Simplex Method Calculator
Simplex Method Calculator solves linear programming problems by optimizing an objective under linear constraints using the classic simplex algorithm efficiently.
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Helping Notes
Write expressions as 3x1+4x2 and constraints like x1+2x2<=8, separated by commas or semicolons.
Both ≤/≥ and <=/>= are supported; spaces don’t matter for parsing.
Enable nonnegativity to impose xi ≥ 0 automatically for all variables.
Results
Answer
Optimal Solution
Status
Method Used
What Is a Simplex Method Calculator?
A Simplex Method Calculator optimizes a linear objective subject to linear constraints by moving from one basic feasible solution to another along the edges of the feasible polytope. In standard notation, we maximize subject to , or equivalently in standard form after adding slack variables. The calculator constructs and updates the simplex tableau, chooses entering and leaving variables via reduced costs and the minimum‑ratio test, and iterates until optimality, infeasibility, or unboundedness is certified. It shows every pivot, keeps exact fractions when desired, and presents human‑readable justifications for classwork, audits, or quick decision support.
About the Simplex Method Calculator
The method evaluates reduced costs . If all (for maximization), the current basis is optimal. Otherwise, an entering column with is chosen (e.g., most positive or Bland’s rule). The direction is , and the leaving row satisfies the ratio test . Zero ratios indicate degeneracy; cycling is avoided with anti‑cycling rules. If , the problem is unbounded. The tableau operations correspond to elementary row operations that maintain feasibility while improving the objective value . Dual information and sensitivity ranges are extracted from the final tableau’s shadow prices.
Standard form:
Reduced costs:
Search direction: Ratio test:
Tableau pivot (schematic):
Optimality (max): Unbounded:
How to Use This Simplex Method Calculator
- Enter your LP in standard or inequality form. The tool adds slack/surplus and artificial variables when necessary (two‑phase or big‑M).
- Choose pivot rule (most‑positive reduced cost, Bland’s rule) and arithmetic mode (fractions or decimals).
- Compute to see the initial tableau, reduced costs, entering/leaving variables, pivot steps, and the updated objective value.
- Stop when optimality is detected, infeasibility is proven (artificial variables remain positive), or unboundedness is certified. Copy the final basis and solution.
Examples
- Feasible maximization: Maximize subject to . Add slacks . The simplex pivots to with .
- Unbounded: Maximize with constraints only. No finite optimum; ratio test finds no positive direction bounds.
- Degeneracy: If the minimum ratio is zero, the basis changes but does not improve; anti‑cycling rules maintain progress.
- Sensitivity (shadow prices): From the final tableau, dual variables give marginal value of increasing each .
Formula Snippets Ready for Rendering
FAQs
What problems does the Simplex Method Calculator solve?
Linear programs: maximize or minimize a linear objective subject to linear equality/inequality constraints with nonnegativity restrictions.
Do I need standard form?
No. The tool can convert ≤, ≥, and = constraints by introducing slack/surplus and artificial variables (two‑phase or big‑M).
How are entering and leaving variables chosen?
Enter a variable with positive reduced cost; leave via the minimum‑ratio test among rows with positive direction entries.
What indicates unboundedness?
If the chosen entering column has no positive entries in the constraint rows, the objective can increase without bound.
What is degeneracy and cycling?
Degeneracy occurs when the minimum ratio is zero; special pivot rules (e.g., Bland’s) prevent cycling and ensure termination.
Can it handle minimization?
Yes—convert to maximization by negating the objective or run simplex on the dual problem.
How are dual prices produced?
They come from the final tableau’s objective row coefficients corresponding to constraints; they measure marginal value of resources.
What precision is used?
You can choose exact fractions for proofs or decimals for quick estimates; computations stay consistent throughout iterations.
Does the method guarantee an optimal solution?
If a feasible optimum exists, simplex will find it; otherwise it certifies infeasibility or unboundedness.
How big a model can I solve?
Limited by memory and time; the tableau grows with variables and constraints. The calculator is optimized for classroom and moderate‑size LPs.
Is sensitivity analysis included?
Yes. Allowable ranges and shadow prices are reported from the final tableau to explain how changes affect optimality.