Blueberries can be harvested using either a "wet" method or a "dry" method. Dry$-$harvested blueberries can be sold at a premium, while wet$-$harvested blueberries are used mainly for blueberry juice and bring in less revenue. Whitesville Cooperative must decide how much of its blueberry crop should be harvested wet and how much should be dry harvested.

Whitesville Cooperative has $10,000$ barrels of blueberries that can be harvested using either the wet or dry method. Dry blueberries are sold for $$34\mathrm{.}50$ per barrel and wet blueberries are sold for $$19\mathrm{.}50$ per barrel.

Once harvested, blueberries must be processed through several operations before they can be sold. Both wet and dry blueberries must go through dechaffing and cleaning operations. The dechaffing and the cleaning operations can each be run continuously for the whole season $($with a total of $2,000$ hours available for dechaffing and $3000$ hours available for cleaning$).$ Each barrel of dry blueberries requires $0\mathrm{.}18$ hours in the dechaffing operation and $0\mathrm{.}32$ hours in the cleaning operation.

Wet blueberries require $0\mathrm{.}04$ hours in the dechaffing operation and $0\mathrm{.}10$ hours in the cleaning operation. Wet blueberries must also go through a drying process. The drying process can also be operated continuously for the whole season $($with a total of $2,000$ hours available$),$ and each barrel of wet blueberries must be dried for $0\mathrm{.}22$ hours.

$($a$)$ Julian, a new intern at Whitesville who works under your supervision, has been tasked with converting the above statement into a linear programming model to determine the optimal number of barrels of blueberries to dry harvest and wet harvest such that sales are maximized. After hours of hard work over the past week, Julian has come up with the following formulation. $(12$ Points$)$

Minimize $34\mathrm{.}5$D $+19\mathrm{.}5$W

Subject to

$1$D $+1$W $<=10,000$ Total Harvest

$0\mathrm{.}18$D $+0\mathrm{.}04$W $<=2000$ Dechaffing

$0\mathrm{.}32$D $+0\mathrm{.}10$W $>=3000$ Cleaning

$0\mathrm{.}22$W $<=2000$ Drying

D $,$ W $<=0$

Double check the above linear programming formulation for accuracy and identify three areas where modifications are required. Re$-$write the modified linear programming formulation below.

$($b$)$ Use Excel Solver to find a solution for the modified linear programming model in part $($a$).(12$ Points$)$

How many barrels of blueberries should be dry harvested? $($Round your answers to the nearest integer.$)$

How many barrels of blueberries should be wet harvested? $($Round your answers to the nearest integer.$)$

What is the optimal value of the objective function?

$($c$)$ Based on your answers in part $($b$),$ which constraints are binding, and which are non$-$binding in the modified linear programming model? Why? Explain. $(8$ Points$)$

$($d$)$ Suppose that Whitesville is considering an increase in its cleaning capacity by hiring a third$-$party contractor. The contractor$$s quote for completing this task is $$50$ per hour. Should Whitesville Cooperative proceed forward with this plan? Why or why not? Explain. $(8$ Points$)$

$($e$)$ Whitesville Cooperative is also considering an increase in its dechaffing capacity by hiring a different third$-$party contractor. The contractor$$s quote for completing this task is $$50$ per hour. Should Whitesville Cooperative proceed forward with this plan? Why or why not? Explain. $(8$ Points$)$

$($f$)$ Add an additional constraint to the modified linear programming formulation in part $($a$),$ requiring that the optimal decision variables take integer values only. Use Excel Solver to find a solution to this linear programming model. $(16$ Points$)$

How many barrels of blueberries should be dry harvested?

How many barrels of blueberries should be wet harvested?

What is the optimal value of the objective function?

Are your answers different compared to part $($b$)?$ Why? Explain.