When tackling optimization problems in mathematics, particularly linear programming, the graphical method is a powerful and easy-to-understand tool. Let’s walk through this method step by step, keeping it simple and intuitive.
What is the Graphical Method?
The graphical method is a way to solve two-variable linear programming problems. It helps you find the best solution (maximum or minimum) for an objective function, subject to a set of constraints. Since it involves only two variables, we can easily represent the problem on a 2D graph.
Real-Life Example
Imagine you run a bakery that makes two types of cookies: chocolate chip and oatmeal raisin. You want to maximize your profit. Each batch of chocolate chip cookies earns $40, and each batch of oatmeal raisin cookies earns $30. However, you have limited resources, such as baking time and ingredients. How many batches of each cookie should you bake to make the most profit?
The Steps of the Graphical Method
1. Define the Variables
- Let x represent the number of chocolate chip cookie batches.
- Let y represent the number of oatmeal raisin cookie batches.
2. Formulate the Objective Function
The objective function represents what you’re trying to optimize. In this case:
Profit = 40x + 30y
Your goal is to maximize this profit.
3. Write the Constraints
Constraints are the limitations you have. For example:
- Baking time constraint: It takes 2 hours to bake a batch of chocolate chip cookies and 1 hour for oatmeal raisin cookies. You have a maximum of 10 hours.→ 2x + y ≤ 10
- Ingredient constraint: Each batch of cookies requires specific amounts of ingredients. Let’s say the ingredients limit is 12 units.→ x + 2y ≤ 12
- Non-negativity constraint: You can’t bake a negative number of batches.→ x ≥ 0, y ≥ 0
4. Graph the Constraints
- Plot each constraint on a graph.
- For inequalities, shade the region that satisfies the constraint.
- The feasible region is the area where all shaded regions overlap. This region contains all possible solutions.
5. Identify the Corner Points
The corner points (or vertices) of the feasible region are where the boundary lines of the constraints intersect. These are potential solutions.
6. Evaluate the Objective Function at Each Corner Point
Substitute the coordinates of each corner point into the objective function to calculate the profit. For example:
If one corner point is (2, 4), then:
Profit = 40(2) + 30(4) = 80 + 120 = $200
7. Choose the Optimal Solution
The corner point that gives the highest (or lowest, depending on the goal) value of the objective function is the optimal solution. This tells you how many batches of each type of cookie to bake.
Illustration
Let’s visualize this with an example graph:
- Draw the lines for 2x + y = 10 and x + 2y = 12.
- Shade the feasible region.
- Label the corner points.
Here are the corner points:
- (0, 0)
- (0, 5)
- (4, 3)
- (5, 0)
Now calculate the profit for each:
- (0, 0): Profit = 40(0) + 30(0) = $0
- (0, 5): Profit = 40(0) + 30(5) = $150
- (4, 3): Profit = 40(4) + 30(3) = $250
- (5, 0): Profit = 40(5) + 30(0) = $200
The maximum profit is $250, achieved when you bake 4 batches of chocolate chip cookies and 3 batches of oatmeal raisin cookies.
Key Points to Remember
- The graphical method works only for problems with two variables.
- The feasible region must be bounded (closed) to ensure a solution exists.
- The optimal solution always lies at a corner point of the feasible region.
Why Use the Graphical Method?
- It’s visual and intuitive.
- It’s perfect for small problems with two variables.
- It helps build a foundational understanding of linear programming concepts.
With this guide, you can confidently use the graphical method to solve two-variable problems. Happy problem-solving!
Photo by Burak The Weekender: https://www.pexels.com/photo/white-android-tablet-turned-on-displaying-a-graph-186464/