Joe is analyzing three alternatives for a new rocket: the HW, the SC, and the PK. The three most important attributes are the speed, number of payloads, and range. The decision maker prefers more of each attribute. The raw data in Table 1 are available. Joe plans to use the AHP to analyze the alternatives in Table 1. After consulting the stakeholders, he determines that, for the payloads attribute, four payloads is worth three times as much as three payloads, four payloads is worth five times as much as two payloads, and three payloads is worth twice as much as two payloads. Create a pairwise comparison matrix for the payloads attribute and find a set of scores that correspond to these pairwise comparisons. (Scale the scores so that the largest score equals 1.)

Table 1: The Attributes of Three Rocket Alternatives (Parnell et al., 2008).

Rocket	Speed (kph)	Number of payloads	Range (km)
HW	66	4	14
SC	45	3	55
PK	30	2	36

2. From the previous question, after consulting the stakeholders, Joe determines that a rocket would be worthless if its speed were less than 15 kph, it carried no payloads, and its range were only 1 km. Specify the attribute values of the three hypothetical alternatives that should be compared to determine the weights for the attributes.

3. Consider the rocket selection example again. Rose is also analyzing the three alternatives listed in Table 1, but she decides to develop a multiattribute utility function. After additional consultation with the stakeholders, Rose determines the relative ratings of the four hypothetical alternatives shown in Table 2. Construct a linear multiattribute utility function that is consistent with this data.

Table 2: The Attributes of Four Hypothetical Rocket Alternatives (Parnell et al., 2008)

Alternative	Speed (kph)	Number of payloads	Range (km)	Rating
1	30	2	14	0
2	66	2	14	15
3	30	4	14	35
4	30	2	55	100

4. Rose needs to assess a utility function for the Range attribute. She wants the utility to be in the range of [0, 1]. After consulting the stakeholders, she determines that they would be indifferent between (a) a rocket with a range of 36 km and (b) randomly choosing between a rocket with a range of 14 km and a rocket with a range of 55 km if the probability of getting the longer-range rocket was 60%. What utility should she assign to the values 14, 36, and 55 km?

5. If Rose uses an increasing exponential utility function over the range (14 and 55 km), what is the appropriate value of (assume that the utility of 34.5 km is approximately the same as the utility of 36 km)? Write out the complete utility function for range. Using this utility function, calculate the utility of 50 km.

6. After more consultation with the stakeholders, Rose determines that the utility functions for speed and number of payloads are linear (and the utility of the worst value equals 0, and the utility of the best value equals 1). Based on this information and the answers to the previous questions, calculate the total utility of each alternative. Under what conditions (combinations of weights) does the optimal alternative remain optimal? (That is, if the weights did not satisfy these conditions, the best choice would be another alternative.)