Problem: E$-$buy, a popular consumer electronics store in Lawrence, follows a base stock policy for its best selling vacuum cleaner. The weekly demand for this brand of vacuum cleaners is normally distributed with a mean of $10,$ and a standard deviation of $4.$ The manufacturer of this vacuum cleaner takes $4$ weeks to replenish an order from E$-$buy. E$-$buy estimates that the holding cost for the vacuum cleaner is $$15/$unit$/$week$,$ while the backorder cost is $$10/$unit$/$week$.$

Part $1$: What should be E$-$buy's target base$-$stock level to balance their backorder and inventory holding cost? If your answer is not an integer, provide at least three decimal places, e$.$g$.,7\mathrm{.}500.$

Part $2$: Given the optimal base$-$stock level you calculated in the previous question, what is E$-$buy's stockout probability for this brand of vacuum cleaners? If your answer is not an integer, provide at least three decimal places, e$.$g$.,7\mathrm{.}500.$

Part $3$:E$-$buy has realized that they have underestimated the backorder cost. What base$-$stock level should they choose to achieve a in$-$stock probability of $0\mathrm{.}75?$ If your answer is not an integer, provide at least three decimal places, e$.$g$.,7\mathrm{.}500.$