Question

In Linear Programming Problem (LPP), the objective function $Z = ax + by$ has the same maximum value at two corner points. The number of points at which $Z_{max}$ occurs is

• A. 1
• B. 2
• C. 0
• D. Infinity

Hint:

This is a standard property to memorize: Optimal value at one point $\rightarrow$ Unique solution. Optimal value at $\ge 2$ adjacent points $\rightarrow$ Infinitely many solutions (the entire edge connecting them).

Answer 1

The Correct Option is D

The given question is about the properties of solutions in a Linear Programming Problem (LPP). Specifically, it concerns a scenario in which the objective function \(Z = ax + by\) reaches its maximum value at two different corner points.

In Linear Programming, if the maximum value of the objective function occurs at more than one corner point, then it implies that the segment connecting these two corner points is a line segment of the feasible region where the objective function has the same value throughout. This is because the objective function is linear, and linearity implies that any linear combination of points with the same function value will also result in the same function value.

Here's a step-by-step explanation:

1. Identify Corner Points: Linear Programming Problems are usually solved by identifying points at the vertices (or corners) of the feasible region. These are candidate points for maximizing or minimizing the objective function.
2. Objective Function Line Segment: If the objective function value, \(Z = ax + by\), is the same at two distinct corner points, say \((x_1, y_1)\) and \((x_2, y_2)\), then all points on the line segment joining these two points also yield the same objective function value.
3. Number of Points with Maximum Value: As every point on the line segment between these two corner points yields the same maximum value for the objective function, there are infinitely many such points, assuming the line segment is not a single point.

Thus, given that the objective function \(Z\) achieves its maximum value at more than one corner, the number of points at which \(Z_{max}\) occurs is indeed infinite. Therefore, the correct answer is Infinity.

Hence, the correct option is Infinity, as none of the other options - 1, 2, or 0 - appropriately reflect the situation described.