INDE 311 Winter 2024 Take-Home Midterm Exam Guide

Name:_________________________ INDE 311 Winter 2024 Midterm Exam (Take-home) Instructions: you are allowed to use lecture notes as well as any other materials on UW Canvas course website, and the course textbook. You may use a calculator, and IOR tutorial/Excel to calculate your solutions (please indicate your use on the exam question and upload your Excel file). All answers and computations must be your own work without communicating with other students nor receiving assistance from others. The minimum penalty for an act of academic dishonesty will be the assignment of a grade of 0 on this exam. Please sign or type your name below that you have acknowledged this honor statement. The exam has four questions, and you have from 10:00 AM to 2:00 PM to solve them. Please show all intermediate steps and box your final answers for each question. Please leave sufficient time for scanning your solutions and uploading them to UW Canvas (in Word, pdf or pictures format). No late midterm exams will be accepted after 2 PM on Feb 14 th , 2024. I have neither given nor received aid on this exam. Signature: ________________________________

INDE 311 Midterm Exam Name __________________________________ 2

INDE 311 Midterm Exam Name __________________________________ 3 1. (30 points) University Bookstore has developed a new product line – a series of books written by engineering professors featuring their cutting-edge research. Management needs to decide whether to produce and market these books. One option is to immediately ramp up production and simultaneously launch an ad campaign. This option would cost $1,000. Based on experience, new book series either take off and do well or fail miserably. Hence, the prediction is for one of two possible outcomes – total sales of 2,500 units or total sales of only 250 units. University Bookstore receives a profit of $2 per unit sold. Management currently thinks that there is about a 40% chance that the production will do well (sell 2,500 units) and a 60% chance that it will do poorly (sell 250 units). Another option is to test market the product locally. The company could print a few books, put up a display in the campus bookstore, and see how they sell without any further advertising. This would require less capital for the production run and no money for advertising. The test market has two possible outcomes, sell 200 units (sell well), or only sell 20 units (sell poorly). The cost for this option is estimated to be $100. University Bookstore receives a profit of $2 per unit sold for the test market as well. The company has often test marketed products in this manner. Products that sell well when fully marketed have also sold well in the test market 80% of the time. Products that sell poorly when fully marketed also sell poorly in the test market 60% of the time. (a) Develop a decision analysis payoff table for immediately ramping up production and launching an ad campaign. Please clearly indicate the decision alternatives, the states of nature, the payoffs (profits) and prior probabilities.

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INDE 311 Midterm Exam Name __________________________________ 4 (b) Recommend a course of action based on 3 different criteria (explain your reasoning) based on the payoffs table from part a). • Maximin payoff • Maximum likelihood • Bayes’ decision rule

INDE 311 Midterm Exam Name __________________________________ 5 (c) Calculate all the posterior probabilities after marketing the product locally using Bayes’ Theorem.

INDE 311 Midterm Exam Name __________________________________ 6 (d) Construct the decision tree and use it to determine the optimal course of action for University Bookstore. [Hint: don ’ t forget to include the profits from local marketing in the branch payoffs]

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INDE 311 Midterm Exam Name __________________________________ 7 (e) Find EVPI & EVE. What is the maximum amount of money University Bookstore should be willing to pay to test market the new books?

INDE 311 Midterm Exam Name __________________________________ 8 2. (20 points) Consider the following three lotteries Lottery 1: Alternative 1: 45% chance of earning $1200 and 55% chance of earning nothing; Alternative 2: earning $40 for sure Lottery 2: Alternative 1: 65% chance of earning $1200 and 35% chance of earning $40; Alternative 2: earning $300 for sure Lottery 3: Alternative 1: 80% chance of earning $1200 and 20% chance of earning $300; Alternative 2: earning $600 for sure a) (10 points) Suppose a decision maker has thus far assigned utilities as follows: U(1200)=1, U(0)=0. Suppose a decision maker is indifferent between the two alternatives offered in each of the above three lotteries. Use this information to compute U(40), U(300), U(600).

INDE 311 Midterm Exam Name __________________________________ 9 b) (10 points) Choose between alternative A1 and A2 in the decision tree by maximizing expected utility.

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INDE 311 Midterm Exam Name __________________________________ 10 3. (25 points) Answer the following short-answer questions. a) (5 points) Suppose {X n } is a Markov chain with transition matrix P. Further, suppose that state i leads to state j, but that state j does not lead to state i. True or false : State j must be transient. If true, explain why; if false, provide a counterexample. b) (5 points) Suppose in a given Markov chain, states i and j communicate with each other. Is it always true that lim 𝑛→∞ P (X n = j|X 0 = i) > 0? If true, justify your answer; if not, provide a counterexample.

INDE 311 Midterm Exam Name __________________________________ 11 c) (15 points) Consider the Markov Chain below. a. (3 points) List all classes in this Markov Chain. b. (3 points) Identify recurrent and transient classes. c. (3 points) Identify the period of each class. d. (3 points) What are the values of the following limits: lim 𝑛→∞ 𝑝 67 (𝑛) when n is odd = lim 𝑛→∞ 𝑝 67 (𝑛) when n is even = lim 𝑛→∞ 𝑝 66 (𝑛) when n is odd = lim 𝑛→∞ 𝑝 66 (𝑛) when n is even = e. (3 points) Assume 𝜋 1 = 0.462, 𝜋 2 = 0.462, 𝜋 3 = 0.077, what are the values of the following limits: lim 𝑛→∞ 𝑝 11 (𝑛) = lim 𝑛→∞ 𝑝 12 (𝑛) = lim 𝑛→∞ 𝑝 13 (𝑛) =

INDE 311 Midterm Exam Name __________________________________ 12 4. (25 points) A salesman travels to four different cities that are located at the vertices of the unit square (i.e., the square with vertices (0, 0), (0, 1), (1, 0), (1, 1)). At each time step, the salesman can jump to one of the two adjacent vertices. He jumps vertically with probability p, and horizontally with probability q. In all cities except (0, 0), with probability r the salesman stays in the same city in the next time step. Furthermore, in (0, 0) (his home office), with probability r the salesman takes a vacation in the next time step at a location outside of the unit square. (Here p + q + r = 1). Once on vacation, the salesman stays on vacation each time step with probability a, and otherwise returns to work at his home office. a) (8 points) Describe the movement of the salesman as a Markov chain. Draw a state transition diagram and define the one-step transition probability matrix.

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INDE 311 Midterm Exam Name __________________________________ 13 b) (8 points) Given r=0.1, q=0.5, p=0.4, and a=0.05, what is the expected number of time steps between consecutive visits to the home office. (Note: you are free to use IOR Tutorial, Excel or other calculators. Please indicate what software/calculator you used so we can understand how you arrived at the answer).

INDE 311 Midterm Exam Name __________________________________ 14 c) (9 points) Suppose the salesman is currently on vacation. Find the probability that, once the salesman returns to work, he goes on vacation again without ever visiting (1, 1). The final answer should be a function of r, p, and q. [Hint: to solve this problem, you may need to modify the Markov chain to add in some absorbing states, then think about absorption probabilities and the idea of conditioning on the first transition.]

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