1. You are the maintenance engineer for your company. You are just about to leave on your vacation for two weeks, and your boss is concerned about certain machines that have been somewhat unreliable, requiring your expertise to keep them running. Your boss has asked you how many of these machines you expect to fail while you are out of town. You have made the following subjective judgments:

1. There is a 0.5 chance that none of the machine will fail;
2. There is an approximately 0.15 chance that two or more will fail; and
3. There is virtually no chance that four or more will fail.

Being impatient with this slow assessment procedure, you decide to try to fit a theoretical distribution.

1. Many operations researcher would use a Poisson distribution in this case. Why might the Poisson be appropriate in this case?
2. Find a Poisson distribution that provides a good representation of your assessed beliefs. Give a specific value for the parameter m. According to the distribution, what is the expected number of machines that will break down during your absence? (Hint: Look in the table for the Poisson distribution)
3. Looking at the information you provide, your boss, however, thinks you are
   somewhat optimistic, so he decides to assign a Poisson distribution with m=1 to the occurrence of machine breakdowns during your two-week vacation. Now the boss has a decision to make. He can either close the plant involving the machines in question at cost of $10,000, or he can leave that plant up and running. Of course, if there is no machine failure, there is no cost. If there is only one failure, he can work with the remaining equipment until you return, so the cost is effectively zero. If there are two or more machine failures, however, there will be assembly time lost, and he will have to call in experts to repair the machine immediately. The cost would be $15,000. What should he do?
4. Find the EVPI concerning the number of machine failures during your absence.

2. [30 Points] Daniel has a research-and-development decision to make based on two alternative projects A and B. However, he can choose only one of the two projects due to budget constraints. The payoffs of these projects depend on the degree of success of their development, as shown in the decision tree in Figure 2.1.

Daniel has made the following assessments of utility values:

1. If he had the choice, he would be indifferent between alternatives M and N shown in Figure 2.2
   .
2. If he had the choice, he would be indifferent between alternatives M’ and N’ shown in Figure 2.3.

1. [20 Points] According to the assessments i) and ii), which project should Daniel choose based on the maximum expected utility. Justify your answer. (Hint: set the utilities of the best and worst consequences in the decision tree to 1 and 0, respectively)
2. [10 Points] What kind of risk attitude is Daniel demonstrating in this decision? Why?

3. [50 Points] Charles has four job offers from Acme Manufacturing (A), Bankers Bank (B), Creative Consulting \((C)\), and Dynamic Decision Making (D). He knows that factors such as location \((\mathrm{L})\), salary \((\mathrm{S})\), amount of challenge in the work \((\mathrm{C})\) and long term prospects \((P)\) are important criteria to him, so he wants to formalize the relative importance among the criteria and evaluate those job offers.

Suppose that Charles, thinking hard about his preferences, comes up with the pairwise comparisons among the criteria and job offers with respect to the criteria, as listed in Tables 3.1-3.5

1. [10 Points] Construct the decision problem as a hierarchy.
2. [30 Points] Use Tables 3.1-3.5 to estimate the relative weights of criteria and the relative weights of the four alternatives with respect to each criterion. Also check the consistency of pairwise comparisons in each table. You can use either the eigenvalue-eigenvector approach or approximation method.

d. [10 Points] Combine the relative importance of criteria and alternatives to obtain a global ranking for each alternative. Which alternative is preferred?

4. [30 Points] Franklin D. Roosevelt High School was recently jarred by an incident in which one student shot and killed another during lunch break, apparently for showing a lack of respect. Parents, members of the community, and administrators were all grieved and determined to do something about violence in the school. The alternatives considered included installing metal detectors at the entrances, teaching dispute resolution, and of course, do nothing.

The final decision lay in the hands of Principal Chan. She identified two objectives: assuring safety within the facility and maintaining an environment conducive to learning, and saving money. She thought an additive utility function is appropriate to model her decision to this problem. In addition, she assessed weights for the three attributes safety (S), learning (L), and cost \((C)\) corresponding to the three objectives using the lottery-assessment method. In considering the lotteries, she concluded that she was indifferent between:

Lottery \(A\) : Win the best possible combination of three attributes with probability 0.35

Win the worst possible combination of three attributes with probability 0.65

and

Lottery \(B\) : Win a combination that is the worst on attributes \(\mathrm{L}\) and \(\mathrm{C}\) and the best on attribute \(\mathrm{S}\)

She also concluded that she was indifferent between

Lottery C: Win the best possible combination of three attributes with probability 0.25

Win the worst possible combination of three attributes with probability 0.75

and

Lottery \(D\) : Win a combination that is the worst on attributes \(\mathrm{L}\) and \(\mathrm{S}\) and the best on attribute \(\mathrm{C}\)

1. [15 Points] Use the above information find the weights \(k_{\mathrm{S}}, k_{\mathrm{L}}, k_{\mathrm{C}}\). (Hint: \(\mathrm{U}(s, l, c)=k_{\mathrm{S}} \mathrm{U}_{\mathrm{S}}(s)+\) \(\left.k_{\mathrm{L}} \mathrm{U}_{\mathrm{L}}(l)+k_{\mathrm{C}} \mathrm{U}_{\mathrm{C}}(c)\right)\)
2. [15 Points] The utilities of the three alternatives on the three attributes are summarized in Table 5.1. Use the information provided in the table to calculate the overall utility of each alternative and identify the preferred alternative.

5. [35 Points] Sam faces an investment decision in which he must think about cash flows in two different years. Regard these two cash flows as two different attributes, and let \(X\) represent the cash flow in Year 1 and \(Y\) the cash flow in Year 2. The maximum cash flow he could receive in any year is $20,000, and the minimum is $5,000. He has assessed his individual utility functions for \(X\) and \(Y\) and has fitted logarithmic utility functions to them.

\(\begin{aligned}

&U_{X}(x)=0.7 \cdot \ln (x)-6 \\

&U_{Y}(y)=1.1 \cdot \ln (y+5000)-10

\end{aligned}\)

Furthermore, he thinks that utility independence holds, so the individual utility functions for each cash flow are appropriate regardless of the amount of the other cash flow. He also has made the following assessments:

1. He would be indifferent between (1) a sure outcome of $7,500 each year for two years and (2) a risky investment with a \(40 \%\) chance at $20,000$ each year and a \(60 \%\) chance at $5,000 each year.
2. He would be indifferent between getting (1) a sure outcome of $10,000 the first year and $5,000 the second and (2) a risky investment with a \(40 \%\) chance at $20,000 each year and a \(60 \%\) chance at $5,000 each year.

Ignore the time value of money for this problem.

1. [20 Points] Use the above assessments to find \(k_{X}\) and \(k_{Y}\). What does the value of \(\left(1-k_{x}-k_{Y}\right)\) imply about the cash flows of the different periods (substitutes or complements)?
   (Hint: \(\left.\mathrm{U}(x, y)=k_{X} \mathrm{U}_{x}(x)+k_{3} \mathrm{U}_{1}(y)+\left(1-k_{X}-k_{Y}\right) \mathrm{U}_{x}(x) \mathrm{U}_{1}(y)\right)\)
2. [15 Points] Use the derived utility function to choose between alternatives A and B in the following decision tree.

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