Two major types of components are being manufactured at a company. These are machined castings $($Product A$)$ and cylinders $($Product B$).$ The machines required for manufacturing include CNC lathes $($Machine $1),$ grinding machines $($Machine $2),$ and milling machines $($Machine $3).$ The shop operates in a single shift of eight hours. There is sufficient demand in the market to sell whatever the manufacturer may choose to produce. The relevant details are given in the table below.

$($a$)(9$ points$)$ Formulate the model $($that is$,$ define the objective and constraints in Excel table format including applicable formulas$)$

$($b$)(9$ points$)$ Set up the Solver model and execute it to answer the following questions.

$($c$)(8$ points$)$ What should be the daily production schedule for the manufacturer that earns them the most money? $($it is okay if the answer shows fractional units of each product$)$

$($d$)(8$ points$)$ What is the marginal value of an hour of CNC lathe time? Over what range is this price value appropriate?

$($e$)(8$ points$)$ The manager obtained additional Grinding machine time through better scheduling. How much additional grinding machine time can be effectively used for this operation? Why?

$($f$)(9$ points$)$ If a change in the market increased the profit on Product B from $$14$ to$$14\mathrm{.}25$ per unit, would the optimal quantities change? Would the value of the objective function change? If yes, what will be the new value?

$($g$)(9$ points$)$ If the amount of CNC lathe machine time available decreased by $10$ mins.$/$day$/$machine$,$ and the Milling machine time decreased by $30$ mins.$/$day$/$machine$,$ would this fall in the range of feasibility for multiple changes? If so$,$ what would the value of the objective function be$?$