**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 = 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 “≥”.

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

vishakha arya says

good concept.

Prabhat says

Very very helpful for students…

Hatprrrab says

I agree with Prabhat.

A. G. Hazra says

It couldn’t be more simplistic. Thanks.

Deepika Pandey says

it could be easier in the table format.

ragul says

thank you

Garini Radha says

Thanks a lot .

bekzod says

good concept and useful thing for students who wants to know about duality method

Anania says

Thanks