Ace Green Windows $($AGW$)$ manufactures environmentally efficient windows as replacements of those in existing homes. It has just received a contract for the next $6$ months, requiring $100,250,190,140,220,$ and $110$ units in months $1$ through $6,$ respectively. Production costs for windows vary with time due to the specialized materials. AGW estimates production will cost $$250,$ $$450,$ $$350,$ $$400,$ $$520,$ and $$500$ per unit in periods $1$ through $6,$ respectively. To take advantage of cost variations, AGW may produce more windows than the contract requires in some months and hold up to $375$ of them in inventory for later months. But holding inventory costs $$30$ per window per month. Assume there is no beginning inventory.

$($a$)$ Define symbolic parameters naming and indexing the constants described above.$($b$)$ Formulate a time$-$phased linear program to compute an optimal $($minimum total cost$)$ finite horizon, manufacturing and inventory plan for the next $6$ months using the parameters of part

$($c$).$ Provide a verbal explanation of the objective function and constraints.

$($d$)$ Enter and solve your model with an optimization software of your choice and submit the output with an explanation of the solution.