Formula Used
The calculator converts a linear programming problem into tableau form.
Maximize Z = c1x1 + c2x2 + ... + cnxn
Each constraint is written as:
a1x1 + a2x2 + ... + anxn ≤ b
Slack variables are added to less-than rows. Surplus variables and artificial variables are added to greater-than rows. Artificial variables are also added to equality rows.
The pivot column is selected from the most negative reduced cost. The pivot row is selected by the smallest valid ratio:
Ratio = RHS / positive entering column value
The pivot row is divided by the pivot value. Other rows are updated until the entering column becomes a unit column. The process stops when no negative reduced cost remains.
Simplex Method Overview
The simplex method solves linear programming problems by moving along corner points. Each move improves the objective value until no better corner remains. This calculator is designed for learners, analysts, and planners who need a clear view of every tableau.
Why Step Tables Matter
A final answer is useful, but the path is often more important. The pivot column shows the entering variable. The ratio test shows the leaving variable. The pivot row creates the next basic solution. Seeing these details helps you check homework, audit models, and explain business decisions.
Inputs That Shape The Model
Start with the goal. Choose maximize or minimize. Enter objective coefficients for every decision variable. Then describe each constraint with coefficients, an inequality sign, and a right side value. The tool also supports greater than and equality rows by adding artificial variables through a large penalty approach.
Reading The Result
The optimal value appears first. Variable values show the recommended decision. Slack values show unused resource. Surplus values show excess above a lower bound. Artificial variables should be zero in a valid final solution. If an artificial value remains positive, the model is infeasible.
Practical Uses
Simplex models appear in production planning, diet mixes, delivery schedules, staffing, portfolio allocation, and resource selection. The method is helpful when choices are limited by time, money, labor, machine hours, or material stock. A small model can still reveal strong decisions.
Good Modeling Habits
Use consistent units. Do not mix kilograms with grams or hours with minutes. Keep constraints realistic. Check whether every variable should stay nonnegative. Review each coefficient before submitting. A tiny entry error can change the pivot path and final answer.
Export And Review
Use CSV export for spreadsheets. Use PDF export for reports or class notes. The graph gives a fast visual summary. For two-variable models, it displays constraint lines and the optimal point. For larger models, it displays a solution bar chart. Save common examples and compare outputs. When a result feels unexpected, inspect the first tableau again. Most mistakes appear in signs, right side values, or objective direction. Careful setup creates reliable optimization work and safer repeatable study results.
FAQs
1. What does the simplex method calculate?
It finds the best value of a linear objective function under linear constraints. The result includes decision variable values, objective value, and supporting slack, surplus, or artificial variable values.
2. Can this calculator solve minimization problems?
Yes. Select minimize. The calculator transforms the objective internally, solves the tableau, and converts the final objective value back to the original minimization direction.
3. Why are artificial variables used?
Artificial variables create an initial basic feasible structure for equality and greater-than constraints. They should leave the final solution or become zero when the model is feasible.
4. What is the pivot column?
The pivot column is the entering variable column. In this calculator, it is selected from the most negative reduced cost in the objective row for the transformed tableau.
5. What is the ratio test?
The ratio test divides each positive entering-column entry into the right side value. The smallest nonnegative ratio chooses the leaving row and keeps the next solution feasible.
6. Why can a model be infeasible?
A model is infeasible when no point satisfies all constraints together. This can happen when resource limits, lower bounds, and equalities conflict with each other.
7. Why can a model be unbounded?
A model is unbounded when the objective can keep improving without a finite limit. The tableau shows this when the entering column has no positive pivot entry.
8. When should I change the penalty value?
Use a larger penalty when objective coefficients are very large. The penalty should dominate normal coefficients so artificial variables are strongly discouraged in the final solution.