Question

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 $36.50 per barrel and wet cranberries are sold for $15.50 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 0.18 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. (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?

(option 1) 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.

(option 2) There are about 416 hours of dechaffing capacity which are not being used, so additional capacity is not necessary.

(option 3) There are about 485 hours of dechaffing capacity which are not being used, so additional capacity is not necessary.

(option 4) 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?

(option 1) Each additional hour of cleaning capacity increases revenues $484.73. This is true for any cleaning capacity value greater than or equal to 523.

(option 2) Each additional hour of cleaning capacity increases revenues $416.00. This is true for any cleaning capacity value between 592 and 1,600.

(option 3) Each additional hour of cleaning capacity increases revenues $5.95. This is true for any cleaning capacity value greater than or equal to 523.

(option 4) Each additional hour of cleaning capacity increases revenues $95.45. This is true for any cleaning capacity value between 592 and 1,600.

Answer 1

Part a.

Subject to:

Let W = barrels of cranberries harvested using wet method and

D = barrels of cranberries harvested using dry method.)

The objective is to maximize the total profit of the product mix.

Max: $15.50W + $36.50D

Subject to:

total harvest: W + D <= 5000

dechaffing: 0.04W + 0.18D <= 1008

cleaning: 0.10W + 0.32D <= 1008

drying: 0.22W <= 1008

W, D ≥ 0

Part b.

The excel model:

Optimal Solution:

W = 2690.909 barrels

D = 2309.091 barrels

Profit = $125,990.91

The sensitivity report:

Variable	Value	Reduced Cost	Original Val	Lower Bound	Upper Bound
W	2690.909	0	15.5	11.406	36.5
D	2309.091	0	36.5	15.5	49.6
	Dual Value	Slack/Surplus	Original Val	Lower Bound	Upper Bound
total harvest	5.955	0	5000	3150	6300
dechaffing	0	484.727	1008	523.273	Infinity
Cleaning	95.455	0	1008	592	1600
drying	0	416	1008	592	Infinity

Part c:

According to sensitivity report slack of dechaffing resource is 484.727, it means the dechaffing capacity is not completely utilized and 484.727 hours are unutilized. Since there are capacity left with dechaffing resource, extra hours are not required.

Correct option: (option 3) There are about 485 hours of dechaffing capacity which are not being used, so additional capacity is not necessary.

Part d.

The dual value of the cleaning resource is 95.455. the lower bound of the cleaning hours is 592 hours and upper bound is 1600 hours. Thus the dual value is valid between the cleaning capacity of 592 hours to 1600 hours.

Correct ans:

(option 4) Each additional hour of cleaning capacity increases revenues $95.45. This is true for any cleaning capacity value between 592 and 1,600.