Agricultural Studies 3300
Agricultural Systems Modeling I
Fall 2008
Assignment # 8
Due Friday, November 14th , 2008, At the start of class.
1.
(ANIMAL FEED MIX) The Cheval Blanc Stable feeds and houses the horses used to pull
tourist-filled calèches through the cobblestoned streets of Quebec’s historic Old City. The stable
owner recognizes the need to set a nutritional diet for the horses in his care. At the same time, he
would like to keep the overall daily cost of feed to a minimum.
The feed mixes available for the horses’ diet are an oat product, a highly enriched grain, and a
mineral product. Each of these mixes contains a certain amount of five ingredients needed daily
to keep the average horse healthy. The table for this problem (below) shows these minimum
requirements, units of each ingredient per kilo of feed mix, and costs for the three mixes.
In addition, the stable owner is aware than an overfed horse is a sluggish worker.
Consequently, he determines that 3 kilograms of feed per day is the most that any horse needs
to function properly. Formulate this problem and solve for the optimal daily mix of the three
feeds.
FEED MIX
DIET
OAT
ENRICHED MINERAL
MINIMUM DAILY
REQUIREMENT
PRODUCT
GRAIN
PRODUCT
REQUIRMENT
(INGREDIENTS)
(UNITS/KG) (UNITS/KG) (UNITS/KG) (UNITS)
A
B
C
D
E
Cost/kg
1
0.25
1.5
0.5
0.25
$0.18
1.5
0.5
2.5
0.75
0.25
$0.28
0.5
0.25
3
1
0.75
$0.34
3
1
4.5
4
2.5
Use Solver to create the Answer and Sensitivity Reports for this LP problem. Now answer the following
questions using these reports. Each question is independent of the others.
(a)
(b)
(c)
(d)
2.
If the price of grain decreases by $0.0l per kilo, will the optimal solution change?
Which constraints are binding? Interpret the shadow price for the binding constraints.
What would happen to the total cost if the price of the mineral product decreased
by 20% from its current value?
For what price range of oats is the current solution optimal?
(AGRICULTURAL PRODUCTION PLANNING) Margaret Kowalchuk’s family owns five
parcels of farmland broken into a southeast sector, north sector, northwest sector, west sector,
and southwest sector. Kowalchuk is involved primarily in growing wheat, alfalfa, and barley
crops and is currently preparing her production plan for next year. The Pennsylvania Water
Authority has just announced its yearly water allotment, with the Kowalchuk farm receiving 7400
acre-feet. Each parcel can only tolerate a certain amount of irrigation per growing season, as
specified in the following table:
PARCEL
Southeast
North
Northwest
West
Southwest
AREA (ACRES)
2000
2300
600
1100
500
WATER IRRIGATION
LIMIT (ACRE-FEET)
3200
3400
800
500
600
Each of Kowalchuk’s crops needs a minimum amount of water per acre, and there is a projected
limit on sales of each crop. Crop data follow:
CROP
MAXIMUM SALES
Wheat
Alfalfa
Barley
110 000 bushels
1 800 tons
2 200 tons
WATER NEEDED PER
ACRE (ACRE-FEET)
1.6
2.9
3.5
Kowalchuk’s best estimate is that she can sell wheat at a net profit of $2 per bushel, alfalfa at $40
per ton, and barley at $50 per ton. One acre of land yields an average of 1.5 tons of alfalfa and
2.2 tons of barley. The wheat yield is approximately 50 bushels per acre.
(a)
(b)
Formulate Kowalchuk’s production plan.
What should the crop plan be, and what profit will it yield?
Use Solver to create the Answer and Sensitivity Reports for this LP problem. Now answer the following
questions using these reports. Each question is independent of the others.
(a)
(b)
(c)
(d)
3.
Is this solution a unique optimal solution? Why or why not?
If there are alternative solutions, use Solver to identify at least one other optimal solution.
Would it help Margaret’s total profit if she could increase barley sales by 10%. If so, how?
How would the availability of more water affect Margaret’s total profit?
A farmer in China has a 100 acre farm on which to plant ginseng and garlic. Every acre planted with
ginseng requires 50 gallons of water per day and must be prepared for planting with 20 pounds of fertilizer.
Every acre of garlic requires 75 gallons of water a day and must be prepared with 15 pounds of fertilizer.
The farmer estimates it will take 2 hours of labour to harvest each acre of ginseng and 2.5 hours of labour
to harvest each acre of garlic. He believes that ginseng will sell for $3 a pound and garlic for about $1.
Each acre of ginseng is expected to yield 90 pounds and each acre of garlic is expected to yield 300 pounds.
The farmer can pump 6000 gallons a day for irrigation purposes from a shallow well. She can buy as much
fertilizer as he needs at a cost of $10 per 50 pound bag. Finally, the farmer can hire all the labour he needs
to help harvest at a rate of $5 per hour. If the farmer sells all she produces, how many acres of each crop
should she put in to maximize her profits?
Write this problem in standard form and solve this problem using the simplex method. Interpret solution,
the dual values and reduced costs. Verify your answers using Excel.
4.
At ag-fairs throughout rural Alberta, A man sells miniature tractors carved from wood. His sole objective
is to maximise profits from the production and sale of three types of toy tractors. The data below describe
the production hours per unit.
Type of Tractor Replica
Hours
Profit per Unit
Carving
Gluing
Painting
Minneapolis Moline
0.2
0.5
0.5
$4.00
John Deere B
0.5
0.5
0.2
$3.00
Massey Harris 55
1.0
0.3
0.2
$2.00
Max Time Available
90
40
60
Write this problem in standard form and solve this problem using the simplex method. Interpret solution,
the dual values and reduced costs. Verify your answers using Excel.