The Full Tableau Simplex Method

Running the simplex method to termination in a single table. The tableau carries the whole system in canonical form against the current basis, each pivot updates it by row operations, and the stopping conditions, optimality and unboundedness, are read off the table rather than computed separately.

Definition

The full tableau is the constraint system written in canonical form against the current basis, with the objective carried as an extra row. For a standard-form minimisation with $m$ constraints and $n$ variables it is an $(m + 1) \times (n + 1)$ array:

\begin{array}{cc} A_{B}^{- 1} A & A_{B}^{- 1} b \\ {\bar{c}}^{T} & - z_{0} \end{array}

The first $m$ rows hold the constraints solved for the basic variables; the last row holds the reduced costs ${\bar{c}}_{j}$ and the negated objective value. Each basic variable's column is a unit vector, so the basic solution is read directly from the right-hand column.

One iteration.

Optimality test. If every ${\bar{c}}_{j} \geq 0$ , the current basis is optimal. Stop.
Entering column. Choose $j$ with ${\bar{c}}_{j} < 0$ . Any such column improves; the most negative is a common rule but not a requirement.
Ratio test. Among rows $i$ with $a_{i j} > 0$ , choose the one minimising $b_{i} / a_{i j}$ . If no entry in the column is positive, the program is unbounded. Stop.
Pivot. Divide the chosen row by its entry in the entering column, then eliminate that column from every other row including the objective row.

The method repeats until one of the two stopping conditions fires.

What the pivot preserves. Row operations do not change the solution set of the constraint system, so every tableau describes the same program. What changes is which basis it is written against, and therefore which solution is displayed in the right-hand column.

Termination. Each pivot with $θ > 0$ strictly decreases the objective, so no basis repeats and the method terminates: there are finitely many bases. When $θ = 0$ the objective does not fall, the basis changes but the point does not, and without an anti-cycling rule such as Bland's the method can in principle cycle.

Assumptions and scope

The method starts from a basic feasible solution. Obtaining one for a program without an obvious starting basis is a separate procedure, and a tableau begun from an infeasible basis reports values that are not solutions.
The program must be in standard form before the tableau is built. Converting inequalities, free variables and a maximisation is prior work, and a tableau built from an unconverted program computes correct arithmetic on the wrong problem.
Any column with a negative reduced cost may enter. The most-negative rule is a convention; different rules give different pivot sequences and the same optimum.
The ratio test uses only rows with a strictly positive entry in the entering column. Including a zero or negative entry produces a step that breaks feasibility.
A tie in the ratio test means the next basic feasible solution is degenerate. Either tied row may leave, and the choice affects the sequence of bases without affecting the optimum.
Termination rests on the objective strictly decreasing at each pivot. With degenerate steps that guarantee lapses, and cycling is prevented by an anti-cycling rule rather than by the tableau itself.

Worked material

Non-example

Four ways a run goes wrong

Including a nonpositive entry in the ratio test. The entering column holds $2$ , $- 1$ and $3$ ; the learner computes all three ratios and takes the most negative as the minimum. A basic variable with a nonpositive entry does not fall as the entering variable rises, so it never limits the step. Including it produces a step that drives another basic variable negative, and the next tableau has a negative right-hand entry. The point is no longer feasible.

Forgetting the objective row. The entering column is eliminated from every constraint row and the objective row is left alone. The tableau now looks canonical, but the bottom row still holds the previous basis's reduced costs. The optimality test reads stale numbers, and the method either stops early or chooses a column that does not improve.

Stopping at the first nonnegative reduced cost. Seeing ${\bar{c}}_{2} = 1$ , the learner concludes optimality. The test is over every nonbasic column, not one: a single positive reduced cost says only that raising that particular variable would not help. Optimality requires no negative entry anywhere in the row.

Reporting only the basic variables. The final tableau shows $x_{1} = 4$ and $x_{4} = 2$ , and the answer is given as $(4, 2)$ . The solution is a vector over all variables, and the nonbasic ones are zero: $x = (4, 0, 0, 2)$ . Dropping them loses the distinction between a variable that is zero and a variable that is absent, and makes the answer impossible to check against the original constraints.

What the four have in common is that each leaves the tableau looking well formed. Three of them are caught by the checks in the previous block; the fourth is caught only by substituting the reported point back into the program as first written.

Contrast

Carrying the table against computing the step

The same iteration can be carried two ways, and they differ in what they store, not in what they decide.

	Full tableau	Computing with $A_{B}^{- 1}$
What is stored	the whole updated array, $(m + 1) (n + 1)$ entries	$A_{B}^{- 1}$ and the current point
Reduced costs	all present, updated every pivot	computed per column, on demand
Optimality test	scan one row	price columns until one is negative, or all are checked
Work per pivot	update every entry	form $p^{T} = c_{B}^{T} A_{B}^{- 1}$ , then one direction $u = A_{B}^{- 1} A_{j}$
Cost on a wide program	every column updated whether consulted or not	only the columns actually priced
Provenance	lost: entries are updated, not derived	explicit: each quantity is the formula that produced it

Why the tableau is taught first. Every number an iteration needs is on the page, so the decisions are visible and the method can be executed and checked by hand. That makes it the right vehicle for learning what the algorithm does.

Why it is not what solvers do. On a program with many more columns than rows, updating every entry each pivot is wasted work, since most columns are never examined. Practical implementations keep a factorisation of $A_{B}$ and price columns selectively; the revised simplex method is the same algorithm organised that way.

What is identical. The entering choice, the ratio test, the step, the sequence of bases, and the optimum. Given the same pivoting rule, both produce the same path. A disagreement between them is an arithmetic error in one, not a difference of method.

The learning risk. Because the tableau updates rather than derives, fluent pivoting can coexist with not knowing what $A_{B}^{- 1}$ currently is or why a reduced cost has its value. Reconstructing one reduced cost from $c_{j} - p^{T} A_{j}$ and checking it against the table is the exercise that keeps the two connected.

Common errors

Common misconception

Every entry of the entering column is a candidate for the ratio test, so the leaving row is the one giving the smallest ratio among all of them, including rows whose entry is zero or negative.

The Full Tableau Simplex Method

Assumptions and scope

Worked material

Four ways a run goes wrong

Carrying the table against computing the step

Common errors

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