Ag Bus 435
Midterm
Section 1
10/27/14
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 note that except for any diagrams or mathematical models, all answers
should be entered into a textbox on your spreadsheet.)
GOOD LUCK!
Problem A:
max
𝑤.𝑟.𝑡.𝑥1 ,𝑥2 ,𝑥3
6𝑥1 + 5𝑥2 + 3𝑥3
Subject to:
2𝑥1 + 2𝑥2 + 4𝑥3 ≤ 24
2𝑥1 + 4𝑥2 + 2𝑥3 ≤ 24
4𝑥1 + 2𝑥2 + 2𝑥3 ≤ 24
2𝑥1 + 2𝑥2 + 2𝑥3 ≤ 18
𝑥1 ≥ 0, 𝑥2 ≥ 0, 𝑥3 ≥ 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 (15
points).
Question 3: In a textbox, please interpret your solution including telling what the Basic
variables, the Non-Basic variable are, and what the maximum achievable is (10 points).
Question 4: What was your pivot number in your second iteration before you normalized it to 1?
Briefly explain how you know this (5 points).
Question 5: In a textbox, explain what is the shadow price is for the x4 variable and what does it
tell you (10 Points).
Question 6: How do you know that there are no iterations left in this problem? Briefly explain
your answer in a textbox (5 points).
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Revised: 10/20/14
Problem B: Solve the following problem without using solver (10 points):
3𝑥1 + 10𝑥2 + 9𝑥3 + 8𝑥4 + 10𝑥5 = 84
3𝑥1 + 7𝑥2 + 10𝑥3 + 8𝑥4 + 6𝑥5 = 72
5𝑥1 + 3𝑥2 + 3𝑥3 + 1𝑥4 + 2𝑥5 = 34
4𝑥1 + 2𝑥2 + 7𝑥3 + 9𝑥4 + 8𝑥5 = 68
4𝑥1 + 2𝑥2 + 5𝑥3 + 4𝑥4 + 6𝑥5 = 51
Problem C:
min
𝑤.𝑟.𝑡.𝑥1 ,𝑥2
30𝑥1 + 20𝑥2
Subject to:
2𝑥1 − 16𝑥2 ≥ 0
1𝑥1 + 1𝑥2 ≥ 90
1𝑥1 + 2𝑥2 ≥ 120
𝑥1 ≥ 0, 𝑥2 ≥ 0
Question 1: Draw the feasible region to this minimization problem making sure that you identify
all corner point feasible solution (30 points).
Question 2: What is the minimum amount that is achieved at the optimal (5 points)?
Problem D: Suppose you raise rhinoceros for the Santa Barbara Zoo. Rhinoceros’s are picky
eaters and will only eat four kinds of food—corn, soybeans, lettuce, wheat, and rye. Each of
these feeds has different nutritional values which are shown in Table 1 below. Each feed is
measured in one kilogram increments. In order to keep the rhinoceros healthy, you need to make
sure that they have a minimum amount of certain nutrients. One of your goals is to make sure
that each of them gets at least 150,000 calories per week.
One of the nutritional components to keep rhinoceros healthy is fat in their diet. It takes at least
360,000 mg of fat per week to ensure their good health. Another component to their diet is to
have carbohydrates where they need 42,000 g per week to stay healthy. A third nutritional
component that must be met is dietary fiber intake. A healthy rhinoceros requires at least 7,000
g per week of fiber to lead a healthy life.
In order to have the weight gain that you would like to see in the rhinoceros, you need a
minimum amount of sugar and proteins. In terms of sugar, the rhinoceros should get at least
45,000 mg per week. For protein, the they should get at least 6,000 grams per week.
Your ultimate goal is to meet the nutritional feeding requirements of the rhinoceros at a
minimum cost where the costs per kilogram of corn, soybeans, lettuce, wheat, and rye are
respectively $4.00, $2.00, $3.00, $5.00, and $6.00.
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Revised: 10/20/14
Table 1: Nutritional Components per 1 Kilogram of Feed Item
Nutritional
Corn
Soybean
Lettuce
Wheat
Component*
Calories
859
454
153
3,111
Fat (mg)
7,859
53,846
2,059
18,519
Carbohydrates (g)
200
85
24
676
Dietary Fiber (g)
24
23
12
102
Sugar (mg)
35,294
4,754
10,718
4,102
Protein (g)
24
108
12
111
*mg = milligrams, g = grams
Rye
3,379
17,751
757
154
12
101
Given the information above please answer the following questions.
1. Please write a mathematical model for this problem (15 Points).
2. Please develop a spreadsheet model for this problem. Use the guidelines for building
good spreadsheets (10 Points).
3. Using solver, please find the cost minimizing solution to the problem, i.e., how much of
each of the feeds do you use? What is the minimum cost of purchasing the cost
minimizing solution (5 Points)? (Please note that the answers may not come out clean
integer values. Please round all answers to two decimal places.)
4. Suppose that you wanted to limit the usage of corn and soybeans because they are both
high in sodium content. To meet this new sodium constraint you want at least 95% of the
total weight to come from lettuce, wheat, and rye. What mathematical constraint would
you need to add to your mathematical model to incorporate this idea (10 Points)?
5. How much more is this new constraint costing you if you feed cost minimizing ration to
the rhinoceros (10 Points)? (Please show your work on the spreadsheet and give a brief
explanation in a textbox on how you found this answer.)
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Revised: 10/20/14