After a grueling year of work, you (Padcha) decide you need absolute peace and q

Question

After a grueling year of work, you (Padcha) decide you need absolute peace and quiet for about a month (February). You visit San Pedro de Atacama in the Atacama dessert of Chile, one of the most pristine and beautiful spots on earth. Every morning you have an espresso with a Swiss veterinarian, Dieter, that has retired there and maintains a small hotel and cafe. After week 2 of peace and quiet you become restless for a more meaningful existence.

In a conversation Dieter you learn that he also raises Llama's, Alpacas, and Vicuñas (all members of the Camel Family) for their wool. The conversation follow--

Dieter: "Padcha, I have this problem related to the nutrition of my animals. I know the nutritional content of the feeds I have available (see the Daily Nutritional Requirements for Animals table). I also know the cost of a kilogram of each feed (see table). I would love to have a way of selecting a mix of Corn, Tankage, and Alfalfa that minimizes my daily cost of providing good nutrition for my animals."

Padcha: "Dieter, I think I might be able to help. In LP optimization, there is a problem that is called the nutrition problem. Of course, it is not only about nutrition, it's also about allocating available resources, in an optimal way (minimal cost), to satisfy requirements. You can image, that there are a lot of problems like that in business. A typical constraint, say for carbohydrates, would read-- the number of kilos of Corn times 90, plus the number of kilos of Tankage times 20, plus the number of kilos of Alfalfa times 40 must equal a minimum 200 units of carbohydrates. The constraint shows how we use the resources/decision variables (kilos of Corn, etc.) to satisfy nutritional needs.

180

a) What is the LP for the nutrition problem?

b) Solve the problem using Solver. What is the value of the optimal solution? What is the number of kilos of Corn, Tankage, and Alfalfa? (Place the Answer Report below)

c) What if the requirements were raised by 1 unit for carbohydrates (from 200 to 201). Without re-solving the problem, what is the new optimal solution value and the units of carbohydrates? (Provide the logic for your answer.)

d) What is the allowable range of values for the coefficient of the Tankage in the objective function, for which a change of the optimal decision variables will not occur?

Daily Nutritional Requirements for Animals Kg. Corn KG. Tankage Kg. Alfalfa Min. Daily Reqs. Carbohydrates 90 20 40 200 Protein 30 80 60

180

Vitamins 10 20 60 150 $Cost of feed/Kg. 35 30 25

Explanation / Answer

a) The Problem is

b)
The problem is converted to canonical form by adding slack, surplus and artificial variables as appropiate

1. As the constraint 1 is of type '' we should subtract surplus variable S1 and add artificial variable A1

2. As the constraint 2 is of type '' we should subtract surplus variable S2 and add artificial variable A2

3. As the constraint 3 is of type '' we should subtract surplus variable S3 and add artificial variable A3

After introducing surplus,artificial variables

subject to

and x1,x2,x3,S1,S2,S3,A1,A2,A30

Positive maximum Cj-Zj is 310M3+1856 and its column index is 1. So, the entering variable is x1.

Minimum ratio is 65 and its row index is 1. So, the leaving basis variable is A1.

The pivot element is 2503.

Entering =x1, Departing =A1, Key Element =2503

R1(new)=R1(old)×3250

R2(new)=R2(old)-20R1(new)

R3(new)=R3(old)-16R1(new)

Continuing in the same way we get the following answer in the 6th iteration

Variable S1 should enter into the basis, but all the coefficients in the S1 column are negative or zero. So S1 can not be entered into the basis.

Hence, the solution to the given problem is unbounded.

Max Z = 35 x1 + 30 x2 + 25 x3 subject to 90 x1 + 20 x2 + 40 x3 200 30 x1 + 80 x2 + 60 x3 180 10 x1 + 20 x2 + 60 x3 150 and x1,x2,x30;

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