Question

The feasible region of a linear programming problem with objective function \( Z = ax + by \), 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.

Hint:

A bounded feasible region guarantees the existence of both a maximum and minimum value for the objective function.

Answer 1

The Correct Option is C

In a linear programming problem, the objective function is represented as \( Z = ax + by \), where \( a \) and \( b \) are fixed coefficients, and \( x \) and \( y \) are variables to be determined. The feasible region encompasses all points \( (x, y) \) that satisfy the problem's constraints.Step 1: Bounded Feasible Region
A "bounded" feasible region signifies a closed and finite area. This means all possible solutions are contained within this specific area, and none lie outside it.
Step 2: Objective Function Behavior in a Bounded Region
The objective function \( Z = ax + by \) is linear, meaning it consistently increases or decreases in a particular direction. Given a bounded feasible region, the linear objective function will achieve its highest and lowest values at the vertices (corner points) of this region.
Step 3: Existence of Maximum and Minimum Values
For a bounded feasible region, the objective function is guaranteed to have both a maximum and a minimum value, occurring at these corner points. This is a fundamental characteristic of linear programming problems with bounded feasible regions.
- If the feasible region is bounded, the objective function's maximum and minimum values will always occur at one of its vertices.
- The maximum value of \( Z \) is the largest value found at a corner point, and the minimum value is the smallest value found at a corner point.
Consequently, a bounded feasible region ensures that both a maximum and a minimum value for the objective function exist.
Step 4: ConclusionTherefore, the correct determination is:\[\boxed{C} \text{ It will have both maximum and minimum values.}\]