$1.$ One item produced by a work center has a setup cost of $$250$ and an annual demand of $20,000.$ Demand is deterministic and occurs at a constant rate. Raw material costs $$1\mathrm{.}50,$ and the completed item is valued at $$2\mathrm{.}25.$ The holding cost rate is $0\mathrm{.}5$ per year. The work center could process $100,000$ of these items per year if it did not make anything else. Historically, batches at the work center spend approximately $80\%$ of the time waiting and $20\%$ of the throughput time in process. Find the batch size that minimizes cost and compute the inventory cost per year. What reorder point should be used to avoid shortages?