Linear Programming Setup Help Please :) Much Appreciated

Question

A company makes three types of candy and packages them into 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, 4 lime 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 .20 for sour, .25 for lemon, and .30 for lime. They can make 5,000 sour, 3,800 lemon, and 5,400 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

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.

 

If you are unfamiliar with the Simplex method, here is a good reference:  http://college.cengage.com/mathematics/larson/elementary_linear/4e/shared/downloads/c09s3.pdf