Q$1-(50$ points$)$ The managers of the Lightning Cycle Company are hoping to develop a new

bicycle braking system. Two alternative systems have been proposed and, although the

mechanics of the two systems are similar, one design will use mainly plastic components while

the other will use mainly metal ones. Ideally, the design chosen would be the lightest and the

most durable but, because some of the technology involved is new, there is some uncertainty

about what the characteristics of the resulting product would be$.$

The leader of Lightning's research and development team has estimated that if the plastic

design is developed there is a $60\%$ chance that the resulting system would add $130$ grams to

a bicycle's weight and would have a guaranteed lifetime of one year. He also reckons that

there is a $40\%$ chance that a product with a $2-$year lifetime could be developed, but this would

weigh $180$ grams.

Alternatively, if the metal design was developed the team leader estimates that there is a $70\%$

chance that a product with a $2-$year guaranteed life and weighing $250$ grams could be

developed. However, he estimates that there is a $30\%$ chance that the resulting product

would have a guaranteed lifetime of $3$ years and would weigh $290$ grams.

It was established that, for the team leader, weight and guaranteed lifetime were mutually

utility independent. The following utilities were then elicited from him:

After further questioning the team leader indicated that he would be indifferent between the

following alternatives:

A: A product which was certain to weigh $130$ grams, but which had a guaranteed

lifetime of only one year; or

B: A gamble which offered a $0\mathrm{.}7$ probability of a product with a weight of $130$ grams and

a guaranteed lifetime of $3$ years and a $0\mathrm{.}3$ probability of a product with a weight of $290$

grams and a guaranteed lifetime of $1$ year.

Finally, the team leader said that he would be indifferent between alternatives C and D below:

C: A product which was certain to weigh $290$ grams, but which had a guaranteed lifetime

of $3$ years;

D: A gamble which offered a $0\mathrm{.}9$ probability of a product with a weight of $130$ grams

and a guaranteed lifetime of $3$ years and a $0\mathrm{.}1$ probability of a product with a weight of

$290$ grams and a guaranteed lifetime of $1$ year.

a$)(5$ points$)$ Draw a decision tree for the team leader's problem.

b$)(20$ points$)$ What do the team leader's responses indicate about the relative weight

which he attaches to the two attributes of the proposed design?

c$)(5$ points$)$ Are the two attributes complements or substitutes?

d$)(20$ points$)$ Which design should the team leader choose, given the above responses?