Bob McDonald owns a 300 acre farm near Roswell. He is planning the operations of his farm for the upcoming year. He has $200,000 available for investment during the coming year, and he estimates that he has 1500 labour hours available for Fall/Winter months (October-March), and 5000 labour hours available for the Spring/Summer months (April-September). If any of these labour hours are not required, they may be used in working at local farmers’ cooperative, at a rate of $16.35 per hour during the Fall/Winter months and $17.50 per hour during the Spring/Summer months/ flora obtains income from the production of wheat, soybeans, peaches, cows, lambs and providing unused labour hours to local farmers’ cooperative.
His estimates of the labour hour requirement, annual investment per acre from three crops are as follows:
|
|
Wheat |
Soybeans |
Peaches |
|
Fall/Winter Labour hr. requirement/acre |
20 |
15 |
not given |
|
Spring/Summer Labour hr. requirement/acre |
40 |
45 |
50 |
|
Annual income/acre |
$1700 |
$1650 |
$1800 |
|
Annual investment/acre |
$140 |
$120 |
$110 |
The corresponding estimates for the two types of livestock are as follows:
|
|
Cows |
Lambs |
|
Fall/Winter Labour hr. requirement/head |
15 |
5 |
|
Spring/Summer Labour hr. requirement/head |
10 |
4 |
|
Annual investment/head |
$350 |
$120 |
|
Annual income/head |
$650 |
$225 |
Additionally, McDonald estimates that each cow raised require 0.25 acres of land, and each lamb 0.15 acres of land. He would like to limit his cow herd to a maximum of 200 cows a year and lamb herd to a minimum of 400 lambs a year. He wants her investment in cows and lambs to be not more than 40% of the total investment, and he wants the lamb herd to be at least twice as big as the cow herd.
Formulate the linear programming model for Bob McDonald’s farm operations planning problem.