Question

Gaermont Sauces is a manufacturer of sauces. This company buys two ingredients on the market (A1 and A2). The price at which A1 buys is $ 20.00 per kilo and the cost of A2 is $ 40.00 per kilo. The supplier supplying these products to Gaermont can only supply 80 kilos of product A1 and 75 kilos of product A2. The ingredients are mixed to form two types of sauce, "Special Spicy" and "The Real Spicy", or they can be sold in the market without the need to process them. A bottle of "Spicy Special" contains 300 grams of ingredient A1 and 400 grams of ingredient A2 and sells for $ 35.00. A bottle of "The Real Hot" contains 500 grams of ingredient A1 and 200 grams of ingredient A2 and retails for $ 30.00. The cost of the containers and other specialties is $ 3 for "Special spicy" and $ 4 for "The real spicy". If the company decides to sell the raw ingredients, the price at which it sells the kilo of A1 is $ 22.00 and the maximum market demand is 35 kilos, while the price at which it could sell the kilo of A2 is $ 45.00 and It could sell up to 20 kilos. Consider that you could only sell 100 bottles of each type of sauce. Formulate this problem as a L.P. that allows the company to maximize its profits. What is the maximum profit that Gaermont Sauces can obtain?

Answer 1

Formulation:

S = No. of bottles of Special-Spicy to be produced and sold
R = No. of bottles of Real-Spicy to be produced and sold
A1 = Ingredient-A1 sold in the market unprocessed (in kilos)
A2 = Ingredient-A2 sold in the market unprocessed (in kilos)

Revenue = 35 S + 30 R + 22 A1 + 45 A2
Cost of the ingredients = 20*(0.3 S + 0.5 R) + 40*(0.4 S + 0.2 R) + 20 A1 + 40 A2 = 22 S + 18 R + 20 A1 + 40 A2
Cost of containers = 3 S + 4 R

So, profit = 35 S + 30 R + 22 A1 + 45 A2 - (22 S + 18 R + 20 A1 + 40 A2) - (3 S + 4 R) = 10 S + 8 R + 2 A1 + 5 A2

Max Z = 10 S + 8 R + 2 A1 + 5 A2

Subject to,

0.3 S + 0.5 R + A1 <= 80

0.4 S + 0.2 R + A2 <= 75

A1 <= 35

A2 <= 20

S <= 100

R <= 100

S, R, A1, A2 >= 0

LINDO Code:

Solution:

So,

No. of bottles of Special-Spicy to be produced and sold = 10
No. of bottles of Real-Spicy to be produced and sold = 100
Ingredient-A1 sold in the market unprocessed = 0 kilo
Ingredient-A2 sold in the market unprocessed = 20 kilos

Total Ingredient-A1 bought = 0.3 S + 0.5 R + A1 = 0.3*10 + 0.5*100 + 0 = 53 kilos
Total Ingredient-A2 bought = 0.4 S + 0.2 R + A1 = 0.4*10 + 0.2*100 + 20 = 44 kilos

Max Profit = $1,000