Consider the linear programming problem(LPP):
Minimize \(Z = x+y\)
\(x + 2y \le 4,\)
\(3x + y \ge 3,\)
\(4x + 3y \ge 6,\)
\(x, y \ge 0.\)
Which of the following is correct for the above linear programming problem (LPP):
(A) The LPP has a bounded feasible region.
(B) The LPP has a unique optimal solution.
(C) The optimal value of the LPP exists at the point (3/2, 0)
(D) The corner points of the feasible region are (3/2, 0), (3/5, 6/5), (2/5, 6/5) and (4, 0)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (C) only
Corner points from the boundary lines:
On the x-axis (y = 0): \(x \le 4\), \(x \ge 1\) and \(x \ge \frac{3}{2}\), so the region runs from \(\left(\frac{3}{2}, 0\right)\) to (4, 0).
On the y-axis (x = 0): \(y \le 2\) but \(y \ge 3\), which is impossible, so the region does not reach the y-axis.
\(3x + y = 3\) and \(4x + 3y = 6\): putting \(y = 3 - 3x\),
\( \displaystyle 4x + 9 - 9x = 6 \)
\( \displaystyle x = \frac{3}{5},\quad y = \frac{6}{5} \)
\(x + 2y = 4\) and \(3x + y = 3\): putting \(y = 3 - 3x\),
\( \displaystyle x + 6 - 6x = 4 \)
\( \displaystyle x = \frac{2}{5},\quad y = \frac{9}{5} \)
So the feasible region is the quadrilateral with corners \(\left(\frac{3}{2}, 0\right)\), (4, 0), \(\left(\frac{2}{5}, \frac{9}{5}\right)\) and \(\left(\frac{3}{5}, \frac{6}{5}\right)\). Values of Z at the corner points:
\( \displaystyle Z\left(\frac{3}{2}, 0\right) = 1.5 \)
\( \displaystyle Z(4, 0) = 4 \)
\( \displaystyle Z\left(\frac{2}{5}, \frac{9}{5}\right) = 2.2 \)
\( \displaystyle Z\left(\frac{3}{5}, \frac{6}{5}\right) = 1.8 \)
So the feasible region is bounded, its corners are \(\left(\frac{3}{2}, 0\right)\), (4, 0), \(\left(\frac{2}{5}, \frac{9}{5}\right)\) and \(\left(\frac{3}{5}, \frac{6}{5}\right)\), and the minimum Z = 1.5 occurs only at \(\left(\frac{3}{2}, 0\right)\).
Checking the statements
(A) The LPP has a bounded feasible region. — Correct. The region is a closed quadrilateral.
(B) The LPP has a unique optimal solution. — Correct. The least value 1.5 occurs at only one corner.
(C) The optimal value of the LPP exists at the point (3/2, 0) — Correct. The minimum, 1.5, is at \(\left(\frac{3}{2}, 0\right)\).
(D) The corner points of the feasible region are (3/2, 0), (3/5, 6/5), (2/5, 6/5) and (4, 0) — Incorrect. The corner is \(\left(\frac{2}{5}, \frac{9}{5}\right)\), not \(\left(\frac{2}{5}, \frac{6}{5}\right)\).