When a basic variable's objective-function coefficient changes, the current basis may stop being optimal. To determine whether the change violates the optimality condition, use the final simplex tableau and recalculate the relevant reduced costs.

A change violates optimality when recalculating the reduced costs causes any non-basic variable to cross its required sign threshold.

To determine if a proposed change to a basic variable's objective function coefficient ($\Delta c_k$) violates optimality, you must evaluate its impact on the reduced costs (or indicators) of all non-basic variables in the final tableau.

Optimality is violated when changing a basic objective coefficient causes a non-basic variable's reduced cost to cross its required sign threshold. This occurs because shifts in basic coefficients ripple through the objective row calculations, altering non-basic indicators. If any non-basic reduced cost violates its required sign condition, the current basis is no longer optimal.

A proposed change to a basic objective function coefficient violates the optimality condition when the resulting reduced cost in the final simplex tableau moves outside the required optimality range. For a maximization problem, the reduced costs must remain non-positive (depending on the tableau convention), so a positive reduced cost indicates that the current solution is no longer optimal. The allowable increase and decrease in the sensitivity analysis show the range within which the coefficient can change without violating the optimality condition.

A change violates optimality when it causes any reduced cost in the final tableau to cross zero, making a non-basic variable attractive enough to enter the basis.

A proposed change violates the optimality condition when the reduced cost of a basic variable moves outside its allowable range, causing it to no longer satisfy the required optimality condition. In simple terms, if the new coefficient makes the reduced cost have the wrong sign, the current solution is no longer optimal.

Check the reduced cost of the basic variable after the coefficient change. If it violates the required sign condition in the final simplex tableau, the current solution is no longer optimal.

In a maximization problem, changing a basic variable's objective coefficient alters all non-basic reduced costs by subtracting the product of that change and the non-basic entries in the basic variable's tableau row. Optimality is violated the moment any non-basic reduced cost drops below zero. In a minimization problem, the update adds that product to the non-basic reduced costs, violating optimality the moment any reduced cost rises above zero.

Recalculate the reduced costs of all non-basic variables using the updated basic coefficient vector. A violation occurs when any non-basic reduced cost changes sign beyond the allowed optimality threshold, indicating that a non-basic variable is now profitable enough to enter the basis.

Check the changed reduced costs in the final tableau: if any reduced cost violates the optimality condition (e.g., becomes negative for a maximization problem using Cj−Zj), the current basis is no longer optimal.

A change violates the optimality condition when it makes a basic variable’s reduced cost fall outside the required sign range in the final tableau.

If the coefficient change makes any reduced cost violate the optimality condition (e.g., become negative in a maximization tableau), the current basis is no longer optimal.

Check the reduced cost (or net evaluation) of the variable in the final simplex tableau. If a change in the objective coefficient makes the reduced cost violate the required optimality sign condition, the current solution is no longer optimal and the simplex method must be continued.

You know the change violates the optimality condition when the new reduced cost of any nonbasic variable no longer has the required sign for the objective (≤ 0 for maximization and ≥ 0 for minimization), meaning the current solution may no longer be optimal.

You can know by checking the reduced cost (or \(C_j-Z_j\)) of the variable in the final simplex tableau after changing its objective coefficient. If the new coefficient causes the reduced cost to violate the required optimality condition—for example, \(C_j-Z_j>0\) in a maximization problem—the current basis is no longer optimal, so the solution may change. If the reduced cost still satisfies the optimality condition, the current basic solution remains optimal.

I know that a proposed change violates the optimality condition when the change makes the reduced cost have the wrong sign. For a maximization problem, if the reduced cost becomes positive, it means the current solution is no longer optimal.

You can know if the change violates the optimality condition by checking the reduced costs (Cj − Zj) in the final simplex tableau. If changing the objective coefficient makes any reduced cost have the wrong sign for a maximization problem, then the current solution is no longer optimal. Basically, you just recalculate the affected reduced costs and check if the optimality condition is still satisfied.

A proposed change to a basic objective-function coefficient violates the optimality condition when the resulting reduced costs no longer satisfy the required optimality signs in the final simplex tableau. In sensitivity analysis, you check the coefficient’s allowable increase and decrease: if the new coefficient stays within that range, the current basis remains optimal; if it goes outside the range, the reduced costs may indicate that another variable should enter the basis, meaning the current solution is no longer guaranteed to be optimal and the problem should be re-solved.