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## 22912 MTH 1007 Operations Research -- Spring 2012 On Line Course Quiz 4

This web page is http://www.bernstein-plus-sons.com/.dowling/MTH1007S12/MTH1007_Quiz_4.html

This is the fourth quiz for MTH 1007 to be taken on Friday, 24 February 2012 after completing the assignment. It should take you between half an hour and an hour to answer these questions.

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

1. Due to local zoning laws, you will only be allowed to build one widget factory this year. It costs \$2,500,000 to set up a small factory to produce up to 10,000 widgets per year. The unit cost of producing one widget in the small factory is \$7,500. It costs \$10,000,000 to set up a medium-sized factory to produce up to 50,000 widgets per year. The unit cost of producing one widget in the medium-sized factory is \$6,000. It costs \$20,000,000 to build a big factory to produce up to 250,000 widgets per year. The unit cost of producing one widget in the big factory is \$5000. You can sell widgets for \$10,000 each. An important factor not under your control is the demand for widgets. Demand may be weak (10,000 units per year), medium (25,000 units per year), or strong (50,000 units per year). Prepare a pay-off table showing profit in columns by factory size and in rows by demand.

2. Due to local zoning laws, you will only be allowed to build one widget factory this year. It costs \$2,500,000 to set up a small factory to produce up to 10,000 widgets per year. The unit cost of producing one widget in the small factory is \$7,500. It costs \$10,000,000 to set up a medium-sized factory to produce up to 50,000 widgets per year. The unit cost of producing one widget in the medium-sized factory is \$6,000. It costs \$20,000,000 to build a big factory to produce up to 250,000 widgets per year. The unit cost of producing one widget in the big factory is \$5000. You can sell widgets for \$10,000 each. An important factor not under your control is the demand for widgets. Demand may be weak (10,000 units per year), medium (25,000 units per year), or strong (50,000 units per year). Using the pay-off table showing profit in columns by factory size and in rows by demand from the prior question, apply the game theory minimax strategy of minimizing maximal losses (or maximizing miminal gain) to decide which size factory to build.

3. Due to local zoning laws, you will only be allowed to build one widget factory this year. It costs \$2,500,000 to set up a small factory to produce up to 10,000 widgets per year. The unit cost of producing one widget in the small factory is \$7,500. It costs \$10,000,000 to set up a medium-sized factory to produce up to 50,000 widgets per year. The unit cost of producing one widget in the medium-sized factory is \$6,000. It costs \$20,000,000 to build a big factory to produce up to 250,000 widgets per year. The unit cost of producing one widget in the big factory is \$5000. You can sell widgets for \$10,000 each. An important factor not under your control is the demand for widgets. Demand may be weak (10,000 units per year), medium (25,000 units per year), or strong (50,000 units per year). The probability of weak demand is is 0.4. The probability of medium demand is 0.4. The probability of strong demand is 0.2. Using the expected value of perfect information, what is the most you should be willing to spend for information about what the actual demand will be.

4. State Bayes' Theorem and explain each term you use.

5. 5% of the population has X disease. A screening test accurately detects the disease for 80% of the people who actually do have it. The test also incorrectly indicates the disease for 30% of the people who actually don't have it. Suppose a randomly selected person screened for the disease tests postive. What is the probability that they have the disease?

6. Explain the Monty Hall problem in the case of 4 doors computing specific probabilities.

7. How can you compute the expected value of perfect information? Give a specific example.

8. Martha's Crystal Ball Service (MCBS) will tell you when the stock market is going up and when it is going down. Without Martha's help, you haven't got a clue as to which way the market will go, you will just toss a fair coin to decide. You are going to go to a bookie and bet \$1000 on whether the stock of MCBS will go up or go down tomorrow. If the market is going to go up, you expect to make \$1000 by betting on it going up and to lose \$1000 by betting on it going down. If the market is going down, you expect to make \$2000 by betting on it going down and to lose \$1000 by betting on it going up. The probability that MCBS will give a prediction of a rising market today given a rising market tomorrow is 0.65. The probability that MCBS will give a prediction of a falling market today given a falling market tomorrow is 0.95. What is the expected value of this imperfect information? Show your work.

9. What is a utility function? Give a specific example.

10. You are contemplating getting a master's degree. You can go to a well-respected institution, WRU, for \$50,000. You can go to a somehat less respected institution, LRU, for \$30,000. The probability of completing your degree ar WRU is 0.75. The probability of completeing your degree at LRU is 0.60. The lifetime value of an WRU degree is \$135,000. The lifetime value of an LRU degree is \$85,000. Explain how to decide whether to go to WRU or LRU.

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Revised 12 February 2012