Question

The city of Asheville, NC is seeking to improve access to its parks. your team will formulate three optimization models to recommend locations for new parks. These conditions are hypothetical and inspired by real-world context on park planning. You will develop optimization models to recommend locations with the following characteristics: 1) Locate new parks to minimize the total demand-weighted distance all people will need to travel to their nearest park. 2) Locate new parks to minimize the furthest distance a person will need to travel to their nearest park. 3) Locate new parks while trading off the two objectives above. Required: Submit a formulation for one of the three optimization models above. Include sections for the following: • Sets • Parameters • Model formulation • Formulation description o The formulation description should be a paragraph that briefly notes what each constraint does. E.g., constraint set (4) limits ... Optional: If you would like to submit more than 1 formulation in this submission, I’ll give you ungraded feedback on these. Formulations for all three models will be submitted in deliverable 4. This is a chance for you to check your work on those before they are graded. Additional Modeling Context The eligible park locations are given in set , and the locations people live are given in set . Within set , there are existing parks within the city ( ⊂ ) and candidate parks ( ⊂ ) that could be located. People can either be closest to an existing park or one of the new parks you will recommend be located. Total demand-weighted distance represents the park distance multiplied by the number of people who have to travel that far. E.g., if 10 people live 1 mile from a park and 5 people live 3 miles from a park, the total demand-weighted distanced is 25 people-miles (= 10*1 5*3). Data is not needed for this submission, however if you would like to take a look, it is provided in the file “InputData.xlsx.” This file includes the elements of each set, the number of people who live in each location and the distance between location and parks .

Answer 1

The optimization model is designed to recommend park locations that minimize the total demand-weighted distance for all residents. It sets binary variables to indicate the closest park to each location and includes constraints for assigning people to parks and limiting the number of new parks.

Step-by-step explanation:

To address the first optimization model for recommending locations for new parks in Asheville, NC to minimize the total demand-weighted distance all people will need to travel to their nearest park, we can consider the following formulation based on sets and parameters:

Sets

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• № - set of possible park locations.
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• ℘ - set of people locations.
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• №existing - subset of existing parks within the city (№existing ⊂ №).
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• №candidate - subset of candidate locations for new parks (№candidate ⊂ №).

Parameters

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• dij - distance between location i and park j.
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• pi - number of people living in location i.

Model Formulation

Minimize ∑i ∈ ℘ ∑j ∈ № (pi × dij × xij), where xij is a binary variable indicating if location i is closest to park j.

Formulation Description

The objective function is designed to minimize total demand-weighted distances to parks. Every person-location has an associated closest park-location. The binary variable xij ensures that each person is only counted once for their closest park. Constraints include ensuring that each person is assigned to one park, and that parks selected from candidate locations do not exceed the number of new parks to be established.