$1.$ Marshall$$s paint store uses a $($Q$,$ R$)$ inventory system to control its stock levels. For a particularly popular white paint, historical data shows that the distribution of monthly demand is approximately normal, with mean $100$ and standard deviation $30.$ Replenishment lead time for this paint is about $8$ weeks. Each can of paint costs the store $$5.$ Although excess demand is back$-$ordered, the store owner estimates that each unit of unfilled demands cost $$12$ in bookkeeping and loss$-$of$-$goodwill costs. Fixed cost of replenishment is $$100$ per order, and holding costs are based on a $12$ percent annual rate of interest. Assume that there are $4$ weeks in a month and $48$ weeks in a year. Determine:

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a$.$ The optimal lot size and reorder point.

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$$ Assuming that the policy in part $($a$)$ is used, determine:

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b$.$ The safety stock.

c$.$ The average annual holding, setup and penalty costs $($separately$).$

d$.$ The average time between placements of orders.

e$.$ The expected demand per cycle.

f$.$ The expected number of units back$-$ordered per cycle.

g$.$ The proportion of order cycles in which no stock out occurs.

h$.$ The proportion of demands that are backordered.

i$.$ The probability that there will not be any stock$-$out during a cycle.