Cranberries can be harvested using either a "wet" method or a "dry" method. Dry$-$harvested cranberries can be sold at a premium, while wet$-$harvested cranberries are used mainly for cranberry juice and bring in less revenue. Fresh Made Cranberry Cooperative must decide how much of its cranberry crop should be harvested wet and how much should be dry harvested.

Fresh Made has $5,000$ barrels of cranberries that can be harvested using either the wet or dry method. Dry cranberries are sold for $$34\mathrm{.}50$ per barrel and wet cranberries are sold for $$19\mathrm{.}50$ per barrel. Once harvested, cranberries must be processed through several operations before they can be sold. Both wet and dry cranberries must go through dechaffing and cleaning operations. The dechaffing and the cleaning operations can each be run $24$ hours per day for the $6-$week season $($for a total of $1,008$ hours$).$ Each barrel of dry cranberries requires $0\mathrm{.}18$ hours in the dechaffing operation and $0\mathrm{.}32$ hours in the cleaning operation. Wet cranberries require $0\mathrm{.}04$ hours in the dechaffing operation and $0\mathrm{.}10$ hours in the cleaning operation. Wet cranberries must also go through a drying process. The drying process can also be operated $24$ hours per day for the $6-$week season, and each barrel of wet cranberries must be dried for $0\mathrm{.}22$ hours.

$($a$)$

Develop a linear program that Fresh Made can use to determine the optimal amount of cranberries to dry harvest and wet harvest. $($Let W $=$ barrels of cranberries harvested using wet method and D $=$ barrels of cranberries harvested using dry method.$)$

Max

s$.$t$.$

total harvest

dechaffing

cleaning

drying

W$,$ D $>=0$

$($b$)$

Solve the linear program in part $($a$).$ How many barrels should be dry harvested? How many barrels should be wet harvested? $($Round your answers to the nearest integer.$)$

wet barrels

dry barrels

$($c$)$

Suppose that Fresh Made can increase its dechaffing capacity by using an outside firm for this operation. Fresh Made will still use its own dechaffing operation as much as possible, but it can purchase additional capacity from this outside firm for $$500$ per hour. Should Fresh Made purchase additional dechaffing capacity? Why or why not?

There are about $416$ hours of dechaffing capacity which are not being used, so additional capacity is not necessary.

There are about $485$ hours of dechaffing capacity which are not being used, so additional capacity is not necessary.

Each additional hour of dechaffing will only improve revenue by about $$485.$ At $$500$ per hour it would not be profitable to purchase additional hours.

Each additional hour of dechaffing will only improve revenue by about $$416.$ At $$500$ per hour it would not be profitable to purchase additional hours.

$($d$)$

Interpret the dual value for the constraint corresponding to the cleaning operation. How would you explain the meaning of this dual value to management?

Each additional hour of cleaning capacity increases revenues $$68\mathrm{.}18.$ This is true for any cleaning capacity value between $592$ and $1,600.$

Each additional hour of cleaning capacity increases revenues $$12\mathrm{.}68.$ This is true for any cleaning capacity value greater than or equal to $523.$

Each additional hour of cleaning capacity increases revenues $$484\mathrm{.}73.$ This is true for any cleaning capacity value greater than or equal to $523.$

Each additional hour of cleaning capacity increases revenues $$416\mathrm{.}00.$ This is true for any cleaning capacity value between $592$ and $1,600.$