3.5-5.docx

3.5-5 Fred Jonasson manages a family-owned farm. To supplement several food products grown on the farm, Fred also raises pigs for market. He now wishes to determine the quantities of the available types of feed (corn, tankage, and alfafa) that should be given to each pig. Since pigs will eat any mix of these feed types, the objective is to determine which mix will meet certain nutritional requirements at a minimum cost. The number of units of each type of basic nutritional ingredient contained within a kilogram of each feed type is given in the following table, along with the daily nutritional requirements and feed cost: Nutritional Ingredient Carbohydrates Protein Vitamins Cost ($)

Kilogram of Corn 90 30 10 84

Kilogram of Tankage 20 80 20 72

Kilogram of Alfafa 40 60 60 60

Minimum Daily Requirement 200 180 150

a. Formulate a linear programming model for this problem b. Display the model on an Excel Spreadsheet c. Use the spreadsheet to check if (x1, x2, x3) = (1, 2, 2) is a feasible solution and, if so, what daily cost would be for this diet. How many units of each nutritional ingredient would this diet provide daily? Given: Nutritional Ingredient Carbohydrates Protein Vitamins Cost ($)

Kilogram of Corn 90 30 10 84

Kilogram of Tankage 20 80 20 72

Kilogram of Alfafa 40 60 60 60

Minimum Daily Requirement 200 180 150

Required: a. Formulate a linear programming model b. Display the model on an Excel Spreadsheet c. Use the spreadsheet to check if (x1, x2, x3) = (1, 2, 2) is a feasible solution and, if so, what daily cost would be for this diet. How many units of each nutritional ingredient would this diet provide daily? d. Take a few minutes to use a trial and error approach with the spreadsheet to develop your best guess for the optimal solution. What is the daily cost for your solution?

Solution: a. Linear Programming Model Let

x1 = quantities of corn feed x 2 = quantities of tankage feed x 3 = quantities of alfafa feed

Minimize

Z = 84x1 + 72x 2 + 60x 3

Subject to constraints: 90x1 + 20x 2 + 40x 3  200 30x1 + 80x 2 + 60x 3  180

10x1 + 20x 2 + 60x 3  150

x1 , x 2 , x 3  0

b. Model on Excel Spreadsheet Obj Function 241.714

Variables x1 x2 x3

Variables/Solution Constraints Constants 200 200 1.143 180 180 0 157.143 150 2.429 1.143 0 0 0 2.429 0

As shown in the result of the Excel Solver, the quantities of corn, tankage and alfafa feed types needed were 1.143, 0, and 2.429, respectively. Since the optimal value is the largest or smallest value of the objective function, it constitutes an optimal solution, which is $241.714. c. (x1, x2, x3) = (1, 2, 2) is a feasible solution with a daily cost of $348. This diet provide 210 kg of carbohydrates, 310 kg of protein, and 170 kg of vitamins daily.