$1.$ The historical data maintained by an electric water pump manufacturer show that the model PL$40$ with flow rate of $\backslash (40\backslash$mathrm$\{$l$\}/\backslash$mathrm$\{$min$\}\backslash )$ has an annual demand of $20,000$ pumps. The company needs to purchase the electric motor required for this pump from a specialised manufacturer and maintain sufficient stock in the inventory to manage the daily production of water pumps without disruptions. The daily need of electric motors for production shows a normal distribution with a standard deviation of $12$ electric motors a day. The supplier takes $14$ days to supply the motors once an order is placed. The estimated ordering cost is $\backslash (\backslash$$ $700\backslash )$ per order and the annual holding cost per motor is $\backslash (\backslash$$ $5\backslash ).$ The company operates $260$ days a year.

$($a$)$ Estimate the economic order quantity $($EOQ$)$ for the electric motors.

$($b$)$ Estimate the reorder point $($ROP$)$ to maintain a service level of $\backslash (95\backslash \%\backslash ).$

$($c$)$ What is the annual holding cost of electric motors?