You have a business heat$-$treating specialty industrial castings. The number of castings you receive for treatment each day is a Poisson random variable with a mean value of $4\mathrm{.}1.$ You process the castings in a super$-$high$-$temperature oven that can hold up to $5$ castings. This oven uses a heating element that sometimes fails; the probability of failure is as follows:

Day of Use

Failure Probability

$1$

$1\%$

$2$

$7\%$

$3$

$9\%$

$4$

$15\%$

$5$

$25\%$

After the fifth day of use, the safety regulations for the oven require that the heating element be replaced even if it is still functioning. On days that the heating element fails, you must wait until tomorrow to reprocess all the castings for that day. Thus, on days that the heating element is working, you have a total processing capacity of up to $5$ castings, but on days that it fails, your capacity is effectively $0$ castings. You process the castings on a first$-$come, first$-$served basis $--$ if you cannot finish all the castings waiting to be processed on a given day, you save them in a queue and try to process as many as possible the next day. You are considering $5$ possible policies, parameterized by a number d $=1,2,3,4,$ or $5.$ At the end of the day, if the heating element has been in use for d days and has not failed, you replace it$.$ On days when the element fails, you also replace it at the end of the day.

The economics of the operation are as follows:

$$ The heating element costs $$800$ to replace if it did not fail

$$ When the element fails, it costs $$1500$ to replace

$$ You receive $$200$ in revenue each time you finish processing a casting

$$ You estimate that each day that each casting spends waiting to be processed costs you $$40$ in loss of goodwill, storage costs, etc.

$$ You may assume all other costs and revenues to be negligible.

Determine by simulation which value of d gives you the highest expected profit over a $60-$day period. You may ignore any costs and revenues from castings left in queue at the end of the period. Use a sample size of at least $500,$ and assume that you start with a new heating element on the first day.

You are also interested in whether the queue of unprocessed castings left at the end of the day exceeds $10$ at any time during the $60-$day period. With the optimal value of d$,$ what is the probability of this event? Use @Risk Excel Software not Python code