A standard maximization problem has the form maximize z = c1x1 + c2x2 + ... + cnxn subject to linear constraints and x1, x2, ..., xn >= 0. Slack variables turn <= constraints into equations, such as a1x1 + a2x2 + s1 = b. In the tableau method, the entering variable is chosen from the most negative objective-row coefficient for the z-row convention, and the leaving variable is chosen by the smallest nonnegative ratio b / positive pivot-column entry.
The solution is optimal when no negative coefficients remain in the objective row.
Key Facts
- For a standard maximization problem, write the objective as z - c1x1 - c2x2 - ... - cnxn = 0 before forming the tableau.
- A <= constraint becomes an equation by adding a slack variable, such as 2x + 3y <= 12 becoming 2x + 3y + s1 = 12.
- In the common z-row tableau convention, the entering variable is the column with the most negative coefficient in the objective row.
- The leaving row is found using the minimum nonnegative ratio RHS / pivot-column entry, using only rows with positive pivot-column entries.
- The pivot entry must be changed to 1 by dividing the entire pivot row by the pivot value.
- All other entries in the pivot column must be changed to 0 using row replacement operations.
- A basic feasible solution is read by setting nonbasic variables to 0 and taking each basic variable from its row RHS value.
- For maximization with the z-row convention, the tableau is optimal when every coefficient in the objective row is greater than or equal to 0.
Vocabulary
- Linear programming
- Linear programming is the process of optimizing a linear objective function subject to linear constraints.
- Simplex tableau
- A simplex tableau is a table that stores the objective equation, constraint equations, variable coefficients, and right-hand-side values.
- Slack variable
- A slack variable is a nonnegative variable added to a <= constraint to turn it into an equality.
- Pivot
- A pivot is the selected entry used to swap one variable into the basis and one variable out of the basis.
- Basic variable
- A basic variable is a variable whose column acts like a unit column and whose value is read from the right-hand side.
- Optimality condition
- The optimality condition is the rule that tells when no further improving pivot is available.
Common Mistakes to Avoid
- Choosing the most negative number from the entire tableau is wrong because the entering variable is selected only from the objective row.
- Using negative or zero pivot-column entries in the ratio test is wrong because only positive entries can safely determine the leaving variable.
- Forgetting to divide the pivot row by the pivot entry is wrong because the entering variable column must become a unit column.
- Changing only the constraint rows and not the objective row is wrong because the objective equation must be updated by the same row-operation logic.
- Reading nonbasic variables from the RHS is wrong because nonbasic variables are set to 0 in the current basic feasible solution.
Practice Questions
- 1 Maximize z = 3x + 2y subject to x + y <= 4, 2x + y <= 5, x >= 0, y >= 0. Write the initial simplex tableau using slack variables s1 and s2.
- 2 For the objective row z - 5x - 3y + 0s1 + 0s2 = 0, which variable enters first under the most negative coefficient rule?
- 3 In a pivot column, the constraint entries are 2, 1, and 4, and the RHS values are 10, 6, and 12. Compute the ratio test values and identify the leaving row.
- 4 Explain why the simplex method moves from one basic feasible solution to another instead of checking every point in the feasible region.
Understanding Simplex Method Worked Walkthrough
The simplex method is easier to understand when it is linked to geometry. Each linear constraint describes a boundary and a permitted side of that boundary. Together, the constraints create a feasible region.
In two variables, this region can be drawn as a polygonal area. The best value of a linear objective occurs at a corner of that region, unless several corners tie. With more than two variables, the picture cannot be drawn easily, but the same rule still applies.
Simplex moves from one corner point to a neighboring corner point. Every move must keep the solution feasible. The method is therefore a controlled search through possible corner solutions rather than a search through every possible value of every variable.
A tableau records one corner solution in a compact form. The variables currently named by row labels are the basic variables. They have values determined by the right hand side.
Variables not assigned to rows are nonbasic, so they are temporarily set to zero. Choosing a pivot changes which variables play these roles. The entering variable is one that can improve the objective from its current value.
The ratio test protects feasibility. A row with a positive entry in the entering column places a limit on how far that variable can increase. The smallest permitted limit reaches zero first, so that row must leave the basis.
This is why using a negative or zero denominator in the ratio test gives a wrong result. Such rows do not create an upper limit in the direction of movement.
Row operations are not just arithmetic cleanup. They rewrite the same system of equations in a new form. After a pivot, the entering variable has a column like a unit column.
That makes its value easy to read from the right hand side. Small arithmetic errors matter because one incorrect sign can change later ratios, select the wrong row, or make a feasible solution appear impossible. Fractions are often safer than rounded decimals.
Students should write each row operation clearly, then check the pivot column before moving on. A useful check is that the new pivot row has a one in the pivot position and every other row has a zero in that column. The objective value should not decrease during a correctly performed maximization pivot.
Some tableaux need special interpretation. A tie in the ratio test can produce a degenerate pivot, where a basic variable becomes zero and the objective value may not improve on that step. This is valid, though it needs careful arithmetic.
If an improving column has no positive entries in the constraint rows, the objective can increase without a bound. The problem is then unbounded rather than optimal. If a nonbasic column has the right objective condition at the final tableau, there can be more than one optimal solution.
Simplex appears in production planning, transport scheduling, diet planning, blending materials, and budget allocation. In real models, the hardest part is often translating words into constraints. Units, resource limits, nonnegativity, and the meaning of each variable deserve as much attention as the tableau work.