AMA542 Advanced Operations Research Methods Assignment 3 | Statistics Homework Help

The Hong Kong Polytechnic University
Department of Applied Mathematics
AMA542 Advanced Operations Research Methods
Assignment 3
Due Date: Apr 15 22:00
1. The number of crimes on each of a city’s three police precincts depends on the number of patrol
cars assigned to each precinct. Three patrol cars are available. Use dynamic programming
to determine how many patrol cars should be assigned to each precinct so as to minimize the
total number of crimes.
No. of patrol cars
0 1 2 3
Precinct 1 14 10 7 4
Precinct 2 25 19 16 14
Precinct 3 20 14 11 8
2. I would like to sell my used car to the highest bidder. Studying the market, I have concluded
that I am likely to receive four types of offers with equal probabilities: very low at about
$1050, low at about $1900, medium at about $2500, and high at about $3000. I decided to
advertise the car for up to 3 consecutive days. At the end of each day, I would decide whether
or not to accept the best offer made that day. What should be my optimal strategy regarding
the acceptance of an offer?
3. Patients arrive at a clinic (with one doctor) according to a Poisson distribution at a rate
of 30 patients per hour. The waiting room does not accommodate more than 14 patients;
arriving patients who cannot get in will leave immediately. Examination time per patient is
exponential with mean rate 20 per hour.
(a) What is the probability that an arriving patient needs not to wait in the waiting room?
(b) What is the expected time until a patient is discharged from the clinic?
4. In an M/M/2 system, mean service time is 5 minutes and mean interarrival time is 8 minutes.
(a) What is the probability of a delay?
(b) What is the probability of at least one of the servers being idle?
(c) What is the probability that both servers are idle?
5. A production line consists of two stations. The product must pass through the two stations
serially. The time the product spends in the first station is constant and equals 30 minutes.
The second station makes an adjustment (and minor changes) and hence its time will depend
on the condition of the item as it is received from station 1. It is estimated that the time
in station 2 is uniform between 5 and 10 minutes. The items are received at station 1 in a
Poisson stream at the rate of one every 40 minutes. Because of the size of the items, a new
unit cannot enter the production line until the one already in the facility clears station 2.
Determine the expected number of items waiting in front of station 1.
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