Problem $1(14$ points$)$ Suppose you need to decide how much capacity to produce in the face of uncertain demand. Assume that it costs $$13$ to build each unit of capacity and each unit of lost sales due to insufficient capacity costs $$37.$ The demand profile is as follows: Demand Probability $80\mathrm{.}1290\mathrm{.}18100\mathrm{.}23110\mathrm{.}2120\mathrm{.}12130\mathrm{.}1140\mathrm{.}051.$ What is the expected demand? $(4$ points$)2.$ Assume your objective is to minimize the total cost $($built capacity$+$lost sales$).$ How much capacity should you build? Round to the nearest whole number. $($Hint: for each possible capacity$-$demand pair, lost sales is max$(0,$Demand$-$Capacity. Then determine expected loss sales for each possible capacity$)(6$ points$)3.$ For the capacity determined above, what is the expected utilization and probability of idle capacity? $(4$ points