LP Optimal Solution Calculator

Example Data Table

Formula Used

The calculator solves a two variable linear programming model.

Objective function:

Z = c₁x + c₂y

Constraint form:

a_ix + b_iy ≤, ≥, or = r_i

Intersection of two boundary lines:

D = a₁b₂ - a₂b₁

x = (r₁b₂ - r₂b₁) / D

y = (a₁r₂ - a₂r₁) / D

Slack for a ≤ constraint: Slack = RHS - LHS

Surplus for a ≥ constraint: Surplus = LHS - RHS

How To Use This Calculator

1. Select maximize or minimize.
2. Enter the objective coefficients for x and y.
3. Add each constraint row with both coefficients.
4. Select the correct operator for each constraint.
5. Enter the right side value.
6. Keep nonnegative variables checked for most planning models.
7. Press the calculate button.
8. Review the optimal point, objective value, vertices, and slack table.
9. Use CSV or PDF export for reports.

Article: LP Optimal Solution Calculator

Why Linear Programming Matters

Linear programming helps choose the best result from limited resources. It is useful when a decision has a clear goal and several restrictions. This calculator focuses on two decision variables. That keeps the method transparent and easy to audit.

How The Calculator Works

The tool checks every constraint line and finds intersection points. It also checks axis points when nonnegative variables are enabled. Each candidate point is tested against every rule. Only feasible points move to the final comparison. The objective value is then calculated for each feasible point.

Graphical Method Logic

This approach follows the graphical method. A two variable linear program reaches its best bounded value at a corner point of the feasible region. That corner may come from two constraint lines. It may also come from an axis and one constraint. When the region is empty, no feasible point exists. When the region is open, the answer may be unbounded.

Common Uses

Use the calculator for production planning, diet mixes, staffing limits, budget allocation, shipping choices, and simple resource problems. Enter the objective coefficients first. Then add each constraint coefficient, operator, and right side value. Choose maximize or minimize. Keep nonnegative variables checked for most business models.

Reading Results

The result panel shows the optimal point, objective value, feasible vertices, and slack details. Slack tells how much unused allowance remains in a less than or equal constraint. Surplus tells how far a greater than or equal constraint exceeds its required minimum. Equality rows show a residual, which should be close to zero.

Export And Review

CSV export is helpful for spreadsheets and records. The PDF export gives a quick report for meetings. Review input units before sharing results. The calculator uses the numbers exactly as entered. It does not convert units automatically.

Model Quality

Linear programming is powerful, but model quality matters. A wrong coefficient can change the answer. A missing constraint can create a misleading profit. Check every assumption with the real process. Then use the optimal solution as a decision guide, not as blind instruction. For larger models, use a simplex or interior point solver. For two variable models, this page gives a clear, inspectable answer. You can also compare several scenarios by changing limits, prices, or capacities. Small changes often reveal which constraint controls the final decision most strongly today.

FAQs