Two Iterations of Full Tableau Simplex
Two Iterations of Full Tableau Simplex
Use tableau simplex when the exam asks for the full tableau method.
Problem
\[\max 3x_1+2x_2\]subject to
\[x_1+x_2+s_1=4,\] \[2x_1+x_2+s_2=5,\] \[x_1+s_3=3.\]Initial basis: $s_1,s_2,s_3$.
Initial tableau
| basis | $x_1$ | $x_2$ | $s_1$ | $s_2$ | $s_3$ | RHS |
|---|---|---|---|---|---|---|
| $s_1$ | 1 | 1 | 1 | 0 | 0 | 4 |
| $s_2$ | 2 | 1 | 0 | 1 | 0 | 5 |
| $s_3$ | 1 | 0 | 0 | 0 | 1 | 3 |
| $z$ | -3 | -2 | 0 | 0 | 0 | 0 |
Pivot 1
$x_1$ enters, $s_2$ leaves.
| basis | $x_1$ | $x_2$ | $s_1$ | $s_2$ | $s_3$ | RHS |
|---|---|---|---|---|---|---|
| $s_1$ | 0 | $1/2$ | 1 | $-1/2$ | 0 | $3/2$ |
| $x_1$ | 1 | $1/2$ | 0 | $1/2$ | 0 | $5/2$ |
| $s_3$ | 0 | $-1/2$ | 0 | $-1/2$ | 1 | $1/2$ |
| $z$ | 0 | $-1/2$ | 0 | $3/2$ | 0 | $15/2$ |
Pivot 2
$x_2$ enters, $s_1$ leaves.
| basis | $x_1$ | $x_2$ | $s_1$ | $s_2$ | $s_3$ | RHS |
|---|---|---|---|---|---|---|
| $x_2$ | 0 | 1 | 2 | -1 | 0 | 3 |
| $x_1$ | 1 | 0 | -1 | 1 | 0 | 1 |
| $s_3$ | 0 | 0 | 1 | -1 | 1 | 2 |
| $z$ | 0 | 0 | 1 | 1 | 0 | 9 |
The solution is
\[x_1=1, \qquad x_2=3, \qquad z=9.\]Exam checkpoint
For simplex questions in minimization form, negative reduced costs indicate possible improvement. Use $u=B^{-1}A_j$, apply the ratio test only to positive components of $u$, then update the basis.