Problem $3.(20$ points$)$

Prof. S$$leyman Kerimov

a$.(10$ points$)$ You have three days to sell a single item. Assume that the willingness to pay for interested buyers are independent random variables. You want to set a price for each day, and you have the following estimation:

On day $1,10$ interested buyers will arrive, where their willingness to pay is drawn from $U[50,100].$

On day $2,5$ interested buyers will arrive, where their willingness to pay is drawn from $U[40,80].$

On day $3,2$ interested buyers will arrive, where their willingness to pay is drawn from $U[30,60].$

Write the optimization problem that you need to solve to maximize your expected revenue. You do not need to find the optimal prices for each day.

b$.(10$ points$)$ After Max "Blessed" Holloway's stunning knockout at UFC $300,$ UFC wants to hold an event at Stan Sheriff Center in Hawaii, which is a $20,000$ capacity venue. They want to use the EMSR$-$a heuristic to find the booking limits$.$ There are $4$ classes with ticket prices $p_1=1000,p_2=750,p_3=500$ and $p_4=250.$ The demands are distributed as follows: $D_{1}N(500,{20}^2),D_{2}N(2500,{30}^2),D_{3}N(7500,{50}^2),$ and $D_{4}N(30000,{50}^2).$ That is$,$ for class $1,$ the demand is normally distributed with mean $500$ and variance of ${20}^2,$ and etc. Let $b_i^{**}$ denote the booking limit for class i under the EMSR$-$a heuristic. Find $b_4^{**}.$