Soft drink Sugar-Right is advertised as having a sugar content of no more than 1

Question

Soft drink Sugar-Right is advertised as having a sugar content of no more than 10%, is blended from five ingredients, each of which has some sugar content. Water may also be added to dilute the mixture. The sugar content of the ingredients and their costs per gallon are given below. At least 1% of the content of Sugar-Right must come from ingredients type3 or type4, 1% must come from ingredients type2 or type5, and 1% must come from ingredients type1 or type4. How much of each ingredient should be used in preparing 15000 gal of Sugar-Right to minimize the cost? What is the minimum cost?

Explanation / Answer

Here

we have to minimize the cost of sugar- right so lets say percentage of type of sugars are A, B, C, D , E respectively from type I to type V and for Water , it is F.

Total cost = 0.48A + 0.32 B +0.53C + 0.28D + 0.43E + 0.31F

Constraint to

A+ B + C + D + E + F = 1 .........(i)

Max (C, D)>=0.01......(ii)

Max(B, E) >=0.01....(iii)

Max (A, D) >=0.01.......(iv)

A + B + C + D + E <= 0.10........(v)

so

Type I sugar required = 0

Type II sugar required = 0

Type III sugar required = 0

Type IV sugar required = 0.172

Type V sugar required = 0.01

Water reuired = 0.818

Total COst = $0.306 per gallon

for 15000 gallon for total cost = 15000 * 0.306 = $ 4590

Type I sugar required = 0

Type II sugar required = 0

Type III sugar required = 0

Type IV sugar required = 0.172 * 15000 = 2580 galleon

Type V sugar required = 0.01 * 15000 = 150 galleon

Water reuired = 0.818 * 15000 = 12270 galleon

Ingredient Type I Type 2 Type 3 Type 4 Type 5 Water Total Percentage content 0 0 0 0.172 0.01 0.818 Sugarcontent 0.28 0.19 0.43 0.57 0.22 0 0.1 Cost 0.48 0.32 0.53 0.28 0.43 0.31 0.306 Constraint A B C D E F i 0 0 0 0.10 0.00 0.10 ii 0 0.17 0.17 iii 0 0.01 0.01 iv 0 0.01 0.01 v 0 0 0 0.17 0.01 0.82 1.00

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