Problem $1(25$ pts$).$ Husni Shreif recently completed his PhD and accepted a job with University of Balamand.

Although he likes her job, he is also looking forward to retiring one day. To ensure that his retirement is

comfortable, Shreif intends to invest $$3,000$ of his salary into a tax$-$sheltered retirement fund at the end of each

year. He is not certain what rate of retum this investment will earn each year, but he expects each year's rate of

return could be modeled appropriately as a normally distributed random variable with a mean of $12\%$ and standard

deviation of $2\mathrm{.}5\%.$

He contacted you as a financial consultant to help him answer the following questions. Design a spreadsheet to

simulate $1000$ scenarios as pictured below:

a$.$ Illustrate what functions go in the following cells: B$6$ through F$6$ and K$5.$

b$.$ Illustrate how will you simulate the data.

c$.$ If he is $30$ years old, how much money should he expects $($on average$)$ to have in his retirement fund at

age $60?$

d$.$ Construct a $95\%$ confidence interval for the average amount he will have at age $60.$ Interpret it$.$

e$.$ What is the probability that he will have more than $$1$ million in his retirement fund when he reaches

age $60?$

Problem $2(25$ pts$).$ ShreifCo can produce microchips on $5$ different machines. The following table summarizes

the manufacturing costs associated with producing the microchips on each machine along with the available

capacity on each machine. If the company has received an order for $2,000$ microchips, how should it schedule

these machines? Formulate an ILP model for this problem. Do NOT design a spreadsheet to solve.

Problem $3(20$ pts$).$ A manufacturer of engine belts uses multipurpose manufacturing equipment to produce a

variety of products. A technician is employed to perform the setup operations needed to change the machines over

from one product to the next. The amount of time required to set up the machines is a random variable that follows

an exponential distribution with a mean of $20$ minutes. The number of machines requiring a new setup is a Poisson

random variable with an average of $2$ machines per hour will require setup $($i$.$e$.$ arrival rate per hour per machine

$=\frac{2}{5}).$ The technician is responsible for setups on $5$ machines.

a$.$ Compute the service rate per hour.

b$.$ What percentage of time is the technician idle, or not involved in setting up a machine?

c$.$ On average, how long is a machine out of operation while waiting for the next setup to be completed?

d$.$ If the company hires another, equally capable technician to perform setups on these machines, how long

on average would a machine be out of operation while waiting for the next setup to be completed?