Problem $1\mathrm{.}1(10$ points$)$

In $2018,$ in Korea, BMW corporation suffered from sharp sales decreases and legal issues. The reason for

the sudden downturn was due to customers experiencing exploding engines in some of BMW$$s sedans. In

$2018-2019$ it was reported that around $60$ BMW sedan$$s engines had exploded because the engine cooling

systems failed to cool the hot vapor from the engines. Consequently, BMW faced serious law suits and

tremendous penalties in Korea.

Since then, steps have been taken to improve the cooling system, and engine$-$system specialists claim that

there is a $80\%$ chance that the improvement program has been successful.

Given that it takes more than five years to design and produce a completely$-$safe cooling system, the

manager of BMW had to decide on his policy for the next several years. He has identified two options.

$1)$ He could use the current cooling system. If the cooling system still has problems, an estimated net

loss of $$20$ M will be incurred. However, if the cooling system has no problem, an estimated net

return of $$90$M will be earned.

$2)$ He could avoid the current cooling system by outsourcing the whole cooling system to suppliers.

Due to the extra adjustment costs incurred if hiring external cooling suppliers, outsourcing would

almost certainly lead to net returns of $$45$ M$.$

Which option would you suggest? To show your answer, you must use a decision tree and clearly show

how you calculate probabilities.

Problem $1\mathrm{.}2$

The BMW manager considered using a test by Mobis, a reputable supplier that specializes in testing

engine cooling systems. Mobis could carry out a test to determine whether or not the cooling system

was problematic. The test by Mobis indicated that there was a $90\%$ chance that the cooling system

was safe. The manager assumed that the test was perfectly accurate. And, BMW paid $$8$M for the

test. The manager has no idea how accurate the indication will be$.$ Your goal is to explain if BMW$$s

$$8$M payment was reasonable. Please answer the following sub problems.

Given that information, please explain if BMW$$s $$8$M payment was reasonable. To show your

answer, you must use a decision tree and clearly show how you calculate probabilities.

$1\mathrm{.}2\mathrm{.}1(10$ Points$)$

Please show a decision tree with full calculation.

$1\mathrm{.}3\mathrm{.}3(10$ Points$)$

Please explain whether BMW should buy the test or not. If your decision tree appearing in $1\mathrm{.}2\mathrm{.}1$ is

wrong, you will get not get any partial point in this problem.

$1\mathrm{.}3\mathrm{.}3(10$ Points$)$

Please explain if BMW$$s $$8$M payment was reasonable. If not please, explain why not. If your

decision tree appearing in $1\mathrm{.}2\mathrm{.}1$ is wrong, you will get not get any partial point in this