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 5000 barrels of cranberries that can be harvested using either the wet or dry method. Dry cranberries are sold for S32.50 per barrel and wet cranberries are sold for $17.50 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 1008 hours). Each barrel of dry cranberries requires 0.18 hours in the dechaffing operation and 0.32 hours in the cleaning operation. Wet cranberries require 0.04 hours in the dechaffing operation and 0.10 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 0.22 hours.
a. Develop a linear program that Fresh Made can use to determine the optimal amount of cranberries to dry harvest and wet harvest.
b. Solve the linear program in part (a). How many barrels should be dry harvested? How many barrels should be wet harvested?
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?
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?