Equation setup help

Question

A company makes three types of candy and packages them in three assortments. Assortment I contains 4 sour, 4 lemon, and 12 lime candies, and sells for $9.40. Assortment II contains 12 sour, 4 lemon, and 4 limelime candies, and sells for $7.60. Assortment III contains 8 sour, 8 lemon, and 8 lime candies, and sells for $ 11.00. Manufacturing costs per piece of candy are $0.20 for sour, $0.25 for lemon, and $ 0.30 for lime. They can make 5000 sour, 4000 lemon, and 5400 lime candies weekly. How many boxes of each type should the company produce each week in order to maximize its profit? What is the maximum profit?

Answer 1

1. Optimization Problem
2. Profit = Selling Price - Cost
3. Assortment 1 (A1) Package Profit = $9.40 - 4($0.20) - 4($0.25) - 12($0.30)  =  $4.00
4. Assortment 2 (A2) Package Profit = $7.60 - 12($0.20)- 4($0.25) -  4($0.30) =  $3.00
5. Assortment 3 (A3) Package Profit = $11.00 - 8($0.20) - 8($0.25) - 8($0.30) =  $5.00
6.  Goal is to maximize the overall profit.  The profit equation is $4(A1) + $3(A2) + $5(A3)
7. Manufacturing capacity limits the amount that can be made each week.
1. For Sour candies:  4(A1) + 12(A2) + 8(A3) ≤ 5000
2. For Lemon candies:  4(A1) + 4(A2) + 8(A3) ≤ 4000
3. For Lime candies:  12(A1) + 4(A2) + 8(A3) ≤ 5400
8. You can reduce each one of the equations above by dividing by 4.  These equations form the constraints.
1. A1 + 3(A2) + 2(A3) ≤ 1250
2. A1 + A2 + 2(A3) ≤ 1000
3. 3(A1) + A2 + 2(A3) ≤ 1350
9. What methods have you been learning?  This is a linear programming problem.  You could solve it using a specialized software package (like Minitab).  You can even use Excel.  It difficult to use graphical method when there are more than two variables, and here we have three (A1, A2, A3).  There are other methods as well, that don't rely graphing or using a software tool.  At any rate, you asked for help with the equation set-up, and the above shows the constraints as well as the equation to maximize.  If you need addition help, let me know.
10. UPDATE:  For the constraint equations, should include that each equation must be greater than or equal to zero.  So, for example, 0 ≤ A1 + 3(A2) + 2(A3) ≤ 1250.  The reason is because the number of packages (A1, A2, A3) cannot be negative.