**A Mixed
Integer Model Example**

Nancy
Smith has $250,000 to invest in three alternative investments – condominiums,
land, and municipal bonds. She wants to
invest in the alternatives that will result in the greatest return on
investment after one year.

Each
condominium costs $50,000 and will return a profit of $9,000 if sold at the end
of one year; each acre of land costs $12,000 and will return a profit of $1,500
at the end of one year; and each municipal bond costs $8,000 and will result in
a return of $1,000 if sold at the end of one year. In addition, there are only 4 condomininums, 15 acres of land,
and 20 municipal bonds available for purchase.

The
linear programming model for this problem is formulated as follows.

maximize Z = $9,000x_{1} +
1,500x_{2} + 1,000x_{3}

subject to

50,000x_{1} + 12,000x_{2} +
8,000x_{3} __<__ $250,000

x_{1} __<__ 4 condominiums

x_{2} __<__ 15 acres

x_{3} __<__ 20 bonds

x_{2} __>__ 0

x_{1}, x_{3} __>__ 0 and integer

where

x_{1} = condominiums purchased

x_{2} = acres of land purchased

x_{3} = bonds purchased