For the linear programming problem Maximize Z = 3x1 + 2x2 subject to x1 − x2 ≤…
2012
For the linear programming problem Maximize Z = 3x1 + 2x2 subject to x1 − x2 ≤ 1, x1 + x2 ≥ 3, x1 ≥ 0, and x2 ≥ 0, the feasible region is:
Answer: B. An unbounded feasible region — CONCEPTA linear-programming feasible region is formed by intersecting all constraint half-planes. A feasible region is the intersection of all half-planes…
- A.
A polygon
- B.
An unbounded feasible region
- C.
A point
- D.
None of these
Show answer & explanation
Correct answer: B
CONCEPT
A linear-programming feasible region is formed by intersecting all constraint half-planes.
A feasible region is the intersection of all half-planes defined by the constraints. It is unbounded if feasible points can move indefinitely in some direction while every constraint remains satisfied.
APPLICATION
Rewrite x1 − x2 ≤ 1 as x2 ≥ x1 − 1, and rewrite x1 + x2 ≥ 3 as x2 ≥ 3 − x1.
Together with x2 ≥ 0, these inequalities require x2 to lie above the lines x2 = x1 − 1 and x2 = 3 − x1 in the first quadrant.
For any x1 ≥ 0, choosing x2 large enough satisfies all three lower bounds. There is no upper bound on x2.
CROSS-CHECK
The ray (0, t) is feasible for every t ≥ 3: −t ≤ 1, t ≥ 3, and both coordinates are non-negative. Since this feasible ray extends indefinitely, the feasible region is an unbounded feasible region.