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Question

# Solve each minimum problem using the duality principle. Minimize$C = x_1 +4x_2 + 2x_3 +4x_4$subject to the constraints 2$\left\{ \begin{array}{ccccccc} x_1 & & +x_3 & \geq 1 \\ & x_2 & & +x_4 \geq1 \\ - x_1 &-x_2 &-x_3 &-x_4 \geq -3 \\ x_1 \geq 0, & x_2 \geq 0, & x_3 \geq 0 & x_4 \geq 0 \end{array} \right.$

Solution

Verified
Step 1
1 of 6

${\bf STEP 1}$. Write the dual problem, a maximum problem.$\\$ Matrix associated with the miniumum problem:$\left[ $\begin{array}{lllll} 1 & 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 1 & 1\\ -1 & -1 & -1 & -1 & -3\\ 1 & 4 & 2 & 4 & 0 \end{array}$ \right]$\\\\ Transposing it, we obtain$\left[ $\begin{array}{llll} 1 & 0 & -1 & 1\\ 0 & 1 & -1 & 4\\ 1 & 0 & -1 & 2\\ 0 & 1 & -1 & 4\\ 1 & 1 & -3 & 0 \end{array}$ \right]$, which translates to a (dual) maximum problem\\\\ {\bf Maximize}$\bf P= y_{1}+y_{2}-3y_{3} $with constraints:\quad$\left{ $\begin{array}{l} y_{1}-y_{3}\leq 1\\ y_{2}-y_{3}\leq 4\\ y_{1}-y_{3}\leq 2\\ y_{2}-y_{3}\leq 4 \\ y_{1}\geq 0,y_{2}\geq 0 \end{array}$ \right.$

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