Linear Programming Optimal Solution Guide
Linear programming helps choose the best result under limits. It is used in production, transport, finance, diet planning, staffing, and scheduling. The model has an objective function. It also has constraints. Each constraint defines what is allowed.
Why It Matters
A manager may want maximum profit. A planner may want minimum cost. A student may need each feasible corner point. This calculator focuses on two decision variables. That format is common in classroom work. It is also useful for quick business checks.
How The Method Works
The tool reads the coefficients for X and Y. It then reads each inequality. Boundary lines are formed from every active constraint. The program finds intersections between those lines. It also checks the axes. Each point is tested against every rule. Only feasible points remain.
The objective value is calculated at each feasible point. For a maximum problem, the largest value is selected. For a minimum problem, the smallest value is selected. This follows a key linear programming idea. When an optimum exists in a bounded polygon, it occurs at a corner point.
Practical Use
You can enter material limits, labor limits, demand limits, budget limits, or capacity limits. Use less than, greater than, or equal signs as needed. Leave unused rows blank. The result table shows every feasible corner. That helps compare choices and explain the answer.
Reading The Result
The best X and Y values show the decision plan. The objective value shows profit, cost, score, or another target. Binding constraints have little or no slack. Nonbinding constraints still have unused capacity. Check slack before changing a plan.
Important Notes
This calculator solves two variable linear programs through corner point enumeration. It is designed for transparent learning. Large industrial problems may need matrix solvers. Still, this page gives a clear first answer and downloadable records. Use the CSV file for spreadsheets. Use the document export for reports.
The example table shows one balanced product mix. You can replace those values with your own problem. Keep units consistent. For example, do not mix hours with minutes unless converted. Better input quality creates stronger decisions, cleaner explanations, and fewer mistakes.
Review each result before using it for final planning.