Question $2(50$ points$)$: Olivia owns a pet supply store called Paws & Claws, where she stocks a special type of organic dog treats made by a small, family$-$owned business in Oregon. Each bag costs Olivia $$10,$ and due to the artisan production process, it takes a full year to receive new orders. To calculate her holding costs, Olivia uses an annual interest rate of $15\%.$ If the treats are out of stock when a customer requests them, Olivia estimates a goodwill loss of $$20$ per bag. Each time she places an order, she incurs a $$25$ processing fee. Based on historical sales, Olivia expects to sell around $300$ bags per year, with a standard deviation of $50$ bags. Assume that the demand follows a normal distribution. a$.(30$ points$)$ Determine the optimal $($Q$,$R$)$ policy. Continue iterating until the gap between successive iterations of the Q value is less than $5\%.($i$.$e$.,(+1)<5\%,$ where $$i$$ represents the iteration number.$)$ b$.(5$ points$)$ Determine the safety stock using the $($Q$,$R$)$ policy determined in part a$.$ c$.(15$ points$)$ Determine the average annual holding, setup, and penalty cost.