Fort Wayne Plastics Inc is trying to develop a plan for its workforce and production levels for the six$-$month period from January to June. The firm produces plastic components. Forecast demands over the next six months are $1,304,633,809,1,175,2,146,$ and $1,373.$ There are currently $($end of December$)347$ workers employed in the plant. Ending inventory in December is expected to be $461$ units, and the firm would like to have $575$ units on hand at the end of June. The cost of hiring one worker is $$546.$ The cost of firing a worker is $$900.$ The cost of holding one unit of inventory for one month is $$60.$

In the past, the plant manager observed that over $21$ working days, with the workforce level constant at $76$ workers, the firm produced $233$ plastic components. The number of working days per month from January to June for this year is $26,25,25,17,22,17.$ Find the optimal production per month as well as the corresponding optimal cost for this plan.

P$1=$

P$2=$

P$3=$

P$4=$

P$5=$

P$6=$

Total Cost $=$