$1.$ Consider the following three$-$period inventory problem. At the beginning of each period a firm must determine how many units should be produced during the current period. During a period in which $\backslash ($ x $\backslash )$ units are produced, a production cost $\backslash ($ c$($x$)\backslash )$ is incurred, where $\backslash ($ c$(0)=0\backslash ),$ and for $\backslash ($ x$>0,$ c$($x$)=3+2$ x $\backslash ).$ Production during each period is limited to at most $4$ units. After production occurs, the period's random demand is observed. Period $1$ demand is $1$ unit, Period $2$ demand is $2$ units Period $3$ demand is equally likely to be $1$ or $2$ units. After meeting the current period's demand out of current production and inventory, the firm's end$-$of$-$period inventory is evaluated, and a holding cost of $\backslash (\backslash$$ $1\backslash )$ per unit is assessed. Because of limited capacity, the inventory at the end of each period cannot exceed $3$ units. It is required that all demand be met on time. Any inventory on hand at the end of period $3$ can be sold at $\backslash (\backslash$$ $2\backslash )$ per unit. At the beginning of period $1,$ the firm has $1$ unit of inventory. Use dynamic programming to determine a production policy that minimizes the expected net cost incurred during the three periods.