Operations Research: Project
A farm family owns 640 acres of land. Its members can produce a total of 9,500 person-hours of labor during the year. If any of these person-hours are not needed, younger members of the family will use them to work on a neighboring farm for $4 per hour during the year. Payment for any working hour on the neighboring farm is paid at the end of the year.
The farm supports two types of livestock (daily cows and laying hens) as well as three crops (soybean, corn, and wheat). All three crops can be harvested to make money, but corn can also be used as feed crop for the cows and wheat can also be used as chicken feed.
Currently, the family has an investment fund of $20,000 coming from previous year that can be used to purchase more livestock. 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 the herd of cows will decrease by about 10 percent and the value of the flock of hens will decrease by about 25 percent due to aging. For instance, if family prefers to buy 10 cows, the value of the whole herd at the end of the year becomes ($35,000+$1,500*10)*0.9=$45,000.
Each cow will require 2 acres of land for gazing and 120 person-hours of work in a year, while producing a net annual cash income of $850. The size of the bam limits the herd to a maximum of 42 cows. The corresponding figures for each hen are as follows: no significant acreage, 0.6 person-hour in a year, and an annual net cash income of $4.25. The chicken house can accommodate a maximum of 5,000 hens.
For each acre planted in each of the three crops, the following table gives the number of person-hours of work that will be required during the year, as well as an estimate of the crop's net value (in either income or savings in purchasing feed for the livestock).
|
|
Soybean
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Corn
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Wheat
|
|
Person-hour required
|
2.4
|
2.1
|
1.3
|
|
Net value ($)
|
70
|
60
|
40
|
To provide the feed for the livestock, the family 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. This means if you spare some portion of the planted crop as animal feed, it cannot be used as a commodity at the same time.
Assuming that all decision are made at the beginning of the year and the revenues are collected at the end of the year, the family wishes to determine how much acreage should be planted in each of the crops and how many cows and hens to have at the beginning of the year to maximize the family's monetary worth at the end of the year (i.e. the sum of the net income from the livestock at the end of the year plus the net value of the crops at the end of the year plus income from working on the neighboring farm plus what remains from the investment fund plus the value of the livestock at the end of the year, minus living expenses of $40,000 for the year).
(a) Formulate a linear programming model for this problem.
(b) Solve the model with Excel Solver. Interpret your solution.
(c) If you have the chance to buy additional land at the price of $50 per acre, would you prefer to do so? What if the price would be $70 per acre?
(d) If you could get more capacity for the bam (for cows) or the chicken house (for hens), how much would you be willing to pay?
Provide the answers for these four questions in your written report.