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Problem 397

Find a solution to the following problem by solving its dual: Minimize \(\quad 9 \mathrm{x}_{1}+12 \mathrm{x}_{2}+15 \mathrm{x}_{3}\) subject to $$ \begin{aligned} &2 \mathrm{x}_{1}+2 \mathrm{x}_{2}+\mathrm{x}_{3} \geq 10 \\ &2 \mathrm{x}_{1}+3 \mathrm{x}_{2}+\mathrm{x}_{3} \geq 12 \\ &\mathrm{x}_{1}+\mathrm{x}_{2}+5 \mathrm{x}_{3} \geq 14 \\ &\mathrm{x}_{1} \geq 0, \mathrm{x}_{2} \geq 0, \mathrm{x}_{3} \geq 0 \end{aligned} $$

Expert verified

The dual of the given linear programming problem is:
Maximize \(W=-10y_1 -12y_2 -14y_3\)
Subject to:
\(-2y_1-2y_2-y_3 \geq -9\)
\(-2y_1-3y_2-y_3 \geq -12\)
\(-y_1-y_2-5y_3 \geq -15\)
\(y_1 \geq 0, y_2 \geq 0, y_3 \geq 0\)
Upon solving the dual problem, we can find the optimal solution for the original problem (i.e., \(x_1, x_2, x_3\)).

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Chapter 17

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Chapter 17

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Chapter 17

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Chapter 17

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