After heated discussions, Mr$.$ Anderson concludes using a deterministic demand curve is unrealistic since demand is uncertain. In the current scenario, marked by uncertain demand, Harmony Haven faces the challenge of balancing its capacity between high and low$-$fare customers. The hotel manager, Mr$.$ Anderson, is determined to test alternative demand distributions and fare strategies.

$3.$ Mr$.$ Anderson sets the low$-$fare price to $$160$ to ensure that there will be plentiful demand. The high$-$class fares, worth $$275,$ have the following discrete demand distribution, and the hotel has $400$ rooms to book in total.

$\backslash$table$[[$Demand$,$Probability$],[70,0\mathrm{.}4],[100,0\mathrm{.}3],[145,0\mathrm{.}3]]$

a$.$ What is the optimal high$-$fare protection level, $Q_H^{**}?(7$ pts$)$

b$.$ How many high$-$fare customers are expected to be rejected under the optimal strategy? $(5$ pts$)$

c$.$ What is the anticipated number of unsold tickets using the optimal strategy? $(5$ pts$)$

d$.$ What is the expected revenue under the optimal strategy? $(3$ pts$)$

e$.$ Can you write a linear program that minimizes the overall costs? $($Minimize cost of the expected high$-$fare lost sales $+$ cost of unsold tickets$)(10$ pts$)$

f$.$ Recalculate the answers for parts a and $b$ if the high$-$fare demand is normally distributed with $=85,$ and $=20.(8$pts$)$

g$.$ After you decide to use $Q_H^{**}($answer from part a$),$ you observe that your demand is actually exponentially distributed with $=85.$ How would you update your answer from part $($b$)?$ DO not solve it$!$ Simply write down the function and plug in what we know. For exponential distribution, use the following if needed: $(7$ pts$)$

$f(x)=e^{-x},F(x)=1-e^{-x},=\frac{1}{}$