6.1. ABC Inc. must make a decision on its current capacity for next year. ABC is
ID: 1721272 • Letter: 6
Question
6.1. ABC Inc. must make a decision on its current capacity for next year. ABC is considering three alternatives. For each alternative, estimated profits (in $000s) based on next year’s demand are shown in the table below. ABC Inc. estimates that the probability of high demand is 0.7 and the probability of low demand is 0.3. Alternative Next Year’s Demand High Low A1: Expand $192 $97 A2: Subcontract $175 $125 A3: Do Nothing $147 $120 [Tip: Do you notice that A2 dominates A3? Because A3 is worse than A2 under both demand scenarios, A3 will never be chosen and we can eliminate A3 from consideration. In other words, we only need to include A1 and A2 in the analysis.] Suppose there is a clairvoyant who can predict the demand perfectly (the clairvoyant’s prediction is always correct). You want to know how much this clairvoyant’s prediction worth (i.e., expected value of perfect information; part c of the problem). In order to do so, you need to first calculate parts a and b.
6.1.a (0.3 pt) Calculate the expected value with perfect information. (Enter your answer in 000s. In other words, calculate the answer with the numbers in the table directly.)
6.1.b (0.3 pt) Calculate the expected value without perfect information. (Enter your answer in 000s.)
6.1.c (0.3 pt) Calculate the expected value of perfect information. (Enter your answer in 000s.)
Now, suppose a clairvoyant’s prediction is not available. Instead, there is an economic consultant who offers to provide his insights regarding whether next year’s demand is 1 BUS2 190 Quantitative Business Analysis, Spring 2016 going to be high or low. Based on this consultant’s track records, his prediction is correct 90% of the time for high demand years, and correct 70% of the time for low demand years. In other words, P(predicts a high demand|high demand year) = 0.9 P(predicts a low demand|low demand year) = 0.7 ABC wants you to calculate how much this consultant’s information (his demand prediction for the next year) worths. In order to do so, you need to obtain the following information. [Hint: Draw a decision tree. Be careful about the sequence of the decision and chance nodes. Label the nodes’ branches and relevant quantities (probabilities or payoffs). The probability for a branch of an uncertain event that occurs after a preceding event is conditioned on the occurrence of the preceding event. To help you go through the process, I have provided detailed calculation for relevant probabilities and expected values in the tree (next page) for this problem.
6.1.d What is the probability that this consultant predicts a high demand year? [Hint: Law of total probability.] Answer 6.1.d. P(predicts a high demand) = P(predicts a high demand ? high demand year) +P(predicts a high demand ? low demand year) = P(predicts a high demand|high demand year) × P(high demand year) +P(predicts a high demand|low demand year)P(low demand year) = 0.9 × 0.7 + 0.3 × 0.3 = 0.72
6.1.e What is the probability that this consultant predicts a low demand year? [Hint: Use rule of complement and the answer in part a.] Answer 6.1.e. P(predicts a low demand) = 1 ? 0.72 = 0.28.
6.1.f Given that the consultant predicts that next year has a high demand, what is the probability that next year’s demand is high? [Hint: Bayes’ Theorem.] Answer 6.1.f . P(high demand year|predicts a high demand) = P(high demand year ? predicts a high demand) P(predicts a high demand) = P(predicts a high demand|high demand year)P(high demand year) P(predicts a high demand) = (0.9)(0.7) 0.72 = 0.875
6.1.g Given that the consultant predicts that next year has a high demand, what is the probability that next year’s demand is low? [Hint: Rule of complement and the answer in part c.] Answer 6.1.g. P(low demand year|predicts a high demand) = 1 ? 0.875 = 0.125.
6.1.h Given that the consultant predicts that next year has a low demand, what is the probability that next year’s demand is high? [Hint: Bayes’ Theorem.] 4 BUS2 190 Quantitative Business Analysis, Spring 2016 Answer 6.1.h. P(high demand year|predicts a low demand) = P(high demand year ? predicts a low demand) P(predicts a low demand) = P(predicts a low demand|high demand year)P(high demand year) P(predicts a low demand) = (0.1)(0.7) 0.28 ? 0.25 (1)
6.1.i Given that the consultant predicts that next year has a low demand, what is the probability that next year’s demand is low? [Hint: Rule of complement and the answer in part e.] Answer 6.1.i. P(low demand year|predicts a low demand) = 1 ? 0.25 = 0.75
6.1.j (0.3 pt) Calculate the expected value with the consultant’s information (i.e. Expected Value with Sample Information.). Based on the decision tree, the expected value with sample information is 168.19.
JUST NEED QUESTIONS J&K
high demand year A1: Expand low demand year high demand year A2: Subcontract Al: Expand (6.18 0.72 predicts high demand 6.3.d A2: Subcontract 168.75 (6.18 168 19 A1: Expand 6.1 predicts low demand 37.5 37.5 (6.1 123Explanation / Answer
Hope you have answers for the other parts!
Answer 6.1.j. The expected value with the consultant’s information is the expected value when choosing optimally based on the information provided by the consultant. Based on the decision tree, the expected value with sample information is 168.19.
Answer 6.1.k. The maximum worth of the consultant’s prediction is the dierence between the expected value with the consultant’s information and the expected value without the consultant’s information. Therefore, the answer is 168.19 - 163.5 = 4.69.
Hope this helps!!
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