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28) A very confused manager is reading a two-page report given to him by his student intern. "She told
me that she had my problem solved, gave me this, and then said she was off to her production
management course," he whined. "I gave her my best estimates of my on-hand inventories and
requirements to produce, but what if my numbers are slightly off? I recognize the names of our four
models W, X, Y, and Z, but that's about it. Can you figure out what I'm supposed to do and why?" You
take the report from his hands and note that it is the answer report and the sensitivity report from Excel's
solver routine.
Explain each of the highlighted cells in layman's terms and tell the manager what they mean in relation to
his problem.
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Answer: Answer Report
Target Cell Max: The target cell should be maximized, so the manager must have provided the intern
with profit information.
Final Value: The final value is the greatest amount possible for the situation. If we are working with profit
figures, this is the best return possible given what we estimate is on hand and how it is to be produced.
This may change if our inventory or recipes are slightly off. The highest profit identified is $88,888.89
Adjustable Cells: The adjustable cells show that we considered any positive quantity of models W - Z as
possible outputs for the week.
Constraints
Shadow Price: This is the marginal return for having one more unit of each resource. Here we have a
shadow price of $8.88, so if we had one more unit of resource in the first constraint, we could make an
additional $8.88. This gives us an idea of the maximum we would be willing to pay for more of that
resource.
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29) The CZ Jewelry Company produces two products: (1) engagement rings and (2) jeweled watches. The
production process for each is similar in that both require a certain number of hours of diamond work
and a certain number of labor hours in the gold department. Each ring takes four hours of diamond work
and two hours in the gold shop. Each watch requires three hours in diamonds and one hour in the gold
department. There are 240 hours of diamond labor available and 100 hours of gold department time
available for the next month. Each engagement ring sold yields a profit of $9; each watch produced may
be sold for a $10 profit.
a. Give a complete formulation of this problem, including a careful definition of your decision variables.
Let the first decision variable, (X1), deal with rings, the second decision variable, (X2), with watches, the
first constraint with diamonds, and the second constraint with gold.
b. Graph the problem fully in the following space. Label the axes carefully, plot the constraints, shade the
feasibility region, plot at least one isoprofit line that reveals the optimal solution, circle the corner points
and highlight the optimal corner point so found, and solve for it algebraically. (Show all your work to get
credit.)
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Answer:
a.
b.
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30) The Really Big Shoe Company is a manufacturer of basketball shoes and football shoes. Ed Sullivan,
the manager of marketing, must decide the best way to spend advertising resources. Each football team
sponsored requires 120 pairs of shoes. Each basketball team requires 32 pairs of shoes. Football coaches
receive $300,000 for shoe sponsorship and basketball coaches receive $1,000,000. Ed's promotional budget
is $30,000,000. The Really Big Shoe Company has a very limited supply (4 liters or 4,000cc) of flubber, a
rare and costly raw material used only in promotional athletic shoes. Each pair of basketball shoes
requires 3cc of flubber, and each pair of football shoes requires 1cc of flubber. Ed desires to sponsor as
many basketball and football teams as resources allow. However, he has already committed to
sponsoring 19 football teams and wants to keep his promises.
a. Give a linear programming formulation for Ed. Make the variable definitions and constraints line up
with the computer output appended to this exam.
b. Solve the problem graphically, showing constraints, feasible region, and isoprofit lines. Circle the
optimal solution, making sure that the isoprofit lines drawn make clear why you chose this point. (Show
all your calculations for plotting the constraints and isoprofit line on the left to get credit.)
c. Solve algebraically for the corner point on the feasible region.
d. Part of Ed's computer output is shown following. Give a full explanation of the meaning of the three
numbers listed below. Based on your graphical and algebraic analysis, explain why these numbers make
sense. (Hint: He formulated the budget constraint in terms of $000.) See the computer printout that
follows.
First Number: The shadow price of 0.0104 for the "Flubber" constraint.
Second Number: The slack or surplus of 6383.334 for the "Budget" constraint.
Third Number: The lower limit of 12.2807 for the "Commitment" constraint.
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b.
Commitments: X1 > 19 → X1 = 19
Budget: 300X1+ 1,000X2 < 30,000
c.
corner point
120X1 + 96X2 = 4,000
(X1 = 19) × -120
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d.
First Number: The shadow price of 0.0104 for the "Flubber" constraint.
Second Number: The slack or surplus of 6,383.334 for the "Budget" constraint.
Third Number: The lower limit of 12.2807 for the "Commitment" constraint.
31) A small oil company has a refining budget of $200,000 and would like to determine the optimal
production plan for profitability. The following table lists the costs associated with its three products.
Marketing has a budget of $50,000, and the company has 750,000 gallons of crude oil available. Each
gallon of gasoline contributes 14 cents of profits, heating oil provides 10 cents, and plastic resin 30 cents
per unit. The refining process results in a ratio of two units of heating oil for each unit of gasoline
produced. This problem has been modeled as a linear programming problem and solved on the
computer. The set up and output follows:
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a. Give a linear programming formulation for this problem. Make the variable definitions and constraints
line up with the computer output.
b. What product mix maximizes the profit for the company using its limited resources?
c. How much plastic resin is produced if profits are maximized?
d. Give a full explanation of the meaning of the three numbers listed following.
First Number: Slack or surplus of 42,500 for the #2 Marketing Budget constraint.
Second Number: Shadow price of 0 for the #1 Refining Budget constraint.
Third Number: An upper limit of "infinity" for the right-hand-side value for the #1 Refining Budget
constraint.
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Answer:
a.
Let X1 = gallons of gasoline refined
X2 = gallons of heating oil refined
32) Lisa lives out in the country with her seven cats and avoids driving into the big city as much as
possible. She has decided to make her own cat food and has the following nutritional guidelines. Each
four-ounce portion must contain 22 units of protein, 15 units of vitamin A, and 8 units of vitamin B. She
has eggs, tomatoes, and chicken meat as possible inputs to her cat food. Each ounce of eggs contains 6
units of protein, 4 units of Vitamin A, and 3 units of Vitamin B. Each ounce of tomatoes contains 1 unit of
protein, 8 units of Vitamin A, and 14 units of Vitamin B. Each ounce of chicken contains 22 units of
protein, 14 units of Vitamin A, and 8 units of Vitamin B. Chicken costs 40 cents per ounce, tomatoes cost 5
cents per ounce, and eggs cost 12 cents per ounce. To make the production process as easy as possible,
she would like to make exactly four ounces of cat food from her recipe. She used POM for Windows and
received the following results. Provide an interpretation.
