Question

A farm family owns 125 acres of land and has $40,000 in funds available for investment. Its members can produce a total of 3,500 person-hours worth of labor during the winter months (mid-October to mid-May) and 4,000 person-hours during the summer. If any of these person-hours are not needed, younger members of the family will use them to work on a neighborhood farm for $5.00/hour, during the winter months and $6.00/hour during the summer. Cash income may be obtained from three crops and two types of livestock: dairy cows and laying hens. No investment funds are needed for the crops. However, each cow will require an investment outlay of $1,200, and each hen will cost $9. These outlays should be considered as a one time capital investment and should not be included in computing the annual profitability of the farm.Each cow will require 1.5 acres of land, 100 person-hours of work from November to April, and 50 person-hours from May to October. 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 from November to April, 0.3 person-hours from May to October, and an annual net cash income of $5. The chicken house can accommodate a maximum of 3,000 hens, and the size of the barn limits the herd to a maximum of 32 cows.Estimated person-hours and income per acre planted in each of the three crops are Soybeans Corn OatsWinter person-hours 20 35 10Summer person-hours 50 75 40Net annual cash income ($) 500 750 350The family wishes to determine how much acreage should be planted in each of the crops and how many cows and hens should be kept to maximize its net cash income?

Answer 1

Explanation:

Obtaining the decision variables, constraints and objective functions, we have:

A. Decision Variables in the problem:

Ls = amount of area allocated for soyabean

Ac = amount of area allocated for corn

Ao = amount of area allocated for oat

Nc = number of cows purchased

Nh = number of hens purchased

E = Excess person-hours in the summer

B. Constraints in the following linear programming model:

As + Ac + Ao + 1.5Nc = 125 acres

1200Nc + 9Nh = 40000

Nc = 32, Nh = 3000

Labor limitation for winters,

20As + 35Ac + 10Ao + 100Nc + 0.6Nh + W = 3500

Labor limitation for summers,

50As + 75Ac + 40Ao + 150Nc + 0.9Nh + S = 4000

Now, the objective function that we need to maximize:

Objective Function, Z = 500As + 750Ac + 350Ao + 1000Nc + 5Hh + 5W + 6S