MHB The column gets out of the basis

  • Thread starter Thread starter evinda
  • Start date Start date
  • Tags Tags
    Basis Column
evinda
Gold Member
MHB
Messages
3,741
Reaction score
0
Hello! (Wave)

Find the basic feasible solutions of the system of restrictions:

$$2x_1+x_2+x_3=10 \\ 3x_1+8x_2+x_4=24 \\ x_2+x_5=2 \\ x_i \geq 0, i=1,2,3,4,5$$

We notice that the rank of the matrix $A=\begin{pmatrix}
2 & 1 & 1 & 0 & 0\\
3 & 8 & 0 & 1 & 0\\
0 & 1 & 0 & 0 & 1
\end{pmatrix}$ is $3$ and obviously for $x_1=x_2=0$ we have the solution $\overline{x_0}=(0,0,10,24,2)$ which is basic feasible non degenerate.
Thus the first tableaux of the algorithm of the searching of the vertices is the following:

$$\begin{matrix}
B & b & P_1 & P_2 & P_3 & P_4 & P_5 & \theta \\ \\
P_3 & 10 & 2 & 1 & 1 & 0 & 0 & 10/2\\
P_4 & 24 & 3 & 8 & 0 & 1 & 0 & 24/3\\
P_5 & 2 & 0 & 1 & 0 & 0 & 5 & -
\end{matrix}$$We pick $P_1$ to get in the basis.

$$\theta_0= \min \{ \frac{10}{2}, \frac{24}{3}\}=5$$

The pivot is the element $2$ so the column $P_3$ gets out of the basis.How do we deduce that $P_3$ gets out of the basis? Since its the only column from $P_3, P_4, P_5$ that contains at the row where the pivot is a positive number? (Thinking)
 
Physics news on Phys.org
Ok, I understood why $P_3$ gets out of the basis.Then we get this matrix:$\begin{matrix}
B & b & P_1 & P_2 & P_3 & P_4 & P_5 & \theta & \\ \\
P_1 & 5 & 1 & \frac{1}{2} & \frac{1}{2} & 0 & 0 & & \Gamma_1'=\frac{1}{2} \Gamma_1\\ \\
P_4 & 9 & 0 & \frac{13}{2} & -\frac{3}{2} & 1 & 0 & & \Gamma_2'=\Gamma_2-3\Gamma_1'\\ \\
P_5 & 2 & 0 & 1 & 0 &0 & 1 & & \Gamma_3'=\Gamma_3-0 \Gamma_1'
\end{matrix}$So the new solution that we found is $(5,0,0,9,2)$ which is basis feasible non degenerate.Then we choose to get in the basis whether the column $P_2$ or $P_3$.
We choose $P_2$.

$$\theta_0= \min \{ \frac{5}{\frac{1}{2}}, \frac{9}{\frac{13}{2}}, \frac{2}{1}\}=\frac{18}{3}$$

The pivot is the element $\frac{13}{2}$, so $P_4$ gets out of the basis.

Then we have this matrix:

$\begin{matrix}
B & b & P_1 & P_2 & P_3 & P_4 & P_5 & \theta & \\ \\
P_1 & \frac{56}{13} & 1 & 0 & \frac{8}{13} & -\frac{1}{13} & 0 & & \Gamma_1'= \Gamma_1-\frac{1}{2} \Gamma_2'\\ \\
P_2 & \frac{18}{13} & 0 & 1 & -\frac{3}{13} & \frac{2}{13} & 0 & & \Gamma_2'=\frac{2}{13} \Gamma_2\\ \\
P_5 & \frac{8}{13} & 0 & 0 & \frac{3}{13} &-\frac{2}{13} & 1 & & \Gamma_3'=\Gamma_3-\Gamma_2'
\end{matrix}$So the new solution that we found is $\left( \frac{56}{13}, \frac{18}{13}, 0,0, \frac{8}{13}\right)$ which is basic feasible non degenerate.Then we choose $P_3$ to get in the basis.

$\theta_0=\min \{ \frac{56}{8}, \frac{8}{3}\}=\frac{8}{3}$The pivot is the element $\frac{3}{13}$ and so the column $P_5$ gets out.
$\begin{matrix}
B & b & P_1 & P_2 & P_3 & P_4 & P_5 & \theta & \\ \\
P_1 & \frac{664}{169} & 1 & 0 & 0 & \frac{1}{3} & -\frac{8}{3} & & \Gamma_1'= \Gamma_1-\frac{8}{13} \Gamma_3'\\ \\
P_2 & 2 & 0 & 1 & 0 & 0 & 1 & & \Gamma_2'= \Gamma_2+\frac{3}{13} \Gamma_3'\\ \\
P_3 & 0 & 0 & 0 & 1 &-\frac{2}{3} & \frac{13}{3} & & \Gamma_3'=\frac{13}{3}\Gamma_3
\end{matrix}$

Thus the new solution is $\left( \frac{664}{169}, 2 , \frac{8}{3}, 0, 0\right)$ which is basic feasible non degenerate.

If we choose $P_5$ we don't get a basic feasible solution, so we have to pick $P_1$.Is it right so far? (Thinking)
 
Hi all, I've been a roulette player for more than 10 years (although I took time off here and there) and it's only now that I'm trying to understand the physics of the game. Basically my strategy in roulette is to divide the wheel roughly into two halves (let's call them A and B). My theory is that in roulette there will invariably be variance. In other words, if A comes up 5 times in a row, B will be due to come up soon. However I have been proven wrong many times, and I have seen some...
Thread 'Detail of Diagonalization Lemma'
The following is more or less taken from page 6 of C. Smorynski's "Self-Reference and Modal Logic". (Springer, 1985) (I couldn't get raised brackets to indicate codification (Gödel numbering), so I use a box. The overline is assigning a name. The detail I would like clarification on is in the second step in the last line, where we have an m-overlined, and we substitute the expression for m. Are we saying that the name of a coded term is the same as the coded term? Thanks in advance.
Back
Top