Prefab, a furniture manufacturer, uses $20,000$ square feet of plywood per month. Its trucking company charges Prefab $$400$ per shipment, independent of the quantity purchased. The manufacturer offers an all unit quantity discount with a price of $$1$ per square foot for orders under $20,000$ square feet, $$0\mathrm{.}98$ per square foot for orders between $20,000$ square feet and $39,999$ square feet, and $$0\mathrm{.}96$ per square foot for orders greater than or equal to $40,000$ square feet. Prefab incurs a holding cost of $20$ percent. What is the optimal lot size for Prefab? What is the annual cost of such a policy? What is the cycle inventory of plywood at Prefab? How does it compare with the cycle inventory if the manufacturer does not offer a quantity discount but sells all plywood at $$0\mathrm{.}96$ per square foot?