Question

Consider the following linear program:

max(3x₁ + 2x₂ + x₃ )

subject to

a. x₁ − x₂ + x₃ ≤ 4
b. 2x₁ +x₂ +3x₃ ≤6
c. −x₁ + 2x₃ = 3
d. x₁ + x₂ + x₃ ≤ 8
e. x₁,x₂,x₃ ≥0

Write the dual problem.

Answer 1

The dual of the given linear programming problem involves creating variables corresponding to the primal constraints and formulating a new objective function and constraints that reflect the primal problem's structure.

Step-by-step explanation:

To write the dual of the given linear program, we first need to recognize that variables in the dual correspond to the constraints in the primal and vice versa. The given primal problem has both inequalities and an equation as constraints. The dual variable associated with an inequality constraint is typically unrestricted in sign, while the one associated with an equation is fixed.

In the case of this primal problem:

1. Variables y1, y2, and y4 correspond to the inequalities a, b, and d, and are therefore ≥ 0.
2. The variable y3 corresponds to the equality constraint c and can take on any real value.

The dual problem is formulated as follows:

min (4y1 + 6y2 + 3y3 + 8y4)

Subject to:

• y1 + 2y2 - y3 + y4 ≥ 3
• -y1 + y2 + y4 ≥ 2
• y1 + 3y2 + 2y3 + y4 ≥ 1
• y1, y2, y4 ≥ 0
• y3 unrestricted