PROBLEM $2$: $(20$ Marks$)$ Bangs Leisure Chairs produces three hand$-$crafted outdoor chairs

famous for beaches, pools, and patios: sling, Adirondack, and hammocks. The unit profit for

these products is $$40,$ $$70,$ and $$80,$ respectively. Each type of chair requires cutting,

assembling, and finishing. The owner is retired and willing to work six hours per day for five

days per week, so he has $120$ hours available each month. He does not want to spend more than

$50$ hours each month on any one activity $($that is$,$ cutting, assembling, and finishing$).$ The

retailer he works with ensures that all products he makes can quickly be sold. Sling chairs are made up of ten wood pieces for the frame and one piece of cloth. The actual cutting of the

wood takes $30$ minutes. Assembling includes sewing the fabric and the attachment of rivets,

screws, fabric, and dowel rods, which takes $45$ minutes. The finishing stage involves sanding,

staining, and varnishing the various parts, which takes one hour. Adirondack chairs take one

and a half an hour to cut and one hour to assemble, and finishing takes one hour. For hammocks,

cutting takes $0\mathrm{.}4$ hours, assembly takes $1\mathrm{.}5$ hours, and finishing takes one hour. How many of

each type of chair should he produce each month to maximize profit?

a$)$ Develop a linear optimization model for the problem on a spreadsheet in Excel. $(10$

marks$)$

b$)$ What is the optimal solution for the problem? $(2$ marks$)$

c$)$ Create a Sensitivity Report and answer the following questions:

$($i$)$ Suppose we forced sling chairs to be in the solution; what would happen to the

objective function$$s value? $(2$ marks$)$

$($ii$)$ What are the maximum and minimum values of the objective function coefficients

for sling chairs, Adirondack chairs, and hammocks for which the current optimal

solution is valid? $(2$ marks$)$

$($iii$)$ What do the shadow prices for the $$Assembling Hours Used$$ and $$Total Hours Used$$

constraints mean? $(2$ marks$)$

$($iv$)$ What are the maximum and minimum values for the cutting, assembly, and finishing

hours used constraints for which the current constraint value does not change? $(2$

marks