Ag Bus 328

Midterm

Section 1

10/29/15

Dr. Hurley

General Instructions: This exam is worth 150 points. You must provide your own paper. You are allowed to use your notes for the exam. You must show all your work when appropriate to get credit. Create a new tab for each question you answer. No cell phones are allowed to be in your possession during the exam. If you are caught with a cell phone, you will receive a zero on the exam. Please email me your spreadsheet answers and save it to AGB 328 exam folder where your name is with the name of the file as your name, e.g. Sean-Hurley. (Please note that except for any diagrams or mathematical models, all answers should be entered into a textbox on your spreadsheet.)

GOOD LUCK!

Problem A:

maxw.r.t. x1,x2,x3,x448x1+48x2+96x3+144x4=Z

Subject to:

2x1+5x2+6x3+7x4≤420

8x1+3x2+5x3+4x4≤480

3x1+4x2+7x3+6x4≤504

x1≥0, x2≥0, x3≥0, x4≥0

Question 1: Set-up the augmented mathematical model for this maximization problem (10 points).

Question 2: Set-up and solve the model given above by Excel using the tabular form. In a textbox, explain briefly the optimal solution including what the maximum achievable is and what are your basic and non-basic variables (15 points).

Question 3: If the RHS of constraint 3 increased from 504 to 505, how much do you expect the Z to increase by? Briefly explain your answer in a textbox or show how you found your answer (5 points).

Question 4: Set-up the dual to the original problem above and in excel set up the augmented form of the model (15 points).

Question 5: What is the optimal solution to the dual including what is the minimum achievable? Briefly explain your answer in a textbox or show how you found your answer. There is no need to tell me what the shadow prices are. There are multiple ways to find this answer and you are free to choose whichever one you like barring not looking at someone else’s answer (10 Points).

Problem B: Suppose you have the following problem below:

minw.r.t. x1,x2,x31x1+2x2+3x3

Subject to:

1x1+2x2+3x3≥270

4x1+3x2+2x3≥540

1x1+2x2+1x3≥210

3x1+1x2+4x3≥360

x1≥0, x2≥0, x3≥0

Question 1: Identify two interior corner point solutions for this problem using three defining equations at a time. You should note that there are four of these solutions and I am only asking you to provide me with two of them. You should also note that an interior solution requires all the x’s in the solution to be non-zero. There is no need for you to draw this solution because it would need to be three dimensional. You can find these solutions using Excel (25 points).

Question 2: Please explain which solutions you found in question 1 are interior feasible corner point solutions and/or which are interior infeasible corner point solutions? Please briefly explain how you know (15 points).

Problem D: Suppose your favorite Labrador dog, Rufus, has been acting sluggish recently. Since you have great concerns for Rufus, you have decided to take him to the vet to see if there was anything wrong. Noticing that Rufus looks to be a bit overweight, the vet decides to do a complete exam on Rufus including a blood test and a weight check. After all the results come back, the vet sits down with you to discuss why she believes Rufus has not been acting like his old self. Based on the blood test, the vet noticed that Rufus seems to be deficient in four nutritional areas—protein, fiber, calcium, and iron. The vet also explains to you that Rufus is also 15 pounds over the upper range of his ideal weight.

To get Rufus healthy again, the vet recommends that you change Rufus’ diet. The goal of this diet change is to make sure that Rufus gets no more than 1,260 calories and for each nutrient at least 4,200 milligrams (mg) of protein, 3,000 mg of fiber, 1,170 mg of calcium, and 1,440 mg of iron. She recommends that you consider four different brands of dog food—Chewy Chunks (CC), Crunchy Kibble (CK), Voracious Veggie (VV), and Meaty Mouthful (MM).

Since you are a price conscious shopper, you ask the vet if it is alright to mix the different brands of foods together in order to achieve the nutrient requirements while making sure that Rufus’ diet does not exceed the calorie suggestion. Upon her approval you go home with Rufus and search on the internet for nutritional information regarding each of the types of foods. Table 1 below lists the nutritional facts for each of the recommended dog food brands.

You next go to your local pet food store to investigate how much each of the brands costs. To your surprise, the store was now selling each brand of food by the gram and you could buy as little or as much of each type that you wanted. Examining the prices of each of the recommended dog foods showed that the most expensive dog food per gram was Meaty Mouthful at $0.16 per gram. Two of the dog food brands were very close in costs per gram—Voracious Veggie was $0.12 per gram, while Chewy Chunks was $0.11 per gram. The cheapest of the four brands was Crunchy Kibble at $0.085 per gram.

Table 1: Nutritional Components per 1 Gram of Feed Item

Nutritional Component* / Chewy Chunks / Crunchy Kibble / Voracious Veggie / Meaty Mouthful / Nutritional Need
Calories / 1 / 2 / 7 / 9 / 1,260
Protein (mg) / 7 / 6 / 5 / 2 / 4,200
Dietary Fiber (mg) / 3 / 9 / 4 / 7 / 3,000
Calcium (mg) / 1 / 1 / 9 / 13 / 1,170
Iron (mg) / 6 / 3 / 8 / 1 / 1,440

*mg = milligrams, g = grams

Given the information above please answer the following questions.

  1. Please write a mathematical model for this problem if your ultimate goal is to minimize cost while meeting the vet’s requirements (15 Points).
  1. Please develop a spreadsheet model for this problem. Use the guidelines for building good spreadsheets. You should have at least three guidelines covered (10 Points).
  1. Using solver, please find the cost minimizing solution to the problem, i.e., how much of each of the dog food brands do you purchase? What is the minimum cost of purchasing the cost minimizing solution (5 Points)?
  1. Suppose after a couple weeks of the diet, Rufus has picked up a little, but is not quite back to normal. You call the vet for advice. Based on hearing your current feeding rations, she recommends that you make sure that the amount of Meaty Mouthful should be at least 7.5% of the total weight of the feed ration you give to Rufus. She believes that based on the ingredients that are in Meaty Mouthful that Rufus will find the ration more palatable and make him happier. What mathematical constraint would you need to add to your mathematical model to incorporate this idea (15 Points)?
  1. How much more is this new constraint costing you if you feed a cost minimizing ration to Rufus (10 Points)? (Please show your work on the spreadsheet and give a brief explanation in a textbox regarding your new answer.)

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Revised: 10/25/15