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Homework answers / question archive / 1 [–/4 Points]DETAILSRAGSMDA9 3

1 [–/4 Points]DETAILSRAGSMDA9 3

Business

1

[–/4 Points]DETAILSRAGSMDA9 3.E.014.
MY NOTES
ASK YOUR TEACHER
PRACTICE ANOTHER
 
Game Day Fashions makes custom t-shirts for fans of college university sports teams. Currently, the company is trying to determine the best way to fill an order it received from Sable University. The official colors of Sable University are orange and maroon, so most of the t-shirts it orders are for one of those two colors. However, one home game each year is designated as a "white out" games where all Sable University fans are encouraged to wear white shirts. Sable University has placed an order for 50,000 orange shirts, 45,000 maroon shirts, and 30,000 white shirts. Under the terms of the contract, the company is permitted to outsource as many as 25,000 of each of the shirts from a competing company. Game Day Fashions can make the orange, maroon, and white shirts for $8.50, $4.00, and $5.00 per unit, respectively. The company handling the outsourcing has agreed to make up to 25,000 of each shirt color at a cost of $10.00, $6.00, and $5.75 per unit for the maroon, orange, and white shirts, respectively. The following table summarizes the amount of time in seconds it takes Game Day's production department to make each shirt color.
  Seconds Required per Unit
Operation Orange Maroon White
Cutting 15 10 12
Sewing 6 4 7
Printing 10 8 6
Game Day has 210 hours of cutting capacity available, 95 hours of sewing, and 160 hours of printing. The company would like to determine the optimal way of filling the order for Sable University. (Let X1, X2, and X3 be the number of orange, maroon, and white shirts respectively that Game Day Fashions should make. Let X4, X5, and X6 be the number of orange, maroon, and white shirts respectively that Game Day Fashions should outsource.)
(a)
Formulate an LP model for this problem to minimize cost (in dollars).
MIN:
 
 
Subject to:Number of orange shirts
 
 
Number of maroon shirts
 
 
Number of white shirts
 
 
Number of orange shirts outsourced
 
 
Number of maroon shirts outsourced
 
 
Number of white shirts outsourced
 
 
Cutting Time (in sec)
 
 
Sewing Time (in sec)
 
 
Printing Time (in sec)
 
 
X1, X2, X3, X4, X5, X6 ≥ 0
(b)
Implement your model in a spreadsheet and solve it. What is the optimal solution?
(X1, X2, X3, X4, X5, X6) = 
 
 
 
 

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2.

[–/5 Points]DETAILSRAGSMDA9 3.E.026.
MY NOTES
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PRACTICE ANOTHER
 
A company is trying to determine how to allocate its $75,400 advertising budget for a new product. The company is considering newspaper ads and television commercials as its primary means for advertising. The following table summarizes the costs of advertising in these different media and the number of new customers reached by increasing amounts of advertising.
Media & # of Ads # of New Customers
Reached		 Cost per Ad
Newspaper: 1–10 900 $1,000
Newspaper: 11–20 700 $900
Newspaper: 21–30 400 $800
Television: 1–5 10,000 $12,000
Television: 6–10 7,500 $10,000
Television: 11–15 5,000 $8,000
For instance, each of the first 10 ads the company places in newspapers will cost $1,000 and is expected to reach 900 new customers. Each of the next 10 newspaper ads will cost $900 and is expected to reach 700 new customers. Note that the number of new customers reached by increasing amounts of advertising decreases as the advertising saturates the market. Assume the company will purchase no more than 30 newspaper ads and no more than 15 television ads. (Let 
N1
 be the number of newspaper ads purchased at $1,000 each. Let 
N2
 be the number of newspaper ads purchased at $900 each. Let 
N3
 be the number of newspaper ads purchased at $800 each. Let 
T1
 be the number of television ads purchased at $12,000 each. Let 
T2
 be the number of television ads purchased at $10,000 each. Let 
T3
 be the number of television ads purchased at $8,000 each.)
(a)
Formulate an LP model for this problem to maximize the number of new customers reached by advertising.
MAX:
 
 
Subject to:total cost (in dollars)
 
 
total number of newspaper ads purchased
 
 
total number of television ads purchased
 
 
maximum number of newspaper ads purchased at $1,000
 
 
maximum number of newspaper ads purchased at $900
 
 
maximum number of newspaper ads purchased at $800
 
 
maximum number of television ads purchased at $12,000
 
 
maximum number of television ads purchased at $10,000
 
 
maximum number of television ads purchased at $8,000
 
 
N1, N2, N3, T1, T2, T3 ≥ 0
(b)
Implement your model in a spreadsheet and solve it. What is the optimal solution?
(N1, N2, N3, T1, T2, T3) = 
 
 
 
 
(c)
Suppose the number of new customers reached by 11–20 newspaper ads is 400 and the number of new customers reached by 21–30 newspaper ads is 700. Make these changes in your spreadsheet and reoptimize the problem. What is the new optimal solution? (Round your answers to the nearest whole number.)
(N1, N2, N3, T1, T2, T3) = 
 
 
 
 
What (if anything) is wrong with this solution and why?
The solution does not make sense since the variable T2 is used before the variable T1.
The solution does not make sense since the variable T3 is used before the variable T2.
    
The solution does not make sense since the variable N2 is used before the variable N1.
The solution does not make sense since the variable N3 is used before the variable N2.
The solution is reasonable given the context of the problem.

