If an LPP admits optimal solution at two consecutive vertices of a joining two points

Options

A. the required optimal solution is at the midpoint of the line joining two points
B. the optimal solution occurs at every point on the line joining these two points
C. the LPP under consideration is not solvable
D. the LPP under consideration must be reconstruted

Answer

To solve the problem regarding the Linear Programming Problem (LPP) that admits an optimal solution at two consecutive vertices of a line joining two points, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We need to analyze a Linear Programming Problem (LPP) where the optimal solution occurs at two consecutive vertices of the feasible region.

**Hint**: Identify what is meant by "consecutive vertices" and how they relate to the feasible region of the LPP.

2. **Graphical Representation**:
- Draw the feasible region on a graph, marking the axes (x-axis and y-axis) and the vertices of the feasible region.

**Hint**: Visualizing the feasible region can help you understand where the vertices are located and how they connect.

3. **Identifying the Optimal Solution**:
- According to the properties of LPP, if an optimal solution exists at two consecutive vertices, it means that the objective function is constant along the line segment joining these two vertices.

**Hint**: Recall that the objective function is maximized or minimized at the vertices of the feasible region.

4. **Analyzing the Midpoint**:
- Check if the optimal solution occurs at the midpoint of the line segment joining the two vertices. In this case, it does not; the optimal solution occurs along the entire line segment.

**Hint**: Remember that the midpoint is just one point, while the optimal solution can be any point along the line segment.

5. **Conclusion**:
- Since the optimal solution occurs at every point on the line joining the two vertices, the correct conclusion is that the optimal solution is not just at the vertices but along the entire line segment connecting them.

**Hint**: Consider the implications of having multiple optimal solutions and how they affect the interpretation of the LPP.

### Final Answer:
The correct option is that the optimal solution occurs at every point on the line joining these two vertices.