$($Inventory Management: QR Model$)$ An automotive warehouse stocks parts that are sold at nearby dealers. One particular part, a popular brand of oil filter, is purchased by the warehouse for $$1\mathrm{.}50$ each. The fixed setup cost $($of order processing and receipt$)$ is $$100$ per order. The inventory carrying charge is based on a $28\%$ annual interest rate. Assume that if a filter is demanded when the warehouse is out of stock, then the demand is backordered, and the cost assessed for each backordered demand is $$1\mathrm{.}28.$ The monthly demand for the filter follows a normal distribution with mean $280$ and standard deviation $77.$ Order lead time is assumed to be five months. Note: Round intermediary results to the second decimal place, while, for the optimal order quantity $($i$.$e$.,$ Q$*)$ and reorder level $($i$.$e$.,$ R$*),$ round the results to the closest integers. When using the L$($z$)$ table, choose the value that is closest to your z$-$score obtained. $.$ a$)$ Find the mean and standard deviation of the demand during the order lead time b$)$ Determine the optimal values of the order quantity and the reorder level $($note: use EOQ approximation for $(*).$ c$)$ Calculate the average annual cost of stockout associated with this item, assuming that the optimal policy $($i$.$e$.,$ Q$*$ and R$*)$ is used. The manager of the warehouse decides that the estimation of the stockout cost of $$1\mathrm{.}28$ may not be very accurate. The manager wants to set the reorder level so that the probability of no stockout is $90\%.$ d$)$ Determine the reorder level to achieve the $90\%$ probability of no stockout. The probability of no stockout is sometimes referred to as the Type $1$ service level. Another commonly used service level, namely, the Type $2$ service level or the fill rate, measures the proportion of demands that are met from stock. Specifically, Type $2$ service level $($fill rate$)=1-$ Expected number of shortages in a year The average annual demand e$)$ Calculate the fill rate as defined above, assuming the reorder level of part d$)$ is used. solve this questionP