In Linear ProgrammingIn the optimal simplex table zj-cj=0 indicatesans.boundedunboundedalternative solutioninfeasible Previous Marked for Review Next

In linear programming, when analyzing an optimal simplex table, the notation $z_j - c_j = 0$ indicates a specific condition regarding the nature of the solution. Here’s a breakdown of the implications:

1. Bounded Solution: The condition $z_j - c_j = 0$ implies that the solution is optimal, and all decision variables have a finite contribution to the objective function at the current vertex of the feasible region. This means the feasible region is bounded and the simplex method has successfully found the maximum or minimum value.

2. Unbounded Solution: If, in the context of the simplex method, instead of equal zero, a situation arises where the $z_j - c_j$ is not limited (i.e., can grow indefinitely), the problem is unbounded. This would mean that you could achieve infinite values for the objective function by extending the feasible region.

3. Alternative Solutions: The case where $z_j - c_j = 0$ for more than one variable indicates the presence of alternative or degenerate solutions. This means there is more than one solution at the corner point of the feasible region that gives the same optimal value.

4. Infeasible Solution: This occurs when none of the constraints can be satisfied simultaneously. If the tableau shows that the solution cannot yield any feasible values (typically indicated by negative values in the solution column), it is classified as infeasible.

Summary

Thus, $z_j - c_j = 0$ specifically indicates that an optimal solution is found, and further investigation is needed to determine if there are alternative solutions or if the problem is bounded/unbounded.