Question

What is the Dual simplex method ?

Answer 1

The first benefit of dual feasibility is that in many circumstances, finding a primal viable basic solution is difficult, but finding a dual feasible basic solution is simple. To obtain a fundamental workable solution for the primal, we'd have to employ the two-phase simplex approach.In principle, whenever we have a dual feasible tableau, we can use the formula $c{B^T}{A^{ - 1}}B$ to find a dual feasible solution, but we don’t always want to. The dual viable solution for the above tableau is $(0,0,0)$. The second advantage of dual feasibility is that by going from dual optimum tableau to dual optimal tableau, we can try to discover an optimal solution.

Complete answer:
Consider the tableau above. It is dual feasible, but primal feasible is a long way off.
We can choose a row with a negative right-hand side, such as the s1 row in the example above. We'll divide by a negative amount if we choose a negative entry in that row as the new pivot in that row, and the right-hand side will turn positive.
Not every option is a good option, much like with the traditional simplex technique. In this example, we have two options for pivots in s1's row: x or y. After row-reducing, the following are the results: x on the left, y on the right.
We must avoid this and maintain dual feasibility. To accomplish so, we must carefully select the pivot entry. The leaving variable was the row we chose, and we were free to choose any leaving variable with a negative value. However, if dual feasibility is to be preserved, the choice of entering a variable is normally made independently.

Note:
Working towards feasibility, the Dual Simplex Method will pivot from dual feasible dictionary to dual feasible dictionary. The Dual Simplex Method is the name given to this new pivoting approach because it is essentially the same as using the Simplex Method on a dual linear issue.