Problem #$1$: Extension of the solved problem in the class

The washing machine plant is interested in determining work force and production levels for the next $8$ months. Forecasted demands for Jan$-$Aug. are: $420,280,460,190,310,145,110,125.$ Starting inventory at the end of December is $200.$ The firm would like to have $100$ units on hand at the end of August. Suppose we are told that over a period of $40$ days, the plant had $38$ workers who produced $520$ units. Assume we are given the following number of working days per month $($January to August$)$: $22,16,23,20,21,22,21,22.$ The holding cost $($per unit per month$)$: $$8\mathrm{.}50,$ hiring cost per worker: $$800,$ firing cost per worker: $$1,250,$ payroll cost: $$\frac{75}{?}$ worker$/$day$,$ and shortage cost: $$50$ unit short$/$month$.$ The company has $40$ workers at the end of December.

We solved this problem in class and obtained a value of $K=0\mathrm{.}3421$ units$/$worker$/$day$.$ In addition, we developed a constant workforce plan with $46$ workers over the planning horizon and estimated the total cost of the plan to be $$599,590\mathrm{.}50.$ We discussed that the cost is very high due to the high inventory level. Hence, we decided to reduce the workforce to reduce the production and the unneeded inventory.

Develop a production plan with constant workforce over January to March, reduce it to be constant over April and May, and finally reduce it again to be constant over June to August without any shortage of demand. Estimate the workforce levels over these three periods and the new total cost of the developed plan.