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# Mathematics

Problems need solving by showing all work in word format or excel format (if excel one page per problem)

1. Evergreen Fertilizer Company produces fertilizer. The Company's fixed monthly cost is \$25,000 and its variable cost per pound of fertilizer is \$0.15. Evergreen sells the fertilizer for \$0.40 per pound. Determine the monthly break-even volume for the company.

2. If the maximum operating capacity of the Evergreen Fertilizer Company described in the previous problem is 120,000 pounds of fertilizer per month, determine the break-even volume as a percentage of capacity.

3. The College of Business at Tech is planning an online MBA program. The initial start-up cost for computing equipment, facilities, course development and staff recruitment and development is \$350,000. The College plans to charge tuition of \$18,000 per student per year. However, the university administration will charge the college \$12,000 per student for the first 100 student enrolled each year for administrative costs and its share of the tuition payments.

a) How many students does the college need to enroll in the first year to break even?
b) If the college can enroll 75 students the first year, how much profit will it make?
c) The college believes it can increase tuition to \$24,000, but doing so would reduce the enrollment to 35. Should the college consider doing this?

Probability and Statistics

4. A manufacturing company has 10 machines in continuous operation during a workday. The probability that an individual machine will break down during the day is 0.10. Determine the probability that during any given day, 3 machines will break down.

5. A metropolitan school system consists of three districts - north, south and central. The north district contains 25%, the south district contains 40% and the central district contains 35%. A minimum - competency test was given to all students; 10% of north district students failed. 15% of south district students failed and 5% of the central district student failed.

a) develop a probability tree showing all marginal, conditional and joint probabilities
b) develop a joint probability table
c) What is the probability that a student selected at random failed the test?

6. The weight of bags of fertilizer is normally distributed with a mean of 50 pounds and a standard deviation of 6 pounds. What is the probability that a bag of fertilizer will weigh between 45 and 55 pounds?

Decision Analysis
7. The owner of the Burger Doodle Restaurant is considering two ways to expand operations: to open a drive-up window or serve breakfast. The increase in profits resulting from these proposed expansions depends on whether a competitor opens a franchise down the street. The possible profits from each expansion in operations, given both future competitive situations are shown in the following payoff table:

Competitor
Decision Open Not Open
Drive Up Window \$ - 6,000 \$ 20,000
Breakfast 4,000 8,000

Select the best decision, using the following decision criteria
A. Maximax
B. Maximin

8. A local real estate investor in Orlando is considering three alternatives investments: a motel, a restaurant or a theater. Profits from the motel or restaurant will be affected by the availability of gasoline and the number of tourists; profits from the theater will be relatively stable under any conditions. The following payoff table shows the profit or loss that could result from each investment:

Gasoline Availability
Investment Shortage Stable Supply Surplus
Motel \$ -8,000 \$15,000 \$20,000
Restaurant 2,000 8,000 6,000
Theater 6,000 6,000 5,000

Determine the best investment, using the following decision criteria.

a) Maximax
b) Maximin
c) Minimax regret
d) Hurwicz (alpha = 0.4)
e) Equal likelihood

9. Microcomp manufacturer of personal computers is planning to build a new manufacturing and distribution facility in South Korea, China, Taiwan, the Philippines or Mexico. It will take approximately 5 years to build the necessary infrastructure (roads etc), construct the new facility and put it into operation. The eventual cost of each facility will differ between countries and will even vary within countries, depending on the financial, labor and political climate, including monetary exchange rates. The company has estimated the facility cost (in \$1,000,000s) in each country under three different future economic and political climates, as follows:

Economic/Political Climate
Country Decline Same Improve
South Korea 21.7 19.1 15.2
China 19.0 18.5 17.6
Taiwan 19.2 17.1 14.9
Philippines 22.5 16.8 13.8
Mexico 25.0 21.2 12.5

Determine the best decision, using the following decision criteria.
a) Minimin
b) Minimax
c) Hurwicz (alpha = 0.4)
d) Equal likelihood

Forecasting
10. The manager of Carpet city outlet needs to make an accurate forecast of the demand for Soft Shag carpet (its biggest seller). If the manager does not order enough carpet from the carpet mill, customers will buy their carpet from one of Carpet City's competitors. The Manager has collected the following demand data for the past 8 months:

Month Demand for Soft Shag Carpet (1,000 yd.)
1 8
2 12
3 7
4 9
5 15
6 11
7 10
8 12

a) Compute a 3 month moving average forecast for month 4 through 9
b) Compute a weighted 3 month moving average forecast for months 4 through 9. Assign weights of 0.55, 0.33 and 0.12 to the month in sequence, starting with the most recent month.
c) Compare the two forecasts by using MAD. Which forecasts appears to be more accurate.

11. The Manager of the Petroco service station wants to forecast the demand for unleaded gasoline next month so that the proper number of gallons can be ordered from the distributor. The owner has accumulated the following data on demand for unleaded gasoline from sales during the past 10 months:
Month Gasoline Demanded (gal.)
Oct 800
Nov 725
Dec 630
Jan 500
Feb 645
Mar 690
Apr 730
May 810
Jun 1,200
Jul 980

a) Compute an exponentially smothered forecast, using an value of 0.30.
b) Compute an adjusted exponentially smoothed forecast (with alpha = 0.30 and & beta = 0.20.
c) Compute the two forecasts by using MAPD and indicate which seems to be more accurate.

Simulation - to be solved in Excel format
12. Every time a machine breaks down at the Dynaco Manufacturing Company, either 1, 2 or 3 hours are required to fix it, according to the following probability distribution:

Repair time (hr) Probability
1 0.30
2 0.50
3 0.20
Total 1.00

a) Simulate the repair time for 20 weeks and then compute the average weekly repair time.
b) If the random numbers that are used to simulate breakdowns per week are also used to simulate repair time per breakdown, will the results be affected in any way? Explain.
c) If it costs \$50 per hour to repair a machine when it breaks down (including lost productivity), determine the average weekly breakdown cost.
d) The Company is considering a preventive maintenance program that would alter the probabilities of machine breakdowns per week as shown in the following table:

Machine breakdowns
Per week Probability
0 0.20
1 0.30
2 0.20
3 0.15
4 0.10
5 0.05
Total 1.00
The weekly cost of the preventative maintenance program is \$150. Using simulation, determine whether the company should institute the preventative maintenance program.

#### Solution Summary

This solution is comprised of detailed step-by-step calculations and analysis of the given problems and provides students with a clear perspective of the underlying concepts.

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