Question

The Delicious Snacks manufactures a snack mix by blending three ingredients: a dried fruit mixture, a nut mixture, and a cereal mixture. Information about the three ingredients (per ounce) is shown below.

Ingredient	Cost	Fat Grams	Protein grams	Calories
Dried Fruit Mixture	1.10	1	1	180
Nut Mixture	1.00	10	8	415
Cereal Mixture	0.70	4	5	130

The company wants to know how many ounces of each mixture to put into the blend. The blend should contain no more than 1250 calories and no more than 20 grams of fat. It should contain at least 15.5 grams of protein. Dried fruit mixture must be at least 25% of the weight of the blend, and nut mixture must be no more than 40% of the weight of the blend.

Formulate a linear programming model that meets these restrictions and minimizes the cost of the blend by determining

(a) The decision variables.

(b) Determine the objective function. What does it represent?

(c) Determine all the constraints. Briefly describe what each constraint represents.

Note: Do NOT solve the problem after formulating

Answer 1

(a)

x = Grams of dried fruit mixture

y = Grams of Nut mixture

z = Grams of Cereal mixture

C = Cost in $

(b)

Minimize C = 1.1x + y + 0.7z. This represents the minimum cost of the blend.

(c)

180x + 415y + 130z ≤ 1250 [Total calories requirement]

x + 10y + 4z ≤ 20 [Maximum fat requirement]

x + 8y + 5z ≥ 15.5 [Minimum protein requirement]

x ≥ 0.25(x + y + z), that is 3x – y – z ≥ 0 [Dried fruit mixture weight to total weight ratio]

y ≤ 0.4(x + y + z), that is x – 1.5y + z ≥ 0 [Nut mixture weight to total weight ratio]

x, y, z ≥ 0 [Non-negativity restrictions]

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