$(20$ points$)$ Consider the same setting as Problem $2($with no backorder$)$ and suppose that the steel supplier offers the auto manufacturer a price of $$1490$ per ton of steel if $Q<mn\; id="EditorElement\_4485923">1</mn><mn\; id="EditorElement\_4485924">2</mn><mn\; id="EditorElement\_4485925">0</mn><mn\; id="EditorElement\_4485926">0</mn>$ tons; $$1220$ per ton if and $$1100$ per ton if $Q2400.$ The annual holding cost rate, $i,$ is $0\mathrm{.}25($as a percentage of the purchase cost$).$ Calculate $Q^{**}$ and $C(Q^{**})$ for the all$-$units discount structure.

$(20$ points$)$ The Pratt & Whitin Company, which manufactures aircraft engines, needs to decide how many units of a particular bolt to order in order to build engines over the next $4$ months. Orders for engines are placed over a year in advance, so the company knows its near$-$term demand exactly; in particular, the number of engines to produce in the next $4$ months will be $150,100,80,$ and $200$ in months $1$ through $4,$ respectively. Each engine requires a single bolt. Orders for bolts incur a fixed cost of $$120,$ and bolts held in inventory incur a holding cost of $$0\mathrm{.}80$ per bolt per month. Find the optimal order quantities in each month and the optimal total cost. $[$Hint$]$: Apply Wagner$-$Whitin Algorithm.