Tableau Simplex on 3-Variable Problems

Extend the simplex tableau method from two decision variables to three. Set up the initial tableau with three decision-variable columns and three slack columns, identify pivot elements via the most-negative-z-row entry and the minimum ratio test, perform pivot iterations using row operations, and read the optimal solution from a terminal tableau.

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Tutorial

Initial Tableau for a Three-Variable LP

For a maximization LP in standard form with three decision variables,

\text{maximize } z = c_1 x_1 + c_2 x_2 + c_3 x_3

subject to $a_{i1} x_1 + a_{i2} x_2 + a_{i3} x_3 \le b_i$ for $i = 1, 2, 3,$ and $x_1, x_2, x_3 \ge 0,$ we introduce one slack variable $s_i$ per constraint, turning each inequality into an equation

a_{i1} x_1 + a_{i2} x_2 + a_{i3} x_3 + s_i = b_i.

The initial simplex tableau now has three decision columns and three slack columns (six variable columns in all), plus the RHS:

\begin{array}{c|cccccc|c} \text{Basic} & x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & \text{RHS} \\ \hline s_1 & a_{11} & a_{12} & a_{13} & 1 & 0 & 0 & b_1 \\ s_2 & a_{21} & a_{22} & a_{23} & 0 & 1 & 0 & b_2 \\ s_3 & a_{31} & a_{32} & a_{33} & 0 & 0 & 1 & b_3 \\ \hline z & -c_1 & -c_2 & -c_3 & 0 & 0 & 0 & 0 \end{array}

The basic variables in the starting tableau are the slacks $s_1, s_2, s_3,$ and the z-row carries the negated objective coefficients $-c_j.$ The procedure is identical to the two-variable case — the tableau is just one column wider.