Question

Consider the following maximization problem:​maximize: x1+3x2subject to the constraints:x1+x2≤6,2x1+x2≤8,x1+2x2≤9x1≥0 and x2≥$$0	ext{maximize: }$$ x_1 + 3x_2\$$	ext{subject to the constraints:}$$\x_1 + x_2 \leq 6, \quad 2x_1 + x_2 \leq 8, \quad x_1 + 2x_2 \leq 9\x_1 \geq 0 $$	ext{ and }$$ x_2 \geq 0maximize: x1​+3x2​subject to the constraints:x1​+x2​≤6,2x1​+x2​≤8,x1​+2x2​≤9x1​≥0 and x2​≥0​​Which of the following is the dual problem?

Accepted Answer

A

Answer

​Minimize:6y1+8y2+9y3subject toy1≥0,y2≥0,y3≥0,y1+2y2+2y3≥1,y1+y2+2y3≥3Minimize:  6y_1 + 8y_2 + 9y_3 \ $$	ext{subject to}$$\  y_1 \geq 0, y_2 \geq 0, y_3 \geq 0, y_1 + 2y_2 + 2y_3 \geq 1 ,  y_1 + y_2 + 2y_3 \geq 3 Minimize:6y1​+8y2​+9y3​subject toy1​≥0,y2​≥0,y3​≥0,y1​+2y2​+2y3​≥1,y1​+y2​+2y3​≥3​​

Answer

​Minimize:6y1+8y2+9y3subject to:y1≥0,y2≥0,y3≥0,y1+2y2+y3≥3,y1+y2+2y3≥1 Minimize: 6y_1 + 8y_2 + 9y_3 \  subject \ to : y_1 \geq 0, y_2 \geq 0, y_3 \geq 0, y_1 + 2y_2 + y_3 \geq 3 ,  y_1 + y_2 + 2y_3 \geq 1 Minimize:6y1​+8y2​+9y3​subject to:y1​≥0,y2​≥0,y3​≥0,y1​+2y2​+y3​≥3,y1​+y2​+2y3​≥1​​

Answer

​6y1+8y2+9y3subject toy1≥0,y2≥0,y3≥0,y1+2y2+y3≤1,y1+y2+2y3≤3 6y_1 + 8y_2 + 9y_3 \ $$	ext{subject to}$$\  y_1 \geq 0, y_2 \geq 0, y_3 \geq 0, y_1 + 2y_2 + y_3 \leq 1 ,  y_1 + y_2 + 2y_3 \leq 36y1​+8y2​+9y3​subject toy1​≥0,y2​≥0,y3​≥0,y1​+2y2​+y3​≤1,y1​+y2​+2y3​≤3​​

Answer

​9y1+8y2+6y3subject toy1≥0,y2≥0,y3≥0,y1+2y2+y3≥1,y1+y2+2y3≥39y_1 + 8y_2 + 6y_3 \ $$	ext{subject to}$$\  y_1 \geq 0, y_2 \geq 0, y_3 \geq 0, y_1 + 2y_2 + y_3 \geq 1 ,  y_1 + y_2 + 2y_3 \geq 39y1​+8y2​+6y3​subject toy1​≥0,y2​≥0,y3​≥0,y1​+2y2​+y3​≥1,y1​+y2​+2y3​≥3​​

Consider the following maximization problem:
$\text{maximize: } x_1 + 3x_2\\\text{subject to the constraints:}\\x_1 + x_2 \leq 6, \quad 2x_1 + x_2 \leq 8, \quad x_1 + 2x_2 \leq 9\\x_1 \geq 0 \text{ and } x_2 \geq 0$
Which of the following is the dual problem?

$Minimize: 6y_1 + 8y_2 + 9y_3 \\ \text{subject to}\\ y_1 \geq 0, y_2 \geq 0, y_3 \geq 0, y_1 + 2y_2 + 2y_3 \geq 1 , y_1 + y_2 + 2y_3 \geq 3$

$Minimize: 6y_1 + 8y_2 + 9y_3 \\ subject \ to : y_1 \geq 0, y_2 \geq 0, y_3 \geq 0, y_1 + 2y_2 + y_3 \geq 3 , y_1 + y_2 + 2y_3 \geq 1$

$6y_1 + 8y_2 + 9y_3 \\ \text{subject to}\\ y_1 \geq 0, y_2 \geq 0, y_3 \geq 0, y_1 + 2y_2 + y_3 \leq 1 , y_1 + y_2 + 2y_3 \leq 3$

$9y_1 + 8y_2 + 6y_3 \\ \text{subject to}\\ y_1 \geq 0, y_2 \geq 0, y_3 \geq 0, y_1 + 2y_2 + y_3 \geq 1 , y_1 + y_2 + 2y_3 \geq 3$

Correct option is A

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