Consider the Linear Program below. Which of the choices represents the best (tightest) lower bound?
Max 2X + Y
Subject to
X + Y ≤ 10
X, Y ≥ 0
A) (∞, -∞)
B) (-∞, -∞)
C) (0, 0)
D) (5, 0)
E) (25, 16)

Question

Quizplus · Accepted Answer

The constraints X + Y ≤ 10, X, Y ≥ 0 define a feasible region, which is a triangular region with vertices at (0,0), (10,0), and (0,10). The objective function 2X + Y is a linear function that increases as we move towards the right and upwards, so the maximum value will be at one of the vertices.

The objective function is largest at the vertex (5,0), where it takes the value 2(5) + 0 = 10. Therefore, 10 is the best (tightest) lower bound.

Note that choice A represents both a lower and an upper bound that are unattainable, since there are no constraints on X and Y that restrict them from taking arbitrarily large or small values. Choice B represents a lower bound of negative infinity, which is not useful. Choice C represents a lower bound of 0, which is not as tight as the bound of 10. Choice E is the maximum value of the objective function, not a lower bound.

Consider the Linear Program Below