Consider an LPP:
Maximize \( Z=5x+y \)
Subject to \( x+3y\le 12 \)
\( 3x+y\le 12 \)
and \( x,y\ge 0 \)
then which of the following statements are TRUE ?
(A) The corner points of the bounded feasible region are (0,0), (4,0), (3,3) and (0,4).
(B) Maximum value of z occurs at (3,3).
(C) Maximum value of z occurs at (4,0).
(D) The maximum value of z is 20.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (C) and (D) only
The line x + 3y = 12 meets the axes at (12, 0) and (0, 4); the line 3x + y = 12 meets them at (4, 0) and (0, 12). The two lines meet where
\( \displaystyle x+3y=12,\ 3x+y=12\ \Rightarrow\ x=y=3 \)
So the feasible region has corner points (0, 0), (4, 0), (3, 3) and (0, 4). Evaluating Z = 5x + y:
\( \displaystyle (0,0):\ Z=0 \)
\( \displaystyle (4,0):\ Z=20 \)
\( \displaystyle (3,3):\ Z=18 \)
\( \displaystyle (0,4):\ Z=4 \)
So the corner points are (0, 0), (4, 0), (3, 3) and (0, 4), and the maximum value of Z is 20, at (4, 0).
Checking the statements
(A) The corner points of the bounded feasible region are (0,0), (4,0), (3,3) and (0,4). — Correct. These are the origin, the two axis points (4, 0) and (0, 4), and the meeting point (3, 3).
(B) Maximum value of z occurs at (3,3). — Incorrect. Z = 18 at (3, 3), which is less than 20.
(C) Maximum value of z occurs at (4,0). — Correct. Z = 20 at (4, 0), the largest corner value.
(D) The maximum value of z is 20. — Correct. The corner values are 0, 20, 18 and 4.