Statistics - Cranberries can be harvested using either a “wet” method or a “dry” method

Linear Programming

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 $32.50 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 0.004 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.

Given the following LP Formulation and Excel solution:

The objective function maximizes the total revenue generated from wet-harvested cranberries and dry-harvested cranberries. Each barrel of wet-harvested cranberries generates $17.50 in revenue; dry-harvested cranberries earn $32.50 per barrel. The first constraint ensures that no more than 5,000 total barrels of cranberries are harvested. The second constraint states that every barrel of wet-harvested cranberries requires 0.04 hours in the dechaffing operation, every barrel of dry-harvested cranberries requires 0.18 hours in the dechaffing operation, and there are 1008 dechaffing hours available. The third and fourth constraints represent similar requirements for the cleaning and drying operations.

Let W = barrels of cranberries harvested using wet method

Let D = barrels of cranberries harvested using dry method

Max 17.5 W + 32.5 D

W + D ≤ 5,000 Total Harvest

0.04 W + 0.18 D ≤ 1,008 Dechaffing

0.10 W + 0.32 D ≤ 1,008 Cleaning

0.22 W ≤ 1,008 Drying

W , D ≥ 0 Nonegativity

Use Excel Solver to Answer

a) How many barrels should be dry harvested? How many barrels should be wet harvested?

b) Supposed 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?

c) Interpret the shadow price for the constraint corresponding to the cleaning operation. How would you explain the meaning of this shadow price to management?