The council wants to construct the facilities that will maximize expected daily

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) 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? Q2) 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? Q3) Define the decision variable for this problem. What tpe of variables must be used in this situation? Write out the objectivie funtion for this problem in terms of the decison variable defined above. Q5) Write out any constraints necessary for this problem in terms of the decision variables defined above.

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

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

Explanation / Answer

Thanks for sharing an interesting case with data to make a decision based on optimal utilization of resources and interests.

Formulate an optimization mode to assist the council in making its decision.

From the given data, Key factors to consider are maximum expected usage by residents that needs to be within budget and available land. I was able to come up with 4 different combinations( listed below) that could be evaluated and excluded to arrive at the optimal and usable decision which answers Q1 and Q2 collectively (explanation given)

Note:Cbnations are taken based on the conditions given in Q1 and Q2 ( Mayor's decision of not more than 3 facilities this year and a group of citizens favouring athletic field over gymnasium)

Decision 1 &3:are eliminated on the grounds of having to include Swimming Pool and tennis Center which is not being preferred and also over utilization of land and under utilization of Budget( Decision 1) & over utilization of Budget and under utilization of Land(Refer Decision 2 working)

Decision 4: Ruled out for over burning the budget and also the requirement for more land than what is alloted despite the fact that it serves the maximum number of residents when combined.

Recommendation on optimal mode -D2: Satisfies the conditions from Q1 & Q2 with a decent occupation with respect to number of people served and optimum usage of land with a need for additional 10,000$ in budget. So the counsil is advised to go with the combination given in Decision 2 by approving the extra in budget.

Q3. Define the decision variable for this problem, What type of variable must be used in this situation?

The Decision Variables are Athletic Club (A  must over Gym) and Swimming Pool &Tennis Center(Not to be combined) Also, Please refer explanations from above that supports the variables listed here.

Ojective Function: A model that would primarily serve the purpose by catering to a maximum number of residents under budget and land alloted with secondary importance given to the conditions asked to be seriously considered by the officials( Conditions from Q1 & Q2)

Constraints: Not to have Swimming pool and Tennis center together in this year's budget and land allotment + The request on prioritising Athletic Field over Gymnasium, are obviously considered quite seriously as the recommendation comes with an additional funding request.

Thanks & Cheers! Happy Learning!

Usage expected Facilities People/day Cost Land Requirement/facility Decision 1: Swimming Pool+ Tennis Center+Athletic Field 1580 140000 26 Decision 2: Tennis Center+Athletic Field + Gymnasium 1280 250000 24 Decision 3: Swimming Pool +Tennis Center + Gymnasium 1080 270000 18 Decision 4: Swimming Pool+Gymnasium +Athletic Field 1700 300000 28

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