Acompany makes three types of candy and packages them in three assortments. assortment i contains 44 cherrycherry, 44 lemonlemon, and 1212 limelime candies, and sells for a profit of $4.004.00. assortment ii contains 1212 cherrycherry, 44 lemonlemon, and 44 limelime candies, and sells for a profit of $3.003.00. assortment iii contains 88 cherrycherry, 88 lemonlemon, and 88 limelime candies, and sells for a profit of $ 5.00$5.00. they can make 4 comma 8004,800 cherrycherry, 4 comma 0004,000 lemonlemon, and 6 comma 0006,000 limelime candies weekly. how many boxes of each type should the company produce each week in order to maximize its profit (assuming that all boxes produced can be sold)? what is the maximum profit?

Let x be the number of assortment I items sold
Let y be the number of assortment II items sold
Let z be the number of assortment III items sold
The cost for making assortment I is .20(4) + .25(4) + .30(12) = $5.40
The cost for making assortment II is .20(12) + .25(4) + .30(4) = $4.60
The cost for making assortment III is .20(8) + .25(8) + .30(8) = $6.00
The equation for total cost is:  cost = 5.4x + 4.6y +6z
The equation for total income is:  income = 9.4x + 7.6y + 11z
The equation for total profit is:  profit = (9.4-5.4)x + (7.6-4.6)y + (11-6)z  --> 4x + 3y + 5z
The total profit equation is the objective function for this example.
The constraint equations are listed below:
4x + 12y + 8z ≤ 5000     (sour candy)
4x + 4y + 8z   ≤ 3800     (lemon candy)
12x + 4y + 8z ≤ 5400     (lime candy)
x ≥ 0
y ≥ 0
z ≥ 0
This system can be solved using the Simplex method:initial simplex tableau:
     x     y     z     s1    s2    s3    n
     4    12    8     1      0     0    5000
     4      4    8     0      1     0    3800
    12     4    8     0      0     1    5400
    -4    -3   -5     0      0     0         0 
The solution to this is listed below:
      x     y     z      s1      s2      s3        n
     0     1     0   .125     -.125     0        150 
    0     0     1  -.0625    .25  -.0625      300
     1     0     0     0          -.125    .125      200
     0     0     0   .0625     .375   .1875     2750 
The maximum profit of $2750 is obtained by selling 200 units of assortment I, 150 units of assortment II, and 300 units of assortment III.

Sis less than or equal to 75 (bottom left)