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 = 300x₁ + 90x₂ + 400x₃ + 150x₄

subject to

$35,000x₁ + 10,000x₂ + 25,000x₃ + 90,000x₄ < $120,000

4x₁ + 2x₂ + 7x₃ + 3x₄ < 12 acres

x₁ + x₂ < 1 facility

x₁, x₂, x₃, x₄ = 0 or 1

where

x₁ = construction of a swimming pool

x₂ = construction of a tennis center

x₃ = construction of an athletic field

x₄ = construction of a gymnasium