Formulate a linear programming problem for the following situation.

Homer Simpson has created two donut assortments for the the Springfield donut shop. The bargain assortment includes 4 pink frosted and 8 glazed donuts and the deluxe assortment contains 9 pink frosted and 3 glazed donuts. The donut shop has 300 pink frosted and 400 glazed donuts to make its assortments. Each bargain assortment makes a profit of $4 and each deluxe assortment makes a profit of $6. How many of each assortment should Homer make in order to maximize the profit?

If there are x bargain boxes and y deluxe boxes, you want to

maximize p = 4x + 6y where
4x+9y <= 300
8x+3y <= 400

Now use your favorite solution method or tool, and you find max p when
x=45
y=13

http://www.zweigmedia.com/RealWorld/LPGrapher/lpg.html

is a nice tool. It will solve the setup, and even graph it for you.

To formulate a linear programming problem for this situation, we need to determine the decision variables, objective function, and constraints.

Decision Variables:
Let's denote the number of bargain assortments produced by "x" and the number of deluxe assortments produced by "y."

Objective Function:
The objective is to maximize the profit. The profit for each bargain assortment is $4, and for each deluxe assortment, it is $6. Hence, the objective function is:
Profit = 4x + 6y

Constraints:
1. The number of pink frosted donuts used in both assortments should not exceed the available quantity of 300.
4x + 9y ≤ 300

2. The number of glazed donuts used in both assortments should not exceed the available quantity of 400.
8x + 3y ≤ 400

3. The number of assortments cannot be negative.
x ≥ 0
y ≥ 0

The complete linear programming problem formulation is as follows:

Maximize: Profit = 4x + 6y
Subject to:
4x + 9y ≤ 300
8x + 3y ≤ 400
x ≥ 0
y ≥ 0

To find the optimal values for x and y that maximize the profit, the problem can be solved using linear programming algorithms like the Simplex method.