MATHIMATICALLY SOLVE FOR EACH IN TERMS OF SUPPLY CHAIN $5-$ A soap manufacturer has dedicated a plant for one major retail chain. Sales at the retail chain average about $10,000$ jugs of soap a month and production at the plant keeps pace with this average demand. Each jug of soap costs the manufacturer $5$ and is sold to the retailer at a wholesale price of $10.$ Both the manufacturer and the retailer use an annual holding cost of $\mathrm{.}25$ percent of unit cost. For each order placed, the retailer incurs an ordering cost of $100.$ The manufacturer incurs the costs of transportation and loading that total $1,600$ per order shipped.

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$).$