Determine the dual of the following linear programming probl

Determine the dual of the following linear programming problem

Minimize 6x_1 + 12x_2 - 18x_3 subject to x_1 - 3x_2 + 6x_3 = 30 2x_1 + 8x_2 - 16x_3 = 70 x_1, x_2 > 0, x_3 unrestricted

Solution

x1, x2 are positive but x3 can be any real number

Objective is to minimise

6x1+12x2-18x3 = 6(x1+2x2-3x3)

Constraints are

x1-3x2+6x3 = 30

2x1+8x2-16x3 =70

2x1-6x2+12x3 =60 (by multiplying first equation by 2)

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14x2-28x3 = 10

x2-2x3 = 5/7 = 0.7143

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x2= 0.7143+2x3 >=0

Hence x3 >=-0.3572

x1 = 30+3x2-6x3 >= 30+2.143+2.1432-0.7144

>=0

Hence solution is

x1>=0

x2>=0

x3>=-0.3572 or x3 =0

minimum z is = 0