Name the region in Linear programming problem that gives the most feasible value of the objective function.

Options

A. Concave set
B. Feasible
C. Convex set
D. Bounded region

Answer

To solve the question regarding the region in a linear programming problem that gives the most feasible value of the objective function, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Objective Function**:
The objective function in a linear programming problem is typically represented as \( f(x, y) = ax + by + c \), where \( a \), \( b \), and \( c \) are constants. The goal is to maximize or minimize this function.

**Hint**: Identify the coefficients in the objective function to understand how they influence the direction of optimization.

2. **Identify the Feasible Region**:
The feasible region is defined by the set of inequalities that represent the constraints of the problem. This region is the area where all the constraints overlap and is typically bounded by the lines representing the inequalities.

**Hint**: Graph the inequalities on a coordinate plane to visualize the feasible region.

3. **Locate the Vertices of the Feasible Region**:
The feasible region is usually a polygon, and its vertices (corners) are critical points where the maximum or minimum values of the objective function can occur.

**Hint**: Use the intersection points of the constraint lines to find the vertices of the feasible region.

4. **Evaluate the Objective Function at Each Vertex**:
To find the most feasible value of the objective function, calculate the value of \( f(x, y) \) at each vertex of the feasible region.

**Hint**: Substitute the coordinates of each vertex into the objective function to find their respective values.

5. **Determine the Optimal Value**:
Compare the values obtained from the previous step. The vertex that yields the highest value (if maximizing) or the lowest value (if minimizing) is the optimal solution.

**Hint**: Keep track of which vertex gives the best value according to the objective of the problem (maximize or minimize).

### Conclusion:
The region in a linear programming problem that gives the most feasible value of the objective function is called the **feasible region**. The optimal solution lies at one of the vertices of this region.