A non-basic variable represents a decision variable currently set to zero in the optimal solution. Its reduced cost measures the net change in the objective function per unit increase of that variable.

When a non-basic variable has a reduced cost of exactly zero, it means you can increase that variable above zero—bringing it into the basis—without changing the objective value at all. Pivoting this variable into the basis shifts the solution to a different corner point (vertex) along an optimal edge or face, thereby confirming the existence of alternate optimal solutions that achieve the exact same objective value.

A zero reduced cost for a non-basic variable indicates alternate optimality because that variable can enter the solution without changing the current objective value. Since the variable is currently zero but has a reduced cost of zero, increasing it can produce another feasible solution with the same optimal objective value. Therefore, there is more than one optimal solution, which is called alternate optimality.

A zero reduced cost means a non-basic variable can enter the solution without changing the optimal value. This shows that another solution can give the same optimal result, which means there is an alternate optimal solution.