1. (WA828/30) Read this problem very carefully. Do whatever analysis you think necessary and then write a short defense for the worker. Include your (computer) analysis in your answer. HINT – The issue is: DID the worker slack off (work slower)? You have to justify why you can use the model(s) you choose! Use the templates (if you want)!
A worker at the State Unemployment Office is responsible for processing a company’s forms when it opens for business. The worker can process an average of 6 forms per work. In 2012, an average of 4.0 companies per week submitted forms for processing, and the worker had a backlog of 0.50 week. . Backlog in this problem is defined as the average time that a customer (i.e. company’s forms) remains at the State Unemployment Office. In 2013, an average of 5.8
companies per week submitted forms for processing, and the worker had a 2.5 week backlog. The poor worker was fired but later sued to get her job back. The court said that because the amount of work submitted to the worker had increased by less than 50%, the backlog should not have been 5 times larger. She must have been slacking off, so the state was justified in firing her. Use queueing theory to defend the worker. (This is based on an actual case!)
2. A young couple has $28,000 (90% of their savings) to invest in either savings bonds or a real estate deal. The (zero coupon) savings bonds return $35,000 ($7,000 interest) in three years. The (completely liquid) real estate investment, after three years, is worth $66,000 if economic conditions are good (70% chance), and worth nothing ($0) if economic conditions are bad (30%). The couple decides to invest in the savings bonds.
a. What do you know about the certainty equivalent (for the couple) of the real estate investment? HINT – Make sure you understand the concept of certainty equivalence.
b. What would you do in these circumstances?
c. Give me an example of a different set of probabilities that would change your decision in b”.
3. (WA 339.94) With a little thought you should be able to “eyeball” the solution. However the question asks to develop a linear integer model to solve the problem. This can be set up in less than 10 minutes. INCLUDE ALL CONSTRAINTS.
An aged merchant of Baghdad was much respected by all who knew him. He had three sons, and it was a rule of his life to treat them equally. Whenever one son received a present, the other two each received a present of equal value. One day this worthy man fell sick and died, bequeathing all of his possessions to his three sons in equal shares. The only difficulty that arose was over the stock of honey. There were exactly 21 barrels. The old man left instructions that each son should not only receive an equal quantity of honey, but each son should receive exactly the same number of barrels, and no honey should be transferred from barrel to barrel on account of the waste involved. Now, as 7 of these barrels were full of honey, 7 were half full, and 7 were empty, this was quite a puzzle, especially because each brother objected to taking more than 4 barrels of the same description (full, half full, or empty). Develop an integer model to solve this puzzle.
4. A robbery has just been committed at the Corner Market in the downtown area of the city. The market owner was able to activate the alarm, and the robber fled on foot. Police officers arrived a few minutes later and asked the owner, “How long ago did the robber leave?” “He left about 9 minutes ago and headed east.”, the owner responded. “He’s probably nine blocks away by now.”, one of the officers said to the other. “Not likely”, said the storeowner. “He was so stoned on drugs that I bet he’s still only within a few blocks of here! He’s probably just running in circles!”
Perform a simulation experiment that will test the storeowner’s hypothesis. Assume the following:
• It takes him 1 minute to walk each block (they are short).
• At the end of each minute interval he is at a corner. There is a 28% chance that he will fall asleep for 1 minute, and an 18% chance that he walk east, or north, or west, or south.
My suggestion is to run at least 1000 trials of the nine-minute journey of the robber.
a. What is the probability he is two blocks or less (walking on streets) from the store?
b. On average how many block has he walked?
c. On average how far away is he (using blocks-no “diagonals”) from the store?
5. In publishing a textbook, the following simplified sequence of activities applies, as shown below:
a) Construct the (PERT/CPM) network representation for this PROJECT. (This problem may also be easier if done with paper and pencil rather than Excel.)
(i) How long should the project take from start to finish?
