The Ploughman family farm owns and operates a 640-acre farm that has been in the family for several generations. The Ploughman’s always have had to work hard to make a decent living from the farm and have had to endure some occasional difficult years. Stories about earlier generations overcoming hardships due to droughts, floods, etc., are an important part of the family history. However, the Ploughman’s enjoy their self-reliant lifestyle and gain considerable satisfaction from continuing the family tradition of successfully living off the land during an era when many family farms are being abandoned or taken over by large agricultural corporations.
John Ploughman is the current manager of the farm while his wife Eunice runs the house and manages the farm’s finances. John’s father, Grandpa Ploughman, lives with them and still puts in many hours working on the farm. John and Eunice’s oldest children, Frank, Phyllis, and Carl, also are given heavy chores before and after school.
The entire family can produce a total of 4,000 person-hours worth of labor during the winter and spring months and 4,500 person-hours during the summer and fall. If any of these person-hours are not needed, Frank, Phyllis, and Carl will use them to work on a neighboring farm for $5 per hour during the winter and spring months and $5.50 per hour during the summer and fall.
The farm supports two types of livestock: dairy cows and laying hens, as well as three crops: soybeans, corn, and wheat. (All three are cash crops, but the corn also is a feed crop for the cows and the wheat also is used for chicken feed.) The crops are harvested during the late summer and fall. During the winter months, John, Eunice, and Grandpa make a decision about the mix of livestock and crops for the coming year.
Currently, the family has just completed a particularly successful harvest which has provided an investment fund of $20,000 that can be used to purchase more livestock. (Other money is available for ongoing expenses, including the next planting of crops.) The family currently has 30 cows valued at $35,000 and 2,000 hens valued at $5,000. They wish to keep all this livestock and perhaps purchase more. Each new cow would cost $1,500, and each new hen would cost $3.
Over a year’s time, the value of a herd of cows will decrease by about 10 percent and the value of a flock of hens will decrease by about 25 percent due to aging.
Each cow will require 2 acres of land for grazing and 10 person-hours of work per month, while producing a net annual cash income of $850 for the family. The corresponding figures for each hen are: no significant acreage, 0.05 person-hours per month, and an annual net cash income of $4.25. The chicken house can accommodate a maximum of 5,000 hens, and the size of the barn limits the herd to a maximum of 42 cows.
For each acre planted in each of the three crops, the following table gives the number of person-hours of work that will be required during the first and second halves of the year, as well as a rough estimate of the crop’s net value (in either income or savings in purchasing feed for the livestock).
Data per Acre Planted
Soybeans Corn Wheat
Winter and spring, person-hours 1.0 0.9 0.6
Summer and fall, person-hours 1.4 1.2 0.7
Net Value $70 $60 $40
To provide much of the feed for the livestock, John wants to plant at least 1 acre of corn for each cow in the coming year’s herd and at least 0.05 acre of wheat for each hen in the coming year’s flock.
John, Eunice, and Grandpa now are discussing how much acreage should be planted in each of the crops and how many cows and hens to have for the coming year. Their objective is to maximize the family's monetary worth at the end of the coming year (the sum of the net income from the livestock for the coming year plus the net value of the crops for the coming year plus what remains from the investment fund plus the value of the livestock at the end of the coming year plus any income from working on a neighboring farm, minus living expenses of $40,000 for the year).
(a) Identify the components of a linear programming model for this problem.
(b) Formulate this model.
(c) Obtain an optimal solution and generate the additional output provided for post-optimality analysis (e.g., the Sensitivity Report). What does the model predict regarding the family’s monetary worth at the end of the coming year?
(d) Find the allowable range to stay optimal for the net value per acre planted for each of the three crops.
The above estimates of the net value per acre planted in each of the three crops assumes good weather conditions. Adverse weather conditions would harm the crops and greatly reduce the resulting value. The scenarios particularly feared by the family are a drought, a flood, an early frost, both a drought and an early frost, and both a flood and an early frost. The estimated net values for the year under these scenarios are as follows:
Net Value per Acre Planted Net Value per Acre Planted Net Value per Acre Planted
Scenario Soybeans Corn Wheat
Drought -$10 -$15 $0
Flood $15 $20 $10
Early frost $50 $40 $30
Drought and early frost -$15 -$20 -$10
Flood and early frost $10 $10 $5
(e) Find an optimal solution under each scenario after making the necessary adjustments to the linear programming model formulated in part (b). In each case, what is the prediction regarding the family’s monetary worth at the end of the year?
(f) For the optimal solution obtained under each of the six scenarios [including the good weather scenario considered in parts (a) and (d)], calculate what the family’s monetary worth would be at the end of the year if each of the other five scenarios occur instead. In your judgment, which solution provides the best balance between yielding a large monetary worth under good weather conditions and avoiding an overly small monetary worth under adverse weather conditions.
Grandpa has researched what the weather conditions were in past years as far back as weather records have been kept, and obtained the following data.
