Replace the March demand forecast $510$ by C$.$

During a $46-$day period when there were $120$ workers, the firm produced $1,700,000$ cookies. Assume that there are $100$ workers employed, and C cookies in inventory at the beginning of month $1($January$).$ Further assume that the inventory carrying cost cI $=$ $$0\mathrm{.}10/$cookie$/$month$,$ employee hiring cost cH $=$ $$100/$work$,$ and cost of firing one worker cF $=$ $$200/$worker$.$ It is required that the inventory level at the end of month $12($December$)$ is $2$C$.$

$(1)$ Mixed strategy $($allowing inventory and workforce changes from month to month$)$: Find an optimal plan $($using linear programming$)$ to meet all monthly demands $($assuming back order not allowed$)$ in order to minimize the total cost including employee hiring and firing costs and inventory carrying cost.

show all solutionThe Mr$.$ Meadows Cookie Company can obtain accurate forecasts for $12$ months based

on firm orders. These forecasts and the number of workdays per month are as follows: