Applied Methods of Operations Research in Agriculture
Assignment Questions
1. Most of the marginal analyses that are important in agricultural economics are dependent onrelationships that involve nonlinearities. Why is linear programming (LP) such an important decision tool in agriculture? Discuss the areas of applications of LP in agriculture.
2.A cash grain farmer has 600 acres of cropland available on which she plans to grow corn andsoybeans in the following season. She has made some budgets, which take into account corn (x) andsoybeans (y). The gross margin for corn is $40 per acre and for soybeans is $45 per acre. She has a maximum of 750 hours of tractor time available at the peak planting periods for both crops. It takes 1 hour per acre for field operations for corn(x)and 1.5 hours per acre for soybeans (y). The maximumacreage she can use for corn is 400 acres. Her sole objective is to select a cropping plan that willmaximize net returns for this set of conditions in the season. Set up the general form of the LP model for this word problem.
3.A farmer owns 500 acres of land, which is suitable for growing corn, soybeans, and sunflowers. Hisexpectations are that the net profit from producing each crop is $55 per acre for corn, $60 per acre forsoybeans, and $50 per acre for sunflowers. He and his family can supply 3,000 hours per year inperforming all the farm operations necessary to grow these crops. In addition, he is endowed with theequivalent of 4,500 hours of tractor time necessary to grow these crops. Assume that the only resources necessary in crop production are land, labor, and tractor time.
|
Resource (unit)
|
Crop
|
Endowment
|
|
Corn
|
Soybeans
|
Sunflowers
|
|
Land (acres)
|
1.0
|
1.0
|
1.0
|
500
|
|
Labor (hours)
|
0.4
|
0.2
|
0.3
|
3000
|
|
Tractor (hours)
|
0.5
|
0.2
|
0.4
|
4500
|
Set up the LP model and prepare a plan that maximizes total profits.
4.A small-scale poultry industry grows broilers, layers, and turkeys, and sells them at a profit of $4, $5, and $6 respectively. The house is divided into three chambers separated by wooden bars to housethe three kinds of birds. The house can accommodate no more than 45 birds. The labor time required for broilers and layers is 3 hours each. The turkeys require 4 hours of labor time. The house can growa maximum of 20 broiler birds, and a maximum of 100 hours of labor are available. Formulate this Problem as an LP model to maximize the total profit. Compute the shadow prices (SP) and range ofoptimality for objective function coefficients.
5.A dairy farmer's cows need three nutrients (A, B, and C) to subsist and produce milk each day.Each cow must receive the equivalent of 100 units of nutrient A, 200 units of nutrient B, and 50 unitsof nutrient C in order to maximize milk output. The farmer can use any combination of three feedsVI, 12, and f3) in meeting these minimum requirements. The local feed dealer sells all three feeds, which have the following cost per pound and nutrient equivalents (for A, B, and C) per pound.
|
Feed
|
Nutrient per pound
|
Cost ($/pound)
|
|
A
|
B
|
C
|
|
f/
|
5
|
22
|
3
|
0.25
|
|
12
|
10
|
25
|
2
|
0.5
|
|
f3
|
7
|
12
|
5
|
0.27
|
Assume that the farmer's objective is to minimize the cost per cow of buying any combination of these three feeds that satisfies the daily nutrient requirement of the cows.
a) Solve this problem using the simplex method.
b) Your optimal solution should indicate that no amount of feed f2 should be purchased and fed tothe farmer's cows. By how much should feedf2's current price of $0.50 per pound decrease inorder for the optimal solution to change?
c) Your optimal solution should indicatethat some amount of feed f3 should be purchased and fedto the farmer's cows. By how much should feedf3's current price of $0.27per pound increasein order for the optimal solution to change?
d) What is the range of feasibility for the minimum requirement for nutrient A?
