Problem $1(20$ points$)$: The Paris Paint Company is in the process of planining its labor force requirements and $($aggregate$)$ production levels for the next four quarters. The marketing department has provided production with the following forecasts of $($aggregate$)$ demand for Paris Paint over the next year:

$\backslash$table$[[$Quarter$,\backslash$table$[[$Demand Forecast$],[($in thousands of gallons$)]]],[1,380],[2,630],[3,220],[4,160]]$

Currently there are $280$ employees with the company. Also, according to the company policies, employees are hired for at least one full quarter, and the corresponding hiring costs amount to $$1,200$ per employee. Firing costs are $$2,500$ per employee. Inventory costs are $$1\mathrm{.}00$ per gallon per quarter. It is estimated that one worker produces $1,100$ gallons of paint each quarter.

Finally, currently Paris has $80,000$ gallons of paint in inventory and would like to maintain an inventory of $20,000$ gallons throughout the coming year.

i$.(10$ pts$)$ Determine the "demand chasing" plan that meets the forecasted quarterly demand with zero inventories by adjusting the come pany labor level every quarter, and the corresponding overall cost.

ii$.(10$ pts$)$ Also, determine an alternative plan $($and the corresponding cost$)$ that seeks to meet the expected demand by maintaining a constant workforce level throughout the year: The level of this workforce should be the minimum possible that will allow the company to meet. its quarterly demand in a timely mamer, p$$ssibly building anticipatory inventories in the process.