Question

Use Simplex method to

Maximize Ζ = 4_x + 10_x2

Subject 2_X+_X2 ≤ 50

2_X + 5_X2 ≤ 100

2_X + 3_X2 ≤ 100

_{X, X2} ≥ 0

Answer 1

$Max Z=4x_1+10x_2\\subject\ to\\2x_1+x_2≤50\\2x_1+5x_2≤100\\2x_1+3x_2≤100\\and\ \ x_1,x_2≥0;$

The problem is converted to canonical form by adding slack, surplus and artificial variables as appropriate

1. As the constraint-1 is of type '≤' we should add slack variable S1

2. As the constraint-2 is of type '≤' we should add slack variable S2

3. As the constraint-3 is of type '≤' we should add slack variable S3

After introducing slack variables

$Max Z=4x_1+10x_2+0S_1+0S_2+0S_3\\subject\ to\\2x_1+x_2+S_1=50\\2x_1+5x_2+S_2=100\\2x_1+3x_2+S_3=100\\and \ x1,x2,S1,S2,S3≥0$

Negative minimum$Z_j-C_j$  is -10 and its column index is 2. So, the entering variable is $x_2$ .

Minimum ratio is 20 and its row index is 2. So, the leaving basis variable is $S_2$ .

∴ The pivot element is 5.

Entering =$x_2$ , Departing =$S_2$ , Key Element =5

Since all $Z_j-C_j≥0$

Hence, optimal solution is arrived with value of variables as :

$x_1=0,x_2=20$

Max Z = 200