So far, we have seen the linear programming constraints with less than type. We come across problems with ‘greater than’ and ‘equal to’ type also. Each of these types must be converted as equations. In case of ‘greater than’ type, the constraints are rewritten with a negative surplus variable s_{1} and by adding an artificial variable a.

Artificial variables are simply used for finding the initial basic solutions and are thereafter eliminated. In case of an ‘equal to’ constraint, just add the artificial variable to the constraint. The co-efficient of artificial variables a_{1}, a_{2},….. are represented by a very high value M, and hence the method is known as BIG-M Method.

*Example : *Solve the following LPP using Big M Method.

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

Subject to constraints

4x_{1} + x_{2} = 4 ....................(i)

5x_{1} + 3x_{2} ≥ 7 ....................(ii)

3x_{1} + 2x_{2} ≤ 6 ....................(iii)

where x_{1} , x_{2} ≥0

*Solution: *Introduce slack and auxiliary variables to represent in the standard form. Constraint 4x_{1} + x_{2} = 4 is introduced by adding an artificial variable a_{1}, i.e., 4x_{1} + x_{2} + a_{1} = 4 Constraint, 5x_{1} + 3x_{2} ≥ 7 is converted by subtracting a slack s_{1} and adding an auxiliary variable a_{2}.

5x_{1}+ 3x_{2} – s_{1} + a_{2} = 7

Constraint 3x_{2} + 2x_{2} ≤ 6 is included with a slack variable s_{2}

3x_{2} + 2x_{2} + s_{2} = 6

The objective must also be altered if auxiliary variables exist. If the objective function is minimization, the co-efficient of auxiliary variable is +M (and -M, in case of maximization)

The objective function is minimization,

Minimize Z = 3x_{1}+ x_{2} + 0s_{1}+ 0s_{2}+ Ma_{1}+ Ma_{2}

z_{min} = 3x_{1} + x_{2}+ Ma_{1}+ Ma_{2}

The initial feasible solution is (Put x_{1}, x_{2}, s_{1} = 0)

a_{1} = 4

a_{2} = 7

s_{2} = 6

Establish a table as shown below and solve:

**Simplex Table**

The solution is,

x_{1} = 5/7 or 0.71

x_{2} = 8/7 or 1.14

z_{min} = 3 x 5 / 7 + 8/7

= 23/7 or 3.29