$16\mathrm{.}4-11.$ The executive search being conducted for Western Bank by Headhunters Inc. may finally be bearing fruit. The position to be filled is a key one$-$Vice

President for Information Processing$-$because this person will have responsibility for developing a state$-$of$-$the$-$art management information system that

will link together Western's many branch banks. However, Headhunters feels they have found just the right person, Matthew Griner who has an

excellent record in a similar position for a midsized bank in New York.

After a round of interviews, Western's president believes that Matthew has a probability of $0\mathrm{.}7$ of designing the management information system

successfully. If Matthew is successful, the company will realize a profit of $$2$ million $($net of Matthew's salary, training, recruiting costs, and expenses$).$

If he is not successful, the company will realize a net loss of $$400,000.$

For an additional fee of $$20,000,$ Headhunters will provide a detailed investigative process $($including an extensive background check, a battery

of academic and psychological tests, etc.$)$ that will further pinpoint Matthew's potential for success. This process has been found to be $90$

percent reliable; i$.$e$.,$ a candidate who would successfully design the management information system will pass the test with probability $0\mathrm{.}9,$

and a candidate who would not successfully design the system will fail the test with probability $0\mathrm{.}9.$

Western's top management needs to decide whether to hire Matthew and whether to have Headhunters conduct the detailed investigative process

before making this decision.

$($a$)$ Construct the decision tree for this problem.

$($b$)$ Find the probabilities for the branches emanating from the event nodes.

$($c$)$ Analyze the decision tree to identify the optimal policy.

$($d$)$ Now suppose that the Headhunters' fee for administering its detailed investigative process is negotiable. What is the maximum amount that

Western Bank should pay?