Question

he 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? Please use Solver(EXCEL) to define Q2 and Q3

	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

1) The objective is to maximize the usage

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

Here X1, X2, X3 and X4 are construction of swimming pool, tennis center, athletic field and gymnasium respectively.

Here X1, X2, X3 and X4 = 0 or 1 (i.e. facility is constructed or not)

Subject to constraints:

70000 X1+ 20000 X2 + 50000 X3 + 180000 X4 240000 (Max budget)

8 X1+ 4 X2 + 14 X3 + 6 X4 24 (Max Land)

X1 + X2 = 1 (i.e. any one of swimming or tennis should be constructed)

2) Athletic field is mandatory thus a new constraint is shown below:

X3 = 1 (i.e. athletic field is mandatory)

3) No more than 3 facilities can be constructed thus a new constraint:

X1+ X2 + X3 + X4

Formulations for question 2.

The solver equations are shown below:

The solution is shown below:

We can see that Swimming pool and athletic field are constructed with :

• Total usage of 1400
• Total cost of $120000
• Total land of 22 acres