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 model is unbounded because a negative RHS appears while all reduced costs are non-negative.
The current basic solution is already optimal, so the algorithm should terminate.
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.
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.
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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