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Herbert J. Bernstein Professor of Mathematics and Computer Science
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90103/93167 MTH 1007 Operations Research -- Fall 2013
On-line Course Quiz 5

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This is a clarified reissue of the fifth quiz for MTH 1007 that was taken on Tuesday, 7 October 2014 after completing the assignment through section 3.2. Use this version to redo the quiz. It should take you between half an hour and two hours 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. You have decided that the only reasonable choices are between a medium-sized factory and a large factory, and in these hard times, demand will either be terrible (5,000 units per year) or weak (12,000 units per year), but you want to be ready for an eventual recovery. 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. The important factor not under your control is the demand for widgets. The probability of terrible demand is 75%. The probability of weak demand is 25%. There is a very good industry analyst, but he is not perfect. The probability that he will say demand will be terrible given the case of actual terrible demand is 75%. The probability that he will say demand will be weak given the case of actual weak demand is 90%. Following the Step 1 of the example on pages 92 -- 93, applying Bayes theorem compute the probability that this analyst will predict terrible demand and the probability that this analyst will predict weak demand.

  2. Following the example in step 2 on page 93 and using your answer to problem 1, compute four more probabilities: the probability of terrible demand given a prediction of terrible demand, the probability of weak demand given a prediction of terrible demand, the probability of terrible demand given a prediction of weak demand and the probability of weak demand given a prediction of weak demand.

  3. Following the example in step 3 and 4 on pages 93, compute the expected profit if you use the imperfect information from the analyst versus the best expected payoff from the two alteratives of the medium and large plant without help from the analyst.

  4. You are trying to choose a reliable insurance company for fire insurance for your factory. There are only two fire insurance companies in your area. 60% of businesses have coverage from company A. 40% of businesses have coverage from company B. 1% of the businesses have coverage from company A and have fire claims that are settled without fuss. 0.2% of the businesses have coverage from company A and have fire claims that are not settled without fuss. 0.8% of businesses have coverage from company B and have fire claims that are settled without fuss. 0.1% of businesses have coverage from company B and have fire claims that are not settled without fuss. Use Bayes's formula to compute the probability of having a fire claim that is not settled without fuss given that your are covered by company A, and to compute the probability of having a fire claim that is not settled without fuss given that you are covered by company B.

  5. Under the assumptions of question 4, what is the probability of having any type of fire claim (settled with or without fuss) given that you are covered by company A, and what is the probability of having any type of fire claim (settled with or without fuss) given that you are covered by company B.

  6. Suppose I have a random number generator that will give me integers ranging from 0 to 99, uniformly distributed. Suppose the probability of selling 0 through 999 widgets is 0.15, the probability of selling 1000 though 4999 widget is 0.40, the probability of seling 5000 through 29999 widgets is 0.25 and the probability of selling 30000 to 59999 widgets is 0.30. Construct as random number conversion table to use the numbers from this random number generator to support simulation using this empirical distribution. In doing this be sure to check if the problem statement is reasonable. If something is wrong, say specifically what, but go ahead with the problem, changing just one of the numbers to a different number so you have a reasonable problem (your choice on which one to change).

  7. Using the table you created for question 6, give the expected sales range to report when the random number generator produces each of the following numbers: 0, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99.

  8. Suppose I am trying to simulate a normal sales distribution with a mean of 20,500 and a standard deviation is 1,500. Suppose I will use a standard normal distribution (what the book calls NRN) that give me the values -4.5, -3, -1, 0, 1.5 and 2.7. What values should my sales simulation use for each of those NRN values.

  9. Suppose the annual sales of widgets are expected to follow the empirical distribution from question 5. As a conservative estimate use the bottom of each sales range when evaluating a sales range. Suppose the fixed costs for the widget factory are $1,000,000. Suppose the unit costs (the variable costs) are normally distributed with a mean of $3,000 per unit and a standard deviation of $300. Suppose the selling price per widget is $12,000. Do a 10 year simulation to determine the expected average annual profit (or loss). Give a complete 10 year simulation table as your answer.

  10. Suppose the annual sales of widgets are expected to follow the empirical distribution from question 5. This time, change your handling of the emprical distribution to use a uniform distribution within each range, i.e. return a sales estimate proportionate to how far into the portion of the random number range you go. Suppose the fixed costs for the widget factory are $1,000,000. Suppose the unit costs (the variable costs) are normally distributed with a mean of $3,000 per unit and a standard deviation of $300. Suppose the selling price per widget is $12,000. Do a 10 year simulation to determine the expected average annual profit (or loss). Give a complete 10 year simulation table with your answer. Compare your results to the ones for the prior question and explain why they are different.

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Revised 19 January 2014