# QT/U3 Topic 5 Duality: Concepts, Significance

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 = 50×1+30×2

Subject to:

4×1 + 3×2 ≤ 100

3×1 + 5×2 ≤ 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:

- 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 y1and 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.
- 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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