## (Solved) Problem 1 An alternative has a probability 0.6 of winning \$25,000, 0.2 of winning \$1,000, and 0.2 of losing \$50,000. a) Determine the expected profit...

Â Problem 1Â

An alternative has a probability 0.6 of winning \$25,000, 0.2 of winning \$1,000, and 0.2 of losing \$50,000. a) Determine the expected profit for this alternative.Â

b) Determine the certainty equivalent for the alternative using the utility function , where x is in thousands of dollars. 10()xuxeâˆ’=âˆ’Â

c) Use the equation CE = to determine an approximate certainty equivalent (where is the variance of the lottery and is the risk tolerance corresponding to the utility function in part b). )2/(2ÏÏƒâˆ’x 2Ïƒ Ï (1/)ÏÎ³=Â

Problem 2Â

You have to choose among three subjects, A, B or C, for a final project in an important senior course. You are concerned about the grade (on a scale of 0 to 4), the time it takes to complete the work (2 to 6 weeks), and how much you are going to learn (on a scale of 1 to 10). Subjects A and B are sure to give you outcomes (grade, time, knowledge) of (3.5, 3, 6) on A and (3, 4, 6) on B. With subject C, however, the outcomes are uncertain and you assess that with probability p the outcomes will be (3.5, 2, 2) and with probability (1-p) the outcomes will beÂ

(3, 6, 8). You want to maximize expected grade and learning, but minimize expected time to completion.Â

ASSUME THAT YOU ARE AN EXPECTED VALUE DECISION MAKER. I.E., YOUR OPTIMAL DECISION IS BASED ON EXPECTED ATTRIBUTE LEVELS OR ON EXPECTED WEIGHTED SUM OF ATTRIBUTE LEVELS.Â

a) For what values of p there is no conflict among the attributes in choosing the subject? What are the optimal choice and the corresponding expected attribute levels when there is no conflict.Â

Assume for the following two parts that p=0.25Â

b) If you do not care at all about learning, what will be your optimal choice of subject.Â

c) Let wt and wk denote your constant linear tradeoffs between time and grade and between knowledge and grade, respectively (the units of these constants are grade point per work week for wt and grade point per learning point for wk). Draw the optimal decision regions on a two dimensional graph whose vertical axis is wt and horizontal axis is wk.Â

d) Is the answer in part b) consistent with your graph in part c) ? Explain.Â

Problem 3Â

Susan, the chief engineer of a computer printer manufacturing company, has to select among three different designs of a new copy machine: A, B, or C. She is concerned about the resulting reliability (measured in estimated average number of months between failures), Manufacturing cost (in hundreds of \$) and speed (pages per minute). Designs A and B are sure to give produce outcomes (2, 1.5, 8) and (3, 1.5, 9) respectively, in (reliability, cost, speed). With design C, however, the assessment is that with probability p the actual machine will be (3, 1, 5) and with probability (1-p) it will be (2, 4, 8). Susan wishes to maximize expected reliability, minimize expected cost and maximize machine speed.Â

a) For what range of p is there no conflict among the attributes with regard to the optimal design? What is the optimal design for that range of p?Â

b) Suppose that p=0.9. And assume that the manufacturer has a multiattribute utility function of the form , where wc denotes the manufacturer's linear tradeoff between reliability and cost, and ws denote the manufacturer's linear tradeoffs between reliability and speed. Draw a graph whose axes are wc and ws showing the regions where each of the design alternatives is optimal (if at all)Â

c) In an interview with the analyst that was hired to help Susan with her decision, she has indicated that she would be indifferent between a sure outcome of ( 2, 1, 5) vs. an uncertain outcome with probability 0.3 of (3, 1, 9) andÂ

uncertain outcome with probability 0.5 of (3, 1, 9) and probability 0.5 of (2,4,5). Find the weights wc and ws that reflect Susan's preferences. Given this information, what Design should Susan select when p=0.9?Â

d) Given the result of part c) how much should be Susan willing to incur in extra manufacturing cost in order to increase the average time between failures by one month?

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