EMGT 501

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EMGT 501

• Final Exam
• Due December 14 (Noon), 2004

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• Note
• Summarize your solutions in a condensed way.
• Send your PPS at toshi_at_nmt.edu .
• Answer on PPS that has a series of slides.
• Do not discuss the Final Exam with other people.

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(1) You are given the opportunity to guess
whether a coin is fair or two-headed, where the
prior probabilities are 0.5 for each of these
possibilities. If you are correct, you win 5
otherwise, you lose 5. You are also given the
option of seeing a demonstration flip of the coin
before making your guess. You wish to use Bayes
decision rule to maximize expected profit.
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1. Develop a decision analysis formulation of this
   problem by identifying the alternative actions,
   states of nature, and payoff table.
2. What is the optimal action, given that you
   decline the option of seeing a demonstration
   flip?
3. Find EVPI.
4. Calculate the posterior distribution if the
   demonstration flip is a tail. Do the same if the
   flip is a head.
5. Determine your optimal policy.
6. Now suppose that you must pay to see the
   demonstration flip. What is the most that you
   should be willing to pay?

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(2) Consider the following blood inventory
problem facing a hospital. There is need for a
rare blood type, namely, type AB, Rh negative
blood. The demand D (in pints) over any 3-day
period is given by
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Note that the expected demand is 1 pint, since
E(D)0.3(1)0.2(2)0.1(3)1. Suppose that there
are 3 days between deliveries. The hospital
proposes a policy of receiving 1 pint at each
delivery and using the oldest blood first. If
more blood is required than the amount on hand,
an expensive emergency delivery is made. Blood is
discarded if it is still on the shelf after 21
days. Denote the state of the system as the
number of pints on hand just after a delivery.
Thus, because of the discarding policy, the
largest possible state is 7.
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1. Construct the (one-step) transition matrix for
   this Markov chain.
2. Find the steady-state probabilities of the state
   of the Markov chain.
3. Use the results from part (b) to find the
   steady-state probability that a pint of blood
   will need to be discarded during a 3-day period.
   (Hint Because the oldest blood is used first, a
   pint reaches 21 days only if the state was 7 and
   then D0.)
4. Use the results from part (b) to find the
   steady-state probability that an emergency
   delivery will be needed during the 3-day period
   between regular deliveries.

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(3) A maintenance person has the job of keeping
two machines in working order. The amount of time
that a machine works before breaking down has an
exponential distribution with a mean of 10 hours.
The time then spent by the maintenance person to
repair the machine has an exponential
distribution with a mean of 8 hours.
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• Show that this process fits the birth-and-death
  process by defining the states, specifying the
  values of the and , and then
  constructing the rate diagram.
• Calculate the .
• Calculate , , , and .
• Determine the proportion of time that the
  maintenance person is busy.
• Determine the proportion of time that any given
  machine is working.

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(4) Consider the EOQ model with planed shortage,
as discussed in our class. Suppose, however, that
the constraint S/Q0.8 is added to the model.
Derive the expression for the optimal value of Q
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(5) A market leader in the production of heavy
machinery, the Spellman Corporation, recently has
been enjoying a steady increase in the sales of
its new lathe. The sales over the past 11 months
are shown below.
Month Sales Month Sales
1 2 3 4 5 6
530 546 564 580 598 570
7 8 9 10 11
614 632 648 670 691
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• Find a linear regression line that fits the data
  set.
• Use both Least Squares (LS) and Least
  Absolute
• Value (LAV) methods.
• (b) Show the formulation for LAV regression.
• (c) Forecast the amount of sales for the 12th
  month,
• based upon LS regression.