Linear Programming Calculator

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Linear Programming Calculator with Steps (up to 21 variables)

Linear Programming Calculator

Solve LP/ILP models with a clear coefficient table. Assumes xᵢ ≥ 0 for all variables; optional upper bounds xᵢ ≤ Uᵢ are supported.

Inputs

Fractions allowed (e.g., 5/3). Decimal mode is faster but approximate.

Beginner mode adds plain‑English narration plus a feasibility verification table (constraint-by-constraint).

Objective coefficients (c)

Upper bounds (optional)

Enter Uᵢ to enforce xᵢ ≤ Uᵢ. Leave blank for “no upper bound”. Nonnegativity xᵢ ≥ 0 is always assumed.

Advanced (integer/binary + step detail + bulk paste)

If you pick integer/binary, the solver switches to Branch & Bound automatically.

Bulk paste (optional)

UB line is optional. Leave a blank entry for “no upper bound”.

Constraints (A x ≤/≥/= b)

Fill coefficients, choose relation, then RHS.

Results

Status
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Method used
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Objective value (Z*)
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Model size
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Planning / educational estimate only. This tool solves an LP/ILP model under simplified assumptions (linearity, nonnegativity xᵢ ≥ 0, and optional upper bounds xᵢ ≤ Uᵢ). Real-world optimization often needs extra constraints (capacities, minimums, fixed charges, logic rules, uncertainty). Do not use this output as a guarantee, contract term, or compliance decision. Confirm important decisions with a qualified professional / production-grade solver. See our Disclaimer.
Run a calculation to see status, objective value, and the solution vector.

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Step-by-step explanation

Steps will appear here after you calculate.

Primal ↔ Dual (optional: generate & solve)

Run a calculation first.

Export full report

Export a complete worked solution (copy/paste friendly).

Choose Word-friendly text to paste into Word. Use HTML for rich formatting.

Chart

If N=2: boundary lines are drawn (no feasible shading). If N>2: a bar chart of top variables is shown after you solve.

Understanding the results
  • Status explains whether the model is feasible and bounded (Optimal / Multiple optimal / Infeasible / Unbounded).
  • Z* is the best objective value (maximum or minimum depending on your selection).
  • x1…xn are decision variables. Their values are the recommended amounts/levels at the optimum.
  • Multiple optimal means there are multiple solutions that achieve the same Z*.
Common mistakes
  • Forgetting the calculator assumes xᵢ ≥ 0.
  • Swapping ≤ and ≥ signs (changes the feasible region completely).
  • Getting Unbounded because real-world caps/limits were not modeled (use upper bounds or additional constraints).
  • Choosing integer/binary without tight bounds (Branch & Bound can grow fast).
Limitations

This is a teaching/utility LP/ILP solver. It does not model nonlinear relationships, uncertainty, or industry-specific feasibility rules. For high-stakes decisions, validate using a production solver and your full constraint set.

FAQs

Use the RankMath FAQ block below.

Sources
  • INFORMS — Linear Programming: Link
  • MIT OCW 15.053 — Optimization Methods in Management Science: Link
  • Bertsimas & Tsitsiklis — Introduction to Linear Optimization: Link
  • Vanderbei — Linear Programming: Foundations and Extensions: Link
How this calculator works (methodology)

1) What you enter (the mathematical model)

You provide: (a) an objective function (maximize or minimize), (b) constraints of the form A x ≤ b, A x ≥ b, or A x = b, and (c) optional upper bounds xᵢ ≤ Uᵢ. The calculator always assumes xᵢ ≥ 0. 

Standard LP form (conceptual):

Max/Min:   Z = cᵀx
Subject to A x (≤, ≥, =) b
           x ≥ 0
Optional:  xᵢ ≤ Uᵢ

2) Standard-form conversion (how constraints become solver-ready)

Simplex methods require a starting “basic feasible solution”. To build that, each constraint is converted:

  • ≤ constraint: add a slack variable (turns ≤ into =).
  • ≥ constraint: subtract a surplus variable and add an artificial variable (to start feasibility).
  • = constraint: add an artificial variable (to start feasibility).
Examples:

1) aᵀx ≤ b   ⇒   aᵀx + s = b,   s ≥ 0   (slack)

2) aᵀx ≥ b   ⇒   aᵀx - s + a = b,  s ≥ 0, a ≥ 0  (surplus + artificial)

3) aᵀx = b   ⇒   aᵀx + a = b,   a ≥ 0   (artificial)

3) Two‑Phase simplex (default)

When artificial variables are required, the calculator uses Two‑Phase simplex:

  • Phase 1: minimize the sum of artificial variables. If the minimum is not 0, the original model is infeasible.
  • Phase 2: optimize your original objective starting from the feasible basis found in Phase 1.

4) Other modes

  • Big‑M (decimal): penalizes artificials heavily; includes a check that artificial variables are near zero.
  • Dual simplex (decimal): works when the tableau is dual-feasible and there are no artificial variables.
  • Branch & Bound (integer/binary): solves LP relaxations, then branches on fractional variables to prove an integer optimum.
  • Graphical (N=2): enumerates corner points. A verification solve is used to detect unbounded/infeasible cases.

5) Exact vs Decimal engine

  • Exact (rational): keeps fractions exactly (best for teaching and accuracy).
  • Decimal: faster but uses tolerances (good for quick checks).

6) Why the “Beginner mode” steps are trustworthy

In Beginner mode, the calculator prints a verification table that plugs the solution into every constraint (and bounds / integrality rules). This is the easiest way for a layperson to confirm “yes, this output really satisfies the model.”