Advanced Simplex Calculator
Enter a linear programming model. The calculator supports maximize and minimize goals,
less than, greater than, and equality constraints. Variables are treated as non-negative.
Example Data Table
This example finds the best mix for two decision variables.
| Part |
x1 |
x2 |
Relation |
RHS |
| Objective Max Z |
40 |
30 |
- |
- |
| Constraint 1 |
2 |
1 |
≤ |
100 |
| Constraint 2 |
1 |
1 |
≤ |
80 |
| Constraint 3 |
1 |
0 |
≤ |
40 |
Formula Used
A linear programming model uses an objective function and linear constraints.
For a maximization model, the common form is:
Maximize Z = c1x1 + c2x2 + ... + cnxn
Subject to a11x1 + a12x2 + ... + a1nxn ≤ b1
The simplex method converts inequalities into equations by adding slack,
surplus, and artificial variables. The calculator uses a Big M penalty
for artificial variables. At each step, it computes Cj - Zj.
The entering column is the largest positive value in a maximization table.
The leaving row is found by the smallest valid positive ratio:
Ratio = RHS ÷ positive pivot column value
Pivot operations continue until no positive reduced cost remains,
or until the model becomes unbounded, infeasible, or reaches the
iteration limit.
Simplex Method Guide
What the Method Does
The simplex method solves linear programming problems. It is useful when a decision has limited resources. A business may want maximum profit. A planner may want minimum cost. A manager may need the best mix of products, labor, time, or materials. The method searches corner points of the feasible region. It moves from one better corner to another. This makes it faster than checking every possible value.
Why the Tableau Matters
A simplex tableau organizes the complete model. It places coefficients, basic variables, right side values, and reduced costs in one table. This structure makes each pivot visible. The pivot column shows the variable that can improve the result. The pivot row shows which current basic variable must leave. The ratio test protects feasibility while the solution improves.
Slack, Surplus, and Artificial Values
Inequality constraints must become equations before solving. A less-than constraint receives a slack variable. A greater-than constraint receives a surplus variable and often an artificial variable. Equality constraints also need an artificial variable. The Big M idea adds a large penalty to artificial variables. This pushes them out of the final basis when a feasible solution exists.
Reading the Final Answer
The final answer gives the best objective value and the decision variable values. Constraint checks show how much capacity remains or how much surplus exists. A zero slack value means the constraint is binding. A positive slack value means unused capacity remains. If artificial variables stay positive, the model is infeasible. If no leaving row exists, the model is unbounded.
Best Practices
Use consistent units before entering data. Keep all right-hand-side values realistic. Avoid mixing hours, minutes, dollars, and units without conversion. Test a small example first. Then add more constraints. Review each pivot step when results look surprising. A clear model creates a reliable result.
FAQs
What is a simplex method calculator?
It is a tool that solves linear programming models. It finds the best objective value while respecting all entered linear constraints.
Can this calculator solve minimization problems?
Yes. It converts minimization into an equivalent maximization form internally. The final objective value is then reported using the original direction.
What does RHS mean?
RHS means right-hand side. It is the limit, demand, capacity, or required value placed after the constraint relation sign.
What is a pivot step?
A pivot step changes the basic variable set. It improves the objective value while keeping the solution feasible when possible.
What is a slack variable?
A slack variable measures unused capacity in a less-than-or-equal constraint. Zero slack usually means the constraint is fully used.
What does infeasible mean?
Infeasible means no solution can satisfy all constraints together. The restrictions conflict with each other or demand impossible values.
What does unbounded mean?
Unbounded means the objective can improve forever in the allowed direction. At least one missing limit usually causes this condition.
Can I export the simplex result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button to save a formatted report with summary tables.