Consider the linear programming problem (LPP):
Maximize \(Z = 6x + 3y\) subject to the conditions,
\(4x + y \ge 80,\ x + 5y \ge 115,\ 3x + 2y \le 150,\ x, y \ge 0\).
In reference to the above LPP, which of the following are correct?
(A) The feasible region is bounded.
(B) The corner points of the feasible region are (15, 20), (40, 15) and (0, 75).
(C) The maximum value of the objective function is 285.
(D) The LPP does not have optimal solution.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (C) only
The corner points are the points where two boundary lines meet and all the constraints hold.
\(4x + y = 80\) and \(x + 5y = 115\): from the first, \(y = 80 - 4x\), so
\( \displaystyle x + 400 - 20x = 115 \)
\( \displaystyle 19x = 285 \)
\( \displaystyle x = 15 \)
\( \displaystyle y = 80 - 60 = 20 \)
\(4x + y = 80\) and \(3x + 2y = 150\): putting \(y = 80 - 4x\),
\( \displaystyle 3x + 160 - 8x = 150 \)
\( \displaystyle 5x = 10 \)
\( \displaystyle x = 2 \)
\( \displaystyle y = 80 - 8 = 72 \)
\(x + 5y = 115\) and \(3x + 2y = 150\): from the first, \(x = 115 - 5y\), so
\( \displaystyle 345 - 15y + 2y = 150 \)
\( \displaystyle 13y = 195 \)
\( \displaystyle y = 15 \)
\( \displaystyle x = 115 - 75 = 40 \)
All three points satisfy every constraint. The points where the lines meet the axes, such as (0, 80), (0, 75), (50, 0) and (115, 0), each break one of the constraints. So the feasible region is the triangle with corners (15, 20), (2, 72) and (40, 15).
Values of Z at the corners:
\( \displaystyle Z(15, 20) = 90 + 60 = 150 \)
\( \displaystyle Z(2, 72) = 12 + 216 = 228 \)
\( \displaystyle Z(40, 15) = 240 + 45 = 285 \)
So the feasible region is the bounded triangle with corners (15, 20), (2, 72) and (40, 15), and the maximum of Z is 285 at (40, 15).
Checking the statements
(A) The feasible region is bounded. — Correct. It is a triangle.
(B) The corner points of the feasible region are (15, 20), (40, 15) and (0, 75). — Incorrect. The third corner is (2, 72); (0, 75) is not feasible, as \(4(0) + 75 = 75 \lt 80\).
(C) The maximum value of the objective function is 285. — Correct. The largest corner value is 285, at (40, 15).
(D) The LPP does not have optimal solution. — Incorrect. The maximum is 285, at (40, 15).