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$8.$ Falcon Tools is a retailer of battery$-$operated handheld power tools for the consumer market $($such as screwdrivers and drills$).$ Let's consider the replenishment$/$stocking decision for a popular classic battery$-$operated screwdriver, the Master$-$Driver. Falcon purchases Master$-$Driver from a supplier and then sell them to the customers. Average monthly demand for Master$-$Driver is normal with mean and standard deviation of $150$ and $50,$ respectively. The lead$-$time for Master$-$Driver is also normally distributed with mean of $1$ month and standard deviation of $15$ days $($Assume $30$ days a month$).$ Falcon purchases Master$-$Driver from the supplier for $\backslash (\backslash$$ $30\backslash )$ per unit and uses an annual holding$/$carrying cost rate of $\backslash (12\backslash \%\backslash ).$ The fixed cost associated with placing an order of Master$-$Driver to the supplier is $\backslash (\backslash$$ $80\backslash ).$ Assuming that Falcon estimates a $\backslash (\backslash$$ $100\backslash )$ cost associated with each stockout $($independent of the units backordered$)$ and targets a fill rate of $98\backslash \%$ for Master$-$Driver, answer the following:

a$)$ What is Falcon's Economic Order Quantity, Reorder point and the associated Average monthly total cost of the operation?

b$)$ Suppose that Falcon has the option to hire a Third$-$Party Logistics $($TPL$)$ to handle the delivery of the orders for Master$-$Driver from the supplier. With the TPL in charge of delivery, the lead$-$time will be exactly $15$ days; however, Falcon will be charged an additional fixed amount of $\backslash (\backslash$$ $20\backslash )$ per order and an additional $\backslash (\backslash$$ $1\backslash )$ per unit in the order.

Determine the Economic Order Quantity and the Reorder point for this alternative. Also find the associated average monthly total cost. Should this plan be adopted from an economic standpoint? $($Assume that a fill rate of $\backslash (98\backslash \%\backslash )$ is desired.$)$