(ii) What activities are on the critical path
b) If the project is well managed, it is possible to produce a book in 20 months or less without crashing any activities. This seems to contradict the findings in “a”. (Please give short answers to this question.)
(i) How do you think publishers get the job done so quickly?
(ii) Is PERT/CPM applicable to publishing?
6. The objective of this question is to learn something about the forecasting methods presented in class and in the book (Forecasting Chapter). Read the original problems; then answer the questions (A-D) below: (You can use the forecasting templates but you will learn a lot more by constructing the spreadsheets to do the analysis by yourself.)
Original Problems (13.31 edition 2, 10.30 edition 3, 10.32 edition 4)
A. After considering seasonal effects, compare both the MAD and MSE values for the
(NOTE – both years of data are given above, use both years to answer questions.)
1. the last-value method (moving average with 1 month)
2. the moving-average method (based on the most recent three months)
3. the exponential smoothing method (with an initial seasonally adjusted estimate of 80 and a smoothing constant of α = 0.2)
4. the exponential smoothing method with trend (with an initial seasonally adjusted estimate of 80, an initial 0 trend, and smoothing constants of α = 0.2, β = 0.2)
when they are applied retrospectively to the past two years.
B. Based on these results, which forecasting method would you recommend that Transcontinental Airlines use in the future
C. Use your method from B to forecast the average number of passengers in the next January.
7. A firm has just hired eight new employees and would like to determine how to allocate their time to four activities. The firm has prepared the following table, which gives the estimated profit for each activity as a function of the number of new employees allocated to it.
a. Use dynamic programming to determine the optimal allocation of new employees to the activities. (Zero credit will be given for eyeball or linear programming solutions.)
b. Suppose only six new employees were hired. What activities would you assign to these employees?
8. Javier Pena has always been interested in financing artistic projects. He has recently been offered two financing opportunities in the fashion industry: financing a new line of avant-garde youth fashions designed by Jorge Vera, and financing a line of business attire designed by Paolo Ricci. Javier has had a lot of past experience with these two designers, and has observed that 20% of Vera’s fashion lines are “hits” and 80% are “misses”. Furthermore, Ricci’s fashion lines are “hits” 30% of the time, and are “misses” 70% of the time.
Javier’s net liquid assets amount to $750,000. As a result he can afford to finance at most one of the two fashion lines. However he does have the option of pre-testing at most one of the fashion lines at the upcoming design show in San Francisco, before deciding which, if any, fashion line he would like to finance for the entire U.S. market for the fall fashion season. The costs and revenue associated with the two fashion lines are given below:
Javier has observed, based on previous years, that of the avant-garde fashion lines that were hits nationwide, 80% were hits in the San Francisco pre-test; of the avant-garde fashion lines that were misses nationwide, 40% were hits in the San Francisco pre-test. Of the business attire fashion lines that were hits nationwide, 90% were hits in the San Francisco pre-test; of the business attire fashion lines that were misses nationwide, 60% were hits in the San Francisco pre-test. While Javier may find pre-test results useful, he knows the accuracy of this kind of test is not high enough to compel him in all cases to act in accordance with the pre-test results. In any event, Javier is willing to act on the basis of expected monetary values (i.e. maximize expected profit).
You have just been hired by Javier to help him analyze his problem so that he can decide what actions to take. He expects a relatively short report of your findings! Include:
a. What should Javier do in all possible situations?
b. How much money will he make on average?
c. What is Javier’s potential maximum loss?
Suggestion – think before you set up the problem
Note that this is a complicated problem. THINK and then do your analysis! Make sure that you include all possible actions. The sequencing of events is important. The cost structure (accounting) is a little complicated. The probabilities can get confusing.
Your grade on this question will depend on both the:
• quality of the analysis
• clarity of the written report
PLEASE SUBMIT A DETACHED 2nd COPY OF YOUR REPORT (Q8) WITHOUT YOUR NAME.