I wrote a pc program that implements the simples method. In most cases, the program output is correct. But there are cases where an error occurs. For example:
max -x-y
s.t.
x<=230
y<=230
x=1
y=0
The problem is converted to:
max -x-y-Ma1-Ma2
s.t.
x+s1=230
y+s2=230
x+a1=1
y+a2=0
And there the simplex table after initialization:
\begin{bmatrix}-&&&x&y&s1&s2&a1&a2&rhs\\\\s1&&&1&0&1&0&0&0&230\\s2&&&0&1&0&1&0&0&230\\a1&&&1&0&0&0&1&0&1\\a2&&&0&1&0&0&0&1&0\\z&&&1&1&0&0&M&M&0\end{bmatrix}
R5-> R5-MR3-MR4
\begin{bmatrix}-&&&x&y&s1&s2&a1&a2&rhs\\\\s1&&&1&0&1&0&0&0&230\\s2&&&0&1&0&1&0&0&230\\a1&&&1&0&0&0&1&0&1\\a2&&&0&1&0&0&0&1&0\\z&&&1-M&1-M&0&0&0&0&-M\end{bmatrix}
The pivot is [a1,x] = 1.
R1 -> R1-R3
R5->R5- (1-M)R3
\begin{bmatrix}-&&&x&y&s1&s2&a1&a2&rhs\\\\s1&&&0&0&1&0&-1&0&229\\s2&&&0&1&0&1&0&0&230\\x&&&1&0&0&0&1&0&1\\a2&&&0&1&0&0&0&1&0\\z&&&0&1-M&0&0&M-1&0&-1\end{bmatrix}
The pivot is [s2,y] = 1.
R4->R4-R2
R5->R5-(1-M)R2
\begin{bmatrix}-&&&x&y&s1&s2&a1&a2&rhs\\\\s1&&&0&0&1&0&-1&0&229\\y&&&0&1&0&1&0&0&230\\x&&&1&0&0&0&1&0&1\\a2&&&0&0&0&-1&0&1&-230\\z&&&0&0&M-1&0&M-1&0&-1-230(1-M)\end{bmatrix}
That's the final table and y=230. But y need to be zero...
I would be happy if someone could tell me where the problem was.
tnx!