Ag Bus 435
Midterm
11/4/13
Dr. Hurley
General Instructions: This exam is worth 160 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!
Question 1: Given the following mathematical set-up and corresponding tabular form, complete
and answer the following questions.
max
𝑤.𝑟.𝑡.𝑥1 ,𝑥2 ,𝑥3
5𝑥1 + 2𝑥2 + 1𝑥3
Subject to:
8𝑥1 + 4𝑥2 + 2𝑥3 ≤ 128
32𝑥1 + 6𝑥2 + 8𝑥3 ≤ 422
18𝑥1 + 12𝑥2 + 6𝑥3 ≤ 972
14𝑥1 + 6𝑥2 + 2𝑥3 ≤ 194
𝑥1 ≥ 0, 𝑥2 ≥ 0, 𝑥3 ≥ 0
Z
x1
1
0
0
0
0
x2
0
0
1
0
0
0
0
0
0
1
x3
x4
-0.22222
1.111111
0.333333
5.333333
-0.44444
0
1
0
0
0
x5
x6
0.018519
0.074074
0.055556
0.555556
-0.12963
0
0
0
1
0
x7
0.314815
-0.74074
-0.05556
-2.55556
0.296296
RHS
68.88889
15.55556
12.66667
710.6667
2.777778
a) Complete the next iteration in Excel (15 points). Note, you should be able to copy and
paste the table into Excel. Also, the answer may not come out clean.
b) Which number was your pivot number? Briefly explain how you know this (5 points).
c) After completing the iteration in part (a), will there be any more iterations to this
problem? Briefly explain your answer in a textbox (5 points).
d) Assume that part (a) was the last iteration to solving the problem, please completely
interpret the results in your final table, i.e., optimal solution to the problem and the
shadow prices (10 points).
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Revised: 5/25/12
Question 2: Given the following mathematical set-up, complete and answer the following
questions.
min
𝑤.𝑟.𝑡.𝑥1 ,𝑥2
3000𝑥1 + 5000𝑥2
Subject to:
7𝑥1 + 12𝑥2 ≥ 3250
65𝑥1 + 10𝑥2 ≥ 6500
1𝑥1 + 1𝑥2 ≥ 375
𝑥1 ≥ 0, 𝑥2 ≥ 0,
a) Set-up the augmented mathematical model for this minimization problem (10 points).
b) Set-up and solve this problem using Solver (15 points).
c) Set-up the dual mathematical model for this problem (10 points).
max
𝑤.𝑟.𝑡. 𝑦1 ,𝑦2 ,𝑦3
3250𝑦1 + 6500𝑦2 + 375𝑦3
Subject to:
7𝑦1 + 65𝑦2 + 𝑦3 ≤ 3000
12𝑦1 + 10𝑦2 + 𝑦3 ≤ 5000
𝑦1 ≥ 0, 𝑦2 ≥ 0, 𝑦3 ≥ 0
d) Set-up the tabular form of this problem in Excel (5 points).
e) Solve the tabular problem in part (d) without using Solver (20 points).
f) Briefly explain your solution in a textbox (5 points).
Question 3: Suppose you are a producer of strawberries in Watsonville. You currently have a
contract with a local packer-shipper to take all of your strawberries as long as you meet the
company’s minimum quality standards. There are three quality standards that you are required
to meet—one for size, one for color, and one for sweetness. The company has developed its own
way of measuring size, color, and sweetness. Based on its measurement of samples from the
berries you send the company, you will need to achieve a point score of at least 750 for size,
13,000 for color, and 32,500 for sweetness.
You have found that the key input that affects the size, color, and sweetness of the berries is
fertilizer. Having done a lot of trials in the past, you have found that there are two fertilizer
brands you would consider using to grow your strawberries. The first fertilizer, known as
Worm-Poo-Plus, costs $300 per ounce. Each ounce that you apply to your field of berries adds
70 points to the sweetness measurement of your berries, 130 points to color, and 2 points to size.
The other fertilizer you can use is called Duck-Poo-Max. This fertilizer costs $500 an ounce.
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Revised: 5/25/12
One ounce of Duck-Poo-Max will increase the sweetness score of your berries by 120 points, the
color score by 20 points, and the size score by 2 points.
Your ultimate goal is to minimize the cost that you spend on fertilizer while meeting the
minimum requirements by the packer-shipper. You are able to use any combination of WormPoo-Plus and Duck-Poo-Max.
a) Please summarize in table form what the data for this problem is (5 points).
b) Please develop a mathematical model for this problem (10 points).
max
𝑤.𝑟.𝑡.𝑊𝑃𝑃,𝐷𝑃𝑀
300𝑊𝑃𝑃 + 500𝐷𝑃𝑀
2𝑊𝑃𝑃 + 2𝐷𝑃𝑀 ≥ 750
70𝑊𝑃𝑃 + 120𝐷𝑃𝑀 ≥ 32500
130𝑊𝑃𝑃 + 20𝐷𝑃𝑀 ≥ 13000
𝑊𝑃𝑃 ≥ 0, 𝐷𝑃𝑀 ≥ 0
c) Please build a spreadsheet model for this problem (10 points).
d) Solve the spreadsheet model using Solver and explain your optimal solution including
how much you should use of each of the fertilizer and what is the lowest achievable cost
(5 points).
e) Please run a sensitivity report and calculate the allowable ranges for the objective
coefficients and the constraints (5 points).
f) Please explain what the shadow price regarding sweetness tells you (5 points).
g) What is the lowest price that Worm-Poo-Plus could go to before your optimal solution
changes? Please briefly explain (5 points).
h) Given that Worm-Poo-Plus increases by $50, while Duck-Poo-Max decreases by $50,
can you definitely say anything about the optimal solution based on the 100% rule for
simultaneous changes? Please briefly explain (5 points).
Question 4: Solve the following problem without using solver (10 points):
35𝑥1 + 7𝑥2 + 9𝑥3 + 37𝑥4 + 31𝑥5 = 379
12𝑥1 + 24𝑥2 + 18𝑥3 + 6𝑥4 + 39𝑥5 = 333
32𝑥1 + 21𝑥2 + 25𝑥3 + 16𝑥4 + 35𝑥5 = 388
10𝑥1 + 32𝑥2 + 16𝑥3 + 20𝑥4 + 12𝑥5 = 262
34𝑥1 + 14𝑥2 + 10𝑥3 + 37𝑥4 + 8𝑥5 = 280
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Revised: 5/25/12
Please briefly explain in a textbox your answer and how you got it. Note, your final answer
should be clean.
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Revised: 5/25/12