Consider the linear programming problem with objective function \(Z = 2x + y\) and constraints \(x + y \ge 4\), \(x - y \ge -2\), \(x \le 6\), \(x \ge 0\), \(y \ge 0\).
(A) The feasible region is bounded
(B) The feasible region has exactly 4 corner points
(C) The minimum value of Z is 5
(D) The maximum value of Z is 12
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
The constraint \(x - y \ge -2\) means \(y \le x + 2\): the region lies below the line y = x + 2. It also lies on the far side of x + y = 4 from the origin, to the left of x = 6 and above the x-axis.
Its corner points:
x + y = 4 with y = 0: (4, 0)
x = 6 with y = 0: (6, 0)
x = 6 with y = x + 2: (6, 8)
y = x + 2 with x + y = 4: (1, 3)
Values of Z = 2x + y:
\(Z(4, 0) = 8\)
\(Z(6, 0) = 12\)
\(Z(6, 8) = 20\)
\(Z(1, 3) = 5\)
So the region is bounded with 4 corner points, the minimum of Z is 5 and the maximum of Z is 20.
Checking the statements
(A) The feasible region is bounded — Correct. x is at most 6, and y is at most x + 2 ≤ 8.
(B) The feasible region has exactly 4 corner points — Correct. They are (4, 0), (6, 0), (6, 8) and (1, 3).
(C) The minimum value of Z is 5 — Correct. \(Z(1, 3) = 2 + 3 = 5\) is the smallest corner value.
(D) The maximum value of Z is 12 — Incorrect. \(Z(6, 8) = 12 + 8 = 20\) is larger; 12 is Z at (6, 0).