A farm family owns 100 acres of land and has $25,000 in funds available for investment. Its members can produce a total of 3500 work-hours worth of labor during the winter months (mid-September to mid-May) and 4000 work-hours during the summer. If any of these work-hours are not needed, younger members of the family will use them to work on a neighboring farm for $4.80 per hour during the winter and $5.10 per hour during the summer.
Cash income may be obtained from three crops (soybeans, corn, and oats) and two types of livestock (dairy cows and laying hens). No investment funds are needed for the crops. However, each cow requires an initial investment outlay of $400, and each hen requires $3. Each cow requires 1.5 acres of land and 100 work-hours of work during the winter months and another 50 workhours during the summer. Each cow produces a net annual cash income of $450 for the family. The corresponding figures for the hen are as follows: no acreage, 0.6 work-hour during the winter, 0.3 more workhour in the summer, and an annual net cash income of $3.50. The chicken house accommodates a maximum of 3000 hens, and the size of the barn limits the cow herd to a maximum of 32 head. Estimated workhours and income per acre planted in each of the three crops are as shown in the following table:

Formulate a mathematical model to assist in determining how much acreage should be planted in each of the crops and how many cows and hens should be kept maximizing net cash income.

a. Identify the decision variables: What decision is to be made?

b. Formulate the objective function: How do these decisions affect the objective?

c. Formulate the constraint set: What constraints must be satisfied? Be sure to consider whether negative values of the decision variables are allowed by the problem, and ensure they are so constrained if required.