Scenario Frequency
Good weather 40%
Drought 20%
Flood 10%
Early frost 15%
Drought and early frost 10%
Flood and early frost 5%
With these data, the family has decided to use the following approach to making its planting and livestock decisions. Rather than the optimistic approach of assuming that good weather will prevail [as done in parts (a) to (d)], the average net value under all weather conditions will be used for each crop (weighing the net values under the various scenarios by the frequencies in the above table).
(g) Modify the linear programming model formulated in part (b) to fit this new approach.
(h) Repeat part (c) for this modified model.
(i) Use a shadow price obtained in part (h) to analyze whether it would be worthwhile for the family to obtain a bank loan with a 10 percent interest rate to purchase more livestock now beyond what can be obtained with the $20,000 investment fund.
(j) For each of the three crops, use the post-optimality analysis information obtained in part (h) to identify how much latitude for error is available in estimating the net value per acre planted for that crop without changing the optimal solution. Which two net values need to be estimated most carefully? If both estimates are incorrect simultaneously, how close do the estimates need to be to guarantee that the optimal solution will not change?
(a) The components of the linear programming model for this problem are:
- Decision variables: The number of acres planted in each of the three crops and the number of cows and hens to have.
- Objective function: To maximize the family's monetary worth at the end of the year.
- Constraints: The total number of person-hours available, the land requirements for the livestock, the maximum capacity of the chicken house and barn, and the feed requirements for the livestock.
- Data: Various data such as net value per acre planted and the net income from the livestock.
(b) Formulate the linear programming model:
Let:
x1 = number of acres planted in soybeans
x2 = number of acres planted in corn
x3 = number of acres planted in wheat
c = number of cows to have
h = number of hens to have
Objective function:
Maximize: 850c + 4.25h + 70x1 + 60x2 + 40x3 + 20000 - 40000
Subject to:
1.0x1 + 0.9x2 + 0.6x3 + 10c + 0.05h <= 4000 (winter and spring person-hours)
1.4x1 + 1.2x2 + 0.7x3 + 10c + 0.05h <= 4500 (summer and fall person-hours)
2x1 + 42c <= 640 (land requirements for cows)
0.05h <= 30 (land requirements for hens)
x2 >= c (feed requirement of corn for cows)
0.05x3 >= h (feed requirement of wheat for hens)
x2 <= 42 (maximum capacity of the barn)
h <= 5000 (maximum capacity of the chicken house)
(c) Obtain an optimal solution and generate the additional output provided for post-optimality analysis:
The optimal solution can be obtained using a linear programming solver. The sensitivity report generated by the solver will provide information about the shadow prices (reduced costs) and allowable ranges for various parameters. The model's prediction regarding the family's monetary worth at the end of the year can be found by evaluating the objective function at the optimal solution.
(d) Find the allowable range to stay optimal for the net value per acre planted for each of the three crops:
The allowable range for the net value per acre planted for each crop can be determined by evaluating the shadow prices (reduced costs) for the corresponding decision variables. A positive shadow price indicates that the optimal solution is not affected by a change in the corresponding net value, while a negative shadow price indicates a change in the net value may affect the optimal solution.
(e) Find an optimal solution under each scenario after making the necessary adjustments to the linear programming model formulated in part (b):
To find the optimal solution under each scenario, the adjusted net values per acre planted need to be substituted into the objective function in part (b). The linear programming model can then be solved for each scenario to obtain the optimal solution. The prediction regarding the family's monetary worth at the end of the year can be found by evaluating the objective function at each optimal solution.
(f) Calculate the family's monetary worth at the end of the year under each of the other five scenarios:
For each of the optimal solutions obtained in part (e), substitute the net values per acre planted for the other five scenarios into the objective function to calculate the family's monetary worth at the end of the year. Compare the results under different scenarios to determine the best balance between a large monetary worth under good weather conditions and avoiding a small monetary worth under adverse weather conditions.
(g) Modify the linear programming model formulated in part (b) to fit the new approach:
In the modified approach, the average net value under all weather conditions will be used for each crop, weighing the net values under the various scenarios by their respective frequencies. The modifications can be made by adjusting the net values per acre planted in the objective function to reflect the weighted averages.
(h) Repeat part (c) for this modified model:
Solve the modified linear programming model to obtain an optimal solution and generate the sensitivity report. Evaluate the objective function at the optimal solution to predict the family's monetary worth at the end of the year.
(i) Use a shadow price obtained in part (h) to analyze whether it would be worthwhile for the family to obtain a bank loan with a 10 percent interest rate to purchase more livestock:
The shadow price for the investment fund can be used to assess whether it would be worthwhile for the family to obtain a bank loan. If the shadow price is greater than the interest rate (10 percent), it indicates that obtaining a loan to purchase more livestock would be beneficial in terms of increasing the family's monetary worth.
(j) Use the post-optimality analysis information to identify the latitude for error in estimating the net value per acre planted for each crop without changing the optimal solution:
The allowable range for each net value can be determined using the shadow prices (reduced costs) for the corresponding decision variables. The net values that have the largest absolute coefficients in the objective function, where the shadow prices have a non-zero value, need to be estimated most carefully. To guarantee that the optimal solution does not change, the estimates for these net values would need to be within a certain range, which can be determined by evaluating the allowable ranges provided in the sensitivity report.