Problem $2.$ A furniture manufacturer produces four products $($Desks$,$ Tables, Chairs, and Bookshelves$).$ The following table shows the amount of time needed in each department to produce the furniture, the available time, and the profit per unit.

Note the units of the required times are in minutes and available time in hours!

$\backslash$table$[[$Departme$,$Time Requirements $($Minutes$),\backslash$table$[[$Availabl$],[$e Hours$]]],[$Desk$,$Table,Chair,Bookshelf$],[$Assembly$,30,40,80,60,200],[$Painting$,80,20,40,20,300],[$Finishing$,40,60,20,40,220],[$Packaging$,22,25,30,35,200],[$Profit Per,$$61\mathrm{.}00,$$$95\mathrm{.}00,$$$60\mathrm{.}00,$$$80\mathrm{.}00,]]$

Write the linear programming model for this problem and use Excel to solve it$.$ Save the sensitivity analysis and answer the following questio

a$.$ What is the optimum solution, and what is the value of the objective function?

b$.$ Which constraints are binding, and which have excess capacity?

c$.$ Explain the meaning of all the values under the Reduced Cost column. Be very specific.

d$.$ If the objective function coefficient of a desk $($profit per unit$)$ is increased by $$3\mathrm{.}50,$ will the optimal solution Change? Fully explain.

e$.$ Suppose the profit per unit for tables is increased by $$1,$ for chairs is decreased by $$2,$ and for bookshelves is increased by $$3.$ Would the optimum solution change? What will be the new value of the objective function? $($Range of Optimality and the $100\%$ Rule$)$

f$.$ Identify the range of feasibility for the right$-$hand$-$side values and fully explain their meanings.

g$.$ Fully interpret the meaning of the Shadow Prices $($Dual Prices$).$

h$.$ A painter has offered his services for painting on this project at a price below the minimum wage. Should we obtain the hours? Why or why not? Fully explain.

i$.$ If you could get additional time in all three departments, which department should get priority and why?

j$.$ Twenty hours of finishing time can be purchased at $25$ cents per minute above the current cost. Should we obtain the additional hours? If yes, what will be our total profit after paying for the time used.

k$.$ Refer to part $"$a$"$ of this problem. One of the company's best clients has requested $10$ chairs. If the company decides to produce the requested units of chairs, what would be the new total profit?