Question

4. Give an example of an LP for which the feasible region is unbounded, but the optimal objective value is finite.

Answer 1

one such example is

Maximise x w.r.t

x>=0 and y>=0 -- eqn 1

x/3+2/3y>=1 --eqn 2

and x/(-1)+2y<=1 --eqn 3

the feasible region extends to the right till infinity but the optimal value is the x coordinate of the intersection of two eq2 and eq3

Answer 2

Here's an example of a linear programming problem where the feasible region is unbounded, but the optimal objective value is finite:

vbnetCopy codeMaximize: 2x + y
Subject to: x + 2y <= 4
           x >= 0
           y >= 0

The feasible region of this problem is unbounded because there is no upper limit on the value of y. As x increases, y can increase without limit, and the feasible region extends infinitely in the positive y direction.

However, the optimal objective value of this problem is finite. The objective function 2x + y can be maximized by setting x = 0 and y = 2, which satisfies the constraints and gives an optimal objective value of 2. Therefore, the feasible region is unbounded, but the optimal objective value is finite.