**A 0-1
Integer Model Example**

A community
council must decide which recreation facilities to construct in its
community. Four new recreation
facilities have been proposed – a swimming pool, a tennis center, an athletic
field, and a gymnasium. The council
wants to construct facilities that will maximize the expected daily usage by
the residents of the community subject to land and cost limitations. The expected daily usage and cost and land
requirements for each facility follow.

Recreation Facility |
Expected Usage (people/day) |
Cost ($) |
Land Requirements (acres) |

Swimming pool Tennis
center Athletic
field Gymnasium |
300 90 400 150 |
35,000 10,000 25,000 90,000 |
4 2 7 3 |

The
community has a $120,000 construction budget and 12 acres of land. The swimming pool and tennis center
must be built on the same part of the land parcel; however, only one of
these
two facilities can be constructed. The
council wants to know which of the recreation facilities to construct in order
to maximize the expected daily usage.
The model for this problem is formulated as follows.

maximize
Z = 300x_{1} + 90x_{2} + 400x_{3} + 150x_{4}

subject
to

$35,000x_{1} + 10,000x_{2}
+ 25,000x_{3} + 90,000x_{4} __<__ $120,000

4x_{1} +
2x_{2} + 7x_{3} + 3x_{4} __<__ 12 acres

x_{1} + x_{2} __<__ 1 facility

x_{1},
x_{2}, x_{3}, x_{4} = 0 or 1

where

x_{1}
= construction of a swimming pool

x_{2}
= construction of a tennis center

x_{3}
= construction of an athletic field

x_{4}
= construction of a gymnasium