Question

A small town wants to build several new recreational facilities. The proposed facilities include an outdoor swimming pool, a fitness center, two basketball courts, a baseball field, a beach volleyball court, and a soccer field. The town council wants to provide the facilities which will be used by the most people, but faces budget and flat land limitations. The town has $470,000 and 10 acres of land. Requirements: The pool requires locker room facilities which would be in the recreation center, so, if the swimming pool is built then the fitness center must also be built. If the soccer field is built the baseball field cannot be built. The council feels that at least two of the ball fields/courts must be built. The daily predicted weekly usage (no. of people) and cost of the facilities (in $1.000), land requirements are shown below. Cost Land requirement Facility Usage ($1,000) (acre) Swimming pool 400 100 Fitness center 500 200 Basketball courts 200 50 Baseball field 200 100 Soccer field 55 75 Beach volleyball court 24 20 Formulate (No Excel work required, do not simplify) Hint: Six variables, no linking constraints.

Answer 1

Let

Y_j = binary integer such that Y_j=1 when the j-th facility is considered, for j=1,2,...,6

['1' denotes Swimming pool; '2' denotes Fitness center, ...., '6' denotes Beach volleyball court]

Max Z = total usage = 400 Y₁ + 500 Y₂ + 200 Y₃ + 200 Y₄ + 55 Y₅ + 24 Y₆

Subject to,

100 Y₁ + 200 Y₂ + 50 Y₃ + 100 Y₄ + 75 Y₅ + 20 Y₆ <= 470 (budeget constraint)

1.0 Y₁ + 1.2 Y₂ + 0.6 Y₃ + 2.0 Y₄ + 4.0 Y₅ + 0.5 Y₆ <= 10 (land constraint)

Y₂ - Y₁ >= 0 (Req-1)

Y₅ + Y₃ <= 1 (Req-2)

Y₃ + Y₄ + Y₅ + Y₆ >= 2 (Req-3)

Y = {0, 1}⁶