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-September 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	Oats
Winter person-hours	20	35	10
Summer person-hours	50	75	40
Net annual cash income ($)	500	750	350

The 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. Formulate the linear programming model for this problem.

Answer 1

In order to formulate the linear programming model, we need to find the decision variables, constraints and objective functions

A. Decision Variables in the problem

Ls = amount of area allocated for soyabean

Lc = amount of area allocated for corn

Lo = amount of area allocated for oat

C = number of cows purchased

H = number of hens purchased

E = Excess person-hours in the summer

B. Constraints in the following linear programming model

Ls + Lc + Lo + 1.5C <= 125 acres

1200C + 9H <= 40000

C<=32, H<=3000 (since the maximum values of hen and cow are given in the problem)

Labor limitation for winters

20Ls + 35Lc + 10Lo + 100C + 0.6H + W = 3500

Labor limitation for summers

50Ls + 75Lc + 40Lo + 150C + 0.9H + S = 4000

C. Now the objective function that we need to maximize

Objective Function (Z) = 500Ls + 750Lc + 350Lo + 1000C + 5H + 5W + 6S