Calculator Inputs
Example Data Table
This sample maximizes profit for two products under material, labor, and demand limits.
| Item | x1 | x2 | Sign | RHS |
|---|---|---|---|---|
| Objective profit | 40 | 30 | Max | - |
| Material | 2 | 1 | <= | 100 |
| Labor | 1 | 1 | <= | 80 |
| Product x1 demand | 1 | 0 | <= | 40 |
Formula Used
Objective: Maximize or minimize Z = c1x1 + c2x2 + ... + cnxn.
Constraints: ai1x1 + ai2x2 + ... + ainxn ≤, ≥, or = bi.
Standard form: Add slack variables for ≤ rows, surplus variables for ≥ rows, and artificial variables when needed.
Reduced cost: Cj - Zj, where Zj = CBB-1Aj.
Ratio test: Leaving row = smallest positive value of RHS divided by entering column value.
Optimality rule: For a maximization table, stop when no positive Cj - Zj remains.
How to Use This Calculator
- Choose maximize or minimize.
- Select the number of decision variables and constraints.
- Enter objective coefficients in the same variable order.
- Enter every constraint coefficient, sign, and right-side value.
- Press the calculate button to create simplex tables.
- Review the result, graph, binding rows, and pivot log.
- Download the CSV or PDF report for records.
Simplex Method for Linear Programming
Linear programming turns a practical goal into a mathematical model. The simplex method then searches the corner points of the feasible region. Each corner represents a possible plan. The best corner gives the highest profit, lowest cost, or strongest score. This calculator helps you build that process in a clear way.
Why the Method Works
A linear objective has no curves. A linear constraint has no curves either. Because of this structure, the optimum occurs at a boundary point when a feasible optimum exists. The simplex table moves from one basic feasible solution to another. It chooses an entering variable with improvement potential. It chooses a leaving variable using the ratio test. This keeps the solution feasible during each pivot.
What This Tool Shows
The calculator accepts maximization or minimization problems. It supports less than, greater than, and equal constraints. Slack variables show unused capacity. Surplus variables show excess above a lower bound. Artificial variables help start difficult models. The iteration table displays basis variables, pivot choices, ratios, and reduced costs. These details make each step easy to audit.
Reading the Answer
The final variable values show the suggested activity levels. The objective value shows the resulting profit, cost, or score. Slack values identify nonbinding limits. A zero slack usually means the constraint is tight. Reduced costs show whether a nonbasic variable can still improve the current plan. The chart gives a quick visual summary.
Practical Uses
Use this method for production planning, diet design, transport allocation, staffing, blending, budgeting, and scheduling. Always check units before solving. Keep coefficients consistent. Review negative right side values carefully. Test several scenarios after the first answer. A clean model gives better decisions, faster reviews, and fewer manual errors.
Modeling Tips
Start with a plain sentence for the objective. Then translate each limit into one row. Place every decision variable in the same order. Use zero when a variable is missing from a constraint. Avoid mixing hours, days, and weeks in one coefficient set. If the output looks unusual, inspect the table before changing the model during the first run. Most errors come from signs, units, or swapped coefficients.
FAQs
What is the simplex method?
The simplex method is an algorithm for solving linear programming problems. It moves between feasible corner points and improves the objective value until no better move remains.
Can this calculator solve minimization problems?
Yes. It converts minimization into an equivalent maximization form internally. The final objective value is then reported using your original minimization coefficients.
What are slack variables?
Slack variables measure unused capacity in less-than-or-equal constraints. A zero slack usually means that resource or limit is fully used.
What are artificial variables?
Artificial variables create a temporary starting basis for equal and greater-than-or-equal constraints. They should become zero in a feasible final solution.
What does binding mean?
A binding constraint is active at the optimum. Its left side equals the right side, so it has no meaningful unused capacity.
Why can a model be unbounded?
A model is unbounded when the objective can improve forever without breaking constraints. This often means a missing limit or wrong inequality direction.
Why can a model be infeasible?
A model is infeasible when no point satisfies every constraint at once. Conflicting limits, negative resources, or incorrect signs can cause infeasibility.
How many variables are supported?
This page supports up to six decision variables and eight constraints. You can raise those limits by adjusting the selector ranges and server-side caps.