1. Tardigrade
  2. Question
  3. Mathematics
  4. The minimum value of Z = 4x + 3y subjected to the constraints 3x + 2y ge 160, 5x + 2y ge 200, x + 2y ge 80; x,y ge 0 is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The minimum value of $Z = 4x + 3y$ subjected to the constraints $3x + 2y \ge 160, 5x + 2y\ge 200, x + 2y \ge 80; x,y\ge 0$ is

Linear Programming

Solution:

Let $l_1 :3x + 2y = 160, l_2 : 5x + 2y = 200$
$l_3 : x + 2y = 80, l_4 : x = 0, l_5 : y = 0$
image
For B : Solving $l_1$ and $l_3$, we get $B(40, 20)$
For C : Solving $l_1$ and $l_2$, we get $C(20, 50)$
Shaded region is the feasible region,
where $A(80,0), B(40, 20), C(20, 50), D(0,100) $
Now minimize $Z = 4x + 3y$
$Z$ at $A(80, 0) = 320$
$Z$ at $B(40, 20) = 40 \times 4 + 20 \times 3 = 220$
$Z$ at $C(20, 50) = 20 \times 4 + 50 \times 3 = 230$
$Z$ at $D(0,100) = 300$
Minimum value of $Z$ is $220$.