a$.$ Given that the retailer is trying to minimize its inventory costs $($ordering cost$+$ holding cost$),$ what lot size will it ask for in each order? What are the annual ordering and holding costs of the manufacturer, retailer and supply chain as a result of this policy?

b$.$ What lot size minimizes the inventory costs across both the manufacturer and the retailer $($supply chain$)?$ How much will this policy change the manufacturer$$s$,$ the retailer$$s$,$ and the supply chain$$s inventory costs relative to $($a$)?$

c$.$ Design an all$-$unit quantity discount that results in the retailer ordering the lot size in $($b$).$