# Dual Formulation: Linear Programming

In this section, we will introduce the concept of Dual Formulation in Linear Programming (LP). Now lets concentrate on the following example:

#### Example: Dual Formulation in Linear Programming

Minimize z = 3x1 + 3x2

subject to

2x1 + 4x2 ≥ 40
3x1 + 2x2 ≥ 50

x1, x2 ≥ 0

Solution.

Maximize z = 40w1 + 50w2

subject to

2w1 + 3w2 ≤ 3
4w1 + 2w2 ≤ 3

w1, w2 ≥ 0

## Primal Dual Relationship in Linear Programming (LP)

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

### Primal Dual Relationship

• 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).