simplex $B^{-1} \cdot A_j$ tableau interpretation

Question

In an iteration of the simplex tableau implementation, what is the interpretation of the columns $B^{-1} \cdot A_j$ underneath each variable $x_j$?

Answer 1

Dantzig & Thapa (Section 3.5) call it the "representation of the sth activity in terms of the basic set of activities" where in your case activity $s$ is $x_j$. In other words the vector describes the change of the values of the basic variables per unit of change of the non-basic variable $x_j$.

Example: solving $$\begin{align} \text{max.}\qquad & z=5x_1+6x_2,\\ \text{s.t.}\qquad & 2x_1+3x_2 \leq 18, & \text{(1)}\\ & 2x_1+x_2 \leq 12, & \text{(2)}\\ & x_1,x_2\geq 0, \end{align} $$ you get the tableau $$\begin{array}{cccccc} \hline x_1 & x_2 & s_1 & s_2 & z & \text{RHS}\\ \hline 0 & 1 & 0.5 & -0.5 & 0 & 3\\ 1 & 0 & -0.25 & 0.75 & 0 & 4.5\\ \hline 0 & 0 & 1.75 & 0.75 & 1 & 40.5 \\ \hline \end{array}$$ with optimal solution $x_1=4.5$, $x_2=3$. Graphically, you can see the vectors of the non-basic variables $s_1$ and $s_2$ pointing into the direction of change: