Simplex Method Calculator | Linear Programming Step-by-Step
Solve two-variable linear programming problems with the simplex method. Enter the objective function (maximize or minimize) and up to four inequality constraints. Displays the initial tableau, each pivot operation with entering and leaving variables, and the final optimal solution with all basic feasible solutions listed.
What Is the Simplex Method Calculator | Linear Programming Step-by-Step?
The simplex method is the foundational algorithm for linear programming. Starting from a basic feasible solution (slack variables as basics), it moves along edges of the feasible polytope toward the optimal vertex. Each iteration selects an entering variable that improves the objective and a leaving variable via the minimum-ratio test to maintain feasibility.
Formula
Maximize Z = c₁x₁ + c₂x₂ subject to Ax ≤ b, x ≥ 0. Convert to standard form by adding slack variables, then pivot: entering variable = most negative objective coefficient; leaving variable = minimum RHS/column ratio.
How to Use
- 1
Choose Maximize or Minimize and enter the objective coefficients c₁ and c₂.
- 2
Add up to 4 constraints. Enter a₁, a₂, the inequality direction (≤ or ≥), and the RHS value b.
- 3
For ≥ constraints (e.g. minimum requirements), the calculator multiplies by −1 to convert to ≤.
- 4
Click Solve with Simplex Method to run the algorithm.
- 5
Read the optimal values of Z, x₁*, and x₂* from the result tiles at the top.
- 6
Scroll through the tableau sequence to follow every pivot step — each shows the entering and leaving variables.
- 7
Load a preset (Classic, Profit, Diet) to explore worked examples instantly.
Select Maximize or Minimize, enter the objective coefficients c₁ and c₂, then build your constraints. Use the ≥ type for lower-bound constraints and the calculator will convert them automatically. Click Solve to see every tableau step.
Example Calculation
Classic problem: Maximize Z = 3x₁ + 5x₂ subject to x₁ ≤ 4, 2x₂ ≤ 12, 3x₁ + 5x₂ ≤ 25, x₁,x₂ ≥ 0. After adding slacks s₁, s₂, s₃, the simplex method finds x₁* = 0, x₂* = 5, Z* = 25 in 2 pivot steps.
Understanding Simplex Method | Linear Programming Step-by-Step
Standard Form Conversion
| Constraint Type | Conversion | Variable Added | Interpretation |
|---|---|---|---|
| a₁x₁ + a₂x₂ ≤ b | Add slack s ≥ 0 | Slack variable | Unused resource; s = b − a₁x₁ − a₂x₂ |
| a₁x₁ + a₂x₂ ≥ b | Multiply by −1 → ≤ | Slack on negated | Converts ≥ to ≤ before adding slack |
| a₁x₁ + a₂x₂ = b | No change needed | Artificial variable | For Big-M or two-phase method |
| Minimize Z = cx | Convert to max −Z | None | Maximizing −Z is identical to minimizing Z |
Simplex Tableau Structure
| Column / Row | Contents | Role in Algorithm |
|---|---|---|
| BV column | Basic variable for each row | Tracks which variable is currently basic (non-zero) |
| Decision columns (x₁, x₂) | Coefficients after pivoting | Entering variable chosen by most negative objective coefficient |
| Slack columns (s₁, s₂, ...) | Identity matrix initially | Become non-identity after pivot operations |
| RHS column | Right-hand side values | Ratio test: RHS ÷ pivot column → selects leaving variable |
| Objective row (Z) | Reduced costs | All non-negative → optimal solution found |
Real-World Applications of Linear Programming
- ▸Production planning — maximize profit subject to resource, labor, and capacity constraints
- ▸Transportation and logistics — minimize total shipping cost across a distribution network
- ▸Portfolio optimization — maximize expected return for a given level of risk
- ▸Diet and nutrition planning — minimize cost while meeting all daily nutrient requirements
- ▸Scheduling — assign tasks or workers to shifts to minimize overtime while meeting demand
- ▸Telecommunications — maximize network throughput subject to bandwidth constraints
- ▸Agriculture — determine optimal crop mix to maximize revenue given land and water limits
Frequently Asked Questions
What is the entering variable rule?
For maximization, the entering variable is the decision variable with the most negative coefficient in the objective row. A negative coefficient means increasing that variable will improve (increase) the objective value.
What is the minimum ratio test?
Once the entering column is chosen, divide each positive entry in that column into the corresponding RHS value. The row with the smallest non-negative ratio becomes the pivot row, and its basic variable leaves the basis. This ensures all RHS values remain non-negative (feasible).
How are ≥ constraints handled?
Multiply both sides by −1 to convert ≥ to ≤, then add a slack variable. For example, x₁ + x₂ ≥ 4 becomes −x₁ − x₂ ≤ −4, then −x₁ − x₂ + s = −4. This keeps the standard form but may require a different phase-1 approach for artificial variables if the basic solution becomes infeasible.
What does "unbounded" mean?
If the entering column has no positive entries, the objective can be increased indefinitely without violating any constraint. This usually signals a modeling error — a missing upper-bound constraint on a resource.
When is the simplex method optimal?
The current basic feasible solution is optimal when all coefficients in the objective row are non-negative (for maximization). No variable can improve the objective, so the current vertex is the optimum.
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