Question

Solve using simplex method

z = 3000x + 5000y

2x + y <= 16

x + 2y <= 11

x + 3y <= 15

Answer 1

Max $z=3000 x+5000 y$ subject to $2 x+y \leq 16 x+2 y \leq 11\ and\ x, y \geq 0$

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

1. As the constraint- 1 is of type ' $\leq$ ' we should add slack variable $S_{1}$

2. As the constraint- 2 is of type ' $\leq$ ' we should add slack variable $S_{2}$

After introducing slack variables

$\operatorname{Max} z=3000 x+5000 y+0 S_{1}+0 S_{2}$

subject to

 $\begin{array}{rr} 2 x+y+S_{1} & =16 \\ x+2 y+S_{2} & =11 \end{array}$

 

and $x, y, S_{1}, S_{2} \geq 0$

Since all $Zj-Cj≥0$

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

$x=7,y=2$

Max $z=31000$