Hercules Steel Company produces three grades of steel: high, good, and regular grade. Each of these products $($grades$)$ has high demand in the market, and Hercules is able to sell as much as it can produce of all three. The furnace operation is a bottleneck in the process and is running at $100\%$ of capacity. Hercules wants to improve steel operation profitability. The variable conversion cost is $$15$ per process hour. The fixed cost is $$200,000.$ In addition, the cost analyst was able to determine the following information about the three products:

$\backslash$table$[[$Category$,$High Grade,Good Grade,Regular Grade$],[$Budgeted units produced,$4,900,4,900,4,900],[$Total process hours per unit,$11,10,9],[$Furnace hours per unit,$4,3,2\mathrm{.}5],[$Unit selling price,$$270,$$$260,$$$240],[$Direct materials cost per unit,$$85,$$$81,$$$75]]$

The furnace operation is part of the total process for each of these three products. Thus, for example, $4\mathrm{.}0$ of the $12\mathrm{.}0$ hours required to process High Grade steel are associated with the furnace.

a$.$ Determine the unit contribution margin for each product.

b$.$ Provide an analysis to determine the relative product profitability, assuming that the furnace is a bottleneck.