Duality in Linear Programming

Definition: The Duality in Linear Programming states that every linear programming problem has another linear programming problem related to it and thus can be derived from it. The original linear programming problem is called “Primal,” while the derived linear problem is called “Dual.”

Before solving for the duality, the original linear programming problem is to be formulated in its standard form. Standard form means, all the variables in the problem should be non-negative and “≥,” ”≤” sign is used in the minimization case and the maximization case respectively.

The concept of Duality can be well understood through a problem given below:

Maximize

Z = 50x₁+30x₂

Subject to:
4x₁ + 3x₂ ≤ 100
3x₁ + 5x₂ ≤ 150
X₁, x₂ ≥ 0

The duality can be applied to the above original linear programming problem as:

Minimize

G = 100y₁+150y₂

Subject to:

4y₁ + 3y₁ ≥ 50
3y₁ +5y₂ ≥ 30
Y₁, y₂ ≥ 0

The following observations were made while forming the dual linear programming problem:

1. The primal or original linear programming problem is of the maximization type while the dual problem is of minimization type.
2. The constraint values 100 and 150 of the primal problem have become the coefficient of dual variables y₁ and y₂ in the objective function of a dual problem and while the coefficient of the variables in the objective function of a primal problem has become the constraint value in the dual problem.
3. The first column in the constraint inequality of primal problem has become the first row in a dual problem and similarly the second column of constraint has become the second row in the dual problem.
4. The directions of inequalities have also changed, i.e. in the dual problem, the sign is the reverse of a primal problem. Such that in the primal problem, the inequality sign was “≤” but in the dual problem, the sign of inequality becomes “≥”.

Note: The dual of a dual problem is the primal problem.