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 of the form \( Z = ax + by \), where \( a \) and \( b \) are constants, and \( x \) and \( y \) are the decision variables. The feasible region of a linear programming problem is the set of all points \( (x, y) \) that satisfy the constraints of the problem. Step 1: Feasible Region is Bounded
The term "bounded" means that the feasible region is a closed and finite region in the plane. This means that there is a well-defined region within which all feasible solutions exist, and no solution lies outside this region.
Step 2: Objective Function Behavior
The objective function \( Z = ax + by \) is a linear function. A linear function either increases or decreases in one direction. Because the feasible region is bounded, it is confined within a certain region of the coordinate plane. The linear objective function will attain its extreme values at the vertices (or corner points) of the feasible region.
Step 3: Existence of Both Maximum and Minimum
Because the feasible region is bounded, the objective function will reach both a maximum value and a minimum value at these corner points. This is a property of linear programming problems with a bounded feasible region.
- If the region is bounded, the objective function will always attain its maximum and minimum at one of the vertices.
- The maximum value corresponds to the largest value of \( Z \) at a corner point, and the minimum value corresponds to the smallest value of \( Z \) at a corner point.
Thus, a bounded feasible region guarantees that both a maximum and a minimum value will exist for the objective function.
Step 4: Conclusion Therefore, the correct answer is: \[ \boxed{C} \text{ It will have both maximum and minimum values.} \]