Optimal Solution Finder Calculator

Example Data Table

Use this sample to test the calculator quickly.

Setting	Value	Meaning
Objective	Maximize Z = 5x + 4y	Profit or score to improve
Constraint 1	6x + 4y ≤ 24	Main capacity limit
Constraint 2	x + 2y ≤ 6	Secondary resource limit
Constraint 3	-x + y ≤ 1	Balance condition
Expected optimum	x = 3, y = 1.5, Z = 21	Best feasible corner point

Formula Used

Objective function: Z = c₁x + c₂y

Constraints: a_ix + b_iy ≤ d_i, with x ≥ 0 and y ≥ 0

Corner-point rule: The optimum for a linear two-variable model occurs at a feasible corner point, if a finite optimum exists.

Line intersection: For two boundaries a₁x + b₁y = d₁ and a₂x + b₂y = d₂, the solver uses determinants to generate candidate corners.

Slack: Slack = RHS − LHS at the chosen solution. A zero slack value means the constraint is binding.

How to Use This Calculator

Choose whether you want to maximize or minimize the objective value.

Enter the objective coefficients for x and y.

Set a decimal precision for displayed results and exports.

Enable the constraints you need, then enter each coefficient and right-side limit.

Click Find Optimal Solution to calculate corner points, slacks, and the optimal vertex.

Review the graph, tables, and binding constraints, then export the report as CSV or PDF.

FAQs

1. What does this calculator solve?

It solves two-variable linear optimization problems with up to three selectable constraints, non-negative variables, and a maximize or minimize objective function.

2. What method does the solver use?

It uses the corner-point method. The page calculates boundary intersections, keeps only feasible vertices, then compares objective values to locate the best point.

3. Why are x and y restricted to non-negative values?

Many planning, production, and allocation models cannot use negative quantities. Non-negativity keeps the solution realistic and matches standard linear programming assumptions.

4. What is a binding constraint?

A binding constraint is fully used at the optimal point. Its slack becomes zero, meaning it directly shapes the best feasible solution.

5. What does slack tell me?

Slack measures unused capacity. A larger slack means that constraint still has room left after applying the chosen optimal solution.

6. Can the calculator show alternate optimal solutions?

Yes. If multiple corner points produce the same best objective value, the page warns that alternate optimal solutions may exist along a feasible edge.

7. What happens if the model is infeasible or unbounded?

The calculator shows a clear status message. Infeasible means no point satisfies every constraint. Unbounded means the objective can keep improving without a finite best value.

8. Can I use decimals in coefficients and limits?

Yes. Every coefficient field accepts decimal values, so the calculator can handle practical business, engineering, and mathematical planning inputs.