$4.$ The demand faced by a company each week is estimated to be normally distributed with a mean of $1,000$ units and a standard deviation of $250$ units. Lead time of delivery is fixed at $2$ weeks. Ordering cost is $$800$ per order, variable cost per unit ordered is $$4,$ the inventory carrying charge is $10\%$ of product value per year. Assume $50$ weeks per year.

a$)$ If the company sets the reorder point to be equal to the average demand during a $2-$ week period, what is the value for R$?$ What service level can the company achieve?

b$)$ If the company would like the probability of running out of stock to be no higher than $10\%,$ specify the inventory policy $($i$.$e$.,$ when and how much to order$)$

c$)$ If the company keeps a safety stock of $300,$ what is the value for R$?$ Is the probability of running out of stock higher or lower than $10\%?$ Please try to intuitively explain based on your answer to part b$).$

d$)$ Discuss the impact of the following changes on the level of safety stock $($assuming only one change occurs at a time$)$:

Average of weekly demand increases to $1,500$ units

Standard deviation of weekly demand decreases to $200.$

Lead time shortens to $1$ week