Linear Programming Problem
Objective Function
Constraints
How to Use the Linear Programming Solver
Problem Setup
- Objective Function: Enter coefficients for the function to optimize
- Constraints: Enter linear inequalities that limit the solution space
- Variables: Currently supports 2-variable problems (x, y)
- Non-negativity: x ≥ 0, y ≥ 0 constraints are automatic
Graphical Method
- Feasible Region: Area satisfying all constraints
- Corner Points: Vertices of the feasible region
- Optimal Solution: Found at a corner point (fundamental theorem)
- Evaluation: Test objective function at each corner
Constraint Types
- ≤ (Less than or equal): Resource limitations
- ≥ (Greater than or equal): Minimum requirements
- = (Equal): Exact specifications
- Non-negativity: Variables cannot be negative
Applications
- Production Planning: Maximize profit with resource constraints
- Diet Problems: Minimize cost while meeting nutritional needs
- Transportation: Minimize shipping costs
- Portfolio Optimization: Maximize return with risk constraints
Frequently Asked Questions
Why is the optimal solution always at a corner point?
The Fundamental Theorem of Linear Programming states that if an optimal solution exists, it occurs at a vertex (corner point) of the feasible region. This is because linear functions achieve their extrema at the boundaries of convex regions.
What if my problem has more than 2 variables?
For problems with more than 2 variables, the graphical method becomes impractical. Use the simplex method or specialized optimization software. This calculator currently supports 2-variable problems for educational purposes.
What does "infeasible" mean?
An infeasible problem has no solution because the constraints are contradictory. For example, requiring x ≤ 5 and x ≥ 10 simultaneously creates an infeasible problem.
Can I have unlimited solutions?
Yes, if the objective function is parallel to a constraint boundary, there can be infinitely many optimal solutions along that edge of the feasible region.
How do I formulate a real-world problem?
Identify decision variables, write the objective function, list all constraints, and ensure all variables are non-negative (or adjust accordingly). Practice with standard problems like production planning or diet optimization.
Related Calculators
- Matrix Calculator - For solving systems of linear equations
- Algebra Solver - For linear equation solving
- Graphing Calculator - To visualize constraints and feasible regions
- Statistical Calculator - For optimization in statistical contexts