Apples can be harvested using either a "wet" method or a "dry" method. Dry-harvested apples can be sold at a premium, while wet-harvested apples are used mainly for apple juice and generate less revenue. Good Apple Cooperative must decide how much of its Apple crop should be harvested wet and how much should be dry harvested.

Good Apple Cooperative has 10,000 barrels of apples that can be harvested using either the wet or dry method. Dry apples are sold for $34.50 per barrel and wet apples are sold for $19.50 per barrel.

Once harvested, apples must be processed through several operations before they can be sold. Both wet and dry apples 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 apples requires 0.18 hours in the dechaffing operation and 0.32 hours in the cleaning operation. Wet apples require 0.04 hours in the dechaffing operation and 0.10 hours in the cleaning operation.

Wet apples 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 apples must be dried for 0.22 hours.

(a) Julian, a new intern at Good Apple Cooperative 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 apples 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.

Minimize 34.5D + 19.5W

Subject to:

1D + 1W 10,000 Total Harvest

0.18D + 0.04W 2000 Dechaffing

0.32D + 0.10W 3000 Cleaning

0.22W 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).

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

How many barrels of apples 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.

(d) Suppose that Good Apple Cooperative is considering increasing its cleaning capacity by hiring a third-party contractor. The contractors quote for completing this task is $50 per hour. Should Good Apple Cooperative proceed forward with this plan? Why or why not? Explain.

(e) Good Apple Cooperative is also considering increasing its dechaffing capacity by hiring a different third-party contractor. The contractors quote for completing this task is $50 per hour. Should Good Apple Cooperative proceed forward with this plan? Why or why not? Explain.

(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.

How many barrels of apples should be dry harvested?

How many barrels of apples should be wet harvested?

What is the optimal value of the objective function?

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