(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 bevera strawberry shake consists of several ingredients. Some information about each of these ingredients is given below. ges. One of their products is a straw berry shake which is designed to be a complete meal. The CaloriesTotalVitamin 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 trawberry 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

The formulation of LP problem could be as follows:

Objective function is:

Minimize Zmin = € (10S + 8C + 25V + 15A + 6T)

Where S,C,V,A,T >= 0 (Non Negativity constraints) And

S stands for one tbsp of Strawberry flavoring,

C stands for one tbsp of Cream

V stands for one tbsp of Vitamin

A stands for one tbsp of Artificial sweetener

And T stands for one tbsp of Thickness Agent

While fulfilling following constraints:

For total calorie between 380 to 420 including both values

50S + 100C + 120A + 80T <=420

50S + 100C + 120A + 80T >= 380

OR instead of two statements write single statement as follows

380 <= 50S + 100C + 120A + 80T <=420

S + 75C + 30T <= 76 (76 20% of 380, THE MINIMUM CALORI REQUIRED)

20S + 50V + 2T >=50 (At least 50 mg of vitamin)

S/A>= 2 ( For taste reason requirement is S =>2 A)

3S + 8C + 1V + 2A + 25T = 15 (Thickness must be at least 15)

Note : You can also use x1, x2, x3, x4and x5 etc in place of S,C,V A and T

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