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Population equilibrium

As explained in Optimization Algorithms: AI techniques for design, planning, and control problems

A condition in which populations are distributed fairly and comparably across geographic districts.

Population equilibrium distributes residents fairly across geographic districts. A district is balanced when its total population falls between an acceptable lower bound and upper bound around the target average. section 8.8

A districting solution evaluates equilibrium by checking whether each district is:

  • Overpopulated when its total exceeds the upper bound. section 8.8
  • Underpopulated when its total falls below the lower bound. section 8.8
  • Appropriately sized when its total remains within both bounds. section 8.8
  • Optimal when the objective counts zero overpopulated or underpopulated districts. section 8.8

East Toronto districting

A genetic algorithm groups 16 East Toronto neighborhoods into three districts using population data, adjacency relationships, a margin of 6,000, and constraints for nonempty and contiguous districts. The resulting groups satisfy the population-balance objective. section 8.8

Listing 8.6 Solving a political districting problem using GA
import geopandas as gpd
import pandas as pd
import folium
 
data_url="https://raw.githubusercontent.com/Optimization-Algorithms-Book/  #1
Code-Listings/main/Appendix%20B/data/PoliticalDistricting/" 
 
toronto = gpd.read_file(data_url+"toronto.geojson")                        #2
neighborhoods = pd.read_csv(data_url+"Toronto_Neighborhoods.csv")          #3
 
range_limit = 16                                                           #4
toronto_sample = toronto.tail(range_limit)                                 #4
values = neighborhoods.tail(range_limit)                                   #4
values = values.join(toronto_sample["cartodb_id"])                         #4

Use the implementation listing to inspect preprocessing, population extraction, adjacency construction, genetic-algorithm parameters, and optimization. section 8.8

How Optimization Algorithms: AI techniques for design, planning, and control problems covers it

How deep it goes
Defines bounds and an objective, then implements population and feasibility checks within a genetic algorithm.
Examples
Uses 16 East Toronto neighborhoods, population records, adjacency data, three districts, and a 6,000-person margin.
What it assumes
Assumes neighborhood population and geometry data, while demonstrating how to derive adjacency relationships before optimization.
What you can do after
Apply it via a genetic algorithm to evaluate and construct geographically connected, population-balanced electoral districts.
How it connects
Links political districting with genetic algorithms, geographic adjacency, constraints, and objective functions.
Cover of Optimization Algorithms: AI techniques for design, planning, and control problems
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