Consider a LPP:
Maximize \( Z=22x+18y \)
Subject to:
\( x+y\le20 \),
\( 360x+240y\le5760 \),
\( x,y\ge0 \).
Which of the following statements are correct ?
A. The maximum value of \( Z \) occurs at (0, 20).
B. The maximum value of \( Z \) occurs at (8, 12).
C. The maximum value of \( Z \) is 352.
D. The maximum value of \( Z \) is 392.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → B and D only
Dividing the second constraint by 120, \( 360x+240y\le5760 \) becomes
\( \displaystyle 3x+2y\le48 \)
The axis corners are (16, 0) and (0, 20), because (20, 0) fails \( 3x+2y\le48 \) and (0, 24) fails \( x+y\le20 \). The two lines meet where
\( \displaystyle 3x+2(20-x)=48 \)
\( \displaystyle x+40=48 \)
\( \displaystyle x=8,\quad y=12 \)
The corner points are (0, 0), (16, 0), (8, 12) and (0, 20). The values of \( Z=22x+18y \) are
\( \displaystyle Z(0,0)=0 \)
\( \displaystyle Z(16,0)=352 \)
\( \displaystyle Z(8,12)=176+216=392 \)
\( \displaystyle Z(0,20)=360 \)
So the maximum value of Z is 392, at (8, 12).
Checking the statements
(A) The maximum value of \( Z \) occurs at (0, 20) — Incorrect. Z = 360 there, which is less than 392.
(B) The maximum value of \( Z \) occurs at (8, 12) — Correct. Z = 392 there, the largest corner value.
(C) The maximum value of \( Z \) is 352 — Incorrect. 352 is the value at (16, 0); the maximum is 392.
(D) The maximum value of \( Z \) is 392 — Correct. Z(8, 12) = 392 is the largest corner value.