Which of the following statements are correct in reference to the linear programming problem (LPP)
Maximise Z = 50x + 40 y
Subject to the constraints: \( 1000x+1200y\le 7600,\ x,y\ge 0,\ 3x+2y\le 18 \).
(A) The LPP has a unique optimal solution at (4, 3) only.
(B) The feasible region is bounded.
(C) The maximum value is unique, but there are an infinite number of optimal solutions.
(D) The feasible region is bounded with corner points (0, 0), (6, 0), (4, 3) and (0, 19/3).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (D) only
Dividing the first constraint by 200:
\( \displaystyle 5x+6y\le 38 \)
Solving \( 5x+6y=38 \) with \( 3x+2y=18 \) (three times the second gives \( 9x+6y=54 \)):
\( \displaystyle 4x=16 \)
\( \displaystyle x=4,\ y=3 \)
The other corners are (0, 0), (6, 0) on \( 3x+2y=18 \) and \( \left(0,\frac{19}{3}\right) \) on \( 5x+6y=38 \). Values of Z = 50x + 40y:
At (0, 0): Z = 0
At (6, 0): Z = 300
At (4, 3): Z = 200 + 120 = 320
At \( \left(0,\frac{19}{3}\right) \): Z = \( \frac{760}{3} \approx 253.3 \)
So the feasible region is a bounded quadrilateral with corners (0, 0), (6, 0), (4, 3) and \( \left(0,\frac{19}{3}\right) \), and the maximum Z = 320 occurs only at (4, 3).
Checking the statements
(A) The LPP has a unique optimal solution at (4, 3) only. — Correct. Z = 320 at (4, 3) is larger than at every other corner.
(B) The feasible region is bounded. — Correct. It is a closed quadrilateral.
(C) The maximum value is unique, but there are an infinite number of optimal solutions. — Incorrect. The maximum 320 occurs at the single point (4, 3).
(D) The feasible region is bounded with corner points (0, 0), (6, 0), (4, 3) and (0, 19/3). — Correct. These are exactly the four corners of the bounded region.