(20 Pts) 5. Slim-Down Manufacturing makes a line of nutritionally complete, weig

Question

(20 Pts) 5. Slim-Down Manufacturing makes a line of nutritionally complete, weight-reduction beverages. One of their products is a strawberry shake which is designed to be a complete meal. The strawberry shake consists of several ingredients. Some information about each of these ingredients is given below. Calories TotalVitamin Ingredient from fat Calories Content Thickeners Cost (per tbsp) (per tbsp) (mg/tbsp) (mg/tbsp) (e/tbsp) Strawberry flavoring 50 20 10 Cream 75 100 Vitamin supplement 50 25 Artificial sweetener 120 15 Thickening agent_ 30 80 25 The nutritional requirements are as follows. The beverage must total between 380 and 420 calories (inclusive). No more than 20% of the total calories should come from fat. There must be at least 50 milligrams (mg) of vitamin content. For taste reasons, there must be at least two tablespoons (tbsp) of strawberry flavoring for each tbsp of artificial sweetener. Finally, to maintain proper thickness, there must be exactly 15 mg of thickeners in the beverage. Management would like to select the quantity of each ingredient for the beverage which would minimize cost while meeting the above requirements. Formulate a linear programming model for this problem.

Explanation / Answer

Let Xj be the quantity (tbsp) of the ingredient-j present. j = 1, 2, 3, 4, 5 where 1 stands for Strawberry flavoring, 2 stands for Cream and so on...

Minimize Z = total cost = 10 X1 + 8 X2 + 25 X3 + 15 X4 + 6 X5

Subject to,

50 X1 + 100 X2 + 0 X3 + 120 X4 + 80 X5 >= 380 (Min total calories)

50 X1 + 100 X2 + 0 X3 + 120 X4 + 80 X5 <= 420 (Max total calories)

1 X1 + 75 X2 + 0 X3 + 0 X4 + 30 X5 <= 20% * (50 X1 + 100 X2 + 0 X3 + 120 X4 + 80 X5)
or, 9 X1 - 55 X2 + 0 X3 + 24 X4 - 14 X5 >= 0 (Max calories from fat)

20 X1 + 0 X2 + 50 X3 + 0 X4 + 2 X5 >= 50 (Min vitamin content)

X1 - 2 X4 >= 0 (Min Strawberry flavoring)

3 X1 + 8 X2 + 1 X3 + 2 X4 + 25 X5 = 15

Xj >= 0

-----------------------------------

So, writing again neatly:

Minimize Z = 10 X1 + 8 X2 + 25 X3 + 15 X4 + 6 X5
Subject to,
50 X1 + 100 X2 + 0 X3 + 120 X4 + 80 X5 >= 380
50 X1 + 100 X2 + 0 X3 + 120 X4 + 80 X5 <= 420
09 X1 - 55 X2 + 00 X3 + 24 X4 - 14 X5 >= 0
20 X1 + 00 X2 + 50 X3 + 00 X4 + 02 X5 >= 50
01 X1 + 00 X2 + 00 X3 - 02 X4 + 00 X5 >= 0
03 X1 + 08 X2 + 01 X3 + 02 X4 + 25 X5 = 15
Xj >= 0, j=1,2,3,4

Related Questions

Navigate

• Browse (All)
• Browse #
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.