$2)$ A supplier always avoids shortages with overtime production. The company follows a base$-$stock policy, but the operations manager must decide what base$-$stock level is optimal. Demand is normally distributed with mean $10000$ and standard deviation $1500.$ The holding cost is only $$0\mathrm{.}05$ per period per unit but if there is a shortage $$ of even just one unit! $$ the supplier must run overtime production with a fixed cost of $$200$ and an extra cost per unit of $$2.$ The product can be produced in any quantity during regular or overtime production. By simulating thousands of times, determine the base$-$stock level $($to the nearest $100)$ that minimizes the average holding$/$overtime production cost per period.

a$)$ What is the best base$-$stock level?

b$)$ What is the average cost per period when using the best base stock level?

c$)$ During what proportion of periods is overtime production required $($for the best base$-$stock level$)?[$Hint: This will be a very small percentage, less than $1\%$ but greater than $0\%.]$