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 = 3x_{1} + 3x_{2}

subject to

2x_{1} + 4x_{2} ≥
40

3x_{1} + 2x_{2} ≥ 50

x_{1}, x_{2} ≥ 0

Solution.

Maximize z = 40w_{1} + 50w_{2}

subject to

2w_{1} + 3w_{2} ≤
3

4w_{1} + 2w_{2 }≤ 3

w_{1}, w_{2} ≥ 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).