Transcript EMGT 501 Final Exam Due: December 14 (Noon), 2004

EMGT 501
Final Exam
Due: December 14 (Noon), 2004
Note
• Summarize your solutions in a condensed way.
• Send your PPS at [email protected] .
• Answer on PPS that has a series of slides.
• Do not discuss the Final Exam with other
people.
(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.
(a) Develop a decision analysis formulation of this
problem by identifying the alternative actions, states of
nature, and payoff table.
(b) What is the optimal action, given that you decline the
option of seeing a demonstration flip?
(c)Find EVPI.
(d)Calculate the posterior distribution if the
demonstration flip is a tail. Do the same if the flip is a
head.
(e)Determine your optimal policy.
(f) Now suppose that you must pay to see the
demonstration flip. What is the most that you should be
willing to pay?
(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
PD  0  0.4,
PD  2  0.2,
PD  1  0.3,
and PD  3  0.1.
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.
(a) Construct the (one-step) transition matrix for this
Markov chain.
(b) Find the steady-state probabilities of the state of the
Markov chain.
(c) 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 D=0.)
(d) 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.
(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.
(a) Show that this process fits the birth-and-death
process by defining the states, specifying the values
of the n and  n , and then constructing the rate
diagram.
(b) Calculate the Pn .
(c) Calculate L , Lq , W , and Wq .
(d) Determine the proportion of time that the
maintenance person is busy.
(e) Determine the proportion of time that any given
machine is working.
(4) Consider the EOQ model with planed
shortage, as discussed in our class. Suppose,
however, that the constraint S/Q=0.8 is added
to the model. Derive the expression for the
optimal value of Q
(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
1
2
3
4
5
6
Sales
530
546
564
580
598
570
Month
7
8
9
10
11
Sales
614
632
648
670
691
(a) 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.