How to add a constraint to the final phase of simplex tableau? How does one go about adding a new constraint in the final phase of the simplex tableau?
In the case where our optimal solution satisfies the constraint, we can conclude that the optimal solution stays the same.
But what if the constraint is not satisfied? Do we have to start over again?

If the previously optimal basic feasible solution violates the new constraint, then the basis (adjusted to include a basic variable in the new constraint) is "superoptimal" (at least as good as, and possible better, than optimal) but infeasible. In that case, you can use the dual simplex algorithm. Whereas the primal simplex algorithm starts with a feasible basis and works toward optimality while maintaining feasibility, the dual simplex algorithm starts with a superoptimal but infeasible basis and works toward feasibility while maintaining superoptimality. Most (all?) text books that discuss the simplex method also discuss the dual simplex method. Dual simplex is the most common way that solvers handle cuts being added to an LP tableau.
