Q16: Bounded feasible region: does Z have a maximum and a minimum?
If the feasible region of a linear programming problem with objective function , is bounded, then which of the following is correct ?
(A) It will only have a maximum value.
(B) It will only have a minimum value.
(C) It will have both maximum and minimum values.
(D) It will have neither maximum nor minimum value.
By the corner point theorem, if the feasible region is bounded, Z has both a maximum and a minimum value, each at a corner point of the region.
Why?
A bounded region can be enclosed in a circle: it is a closed polygon with its edges included. Z is evaluated at every corner point. The largest of these values is the maximum of Z and the smallest is the minimum, so both always exist.
So (A), (B) and (D) are ruled out.
Why?
Watch out: "only a maximum" or "only a minimum" can happen only when the region is unbounded. For example, on the region , , has a minimum, 0, but no maximum.
Answer: (C) It will have both maximum and minimum values.
Chapter 12, Linear Programming: Bounded feasible region: maximum and minimum.
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