A data scientist is minimizing the total cost of CPU-hours on a cluster. After several primal simplex iterations, the current tableau shows that every non-basic variable has a reduced cost that is zero or positive, yet one basic slack variable has a right-hand-side value of -4. What does this situation imply about the current basis, and what is the appropriate next step to continue solving the problem without restarting from scratch?
The current basic solution is already optimal, so the algorithm should terminate.
The model is unbounded because a negative RHS appears while all reduced costs are non-negative.
The basis is dual-feasible but primal-infeasible; pivot on the basic variable with the negative RHS (i.e., switch to the dual simplex or an equivalent negative-RHS pivot) to restore feasibility before proceeding.
The solution is feasible but not optimal; choose the non-basic variable with the most negative reduced cost to enter the basis and continue with the primal simplex method.
A minimization tableau in which all reduced costs for non-basic variables are non-negative satisfies the dual (optimality) condition, but primal feasibility also requires every basic variable's RHS to be non-negative. The negative RHS therefore means the present basic solution is primal-infeasible even though it is dual-feasible. In such cases the standard (primal) simplex cannot proceed, because no entering variable can fix the infeasibility while preserving the non-negativity of basic variables. The dual simplex method (or an equivalent "negative-RHS" pivot rule) is designed for exactly this scenario: it selects the basic variable with the most negative RHS to leave the basis and pivots so that feasibility is restored while keeping the reduced-cost pattern intact. Stopping, performing an ordinary ratio test, or declaring the model unbounded all ignore the fact that the basis can be repaired systematically, so they are incorrect.
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