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## 90103 MTH 1007 Operations Research -- Fall 2013 On Line Course Quiz 6

This web page is http://www.bernstein-plus-sons.com/.dowling/MTH1007F13/MTH1007_Quiz_6.html

This is the sixth quiz for MTH 1007 to be taken on Tuesday, 22 October 2013 after completing the assignment. It should take you between half an hour and two hours to answer these questions.

You will need to generate random numbers for this quiz. When you use the tables of random numbers in the book on pages 107 and 109, that is your source of random numbers. When you spin a pencil, that is your source of random numbers. When you use the Excel functions RAND or RANDBETWEEN, that is your source of random numbers.

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Please fill in the following information:

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Please answer the following questions on this form (or on a paper copy of this form).

1. You have a widget factory that cost you \$10,000,000 to build, has a unit cost of \$5,000 per widget and a unit revenue of \$10,000 per widget. There is a 10% chance of selling no widgets. There is a 10% chance of selling 10,000 widgets, a 10% chance of 20,000 widgets, a 10% chance of selling 30,000 widgets, a 10% chance of selling 40,000 widgets, a 10% chance of selling 50,000 widgets, a 10% chance of selling 60,000 widgets, a 10% chance of selling 70,000 widgets, a 10% chance of selling 80,000 widgets, and a 10% chance of selling 90,000 widgets. Use the table of random digits from 0-9 on page 107 as follows: take whatever digit you get next and multiply it by 10,000 and use that as your next estimate of sales. Write a simulation table of 20 rows using those sales estimates as the input and giving the profit as the output. Then compute the mean, standard deviation, 5th and 95th percentiles of sales and of profits.

2. You have a widget factory that cost you \$10,000,000 to build, has a unit cost of \$5,000 per widget and a unit revenue of \$10,000 per widget. There is a 50% probability that you will sell 1000 widgets and a 50% probability that you will sell 10000 widgets. Use a coin toss or spin a pencil to generate your sales estimates. Write a simulation table of 20 rows using those sales estimates as the inout and giving the profit as output. Then compute the mean, standard deviation, 5th and 95th percentiles of sales and of profits. For extra credit, comment on the meaning of the percentiles.

3. You are trying to plan for retirement. You have a choice of fixed rate investments that return a real annual rate of return normally distributed with a mean of 2% and a standard deviation of 5%, or equity investments that return a real annual rate of return normally distributed with a mean of 11% and a standard deviation of 25%. When you run your simulations, you are to take the total value of what you have at the end of each year and re-invest it for the next year. Your simulations will run for 40 years. You are to run each 40 year simulation at least 20 times and then collect statistics on the mean final value of your investment and the standard deviation of the final value of your investment for each of three scenarios: Investing \$20,000 entirely in fixed rate of return, versus investing \$10,000 in fixed rate and \$10,000 in equities, versus investing \$20,000 in equities.

4. Now take the simulation for question 2 and redo it investing the \$20,000 in equities for 10 years, then in year 11 changing to 3/4 in equities and 1/4 in fixed rate through year 20, then in year 21 changing to 1/2 in equities and 1/2 in fixed rate through year 30, then in year 31 changing to 1/4 in equities and 3/4 in fixed rate through year 40. Compare the results of this scenario, compare it to the 3 scenarios from question 2 in terms is return and risk.

5. You have a production line which takes product through a sequence of 3 work stations. Each work station has to wait until the next workstation is done before it can work on its next item. Each station has an approximately normally distributed processing time of 10 minutes with a standard deviation of 3 minutes, but the negative tail is cut off becuase the processing time is never less than 3 minutes. Run a simulation to estimate the mean time to get through all three work stations and estimate the standard deviation. Show your work.

6. Redo question 5 assuming unform distribution of processing time for each stations from 7 minutes to 13 minutes. Do at least 50 passes. Show your work. Compare your results with the normal distribution and comment on the differences, if any.

7. The United Brick company produces pavers. Each paver has to be at least 3.95 inches wide, 7.95 inches long and 1.49 inches thick, and no more than 4.05 inches wide, 8.05 inches long and 1.51 inches thick or it has to be discarded. They have an automated factory that produces pavers with a normally distributed width of mean 4 inches and a standard deviation of .05 inches, a normally distributed length of mean 8 inches and a standard deviation of .05 inches, and a normall distributed thickness of mean 1.5 inches with a standard deviation of .01 inches. They have a manual production facility that produces pavers with a uniformly distributed width from 3.94 inches to 4.06 inches, a uniformly distributed length from 7.94 inches to 8.06 inches, and a uniformly distributed thickness from 1.487 to 1.513 inches. Run simulations on both facilities for 500 pavers and estimate the percentage of pavers that have to be discarded from each. Do a second estimate of the percentage of pavers that stay within middle half of the permitted band.

8. Simulate a pair of dice. Each die has six faces numbered 1 through 6, allowing results for the pair from 2 (snake-eyes) through 7 (a natural) through 11 (another natural) to 12 (box-cars). Run you simulation an estimate the probability of either snake-eyes of box cars and the probability of 7 or 11. Show your work.

9. Explain, step-by-step, how to create an Excel Data Table.

10. Explain, step-by-step, how to use the Calc Multiple Operations feature.

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Revised 7 October 2013