Which of the following statements are correct in reference to the linear programming problem(LPP):
Maximize \(Z = 5x + 2y\)
subject to the following constraints
\(3x + 5y \le 15,\quad 5x + 2y \le 10,\quad x \ge 0, y \ge 0.\)
(A) The LPP has a unique optimal solution at (2, 0) only.
(B) The feasible region is bounded with corner points (0, 0), (2, 0), (20/19, 45/19) and (0, 3).
(C) The optimal value is unique, but there are an infinite number of optimal solutions.
(D) The feasible region is unbounded.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B) and (C) only
The lines \(3x + 5y = 15\) and \(5x + 2y = 10\) meet the axes at (5, 0), (0, 3) and (2, 0), (0, 5). Both constraints are of the \(\le\) type, so the feasible region lies below both lines in the first quadrant. The two lines meet where:
\(15x + 25y = 75\)
\(15x + 6y = 30\)
\(19y = 45\)
\( \displaystyle y = \frac{45}{19} \)
\( \displaystyle x = \frac{20}{19} \)
The corner points and the values of Z:
\(Z(0, 0) = 0\)
\(Z(2, 0) = 10\)
\( \displaystyle Z\left(\frac{20}{19}, \frac{45}{19}\right) = \frac{100}{19} + \frac{90}{19} = 10 \)
\(Z(0, 3) = 6\)
The maximum 10 occurs at two corners. The objective \(5x + 2y\) is parallel to the constraint line \(5x + 2y = 10\), so every point of the edge joining them is optimal.
So the feasible region is bounded with corners (0, 0), (2, 0), \(\left(\frac{20}{19}, \frac{45}{19}\right)\) and (0, 3), and the maximum Z = 10 is reached at every point of the edge from (2, 0) to \(\left(\frac{20}{19}, \frac{45}{19}\right)\).
Checking the statements
(A) The LPP has a unique optimal solution at (2, 0) only. — Incorrect. The point \(\left(\frac{20}{19}, \frac{45}{19}\right)\) also gives Z = 10.
(B) The feasible region is bounded with corner points (0, 0), (2, 0), (20/19, 45/19) and (0, 3). — Correct. The axis points (0, 0), (2, 0), (0, 3) and the meeting point \(\left(\frac{20}{19}, \frac{45}{19}\right)\) of the two lines are the corners, and they enclose the region.
(C) The optimal value is unique, but there are an infinite number of optimal solutions. — Correct. The maximum value 10 is unique, and it is reached at infinitely many points of the edge from (2, 0) to \(\left(\frac{20}{19}, \frac{45}{19}\right)\).
(D) The feasible region is unbounded. — Incorrect. The region is a bounded quadrilateral.