Linear Programming Graphs
Solving optimization problems graphically
Explore linear programming problems graphically by adjusting constraints and observing how the feasible region and optimal solution change. Select different problem scenarios to see real-world applications of linear programming!
Production Planning
Diet Problem
Transportation
Optimal Solution:
x = 4, y = 6
Maximum Profit: $260
Current Problem:
A factory produces two products (x and y). Product x gives $30 profit per unit, product y gives $20 profit per unit. Find the optimal production mix to maximize profit given resource constraints.
Objective: Maximize Z = 30x + 20y
Understanding Linear Programming
Key Concepts:
Linear programming is a method to achieve the best outcome in a mathematical model:
- Objective Function: What needs to be maximized or minimized
- Decision Variables: Variables that represent choices
- Constraints: Limitations on resources or requirements
- Feasible Region: All possible solutions that satisfy constraints
- Optimal Solution: The best solution within the feasible region
Graphical Method:
For problems with two variables, we can solve graphically:
- Plot each constraint as a line on the graph
- Identify the feasible region where all constraints are satisfied
- Find the corner points of the feasible region
- Evaluate the objective function at each corner point
- The optimal solution is at the corner point with the best objective value
