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) Define the decision variables for this problem. What type of vaiables must beused in this situation? Q2) Write out the objective function for this problem in terms of the decision variables defined above. Q3?Write out any constraints necessary for this problem in terms of the decision variables defined above. Q4) 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? Q5) 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

Q.1

Decision variables

We must maximize the expected usage (people / day) by constructing facilities subject to the cost and land area constraints.

Hence the decision variables would be number of facilities of each type.

Let

X1 = swimming pool,

X2 = tennis center,

X3 = athletic field,

X4 = gymnasium

These variable (X1, X2, X3, X4) must be non-zero integers.

Q2.

The objective is to maximize the expected usage (people / day), Hence the objective function is

Maximize Z = 600*X1 + 180*X2 + 800*X3 + 300*X4

Q3.

Constraints

70,000*X1 + 20,000*X2 + 50,000*X3 + 180,000*X4 <= 240,000

8*X1 + 4*X2 + 14*X3 + 6*X4 <= 24

X1 + X2 = 1

X1, X2, X3, X4 >= 0 and integer

Q4.

Modified model will be

Maximize Z = 600*X1 + 180*X2 + 800*X3 + 300*X4

subject to

70,000*X1 + 20,000*X2 + 50,000*X3 + 180,000*X4 <= 240,000

8*X1 + 4*X2 + 14*X3 + 6*X4 <= 24

X1 + X2 = 1

X4 >= X3 (this indicates the condition that gymnasium would be approved only if athletic field is approved)

X1, X2, X3, X4 >= 0 and integer