Final answer:
The existence of an unbounded solution in a simplex method occurs when there are no positive entries in the pivot column below the objective function, indicating that the function can be optimized indefinitely due to an unbounded feasible region.
Step-by-step explanation:
The condition to determine the existence of an unbounded solution of a linear programming problem using the simplex method arises when, during the pivot operations, there is no positive entry in any pivot column of the tableau under the objective function row, meaning we cannot find a minimum ratio to choose a pivot row.
This situation indicates that the feasible region is unbounded in the direction of the corresponding variable, and as such, the objective function can be increased indefinitely.
In practical terms, if we are maximizing, we can make the value as large as desired; if we are minimizing, we cannot find a minimum value due to the lack of lower bounds in the direction of descent.