$3.(40$ points$)$ An automotive warehouse employs a continuous review $($QR$)$ policy to control their inventory for a speci c fuel lter. The inventory holding cost per unit per year is $$20,$ the xed ordering cost is $$2000,$ the unit proportional cost is $$100,$ and the expected annual demand is $3000$ units. The lead time demand is normally distributed with a mean of $500$ and a standard deviation of $100.$ All shortages are backordered at a cost of $$40$ per unit. $($a$)(10$ points$)$ Find the lot size that minimizes the expected annual cost if a reorder point of $600$ is to be used. Calculate the expected annual cost in this case. $($b$)(15$ points$)$ Find the optimal values of the lot size and the reorder point. Perform at most three iterations $($including iteration $0).$ Calculate the optimal expected annual cost. $($c$)(15$ points$)$ Now ignore the shortage cost. Find the optimal values of the lot size and the reorder point when a Type $1$ service level of $0\mathrm{.}95$ is required. Also, nd the optimal values of the lot size and the reorder point when a Type $2$ service level of $0\mathrm{.}95$ is required. Perform at most three iterations $($including iteration $0).$