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

Step 1: Understanding the Concept:
This question tests a theoretical property of Linear Programming Problems (LPP). The feasible region in an LPP is a convex polygon. The objective function represents a family of parallel lines.

Step 2: Key Formula or Approach:
Recall the Multiple Optimal Solutions theorem. If an objective function reaches its maximum (or minimum) value at two distinct corner points of the feasible region, then every point on the line segment joining these two corner points will also give the same maximum (or minimum) value.

Step 3: Detailed Explanation:
Let the objective function be $Z = ax + by$. Suppose the maximum value $Z_{max}$ occurs at two corner points, say $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$. This means $Z(P_1) = ax_1 + by_1 = Z_{max}$ and $Z(P_2) = ax_2 + by_2 = Z_{max}$. The line representing the objective function $ax + by = Z_{max}$ passes through both $P_1$ and $P_2$. Since the feasible region is convex, the entire line segment connecting $P_1$ and $P_2$ lies on the boundary of the feasible region. Any point $P(x, y)$ on this line segment can be represented as a convex combination of $P_1$ and $P_2$: \[ P = tP_1 + (1-t)P_2 \text{ for } 0 \le t \le 1 \] Let's find the value of $Z$ at any such point $P$: \[ Z(P) = Z(tP_1 + (1-t)P_2) \] Since $Z$ is a linear function, we can distribute it: \[ Z(P) = t \cdot Z(P_1) + (1-t) \cdot Z(P_2) \] Substitute $Z(P_1) = Z_{max}$ and $Z(P_2) = Z_{max}$: \[ Z(P) = t \cdot Z_{max} + (1-t) \cdot Z_{max} \] \[ Z(P) = Z_{max}(t + 1 - t) = Z_{max}(1) = Z_{max} \] This shows that every single point on the line segment joining the two corner points gives the same maximum value. Since a line segment contains an infinite number of points, the maximum value occurs at infinitely many points.

Step 4: Final Answer:
The number of points at which $Z_{max}$ occurs is infinity.