Question

The council wants to construct the facilities that will maximize expected daily usage by the residents of the community subject to land and cost llimitations. The usage, cost, and land data for each facility are listed below. The community has $240,000 construction budget and 24 arces of land. Because the swimming pool and tennis center must be built on the same part ot the land parcel, however, only one of these two facilities can be constructed. Formulate an optimization mode to assist the council in making its decision. Q1)Write out any constraints necessary for this problem in terms of the decision variables defined above. Q2) A very inluential citizens group has made it clear that it strongly prefers the athletic fielld to the gynnasim; thus the council will not approve the gymnasium unless the athletic field is also approved. How can the model be modified to incorporate this condition? Q3) The mayor has deicded that no more than three of the facilities may be constructed wiht funds from this year's budget. How can the model be modified to incorporate this comdition?

	expected usage
facility	people/day	cost($)	Land requirements (acres)
swimming pool	600	70,000	8
tennis center	180	20,000	4
athletic field	800	50,000	14
gymnassium	300	180,000	6

Answer 1

Q1)

LP model including the decision variables, objective function and constraints is following

Let S, T, A, G represent the binary variable such that it takes a value of 1, if Swimming pool, Tennis center, Athletic field or Gymnasium facility is to be constructed.

Objective: Max 600S+180T+800A+300G

s.t.

70000S+20000T+50000A+180000G <= 240000

8S+4T+14A+6G <= 24

S+T <= 1

S,T,A,G {0,1}

Q2) Following constraint has to be added for this requirement

G - A <= 0

Q3) Following constraint must be added for this requirement

S+T+A+G <= 3