Bases are looping using simplex method

Question

The question is:

Maximize $x_1 − 2x_2 − 3x_3 − x_4$ subject to the constraints $x_j ≥ 0$ for all $j$ and

\begin{align} x_1 − x_2 − 2x_3 − x_4 \le& 4 \\ 2x_1 + x_3 − 4x_4 \le& 2 \\ −2x_1 + x_2 + x_4 \le& 1. \end{align}

Adding slack variables I get:

\begin{align} x_1 − x_2 − 2x_3 − x_4 + s_1 + 0s_2 + 0s_3 =& 4 \\ 2x_1 + 0x_2 + x_3 − 4x_4 + 0s_1 +s_2 + 0s_3 =& 2 \\ −2x_1 + x_2 + 0x_3 + x_4 + 0s_1 + 0s_2 + s_3 =& 1. \end{align}

Doing out my table I get:

\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline BASES&x_1&x_2&x_3&x_4&s_1&s_2&s_3&RHS \\ \hline &1&-2&-3&-1&0&0&0&0\\ \hline s_1&1&-1&-2&-1&1&0&0&4\\ \hline s_2&2&0&1&-4&0&1&0&2\\ \hline s_3&-2&1&0&1&0&0&1&1\\ \hline \end{array}

I choose to swap bases $x_1$ and $s_2$ since $x_1$ takes on the greatest coefficient in the objective function and $s_2$ basis row takes on the smallest non-negative ratio between RHS and and $x_1$. I get:

\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline BASES&x_1&x_2&x_3&x_4&s_1&s_2&s_3&RHS \\ \hline &0&2&\frac{7}{2}&-1&0&\frac{1}{2}&0&1\\ \hline s_1&0&1&\frac{5}{2}&-1&-1&\frac{1}{2}&0&-3\\ \hline x_1&1&0&\frac{1}{2}&-2&0&\frac{1}{2}&0&1\\ \hline s_3&0&1&1&-3&0&1&1&3\\ \hline \end{array}

Now $x_3$ holds the greatest coefficient and $x_1$ the smallest ratio since $1/\frac{1}{2} = 2 < 3$

Swapping those we get

\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline BASES&x_1&x_2&x_3&x_4&s_1&s_2&s_3&RHS \\ \hline &7&-2&0&-13&0&3&0&6\\ \hline s_1&5&-1&0&-9&1&2&0&13\\ \hline x_3&2&0&1&-4&0&1&0&2\\ \hline s_3&-2&1&0&1&0&0&1&1\\ \hline \end{array}

Now we choose to swap $x_1$ in for the basis of $x_3$ since $x_1=7$ holds the largest value and $\frac{2}{2} = 1$ holds the smallest ratio.

From here we just keep swapping $x_1$ and $x_3$ indefinitely and the coefficients of the objective function never reach all-negative values.

Yet this seems to have a solution on linear programming calculators, so I must be doing something wrong. Does anyone know what?

Answer 1

Here's some errors in the OP's steps.

1. In the initial tableau, at the row representing the objective function $z = x_1 − 2x_2 − 3x_3 − x_4$ (a.k.a. $z$-row), always change the sign of coefficients \begin{array}{|r|r|r|r|r|r|r|r|r|r|} \hline \text{bases} &x_1&x_2&x_3&x_4&s_1&s_2&s_3& \text{RHS} \\ \hline &\bf-1&\bf2&\bf3&\bf1&0&0&0&0\\ \hline s_1&1&-1&-2&-1&1&0&0&4\\ \hline s_2&2&0&1&-4&0&1&0&2\\ \hline s_3&-2&1&0&1&0&0&1&1\\ \hline \end{array} because
   • $z -(x_1 - 2x_2 - 3x_3 - x_4) = 0$ (To understand the row of the rightmost 0, you may think about $A {\bf x} + I_3 {\bf s} = {\bf b}$ on RHS.) \begin{array}{|r|r|r|r|r|r|r|r|r|r|r|} \hline \text{bases} &z&x_1&x_2&x_3&x_4&s_1&s_2&s_3& \text{RHS} \\ \hline &\bf1&\bf-1&\bf2&\bf3&\bf1&0&0&0&0\\ \hline s_1&\bf0&1&-1&-2&-1&1&0&0&4\\ \hline s_2&\bf0&2&0&1&-4&0&1&0&2\\ \hline s_3&\bf0&-2&1&0&1&0&0&1&1\\ \hline \end{array}
   • the coefficient of $z$ is always one, and it's never changed, so it's omitted (to save ink).
2. Choose only negative elements in the $z$-row.
   • Think of $z$ as an affine function of $x_1,\dots,x_4$. In $z = x_1 − 2x_2 − 3x_3 − x_4$, it's natural to increase $x_1$ in order to maximize $z$.
   • This corresponds to choosing a negative entry at the $z$-row in the simplex tableau. (We often choose the most negative one.) \begin{array}{|r|r|r|r|r|r|r|r|r|r|} \hline \text{bases} &x_1&x_2&x_3&x_4&s_1&s_2&s_3& \text{RHS} \\ \hline &\bf-1&2&3&1&0&0&0&0\\ \hline s_1&1&-1&-2&-1&1&0&0&4\\ \hline s_2&\bf2^*&0&1&-4&0&1&0&\bf2\\ \hline s_3&-2&1&0&1&0&0&1&1\\ \hline \end{array}
   • In this case, we replace $s_2$ with $x_1$. (Denote this as $\{s_1,s_2,s_3\}\to\{s_1,x_1,s_3\}$ for short.)
3. As the RHS of a simplex tableau represents the current basic feasible solution (BFS, denoted as $\bf x_B$), it can never take a negative value. (OP's second tableau row $s_1$ should be multiplied by -1.) \begin{array}{|r|r|r|r|r|r|r|r|r|r|} \hline \text{bases} &x_1&x_2&x_3&x_4&s_1&s_2&s_3& \text{RHS} \\ \hline &0&2&\frac{7}{2}&-1&0&\frac{1}{2}&0&1\\ \hline s_1&0&-1&-\frac{5}{2}&1^*&1&-\frac{1}{2}&0&\bf3\\ \hline x_1&1&0&\frac{1}{2}&-2&0&\frac{1}{2}&0&1\\ \hline s_3&0&1&1&-3&0&1&1&3\\ \hline \end{array}
4. At the $z$-row in the above tableau, the only negative entry -1 under $x_4$ is chosen, as well as the only positive entry 1 in the $x_4$ column. Do a pivot operation at the entry marked with *. \begin{array}{|r|r|r|r|r|r|r|r|r|r|} \hline \text{bases} & x_1 & x_2 & x_3 & x_4 & s_1 & s_2 & s_3 & \text{RHS} \\ \hline & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 4 \\ \hline x_4 & 0 & -1 & -\frac52 & 1 & 1 & -\frac12 & 0 & 3 \\ \hline x_1 & 1 & -2 & -\frac92 & 0 & 2 & -\frac12 & 0 & 7 \\ \hline s_3 & 0 & -2 & -\frac{13}{2} & 0 & 3 & -\frac12 & 1 & 12 \\ \hline \end{array} Since every entry in the $z$-row is non-negative, we've done.

Hence, the optimal solution is $(x_1,x_2,x_3,x_4) = (7,0,0,3)$ with optimal value 4.