6. A company has three plants: a, b,and c,and there are two major distribution centers, dand e.In thecurrent quarter, factories a, b,and chave the capacities 1,000, 1,500, and 1,200 units, respectively.The demands of distribution centers d and eare 2,300 and 1,400 items, respectively. Thetransportation is conducted by truck at the cost of 8 cents per item per kilometer. Design atransportation problem to minimize transportation costs. The distances are listed (in kilometers):
|
d
|
e
|
SUPPLY
|
|
A
|
80
|
215
|
1000
|
|
B
|
100
|
108
|
1500
|
|
C
|
102
|
68
|
1200
|
|
DEMAND
|
2300
|
1400
|
3700
|
7. A producer wholesaler has three factories, and the fruit and vegetables it ships are supplied to fourdifferent distribution centers. The table below gives unit shipping costs ($) to each warehouse, alongwith factory capacities and warehouse demands
|
Factory
|
Warehouse
|
|
|
1
|
2
|
3
|
4
|
Capacity
|
|
Factory 1
|
0.40
|
0.80
|
0.30
|
0.60
|
800
|
|
Factory 2
|
1.60
|
0.40
|
1.20
|
1.00
|
1000
|
|
Factory 3
|
1.20
|
0.20
|
0.80
|
0.40
|
600
|
|
Demand
|
400
|
400
|
600
|
1000
|
|
Write the LP problem and determine a shipping schedule to minimize transportation costs.
8. Consider an agricultural input company which sells farm inputs such as fertilizer, feed, herbicide,seed, and so on to farmers. It has a team of nine salespeople who can cover nine territories. The company needs to assign each salesperson to one and only one of the nine territories. The ninesalespeople have different sales ability, and the president of the company estimates the following profitability for each salesperson in each territory:
|
Territory
|
Salespeople profits ($1000 per month)
|
|
S1
|
S2
|
S3
|
S4
|
S5
|
S6
|
S7
|
S8
|
S9
|
|
1
|
25
|
22
|
18
|
23
|
19
|
15
|
14
|
22
|
20
|
|
2
|
19
|
20
|
17
|
14
|
16
|
10
|
9
|
13
|
11
|
|
3
|
19
|
18
|
23
|
21
|
22
|
20
|
24
|
10
|
19
|
|
4
|
7
|
9
|
15
|
10
|
11
|
7
|
8
|
4
|
14
|
|
5
|
6
|
8
|
6
|
9
|
9
|
15
|
5
|
17
|
11
|
|
6
|
12
|
13
|
10
|
16
|
17
|
11
|
19
|
15
|
8
|
|
7
|
15
|
10
|
6
|
8
|
12
|
13
|
11
|
9
|
19
|
|
8
|
21
|
15
|
16
|
19
|
18
|
10
|
8
|
14
|
13
|
|
9
|
9
|
3
|
6
|
8
|
5
|
10
|
3
|
15
|
12
|
Set up the assignment model and determine the optimal solution to maximize total profits.
9.Suppose a farmer has four equally ranked goals: (1) spend 50 hours per week with his family, (2) work enough to earn $3,000 per week, (3) spend 15 hours per week volunteering at the local foodpantry, which supplies free food to low-income families, and (4) sleep for 70 hours per week. Assume that the farmer earns $50 per hour of work. Assuming that each goal is equally ranked, setup this exercise as a GP problem and solve it using Solver.
10. A farmer produces corn, wheat, and soybeans using three resources: hired labor, family labor,and machine time. Over a three-month production period, the farm is endowed with 1,200 hours of hired labor, 800 hours of family labor, 2,000 hours of machine time, and 1,000 acres of land. The resource requirements for each commodity are summarized below:
Resource Soybeans Wheat Corn Endowment
|
(Hours/unit of good)
|
|
|
|
|
Hired Labor
|
1.0
|
1.1
|
1.3
|
1,200 hours
|
|
Family Labor
|
0.7
|
0.6
|
0.8
|
800 hours
|
|
Machine Time
|
2.2
|
2.8
|
3.0
|
2.000 hours
|
|
Land
|
1.0
|
1.0
|
1.0
|
1,000 acres
|
Assume that the per acre unitprofit of the three commodities over the previous 10 periods is the following:
|
Observation
|
Soybeans
|
Wheat
|
Corn
|
|
1
|
420
|
186
|
400
|
|
2
|
390
|
198
|
200
|
|
3
|
194
|
194
|
300
|
|
4
|
188
|
220
|
90
|
|
5
|
182
|
222
|
400
|
|
6
|
170
|
240
|
380
|
|
7
|
184
|
200
|
150
|
|
8
|
180
|
242
|
50
|
|
9
|
172
|
254
|
420
|
|
10
|
160
|
258
|
384
|
|
Average
|
224
|
221
|
227
|
a. Use Excel or a statistical package to derive the variance-covariance matrix from the data above.
b. Formulate and solve the following QP problem using Solver minimize risk subject to a minimum expected profit constraint and all the other structural constraints given in this example. Trace out an E-V frontier by parametrically varying the RHS value for the minimumexpected profit constraint.