MATH2070 Chap.2 Simplex Algorithm and Tableau Decisions
Simplex Algorithm and Tableau Decisions
Simplex represents feasible corners algebraically as basic feasible solutions. Slack variables turn upper-bound inequalities into equations, non-basic variables are set to zero, and a basis determines the remaining values. Each iteration separates two decisions: reduced cost identifies an improving entering variable, while the minimum positive ratio identifies the leaving variable that preserves feasibility.
The chapter also diagnoses degeneracy, alternative optima and unbounded improving columns. A pivot changes the basis while maintaining equivalent constraint equations, so basis labels and right-hand sides are part of the reasoning, not clerical decoration. Eligible ratios identify the first basic variable to reach zero and must exclude coefficients that do not limit the proposed movement.
Degenerate pivots may change the basis without changing the current corner or objective. At termination, a feasible dictionary with no improving reduced cost supports optimality, whereas an improving column with no eligible leaving row identifies unbounded improvement. Reading these structural signals is as important as completing row operations accurately.
What this chapter covers
- 01
Standard form
- 02
Slack variables
- 03
Basic feasible solutions
- 04
Entering variables
- 05
Ratio tests
- 06
Pivot updates
- 07
Stopping and special cases
Worked example · free
Perform one justified simplex pivot
- 1Choose an improving entering variable.
- 1Show every eligible ratio.
- 1Pivot all rows including the objective.
- 1Read and check the new basis.
Key terms
- Basis
- A selected set of columns used to solve for the basic variables.
- Non-basic variable
- A variable fixed at zero for the current basic solution.
- Pivot
- A Gauss-Jordan update that exchanges one basic and one non-basic variable.
- Degeneracy
- A basic feasible solution with at least one basic variable equal to zero.
- Reduced cost
- The objective-row quantity indicating how the objective changes when a non-basic variable begins to enter, under the dictionary's stated sign convention. Its sign guides improvement but does not choose the leaving row. That separate feasibility decision comes from eligible ratios in the entering column.
- Degenerate pivot
- A basis change with zero step length, often caused by a zero minimum ratio or a tie. The basic labels may change while the represented corner and objective value stay fixed.
Simplex Algorithm and Tableau Decisions FAQ
What is the difference between entering and leaving choices?
The entering variable is chosen because moving it from zero can improve the objective. The leaving variable is chosen by the minimum positive ratio so that no current basic variable becomes negative.
How does simplex detect an unbounded objective?
An improving entering variable has no eligible positive coefficient in the constraint rows, so no basic variable reaches zero as it increases. Feasible objective improvement can continue without limit.
Does a ratio tie prove multiple optima?
No. A ratio tie may produce degeneracy. Multiple optima are indicated at an optimal tableau by an eligible non-basic variable with zero reduced cost, which permits movement without changing the objective.
Exam move
Treat each dictionary as a corner rather than as a matrix of numbers. Before choosing a pivot, list the basic variables, set the non-basic variables to zero, read the current objective value and confirm that every basic right-hand side is non-negative. This makes feasibility visible and gives the row operations a geometric meaning.
Keep one sign convention for the objective row throughout a problem and write it at the top of the page. Separate entering and leaving decisions in your spoken explanation. The entering variable is chosen because its reduced cost offers improvement; the leaving variable is chosen by eligible ratios to protect feasibility.
Calculate ratios in a small side table, omit zero or wrong-sign coefficients under the active convention, and state any tie. After pivoting, verify that the new basic column has the required unit pattern and that the objective moves as predicted unless the step is degenerate. Build diagnosis sets containing optimal, unbounded, tied and degenerate dictionaries.
Do not complete long iterations at first: identify the case and justify it from the objective row, column signs and ratios. Next, run two full problems while preserving exact arithmetic and labelling the basis after every iteration. If an entry is wrong, trace backward to the first tableau where a pivot invariant fails instead of restarting from the final answer.
For retrieval practice, hide the procedural notes and reconstruct the algorithm from three questions: what improves, what limits movement, and what certifies stopping? Translate the final basic solution back to the original variables, compute slacks and interpret the objective in context. Revisit one failed problem after two days and again after a week.
Progress is shown not by faster row operations alone, but by reliably distinguishing an optimal tableau from an infeasible start or an improving direction with no leaving row.
Working through Simplex Algorithm and Tableau Decisions in MATH2070? Sia is AskSia’s AI Mathematics tutor — ask any MATH2070 Simplex Algorithm and Tableau Decisions question and get a clear, step-by-step explanation grounded in how MATH2070 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
