Crunchy, a cereal manufacturer has dedicated a plant for one major retail chain. Sales at the retail chain average about $40,000$ boxes a month and production at the plant keeps pace with this average demand. Each box of cereal costs Crunchy $$4$ and is sold to the retailer at a wholesale price of $$6.$ Both Crunchy and the retailer use a holding cost of $20$ percent. For each order placed, the retailer incurs an ordering cost of $$300$ per order placed. Crunchy incurs the cost of transportation and loading that totals $$1,500$ per order shipped.

a$)$ Given that it is trying to minimize its ordering and holding cost, what lot size will the retailer ask for in each order? What is the annual ordering and holding cost for the retailer as a result of this policy? What is the annual ordering and holding cost for Crunchy as a result of this policy? What is the total inventory cost across both parties as a result of this policy?

b$)$ What lot size minimizes the inventory costs $($ordering$,$ delivery, and holding$)$ across this entire supply chain $($both Crunchy and the retailer$)?$ What is the total inventory cost of this supply chain?

c$)$ Compare the results obtained in a$)$ and b$)$ and discuss your observations.