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This pandemic, Abheedette learned to bake while on home quarantine.
She also realized that she will be able to make P60.00 profit per tray of banana muffins
and P120.00 profit per tray of blueberry muffins.
She needs 2 cups of milk and 3 cups of flour to bake a tray of banana muffins.
And, baking a tray of blueberry muffins takes 4 cups of milk and 3 cups of flour.
She has 16 cups of milk and 15 cups of flour.
How many trays of each flavor must be baked to maximize the profit?
a. Define the variable used:
b. LP Model:
c. Identify the feasible region:
d. Corner Points and the objective functions:
e. Optimal Solution (final answer):
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The solution in the post by @CPhill is INCORRECT.
His answer x= 3, y= 3 does not satisfy the constraints.
So, I came to bring you a correct solution.
**a. Define the variables used:**
* Let `x` be the number of trays of banana muffins.
* Let `y` be the number of trays of blueberry muffins.
**b. LP Model:**
* **Objective function:** Maximize profit (P) = 60x + 120y
* **Constraints:**
* Milk: 2x + 4y ≤ 16
* Flour: 3x + 3y ≤ 15
* Non-negativity: x ≥ 0, y ≥ 0
**c. Identify the feasible region:**
To graph the feasible region, we'll first find the intercepts of the constraint lines:
* **Milk constraint (2x + 4y ≤ 16):**
* x-intercept (y = 0): 2x = 16 => x = 8
* y-intercept (x = 0): 4y = 16 => y = 4
* **Flour constraint (3x + 3y ≤ 15):**
* x-intercept (y = 0): 3x = 15 => x = 5
* y-intercept (x = 0): 3y = 15 => y = 5
Now, plot these lines and shade the region that satisfies all constraints (including non-negativity).
Find the intersection point (x,y) = (2,3) of the sloping lines.
The feasible region is a polygon with vertices at (0, 0), (0, 4), (2, 3), and (5, 0).
See my plot under this link
https://www.desmos.com/calculator/hxbuw3toqz
**d. Corner Points and the objective functions:**
Evaluate the objective function (P = 60x + 120y) at each corner point:
* (0, 0): P = 60(0) + 120(0) = 0
* (0, 4): P = 60(0) + 120(4) = 480
* (2, 3): P = 60(2) + 120(3) = 480
* (5, 0): P = 60(5) + 120(0) = 300
**e. Optimal Solution (final answer):**
There are two optimal points: (0,4) and (2,3). It means that the maximum profit of P480.00 is achieved
when Abheedette bakes 0 trays of banana muffins and 4 trays of blueberry muffins,
or 2 trays of banana muffins and 3 trays of blueberry muffins.
It is a rare case in solving minimax linear problems, when where are more that one solution.
It may happen when the slope of one of boundary lines coincides with the gradient of the objective function.
It is a rare case, but, nevertheless, it may happen, as you see it in this problem.
Solved.
For plotting, I used free of charge plotting tool at this site http:\\www.desmos.com/calculator/