A company makes three types of candy and packages them in three assortments. Ass

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 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 $0.20 for sour, $0.25 for lemon, and $0.30 for lime. They can make 4,800 sour, 4,000 lemon, and 5,600 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? The maximum profit is $ when boxes of assortment I, boxes of assortment II and boxes of assortment III are produced.

Explanation / Answer

Let boxes produces are x, y and z for Assortment1, Assortment2 and Assortment3 respectively

Cs = cost of one sour = $0.20

Cle = cost of one lemon = $0.25

Cli = cost of one lime = $0.30

hence total sell will be

S = 9.40x + 7.60 y + 11.0 z

total manufacturing cost for each single assortment1

C1 = 4 * 0.20 + 4 * 0.25 + 12 *0.30 = $5.4

Similarly

total manufacturing cost for each single assortment2

C2 = 12*0.20 + 4 *0.25 + 4 * 0.30 = $4.6

and

total manufacturing cost for each single assortment3

C3 = 8 * 0.20 + 8 * 0.25 + 8 * 0.30 = $6.0

so toatl profit will be

P = S - C1 *x + C2 *y + c3 *z

   =(9.40x + 7.60 y + 11.0 z) - 5.4x - 4.6y -6z

P = 4.0x +3.0y + 5.0 z -----------------------------------(1)

we have to maximize P contained to following conditions

4x +12y +8 z <= 4800   -------------------------------------(2) (maximum number of sours produced in 1 week)       

4x +4y + 8z <= 4000 -----------------------------------------(3) (maximum number of lemons produced in 1 week)

12x + 4y + 8z <= 5600 --------------------------------------(4)        (maximum number of lime produced in 1 week)

x > = 0, y >=0 , z=> 0 -------------------------------------(5)                number of assortment can't be negative

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