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 variables must be used in this situation? Q2) Write out the objective function for this problem in terms of the decision variables deine above. Q3) Write out any constraints necessary for this problem in terms of the decision variables defined above. Q4)Write out any constraints necessary for this problem in terms of the decision variables defined above. Q5) 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? Q6) 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

Ans 1> This problem is basically an assignment type of problem. Thus, we will be using binary decision variables.

So, the decision variables will be:

X1 = 1 or 0 if swimming pool is selected or not selected

X2 = 1 or 0 if tennis center is selected or not selected

X3 = 1 or 0 if the athletic field is selected or not selected

X4 = 1 or 0 if gymnasium is selected or not selected

Ans 2> The objective of this problem is to maximize the daily usage by the residents. Thus the objective function is:

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

Ans 3 or Ans 4> The constraints for the objective function are:

• (70000*X1)+(20000*X2)+(50000*X3)+(180000*X4)<=240000 { Budget Constraint}
• (8*X1)+(4*X2)+(14*X3)+(6*X4)<=24 {Land Constraint}
• X1+X2 = 1 { Either Swimming pool or tennis center can be constructed}

Ans 5> The statement means approval or disapproval of gymnasium and athletic facility happens together. Thus, in this case, we can have a common decision variable for both { i.e X3=X4}. And so, the total decision variable now becomes 3 rather than 4.

Ans 6> The limitation of constructing no more than 3 facilities can be incorporated in the above model by just adding one more constraint to the model. The constraint will be: X1+X2+X3+X4 <=3