# Duality

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 = 50x_{1}+30x_{2}

Subject to:

4x_{1 }+ 3x_{2 }≤ 100

3x_{1} + 5x_{2 }≤ 150

X_{1}, x_{2 }≥ 0

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

**Minimize**

G = 100y_{1}+150y_{2}

Subject to:

4y_{1 }+ 3y_{1 }≥ 50

3y_{1 }+5y_{2 }≥ 30

Y_{1}, y_{2} ≥ 0

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

- The primal or original linear programming problem is of the maximization type while the dual problem is of minimization type.
- The constraint values 100 and 150 of the primal problem have become the coefficient of dual variables y
_{1}and y_{2}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. - 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.
- 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 “≥”.

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