A farmer in the Midwest has $1,000$ acres of land on which she intends to plant corn, wheat, and

soybeans. Each acre of corn costs $$100$ for preparation, requires $7$ worker$-$days of labor, and

yields a profit of $$30.$ An acre of wheat costs $$120$ to prepare, requires $10$ worker$-$days of labor,

and yields $$40$ profit. An acre of soybeans costs $$70$ to prepare, requires $8$ worker$-$days, and

yields $$20$ profit. The farmer has taken out to loan of $$80,000$ for crop preparation and has con$-$

tracted with a union for $6,000$ worker$-$days of labor. A midwestern granary has agreed to pur$-$

chase $200$ acres of corn, $500$ acres of wheat, and $300$ acres of soybeans. The farmer has

established the following goals, in order of their importance:

$(1)$ To maintain good relations with the union, the labor contract must be honored; that is$,$ the

full $6,000$ worker$-$days of labor contracted for must be used.

$(2)$ Preparation costs should not exceed the loan amount so that additional loans will not have

to be secured.

$(3)$ The farmer desires a profit of at least $$105,000$ to remain in good financial condition.

$(4)$ Contracting for excess labor should be avoided.

$(5)$ The farmer would like to use as much of the available acceage as possible.

$(6)$ The farmer would like to meet the sales agrecment with the granary. However, the goal

should be weighted according to the profit returned by each crop.

a$.$ Formulate a goal programming model to determine the number of acres of each crop the

farmer should plant to satisfy the goals in the best possible way.

b$.$ Solve this model by using the computer.