Quarks sells three types of muesli, the Shoko, the Fruit & Nut, and the Berries Mix. Annual demands for the three products are DS $=50,000$ for the Shoko, DF $=40,000$ units for the Fruit & Nut, and DB $=30,000$ units for the Berries Mix. Each type costs Quarks $$2$ per package. A fixed transportation cost of $$1,000$ is incurred each time an order is delivered. For each type ordered and delivered on the same truck, an additional fixed cost of $$100$ per type is incurred for receiving and storage. Quarks incurs an annual unit holding cost of $20$ percent. Suppose Quarks$$s Shoko supplier now offers quantity discount. If Quarks orders more than $20,000($including $20,000)$ packages in a single order, it will be charged a discounted price for all units ordered.

a$.$ If the discount rate is $1\%,$ find the optimal order quantity and total annual cost including fixed ordering cost, variable ordering cost, and inventory holding cost. $(3$pt$)$

b$.$ If the discount rate is $0\mathrm{.}1\%,$ find the optimal order quantity and total annual cost including fixed ordering cost, variable ordering cost, and inventory holding cost. $(3$pt$)$