Linear Programming Problem Solver Calculator

Enter objective coefficients and every linear limit now. Test feasible corner points using reliable logic. Download results after solving the programming problem instantly online.

Calculator

Objective form: Z = c₁x + c₂y

Constraints

Enter each row as ax + by sign right side.

Example Data Table

Item Value Meaning
Objective Maximize Z = 40x + 30y Profit from two products
Constraint 1 2x + y ≤ 100 Material limit
Constraint 2 x + y ≤ 80 Labor limit
Constraint 3 x ≤ 40 Demand limit
Nonnegative rules x ≥ 0, y ≥ 0 No negative production

Formula Used

The calculator solves a two variable linear programming model with this objective form:

Z = c₁x + c₂y

Each restriction is entered in this form:

aᵢx + bᵢy ≤ rᵢ, aᵢx + bᵢy ≥ rᵢ, or aᵢx + bᵢy = rᵢ

Corner points are found by intersecting pairs of boundary lines:

x = (r₁b₂ − r₂b₁) / (a₁b₂ − a₂b₁)

y = (a₁r₂ − a₂r₁) / (a₁b₂ − a₂b₁)

The calculator keeps feasible points only. Then it evaluates Z at every feasible corner. The highest value is selected for maximization. The lowest value is selected for minimization.

Slack for a less than constraint is right side − left side. Surplus for a greater than constraint is left side − right side.

How To Use This Calculator

  1. Select maximization or minimization.
  2. Enter the x and y coefficients for the objective function.
  3. Choose whether x and y must stay nonnegative.
  4. Enter each constraint row with coefficients, sign, and right side value.
  5. Leave unused constraint rows empty.
  6. Press the solve button.
  7. Review feasible corner points, objective values, slack, and binding status.
  8. Download the result as CSV or PDF when needed.

Linear Programming For Better Decisions

Linear programming helps you choose the best result under limits. Many business, school, and planning tasks use this method. The calculator supports two decision variables, so the solution can be checked with the corner point method. You enter an objective function, such as profit or cost. Then you add each restriction. The tool tests every feasible intersection and compares objective values.

Why Corner Points Matter

A linear objective reaches its best value at a feasible corner point when a bounded solution exists. That idea makes the process reliable and easy to audit. This calculator lists every tested point. It also reports whether each constraint is binding. A binding constraint has no remaining slack. A nonbinding constraint still has unused capacity. These details help you explain the answer, not just copy a number.

Useful Inputs For Real Problems

You can solve maximization or minimization models. You can use less than, greater than, or equality signs. Nonnegative variable options are included because many models cannot produce negative products, hours, or units. Advanced users can compare slack, surplus, feasibility, and objective value. The result area appears above the form after submission. That keeps the answer visible while inputs remain available for editing.

Where This Calculator Helps

Students can check homework steps for graphical linear programming. Teachers can prepare examples with clear vertex tables. Managers can estimate a product mix, labor plan, diet blend, or resource allocation. The method is also useful when you need a transparent answer. Every step is shown in a table, so mistakes in signs or coefficients are easier to find.

Exporting Your Work

After solving, use the CSV option for spreadsheets. Use the PDF option for reports or class notes. Both exports include the important result values. Review your units before sharing the output. A model is only as good as the assumptions behind its coefficients and limits.

Important Modeling Habits

Write each constraint from the same viewpoint. Keep units consistent across every row. Use realistic upper or lower bounds when a situation has them. Check signs before solving. A greater than sign can change the feasible region. If the result seems unusual, adjust the model and solve again. Save each tested result too.

FAQs

What does this calculator solve?

It solves two variable linear programming problems. It checks feasible corner points and compares objective values. It supports maximization, minimization, nonnegative variables, and common inequality signs.

Can I use more than two variables?

No. This page is designed for two decision variables. That keeps the corner point method clear and table based. Problems with more variables need simplex or matrix methods.

What is a feasible point?

A feasible point satisfies every constraint and selected nonnegative rule. The calculator tests intersections and keeps only points that meet all conditions.

What is a binding constraint?

A binding constraint is exactly active at the selected solution. Its slack or surplus is zero, or very close to zero after rounding.

Why do corner points matter?

For a bounded two variable linear model, the best objective value occurs at a feasible corner point. Checking those points gives the optimal answer.

What does unbounded mean?

Unbounded means the objective can keep improving in a feasible direction. In that case, there is no finite best value for the model.

Can I enter equality constraints?

Yes. Choose the equality sign for any row. The calculator treats that boundary as a required line when it checks feasibility.

Why should I export results?

CSV is useful for spreadsheets. PDF is useful for reports, assignments, and records. Exporting also helps review the solution later.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.