$(35$ pts$.)$ In Preparation for winter season, a clothing company is manufacturing goose overcoats. The selling season for the company is only $4$ months long, and lasts from November $1$ through February $28.$ The sales division has forecasted the following demands for next year.

$\backslash$table$[[$Month$,$Demand$],[$November$,30],[$December$,40],[$January$,30],[$February$,20]]$

All overcoats sold by this store are purchased from outside sources. The unit purchasing cost is $$4$ per overcoat; however, the supplier will only sell in lots of $10,20,30,40,$ or $50$ units. Any orders for more than $50$ or less than $10$ will not be accepted. The ordering cost is a fixed cost of $$2.$ Due to large in$-$process inventories, the store will carry no more than $40$ units of coats in inventory at the end of any one month. Carrying charges are $$0\mathrm{.}2$ per overcoat per month. Assuming that it is desired to have both the beginning and ending inventory at zero, find an ordering policy that will minimize total seasonal costs using Dynamic Programming.

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