Q. Solve the linear programming problem: max $ Z=x+2y $ subject to constraints: $ x-y\le 10, $ $ 2x+3y\le 20, $ $x \le 0, y \le 0$
J & K CETJ & K CET 2014Linear Programming
Solution:
We have, max $ Z=x+2y $ Subject to constraints, $ x-y\le 10 $ $ 2x+3y\le 20 $ $ x\ge 0,\,\,y\ge 0 $ On taking given constraints as equation, we get The following graph.
Here, OAB is the required feasible region whose comer points are $ O(0,\,\,0),\,\,\,A(10,\,\,0) $ and $ B\left( 0,\frac{20}{3} \right) $ .
Comer point
$ z=x+2y $
at $ O(0,\,\,0) $
$ Z=0 $
at $ A\,(10,\,\,0) $
$ Z=10 $
at $ B\left( 0,\frac{20}{3} \right) $
$ Z=0+2\times \frac{20}{3}=\frac{40}{3} $
Hence, maximum value of Z is $ \frac{40}{3}, $ which is obtained at $ B\left( 0,\frac{20}{3} \right). $
