﻿ 1 The Charm City Snacks manufactures a snack mix by blending three ingredients a dried fruit mixtu
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### 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.

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

### Solution

xxxxxxxx :-

(x) xxxxxxxx xxxxxxxxx :-

xx is xxxxx xxxx; xxx company xxxxx to xxxx xxx xxxy ounces xx each xxxxxxx xx xxx into xxx blend. xxxxx xxy xxxxxxxx is xx select xxxxxxxy xx xxxx mixture xxxx that xxx xxxx xx the xxxxx is xxxxxxx. xxxxxxxxx xxx,

X1 = Ounces xx xxxxx xxxxx Mixture xxx into xxxxx Xx = Ounces xx Nut xxxxxxx xxx xxxx blend

Xx = xxxxxx xx xxxxxx Mixture xxx into xxxxx (x) xxxxxxxxx Function :-

The xxxx xx xxxx mixture xxx ounce xx xxxxx xx follows,

xxx objective xx xx xxxxxxzx cost xx the xxxxx xy xxxxxxx appropriate xxxxxx of Xx, Xx xxx X3.

xxxxxxxxx, objective xxxxxxxx xx xxxxx as,

Minimize            z=x.xxXx+x.xxXx+x.xxXx (x) xxxxxxxxxxx :-

xxxxxxxxxxx on xxxxxxxx xx xxxx mixture xx given xxx xx xx follows,

xx is xxxxx xxxx, xxx blend xxxxxx contain xx xxxx xxxx 1250 xxxxxxxx. Hence,

xxxXx+xxxXx+xxxXx xxxx

It xx given xxxx, xxx xxxxx should xxxxxxx no xxxx xxxx xx grams xx fat. xxxxx, Xx+xxXx+xXx 25

xx is xxxxx xxxx, xxx blend xxxxxx contain xx xxxxx xx grams xx protein. xxxxx, Xx+xXx+xXx 15

xx is xxxxx xxxx, xxxxx fruit xxxxxxx must xx xx xxxxx 25% xx the xxxxxx xx xxx blend. xxxxx,

X1 x.xx(Xx+Xx+Xx) xx

0.75X1-0.25X2-0.25X3 0

xx xx xxxxx that, xxx mixture xxxx xx xx more xxxx 45% xx xxx xxxxxx of xxx blend. xxxxx, Xx x.xx(Xx+Xx+Xx) OR

-x.xxXx+x.xxXx-x.xxXx x xxxxxxxxxx Cost
 xxxxx Fruit xxxxxxx x.xx xxx Mixture x.xx Cereal xxxxxxx x.xx 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
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