-equation-content=" \displaystyle 31 " /> . The max value is <img class="equation_image" title=" \displaystyle p = 558 " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%20558%20" alt="LaTeX: \displaystyle p = 558 " data-equation-content=" \displaystyle p = 558 " /> </p> </p>