Minimize $z = 2x_1 + 2x_2 + 4x_3$

Subject to $2x_1 + 3x_2 + 5x_3 ≥ 2 \\ 3x_1 + x_2 + 7x_3 ≤ 3 \\ X_1 + 4x_2 + 6x_3 ≤ 5 \\ X_1, X_2, X_3 ≥ 0. $

Mumbai University > COMPS > Sem 4 > Applied Mathematics 4

Marks : 06

Year : DEC 2015

All the constraint must be in less than form thus multiplying -1 in 1st constraint.

Minimize $Z= 2x_1+2x_2+4x_3 \\ -2x_1-3x_2-5x_3 ≥ -2 \\ 3x_1+x_2+7x_3 ≤ 3 \\ x_1+4x_2+6x_3 ≤ 5$

For three constraint three slack variable:

Minimize $Z= 2x_1+2x_2+4x_3-O_{s1}-O_{s2}-O_{s3} \\ i.e.\space Z-2x_1-2x_2-4x_3+O_{s1}+O_{s2}+O_{s3}=0 $

Subject to $-2x_1-3x_2-5x_3+S_1+O_{s2}+O_{s3}=-2 \\ 3x_1+x_2+7x_3 +O_{s1}+S_2+O_{s3}=3 \\ x_1+4x_2+6x_3+O_{s1}+O_{s2}+S_3=5 $

Selecting the lowest value i.e. $2/3$

Thus, $S_1$ Beans And $x_2$ Enters

Dividing $S_1$ by 3 we get $x_2$

$x_2 \space\space \dfrac 23 \space\space 1 \dfrac 53 \space\space \dfrac {-1}3\space\space 0\space\space 0 \space\space \dfrac 23$

∴ Now making values above and below using $x_2 \\ i.e. \space z+2x_1 \\ S_2-x_2 \\ S_3-4x_2$

$X_1 =0\space \space X_2 = 2/3 \space \space X_3 = 0 Z_{min}=4/3$

∴ All the values at R.H.S. is positive