Heuristic Search and Information Visualization Methods for School

Redistricting

University of Maryland Baltimore County

Marie desJardins, Blazej Bulka, Ryan Carr, Andrew Hunt, Priyang Rathod, and Penny Rheingans

This work was partially supported by NSF # IIS-0414976.Thanks to David Drown and the Howard County Public School System for data and valuable inputs.

July 18, 2006 2

Overview

The Problem: School Redistricting

Searching for Good Plans

Results

Future Work and Conclusions

July 18, 2006 3

The Problem: School Redistricting

July 18, 2006 4

School Redistricting

Assign neighborhoods (or planning polygons) within a school district to schools while considering multiple factors, such as busing costs, test score distribution, and school utilization

Finding the best assignment (or plan) is a multiattribute optimization problem

Also want to generate qualitatively different plans that represent tradeoffs among the criteria, and help users visualize these tradeoffs

Search space is very large: O(sp), where s is the number of schools (12 high schools; 30 elementary) p is the number of polygons (~263)

Currently in Howard County, Maryland, the process is almost entirely manual

July 18, 2006 5

Evaluation Criteria

Educational benefits for students Frequency with which students are redistricted Number and distance of students bused Total busing cost Demographics and academic performance of schools Number of students redistricted Maintenance of feeder patterns Changes in school capacity Impact on specialized programs Functional and operational capacity of school infrastructure Building utilization

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Evaluation Criteria

Educational benefits for students Frequency with which students are redistricted Number and distance of students bused Total busing cost Demographics and academic performance of schools Number of students redistricted Maintenance of feeder patterns Changes in school capacity Impact on specialized programs Functional and operational capacity of school infrastructure Building utilization

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Evaluation Criteria

1. Number of students bused Students who can walk to a school should be assigned to that school

2. Busing cost Estimated as a population-weighted sum of polygon-school distances

3. Demographics FARM (Free and Reduced Meal) ratio at each school should ideally be the

same as that of the county as a whole

4. Academic performance MSA (Maryland State Assessment) scores at each school should ideally be

the same as those of the county as a whole

5. Capacity Each school should be between 90% and 110% of available capacity

Penalty functions are defined for each of the five criteria above per-polygon, per-school, per-plan cost measure from 0 (good) to 1 (bad)

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Selecting Multiple Plans: Diversity

One plan dominates another if it is better along all dimensions

Two plans are incomparable if each is better than the other along at least one dimension

A good set of plans should: contain no dominated plans consist of qualitatively different plans

We measure “qualitatively different” using Euclidean distance in the evaluation space: Div(P) = 1/|P| pi, pj P Dist(pi, pj)

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Closest-School Plan

Marriottsville High(new)

July 18, 2006 10

Closest [outer] vs. Recommended [inner]

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Searching for Good Plans

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Multiattribute Optimization

Previous approaches to multiattribute optimization: Weighted methods: Combine attributes into a single weighted sum Priority-based methods: Optimize one attribute, then perform

constrained optimization on the other attributes MOA* (and variations): Find all nondominated solutions using

heuristic search Evolutionary methods: Use genetic search to explore the

population space using recombination and fitness-based selection

Redistricting domain: No single set of weights or prioritization scheme Very large search space can’t find all (or even most)

nondominated solutions Use local search

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Basic Hill-Climbing

Baseline: Choose an initial plan as a starting point (seed) then hill-climb through “[weighted] sum of criteria” space

Seed options: closest-school plan current plan random plan “breadth-first” assignment “minimum-spanning-tree” assignment

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Biased Hill-Climbing

General approach: Choose an initial plan as a seed Hill-climb through “dominated plan” space Save “incomparable plans” as they are encountered Stop at a local maximum

Restart search starting from a plan in the incomparable list

Blind bias: At restart, choose a plan from the incomparable list at random

Diversity bias: At restart, choose the plan that is farthest in evaluation space

from the solutions found so far

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Results

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Quality of Generated Plans

Quality of generated plans is better than manually generated plans ...with respect to the particular evaluation criteria we’ve defined

Original-plan seed does better than closest-plan seed leads to the “wrong” local maximum

Compared to recommended plan, generated plans generally perform: ...better with respect to capacity ...comparably with respect to socioeconomic and academic

measures ...better with respect to busing costs ...comparably with respect to walk utilization

Outer: RecommendedInner: Best generated

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Diversity of Generated Plans

Baselines: Manual plans: closest, recommended, and alternative

(diversity measure = 0.223 ) Unweighted hill-climbing: multiple runs of basic hill-climbing with

different initial seeds (diversity measure = 0.044 ) Weighted hill-climbing: multiple runs of basic hill-climbing with

different weight vectors (diversity measure = 0.048 )

Biased hill-climbing: with blind bias: diversity measure = 0.032 with diversity bias: diversity measure = 0.389

Note: Biased hill-climbing yields a somewhat worse average unweighted sum, so the plans are not quite “as good” in a direct comparison

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Future Work and Conclusions

July 18, 2006 19

Future Work

Modeling additional evaluation criteria: Feeder statistics Redistricting frequency

Incorporating projected future demographic shifts into evaluation, search, and visualization

Extensions to search methods: Other definitions of diversity (e.g., dispersion, similar to k-means

mean-squared error) Other multiattribute optimization methods (particularly genetic

methods) Visualization extensions:

Visualizing feeder patterns Computing and visualizing gradients in search space

Deployment and user testing

July 18, 2006 20

Conclusions

School redistricting is an important and challenging problem

The multiattribute optimization framework is a good paradigm for this application

Novel search techniques and evaluation methods are needed

Diversity-biased hill climbing is a promising initial approach