BZoom sells toy bricks that can be used to construct a wide range of machines, animals, buildings, and so on$.$ It purchases a red dye powder to include in the resin it uses to make the bricks. The powder is purchased from a supplier for $$1\mathrm{.}30$ per kg$.$ At one production facility, BZoom requires $400$ kg of this red dye powder each week. BZoom's annual holding costs are $30$ percent, and the fixed cost associated with each order to the supplier is $$50.$

a$.$ How many kilograms should BZoom order from its supplier with each order to minimize the sum of the ordering and holding costs?

b$.$ If BZoom orders $4000$ kg at a time, what would be the sum of the annual ordering and holding costs?

c$.$ If BZoom orders $2000$ kg at a time, what would be the sum of the ordering and holding costs per kilogram of dye?

d$.$ If BZoom orders the quantity from part $(A)$ that minimizes the sum of the ordering and holding costs, what is the annual cost of the EOQ expressed as a percentage of the annual purchase cost?

e$.$ BZoom's purchasing manager negotiated with its supplier to get a $2\mathrm{.}5$ percent discount on orders of $10,000kg$ or greater? What would be the change in BZoom's annual total cost $($purchasing$,$ ordering, and holding$)$ if it took advantage of this deal instead of ordering smaller quantities at the full price?

i$.$ It would decrease by more than $$1000.$

ii$.$ It would decrease by less than $$1000.$

iii. It would increase by less than $$1000.$

iv$.$ It would increase by more than $$1000.$