1) The Charm City Snacks manufactures a snack mix by blending three ingredients:
ID: 3109596 • Letter: 1
Question
1) The Charm City 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.
PLEASE TYPE ALL ANSWERS - PLEASE NO WRITTEN ANSWERS
Ingredient
Cost
Fat Grams
Protein grams
Calories
Dried Fruit Mixture
1.10
1
1
170
Nut Mixture
0.90
14
8
400
Cereal Mixture
0.60
7
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 25 grams of fat. It should contain at least 15 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 45% 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
Ingredient
Cost
Fat Grams
Protein grams
Calories
Dried Fruit Mixture
1.10
1
1
170
Nut Mixture
0.90
14
8
400
Cereal Mixture
0.60
7
5
130
Explanation / Answer
Solution :-
(a) Decision Variables :-
It is given that; the company wants to know how many ounces of each mixture to put into the blend. Hence key decision is to select quantity of each mixture such that the cost of the blend is minimum.
Therefore let,
X1 = Ounces of Dried Fruit Mixture put into blend
X2 = Ounces of Nut Mixture put into blend
X3 = Ounces of Cereal Mixture put into blend
(b) Objective Function :-
The cost of each mixture per ounce is given as follows,
Our objective is to minimize cost of the blend by finding appropriate values of X1, X2 and X3.
Therefore, objective function is given as,
Minimize z=1.10X1+0.90X2+0.60X3
(c) Constraints :-
Information on contents of each mixture is given and is as follows,
It is given that, the blend should contain no more than 1250 calories. Hence,
170X1+400X2+130X3 1250
It is given that, the blend should contain no more than 25 grams of fat. Hence,
X1+14X2+7X3 25
It is given that, the blend should contain at least 15 grams of protein. Hence,
X1+8X2+5X3 15
It is given that, Dried fruit mixture must be at least 25% of the weight of the blend. Hence,
X1 0.25(X1+X2+X3) OR
0.75X1-0.25X2-0.25X3 0
It is given that, nut mixture must be no more than 45% of the weight of the blend. Hence,
X2 0.45(X1+X2+X3) OR
-0.45X1+0.55X2-0.45X3 0
Ingredient Cost Dried Fruit Mixture 1.10 Nut Mixture 0.90 Cereal Mixture 0.60Related Questions
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