How to use Big M Simplex method to solve LP problem?

Question

I've given the following LP problem:

P(x) = 4x1 + 5x2 -> max;
  x1 - 2x2 <= 15;
 4x1 + 3x2 <= 24;
-2x1 + 5x2 >= 20;
x1 >= 0; x2 >= 0;

I have to perform 3 tasks:

1. Convert this problem to Normal form and check how many variables and constraints there are
2. Convert the normal form to a Big M problem and perform a Big M simplex for the first iteration
3. Create a dual problem for the above LP problem

I can do the 1st task and maybe the 3rd, but I've no clue how the Big M method works. I tried to search, but I couldn't find an actual example. I cannot understand the usual mathematical formulas, but I can instantly lear from even one step-by-step guide. Any link where I can find a step-by-step solution with actual numbers would be fine.

The normal form I got for task 1 is as follows:

P(x) = 4x1 + 5x2 -> max;
  x1 - 2x2 + x3 = 15;
 4x1 + 3x2 + x4 = 24;
-2x1 + 5x2 - x5 = 20;
x1 >= 0; x2 >= 0; x3 >= 0; x4 >= 0; x5 >= 0;

Is the above Normal form correct? It has 5 variables and 3 constraints, am I correct?

Answer 1

First, write the constraints as equations:

(1) $x_1 - 2x_2 \leq 15$ we need to add a slack variable: (1)* $x_1 - 2x_2+x_3 = 15$

(2) $4x_1 + 3x_2 \leq 24$ here we need a slack variable: (2)* $4x_1 + 3x_2 + x_4= 24$

(3)$-2x1 + 5x2 \geq 20$ here we need a surplus variable: (3)*$-2x_1 + 5x_2 -x_5 = 20$

Note that we use a slack variable($+x_i $) for "$\leq$" restrictions and a surplus variable($-x_i$) for "$\geq$ " restrictions. Now we need to add an artificial variable in the constraint (3)* because we can't use surplus variables as basic variables and we need a basic variable for each contraint. The problem given is equivalent to:

max $P = 4x_1 + 5x_2-Mx_6$;

subjecto to

(1)* $x_1 - 2x_2+x_3 = 15$

(2)* $4x_1 + 3x_2 + x_4= 24$

(3)*$-2x_1 + 5x_2 -x_5+x_6 = 20$

so tht initial simplex table:

From here you only need to sum M times the line of $X_6$ in the line of Z and apply the simplex method.