Answer to Question #182594 in Operations Research for mike

Question

Healthy Snacks Co. produces snack mixes. Recently, the company has decided to introduce a new snack mix that has peanuts, raisins, pretzels, dries cranberries, sunflower seeds and pistachios. Each bag of the new snack is designed in order to hold 250 grams of the snack. The company has decided to market the new product with an emphasis on its health benefits. After consulting nutritionists, Healthy Snacks decides to mix the ingredients so that the snack has the following specifications:

Answer 1

Total problem:

Healthy Snacks Co. produces snack mixes. Recently, the company has decided to introduce a new snack mix that has peanuts, raisins, pretzels, dries cranberries, sunflower seeds and pistachios. Each bag of the new snack is designed in order to hold 250 grams of the snack. The company has decided to market the new product with a emphasis on its health benefits. After consulting nutritionists, Healthy Snacks decides to mix the ingredients so that the snack has the following specifications:

1 - Total calories ≤ 800

2 - Fat ≤ 20

3 - Sodium ≤ 200

4 - Potassium ≤ 700

5 - Carb ≤ 140

6 - Protein ≥ 10

7 - Vitamin A ≥ 3

8 - Vitamin B6 ≥ 25

9 - Vitamin C ≥ 2

10 - Iron ≥ 15

11 - Calcium ≥ 5

Table below shows the amount of each nutrition in each snack's ingredient per 1 gram:

Table below shows the cost of each ingredients per 1 gram:

Per marketing department research, each bag of snack should have at least 10 grams of each ingredient.

The company wants to minimize the cost of the snack mix. This is a minimization problem.

Let the company decided to mix X1 gram of peanut, X2 gram of raisin, X3 gram of pretzel, X4 gram of dried cranberries, X5 gram of sunflower seed and X6 gram of pistachio.

The Decision variables are,

$X_1,\ X_2,\ X_3,\ X_4,\ X_5,\ X_6$

Now the objective function is defined as,

$Minimize\ \ 0.005X_1+0.020X_2+0.013X_3+0.013X_4+0.022X_5+0.031X_6$

The constrains are defined as,

1 - Total calories ≤ 800

$5.67X_1+3X_2+X_3+3.08X_4+5.84X_5+5.62X_6\leq 800$

2 - Fat ≤ 20

$0.49X_1+0.01X_3+0.01X_4+0.51X_5+0.06X_6\leq 20$

3 - Sodium ≤ 200

$0.18X_1+0.12X_2+3.32X_3+0.03X_4+0.09X_5+0.01X_6\leq 200$

4 - Potassium ≤ 700

$7.05X_1+7.49X_2+0.36X_3+0.40X_4+6.45X_5+10.25X_6\leq 700$

5 - Carb ≤ 140

$0.16X_1 + 0.79X_2 +0.21X_3 + 0.83X4 + 0.20X_5 + 0.28X_6 \leq 140$

6 - Protein ≥ 10

$0.26X_1+0.03X_2+0.03X_3+0X_4+0.21X_5+0.20X_6\geq 10$

7 - Vitamin A ≥ 3 (in %)

$0X_1+0X_2+0X_3+0X_4+0.0021X_5+0.0020X_6\geq 3$

8 - Vitamin B6 ≥ 25 (in %)

$0.0017X_1+0.0012X_2+0X_3+0X_4+0.0068X_5+0.0085X_6\geq 25$

9 - Vitamin C ≥ 2 (in %)

$0X_1+0.0002X_2+0X_3+0X_4+0.0002X_5+0.0009X_6\geq 2$

10 - Iron ≥ 15

$0.25X_1+0.09X_2+0.07X_3+0.03X_4+0.29X_5+0.21X_6\geq 15$

11 - Calcium ≥ 5

$0.09X_1+0.05X_2+0X_3+0X_4+0.07X_5+0.10X_6\geq 5$

12 - $X_1,\ X_2,\ X_3,\ X_4,\ X_5,\ X_6 \ \geq 0$