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A company 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?

Respuesta :

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.

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