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3.

[–/5 Points]DETAILSRAGSMDA9 3.E.031.
MY NOTES
ASK YOUR TEACHER
PRACTICE ANOTHER
 
Virginia Tech operates its own power generating plant. The electricity generated by this plant supplies power to the university and to local businesses and residences in the Blacksburg area. The plant burns three types of coal, which produce steam that drives the turbines that generate the electricity. The Environmental Protection Agency (EPA) requires that for each ton of coal burned, the emissions from the coal furnace smoke stacks contain no more than 2,500 parts per million (ppm) of sulfur and no more than 2.8 kilograms (kg) of coal dust. The following table summarizes the amounts of sulfur, coal dust, and steam that result from burning a ton of each type of coal.
Coal Sulfur (in ppm) Coal Dust (in kg) Pounds of Steam Produced
1 1,100 1.7 25,000
2 3,600 3.2 37,000
3 1,350 2.4 27,000
The three types of coal can be mixed and burned in any combination. The resulting emission of sulfur or coal dust and the pounds of steam produced by any mixture are given as the weighted average of the values shown in the table for each type of coal. For example, if the coals are mixed to produce a blend that consists of 35% of coal 1, 40% of coal 2, and 25% of coal 3, the sulfur emission (in ppm) resulting from burning one ton of this blend is:
0.35 ? 1,100 + 0.40 ? 3,600 + 0.25 ? 1,350 = 2,162.5
The manager of this facility wants to determine the blend of coal that will produce the maximum pounds of steam per ton without violating the EPA requirements. (Let Pi = proportion of coal i to include in the mix.)
(a)
Formulate an LP model for this problem to maximize steam per ton in pounds.
MAX:
 
 
Subject to:Sulfur Amount (in ppm)
 
 
Coal Dust Amount (in kg)
 
 
Sum of the Proportions
 
 
P1, P2, P3 ≥ 0
(b)
Create a spreadsheet model for this problem and solve it using Solver. What is the optimal solution? (Round your answer to four decimal places.)
(P1, P2, P3) = 
 
 
 
 
(c)
If the furnace can burn up to 30 tons of coal per hour, what is the maximum amount of steam (in pounds) that can be produced per hour? (Round your answer to the nearest integer.)
 pounds

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4.

[–/5 Points]DETAILSRAGSMDA9 3.E.034.
MY NOTES
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PRACTICE ANOTHER
 
The Sentry Lock Corporation manufactures a popular commercial security lock at plants in Macon, Louisville, Detroit, and Phoenix. The per unit cost of production at each plant is $35.50, $37.50, $39.00, and $36.25, respectively, while the annual production capacity at each plant is 18,000, 15,000, 25,000, and 20,000, respectively. Sentry's locks are sold to retailers through wholesale distributors in seven cities across the United States. The unit cost of shipping from each plant to each distributor is summarized in the following table along with the forecasted demand from each distributor for the coming year.
Unit Shipping Cost to Distributor in
Plants Tacoma San Diego Dallas Denver St. Louis Tampa Baltimore
Macon $2.50 $2.75 $1.75 $2.00 $2.10 $1.80 $1.65
Louisville $1.85 $1.90 $1.50 $1.60 $1.00 $1.90 $1.85
Detroit $2.30 $2.25 $1.85 $1.25 $1.50 $2.25 $2.00
Phoenix $1.90 $0.90 $1.60 $1.75 $2.00 $2.50 $2.65
Demand 8,500 14,000 13,500 13,400 18,000 15,000 9,000
Sentry wants to determine the least expensive way of manufacturing and shipping locks from its plants to the distributors. Because the total demand from distributors exceeds the total production capacity for all the plants, Sentry realizes it will not be able to satisfy all the demand for its product, but wants to make sure each distributor will have the opportunity to fill at least 80% of the orders received. (Assume all production capacity will be used.)
Create a spreadsheet model for this problem and solve it. What is the optimal solution? Fill in the table below with the number of locks that should be sent from each of the plants to each of the distributors.
Number of Units Shipped
From/To Tacoma San Diego Dallas Denver St. Louis Tampa Baltimore
Macon
Louisville
Detroit
Phoenix

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5.

[–/5 Points]DETAILSRAGSMDA9 3.E.020.
MY NOTES
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PRACTICE ANOTHER
 
A bank has $682,000 in assets to allocate among investments in bonds, home mortgages, car loans, and personal loans. Bonds are expected to produce a return of 11%, mortgages 9.5%, car loans 10.5%, and personal loans 14.5%. To make sure the portfolio is not too risky, the bank wants to restrict personal loans to no more than the 25% of the total portfolio. The bank also wants to ensure that at least as much money is invested in mortgages as is invested in personal loans. The bank also wants to invest at least as much in bonds as they do in personal loans. (Let 
X1, X2, X3, and X4
 be the amount (in dollars) invested in bonds, mortgages, car loans, and personal loans, respectively.)
(a)
Formulate an LP model for this problem with the objective of maximizing the expected return (in dollars) on the portfolio.
MAX:
 
 
Subject to:total amount spent
 
 
amount for personal loans
 
 
mortgages and personal loans
 
 
bonds and personal loans
 
 
X1, X2, X3, X4 ≥ 0
(b)
Implement your model in a spreadsheet and solve it. What is the optimal solution?
(X1, X2, X3, X4) = 
 
 
 
 

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