Answer to Question #237068 in Operations Research for opr

Question #237068

Find the dual program of the following linear programming problem.

𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = 5𝑥1 − 2𝑥2

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

3𝑥1 + 2𝑥2 ≥ 16

𝑥1 − 𝑥2 ≤ 4

𝑥1 ≥ 5

𝑥1 ≥0,𝑥2 𝑖𝑠 𝑢𝑛𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑒𝑑 


1
Expert's answer
2021-09-27T03:06:57-0400

All constraints can be converted to "\\leq" by multiplying by -1. So we have;

"\\text{Max} ~~z=5x_1-2x_2\\\\\n\\text{subject to}\\\\\n~~~~~-3x_1-2x_2\\leq-16\\\\\n~~~~~~~~~~~x_1-~~x_2\\leq4\\\\\n~~~~~~~-x_1~~~~~~~~~~~\\leq-5\\\\\n\\text{and } x_1\\leq 0; x_2 \\text{is unrestricted}."

Since the primal has two variables and three constraints, then the dual will have three variables and two constraints. Also, the "x_2" variable is unrestricted in the primal, therefore the second constraint in the dual shall be equality.

Dual program is;

"\\text{Min } ~~z=-16y_1+4y_2-5y_3\\\\\n\\text{subject to }\\\\\n~~~~~-3y_1+y_2-y_3\\geq5\\\\\n~~~~~-2y_1-y_2~~~~~~~~=-2\\\\\n\\text{and } y_1,y_2\\geq0."



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