Difference between two phase method and big M I am searching for differences between two phases and big M method for finding the solution for a linear problem with simplex method.
Also I realized that two phases method is algebraically more easier than big M method and as you see here, the two phase method breaks off big M function in two parts, first the real coefficients and second coefficients the the M's amount.
But I cant find any right answer for my question:" Why two phases method is algebraically more easier than big M  method?" I have just an intuitive answer and not any thing else...
Thanks for your help.
 A: Big-M Method
The Big-M method of handling instances with artificial variables is the “commonsense approach”. Essentially, the notion is to make the artificial variables, through their coefficients in the objective function, so costly or unprofitable that any feasible solution to the real problem would be preferred, unless the original instance possessed no feasible solutions at all. But this means that we need to assign, in the objective function, coefficients to the artificial variables that are either very small (maximization problem) or very large (minimization problem); whatever this value,let us call it Big M.
In fact, this notion is an old trick in optimization in general; we simply associate a penalty value with variables that we do not want to be part of an ultimate solution(unless such an outcome is unavoidable). Indeed, the penalty is so costly that unless any of the  respective variables' inclusion is warranted algorithmically, such variables will never be part of any feasible solution. This method removes artificial variables from the basis. Here,  we assign a large undesirable (unacceptable penalty) coefficients to artificial variables from the objective function point of view. If the objective function (Z) is to be minimized, then a very large positive price (penalty, M) is assigned to each artificial variable and if Z is to be minimized, then a very large negative price is to be assigned. The penalty will be designated by +M for minimization problem and by –M for a maximization problem and also M>0.
Two-Phase Method
This method differs from Simplex method that first it is necessary to accomplish an auxiliary problem that has to minimize the sum of artificial variables. Once this first problem is resolved and reorganizing the final board, we start with the second phase, that consists in making a normal Simplex. Steps to solve a problem using two-phase simplex method:
n   Step 1
Modify the constraints so that the right-hand side of each constraint is nonnegative. This requires that each constraint with a negative right-hand side be multiplied through by -1.
n  Step 2
Identify each constraint that is now an = or ≥ constraint. In step 3, we will add an the artificial variable to each of these constraints.
n  Step 3
Convert each inequality constraint to standard form.
For ≤ constraint i, we add a slack variable si;
For ≥ constraint i, we add an excess variable ei;
n  Step 4
For now, ignore the original LP’s objective function. Instead solve an LP whose objective function is min w’= (sum of all the artificial variables). This is called the Phase I LP. The act of solving the phase I LP will force the artificial variables to be zero.
Since each ai≥0, solving the Phase I LP will result in one of the following three cases:
n  Case 1
The optimal value of w’ is greater than zero. In this case, the original LP has no feasible solution.
n  Case 2
The optimal value of w’ is equal to zero, and no artificial variables are in the optimal Phase I basis. In this case, we drop all columns in the optimal Phase I tableau that correspond to the artificial variables. We now combine the original objective function with the constraints from the optimal Phase I tableau. This yields the Phase II LP. The optimal solution to the Phase II LP is the optimal solution to the original LP.
