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13. A farm family owns 125 acres of land and has $40,000 in found available for investment. Its members can produce a total of

3,500 person-hours worth of labor during the winter months (mid-September to mid-May) and 4,000 person-hours during the
summer. If any of these person-hours are not needed to farmer family, younger members of family will use their capacity to
work on a neighboring farm for $5/hour during the winter months and $6/hours during the summer. Cash income may be
obtained from three crops and two types of livestock: diary cows and lying hens. No investment founds are needed for corps.
However, each cow will require an investment outlay of $1,200, and each hen will cost $9.

Each cow will require 1.5 acres of land, 100 person-hours of work during the winter months, and other 50 person-hours during
the summer. Each cow will produce a net annual cash income of $1,000 for the family. The corresponding figures for each hen
are: no acreage, 0.6 person-hours during the winter, 0.3 more person-hours during the summer, and an annual net cash income
of $5. The chicken house can accommodate a maximu m of 3,000 hens, and the size of the barn limits the herd to a maximum
32 cows.

Estimated person-hours and income per acre planted in each of the three corps are as folows

Crop Soybeans Corn Oats
Winter person-hours 20 35 10
Summer person-hours 50 75 40
Net annual cash income ($) 600 900 450

The family wishes to determine how much acreage should be planted in each of the corps and how many cows and hens
should be kept to maximize its net cash income. Formulate the Linear Programming model for this problem.
A. Decision Variables: SL = Amount of acres allocated for soybean production
cL = Amount of acres allocated for corn production

oL = Amount of acres allocated for oat production
C = # of cows purchased
H = # of hens purchased
W = Excess person-hours in the winter
S = Excess person-hours in the summer

B. Constraints:

1) Total land allocated for crop production and cows limited by the available land of 125 acres
1255.1 ≤+++ CLLL ocS

2) Total cost of purchasing cows and hens must be less than $40,000
000,409200,1 ≤+ HC

3) Limitations on the size of barn and chicken house

4) Labor limitations
500,36.0100103520 =+++++ WHCLLL ocS è For winters
000,49.0150407550 =+++++ SHCLLL ocS è For summers

4) Non negativity constraints
0,,,,,, ≥SWHCLLL ocS

C. Objective function:

Z = total of crop income, income from animals, and income from neighbor farm
Maximize SWHCLLLZ ocS 655000,1450900600 ++++++=

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