Answer to Question #242465 in Operations Research for opr

Question

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

Answer 1

All constraints can be converted to ≤ by multiplying by -1. So we have;

Maximize: z= 5x₁ - 2x₂

subject to constraints:

-3x₁ - 2x₂ ≤ -16

x₁ - x₂ ≤ 4

-x₁ ≤ -5

and x₁ ≤ 0; x₂ is unrestricted.

Since the primal has two variables and three constraints, then the dual will have three variables and two constraints. Also, the x₂ variable is unrestricted in the primal, therefore, the second constraint in the dual shall be equality.

Dual program is;

Minimize: z = -16y₁ + 4y₂ - 5y₃

Subject to constraints

-3y₁ + y₂ - y₃ ≥ 5

-2y₁ - y₂ = -2

and y₁,y₂ ≥ 0