Duality & Sensitivity Testing

Example

Minimize z = 3x₁ + 3x₂

subject to

2x₁ + 4x₂ ³ 40
3x₁ + 2x₂ ³ 50

x₁, x₂ ³ 0

Solution.

Maximize z = 40w₁ + 50w₂

subject to

2w₁ + 3w₂ £ 3
4w₁ + 2w₂ £ 3

w₁, w₂ ³ 0

The Primal-Dual construction relationship

"One picture is worth more than ten thousand words." - Chinese Proverb

Relationship Between Primal And Dual Problem

• The number of constraints in the primal problem is equal to the number of dual variables, and vice versa.
• If the primal problem is a maximization problem, then the dual problem is a minimization problem and vice versa.
• If the primal problem has greater than or equal to type constraints, then the dual problem has less than or equal to type constraints and vice versa.
• The profit coefficients of the primal problem appear on the right-hand side of the dual problem.
• The rows in the primal become columns in the dual and vice versa..

All primal and dual variables must be non-negative (>0).