The optimal value of the objective function Z = ax + by may or may not exist, if the feasible region for a LPP is unbounded.

Explanation:
To understand why the optimal value of the objective function may or may not exist when the feasible region for a Linear Programming Problem (LPP) is unbounded, let's first define what an unbounded feasible region is.

An unbounded feasible region is a region in the graph of the constraints where there are no restrictions on the values of the decision variables. This means that the feasible region extends infinitely in one or more directions.

Optimal Value of the Objective Function:
The optimal value of the objective function refers to the maximum or minimum value that can be achieved for the given objective function within the feasible region. In other words, it represents the best possible value that can be obtained for the objective function.

Possible Scenarios:
When the feasible region is unbounded, there are two possible scenarios in terms of the optimal value of the objective function:

1. Optimal value exists: In some cases, even though the feasible region is unbounded, there may still exist an optimal value for the objective function. This occurs when the objective function is bounded and has a maximum or minimum value within the feasible region. The optimal value can be found at the extreme point(s) of the feasible region.

2. Optimal value does not exist: In other cases, the objective function may not have an optimal value within the unbounded feasible region. This occurs when the objective function is unbounded as well, meaning it can increase or decrease indefinitely without reaching a maximum or minimum value. In such cases, the objective function does not have a finite optimal value.

Conclusion:
Therefore, the optimal value of the objective function Z = ax + by may or may not exist when the feasible region for a LPP is unbounded. It depends on whether the objective function is bounded or unbounded within the feasible region. If the objective function is bounded, an optimal value exists, whereas if the objective function is unbounded, an optimal value does not exist.