UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, DECEMBER 2018
MIE566H1S - DECISION ANALYSIS Exam Type: A, Calculator Type: 2
Examiner: D M. Frances
Write ALL your work and results in the appropriate section of the QR coded sheets Create and work with files on your Desktop and save your work often.
Refer to S:\courses\M1E566\source for useful information Submit files to S:\courses\M1E566\submit with drag-and-drop.
Before lo-off see the TA to ensure proper submission, and initial the list
Question 1
-40 marks
In the magic world, there's a popular website called Potterless, where wizards and witches can enjoy a variety of TV shows. The production team at Potterless is considering to produce a new show,
Aihus C'rindelwald and the Godric's Hollow. It would cost them 820.000 to prepare and film the
show. Their advertising revenue depends on how many users watch the show, (audience rating). If more than 50% of the users watch the show, they will get $50,000 advertising fee: if 20% to 50% of the users watch the show, they get 830.000: if less than 20% of the users watch the show, they only get $15,000. Instead of producing the show, they canalso choose to sell their idea to Mundungus, a freelancer, for $8,000.
The team thinks that there is a 15% chance thatA50% of users would watch the show and 20% chance that less than 20% of users would watch it. Before making the decision whether to produce or sell the show, the team can choose to do an audience survey, which costs $2,000. If the show has a more than 50% rating, there is a 70% chance that the sample audience will say they like the show. If the show has a rating between 20% and 50%, there is a 40% chance that the sample audience will like it. If the show has a rating that is less than 20%, there is 75% chance that the sample audience will not like it. (5 marks) Show the calculation of the posterior probability for audience rating using Bayes' rule. (5 marks) Solve the problem (use any method that YOU prefer) and answer the question: what
should be their decision on the show (produce or sell)?
(15 marks) If the team believes that the probability p that a user would watch the show has a prior beta distribution and the survey result for whether audience likes the show and decides
to watch it follows the binomial distribution. Show that the Beta distribution is conjugate to the Binomial distribution, and the updated Beta parameters are:
= + n±
= 3O + ni - nx
d) (15 marks) Suppose that producing the TV show still costs $20,000 and selling the idea makes $8,000. The production team can receive $1,000 advertising fee for each 10,000 views (we assume that if one user watches the show, it counts as one view). They are unsure about the probability that the audience would watch the show on the website. From past experience, the team believes that the audience rating probability (denoted as p) has a prior beta distribution with a = 30 and b = 70. The result from the audience survey (costs $2,000) is normally distributed with mean p. = p
and precision r
= (0 2)2. Assume that the website has 1 million users. What's the team's optimal
decision if the survey result says that 60% of the audience likes the show. (Use Open BUGS to solve the problem.)
Question 2
-50 marks
The Gardiner Expressway is well past its designed lifespan and Toronto City Council needs to decide on its future. A committee consisting of urban planners and engineers have suggested the follow-ing alternatives: make extensive repairs to existfollow-ing highway structures: replace the structures with a tunnel: or demolish the highway without replacing it. (instead turning the highway into a large boulevard). The committee estimates that repairing the highway will cost $850 million; replacing the highway will cost $1050 million; and demolishing the highway will cost $950 million. All options would take 10 years to complete. The repairing option includes a possible construction delay which is exponentially distributed with scale = 0.8 years, but is unlikely to be greater than 5 years.
Some city councillors have suggested implementing a road toll of $2 per vehicle-trip to help finance costs of construction. A preliminary study showed that this stretch of road would likely have 25 million to 35 million vehicle-trips (uniformly distributed) over its lifespan. Assume there is no cost associated with installing a toll system.
City Council is also concerned about their approval rating, which is currently at. 65 points. Council members believe that the approval rating would decrease by 20 to 30 points uniformly distributed if a toll is implemented. Due to the complexity of construction, the repair option comes with a fixed 10-point drop in approval, while the rebuild option has a fixed 5-point, drop in approval rating. Council also believes the local newspaper could potentially publish a "hit-piece", or an ar-ticle accusing them of mismanaging the project, which would cause a fixed 15-point drop in approval. Additionally, City Council is concerned with the possibility of a construction-related accident. If an accident does occur, assume that construction will take 3 additional years, approval will decrease by 10 points, and that monetary compensation of $200 million will have to be paid.
