Linear Programming: The Diet Plan
Explore linear programming concepts through a diet optimization problem. Find the optimal combination of foods that meets nutritional requirements while minimizing cost by graphing constraints and identifying the feasible region.
Help & Instructions
▼- View Constraints: See the nutritional constraints for the diet problem
- Graph Visualization: Observe how constraints create a feasible region
- Find Optimal Solution: Click to find the optimal combination of foods
- Adjust Constraints: Modify constraints to see how the solution changes
- Analyze Results: View the optimal values and cost calculation
- Understand how to graph linear inequalities
- Learn to identify the feasible region of a linear programming problem
- Discover how to find the optimal solution at corner points
- Recognize real-world applications of linear programming
Linear Programming Graph
Visualization of constraints and feasible region:
Optimal Solution
Best combination for minimum cost:
Problem Constraints
Inequalities defining the feasible region:
Feasible Region
Visual representation of solution space:
Linear programming is a mathematical method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. The feasible region is the set of all possible points that satisfy the problem's constraints.
The Mathematics of Linear Programming
Linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set formed by the intersection of finitely many half spaces.
a₂₁x₁ + a₂₂x₂ + ... + a₂ₙxₙ ≤ b₂
...
x₁, x₂, ..., xₙ ≥ 0
- If a solution exists to a linear programming problem, then it occurs at one of the corner points of the feasible region.
- If two adjacent corner points give the same objective value, then every point on the line segment connecting them also gives that value.
- The feasible region is always a convex set.
Linear programming concepts are used in:
- Business: Profit maximization and cost minimization
- Agriculture: Determining optimal crop mix
- Manufacturing: Production planning and inventory control
- Transportation: Route optimization and logistics
- Finance: Portfolio optimization and risk management
