Question

2.22. Ralph Edmund loves steaks and potatoes. Therefore, he has decided to go on a steady diet of only these two foods (plus some liquids and vitamin supplements) for all his meals. Ralph realizes that this isn't the healthiest diet, so he wants to make sure that he eats the right quantities of the two foods to satisfy some key nutritional requirements. He has obtained the following nutritional and cost information Grams of Ingredient per Serving Ingredient Carbohydrates Protein Fat Cost per serving Steak Potatoes Daily Requirement (grams) 5 20 15 $4 5 2 $2 250 240 K60 Ralph wishes to determine the number of daily servings (may be fractional) of steak and potatoes that will meet these requirements at a minimum cost a. Identify verbally the decisions to be made, the constraints on these decisions, and the overall measure of performance for the decisions b. Convert these verbal descriptions of the constraints and the measure of performance into quantitative expressions in terms of the data and decisions c. Formulate and solve a linear programming model for this problem on a spreadsheet d. Formulate this same model algebraically

Answer 1

2.22

(a)

The decision to be made is the number of servings for the two foods to eat. Constraints are to fulfill the minimum required carbohydrates and protein without crossing the maximum limit for fat. The measure of the performance is the total cost which needs to be minimized.

(b)

Let X1 and X2 be the servings of Streak and Potato. The carbohydrates acquired will be equal to 5X1 + 15X2 and this must be more than or equal to 50 grams. Similarly, the protein acquired will be equal to 20X1 + 5X2 and this must be more than or equal to 40 grams. Finally, the fat acquired will be 15X1 + 2X2 and this must be less than or equal to 60 grams. The total cost is 4X1 + 2X2 and this must be minimized.

(c)

Answer report

(d)

Minimize 4X1 + 2X2
Subject to,
5X1 + 15X2 >= 50
20X1 + 5X2 >= 40
15X1 + 2X2 <= 60
X1, X2 >= 0