Question

What is difference between two phase method and dual simplex method?

Answer 1

Put succinctly (at least by my standards), the simplex method starts with a feasible but suboptimal solution and generates a sequence of feasible but less suboptimal ones until it reaches an optimal solution and stops. The dual simplex method starts with a superoptimal (too good to be true) but infeasible solution and generates a sequence of progressively less infeasible (and less superoptimal) ones until it arrives at a feasible solution (which will be optimal).
For some LPs primal simplex is faster, while for some other LPs dual simplex is faster. Dual simplex is popular in situations where you start with an optimal solution and then add a few constraints (or modify the right hand sides of a few existing constraints), rendering the previous solution infeasible but superoptimal. That describes the branching process in branch-and-bound/branch-and-cut algorithms for mixed integer linear programs.