Good day.
Given the following notation for an initial canonical tableau for a linear program in standard form:
$$ T_1 = \begin{bmatrix} I & B^{-1}N & \bar{x}_{B} \\ 0^\intercal & \hat{x}_{N}^\intercal & -\bar{z} \end{bmatrix} $$
with:
- $B$ representing the matrix of the basic technological coefficients ($x_{ij})$,
- $N$ representing the matrix of the non-basic tech. coefficients,
- $\bar{x}_{B}$ representing the
B
-partition of the given feasible basic solution, - $\hat{x}_{N}^\intercal$ representing the vector of the non-basic reduced costs,
- $\bar{z}$ representing the value of the objective function at the given $\bar{x}$ solution.
how would you go about representing the following standard linear program in tableau form?
$$\begin{alignedat}{4} \min_x{z} = & \; -4x_1 & \; -3x_2 & \; +7x_3 \\ & \; \quad\ 3x_1 & \; -5x_2 & \; +4x_3 & \; + x_4 && = & \; 3 \\ & \; \quad\ 6x_1 & \; +4x_2 & \; -5x_3 & & \; +x_5 \ & = & \; 2 \\ &&&&& \quad\ \ x & \ge & \; 0 \end{alignedat}$$
Update (adding what I have already tried/already know and tried to apply): looking for help on manuals and here on OR SE alike, the tableau seems to be generally in the form:
$$ T_2 = \begin{bmatrix} A & b \\ c^\intercal & -\bar{z} \end{bmatrix} $$
with $A$ being the matrix of all technological coefficients, $b$ the resources vector and $c^\intercal$ the transpose costs vector.
The problem does not give you an initial $\bar{x}$ solution and I don't personally know how to represent the program above in tableau form.
Is the following correct? If so, why is $T_1 = T_2$?
$$\bar{T} = \begin{bmatrix} 3 & -5 & 4 & 1 & 0 & 3 \\ 6 & 4 & -5 & 0 & 1 & 2 \\ -4 & -3 & 7 & 0 & 0 & 0 \end{bmatrix} $$
The tableau above is built supposing that $x_4, x_5$ is the feasible basic solution vector and that, then, the tableau is in canonical form with respect to $\bar{x} = \begin{bmatrix} 0 & 0 & 0 & 3 & 2 \end{bmatrix}$.