Joshua, who serves as the manager at Enterprise, a car distributor, is currently in the process of determining the best quantity of cars of a new model to procure for the CNY$.$ The current cost of each ordered car stands at $$25,000.$ An increase in car prices next week, estimated to be in the range of $5-10\%,$ is expected. The demand for cars for this model has the probability distribution below:

The anticipated revenue per car for this model during the new year week is projected to be an average of $$35,000.$ The company wants to satisfy all the demand, but should Joshua decide to acquire additional cars just before the new year, there is an additional cost of $$2,000$ per car. In the event of surplus cars at the company, he can sell these extra cars at a rate between $$20,000$ and $$22,000$ per car.

$($a$)$ Analyse to find out the number of cars Joshua should order now to maximise the expected profit. The management of the company wants him to buy the cars now or just before the new year. Use the built$-$in Excel functions for your analysis and check at least $5$ different options. Explain your answer.

$($b$)$ Determine the $95\%$ confidence interval for the expected profit.

$($c$)$ If the demand for the car follows a normal distribution instead, with a mean of $35$ and a standard deviation of $10,$ determine the $$best$$ order quantity with the largest mean profit. Use @ RISK functions to develop your simulation model. Check at least $5$ different options for order quantity.

$($d$)$ Discuss whether this order quantity would be considered best by the car dealer based on the idea that the decision$-$maker may use criteria other than profit to make the decision. You may use any statistics and$/$or graphs from @RISK.

In your answer, clearly show the input parameters, decision variable$($s$),$ output variable$($s$)$ and any other simulation parameters and formula you used for your analysis.$\backslash$table$[[$Demand$,$Probability$],[20,0\mathrm{.}15],[25,0\mathrm{.}20],[30,0\mathrm{.}30],[35,0\mathrm{.}20],[40,0\mathrm{.}15]]$