City Council has the following utility functions:
ucost = 1
cost in millions maximum cost outcome
approval in % Ua ppr ovai =
65
= (0.5xtimein years)-10 - C'5i' •.I LCL<) -- 1c)
(5 points) Assume that the probability of an accident occurring is fixed at 3%. If City Council is only concerned with cost, what is the optimal decision set?
(5 points) b) Assume that the probability of an accident occurring is uniformly distributed between 0% and 6% and the probability of a hit-piece being published is uniformly distributed be-tween 0% and 5%. If City Council is only concerned about their approval rating, what is the optimal decision set?
(15 points) () Suppose that City Council has the following indifference points:
Certainty Equivalent Gamble
($780 million, 0% approval, 18 years) 30% ($780 million, 65% approval, 10 years) 70% ($ L
2
.00- million, 0% approval, 18 years)($ million, 65% approval, 18 years) 45% ($780 ui11ion, 65% approval. 10 years) . 5G% ($106 million, 0510 approval, 18 years) ($161& million, 0% approval, ig years) 70% ($780 million, 65% approval, 10 years)
Rs~
30% ($140mil1ion. 0% approval. 18 years)What is the multi-attribute utility function?
(10 points) Considering the multi-attribute utility function, what is the optimal decision set? (15 points) You overheard a conversation between two traffic engineers. One said that 27 million vehicle-trips were recorded in the most recent traffic flow study, and the other said that 1 in the last 20 construction projects had a major accident. Knowing this information and the multi-attribute utility function, what is the optimal decision set?
Some useful information:
step(x) returns 1 if x > 0, and 0 otherwise exp(x) = CX pow(x,y) = XY Distribution PDF Mean Uniform .f(x) = Exponential f(r) = Gamma f(x) = 177) 3
Question 3
-10 marks
Which is the best answer for each of the following multiple choice questions? 1. In the 1960's there were
an abundance of Bayesian applications a fair number of Bayesian applications a paltry number of Bayesian applications almost no Bayesian applications
2. Frequentists make statements like "there is a 0.92 probability that p > 0.25" all the time
sometimes never it depends
3. Conjugate distributions were introduced by Howard Raiffa and Robert Schlaiffer Jimmy Savage and Bruno de Finetti Robert Schlaiffer and John Pratt None of the above
4. The Federalist Papers authorship problem is a classification problem
regression problem an inference problem a decision problem
5. The biggest surprise lesson from the Federalist Papers project was that it took a decade to complete
they had to develop new theories and techniques prior distributions were of no major importance they had to check their data using slide rules
6. Which of the following projects publicly acknowledged the use of Bayes predicting who would win the US elections in 1960
search for lost H-bomb off the Spanish coast in 1966 government nuclear power station safety in 1974 industry nuclear power station safety in 1981
7. Conjugate distributions were used while
helping Egypt clear the Suez canal of unexploded bombs in 1975 find the lost Scorpion submarine in 1968
locate a sunken tuna fishing boat off Long Beach in 1974 locate survivors off a capsized boat in the Pacific in 1976
8. Which of the following events was assigned a probability using Bayesian Analysis Nuclear accident at 5-mile Island
Rocket boost failure of the Challenger in 1986 Soviet submarine in the west Mediterranean in 1976 all of the above
9. Bayesian methods were found to be better than frequentist methods for conjugate distributed data
data involving abrupt changes exponentially distributed data gradually changing data
10. The same Bayesian methods that were used when looking for the Scorpion submarine in 1968 (without credit to Bayes) were used in 2011 (with credit to Bayes) to find
stranded skiers in the Alps sunken submarine near Hawaii a lost flight in the Atlantic none of the above
I
