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An Optimization Model for School Redistricting


An Optimization Model for School Redistricting

Rick Kohn

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A previous discussion proposed certain areas for improvement in a potential redistricting reform. The redistricting proposals had various flaws that were especially detrimental to the less advantaged population of the county.

The flaws resulted in plans that were characterized as maintaining or increasing the already high level of segregation in the county by income and race.

More low-income students and people of color were concentrated in the lowest income schools. In fact, length of time students ride buses was increased in order to maintain or increase the degree of segregation.

In order to test the feasibility of at least partially balancing for income, as well as capacity, we developed an alternative plan by hand.  This alternative plan and more examples of how the process increased segregation among schools were presented here.

Now we present a high school redistricting plan developed through computer optimization to demonstrate what districts would look like if balanced simultaneously for several factors (capacity utilization, income, distance).

The percentage of students receiving Free And Reduced Meals (FARM) was used as an indicator of the concentration of lower-income students. The variation in FARM% from school to school was minimized, at the same time that capacity utilization was distributed, and travel time to schools was minimized.

This plan implemented the mathematical modeling methodology described in the scientific literature previously prepared for Howard County Schools (AI Magazine, 2007, Volume 28)

This model minimized variance among schools in capacity utilization, FARM percentage, and minimized travel distance in miles by fastest route (according to Google Maps for travel from each polygon center to the best option of the closest 6 schools).

Capacity was calculated for the average over the five-year period from 2018 to 2023. This plan is shown in detail in the following graphs. Generally, this optimized plan does not always aggregate adjacent neighborhoods, and it results in some islands. These issues could be corrected by adjustment by hand, or by adding additional parameters to penalize for islands.

Table 1: Capacity Utilization, %FARM, Average Travel Time to School, and Average Distance from School for Optimized Plan

The results for this plan are shown by school district in Table 1, and the quantitative comparison to other plans is shown in Table 2. 

This table shows that one can develop a plan that is quantitatively superior to all recently proposed plans by objectively balancing for capacity utilization, FARM% distribution, and minimizing distance from schools.

Table 2: Summary of Results from Four High School Redistricing Plans

The various plans are compared in Table 2. The last row presents the optimized plan. Standard deviation (SD) represents the variation from school to school. Lower SD indicates less variation as desired. 

For example, the capacity standard deviation will be nearly 20% for the average of the next 5 years if nothing is done. This high standard deviation represents a large amount of variation from school to school with some schools having more than 120% of capacity and other at 80%. 

The figures for the plan are provided in the following series of Figures 1(A) to 1(L).

The Feasibility study resolved the capacity utilization problem, but left the standard deviation among FARM% at about 16%, with schools like Oakland Mills increasing in FARM percentage while other schools decreased. One can easily find examples of schools that become more balanced while others become less so, but the standard deviation shows the overall consistency in maintaining an imbalance among schools.

The computer optimized plan also decreased standard deviation among capacity and FARM percentage, and decreased the average time it takes to drive to the school compared to the Feasibility Study. Thus, these results show that it is feasible to decrease the standard deviation in FARM percentage among schools to no greater than 12%, and doing so could decrease travel time to school for the average student compared with current plans.

The final summary of the AAC report includes this statement:

“In terms of FARM distribution, our plan reduces the standard deviation across schools from 19.2 to 18.8.  In reviewing the policy target of driving a 25% reduction in deviation we feel that is not a feasibly achievable target. In fact, to achieve that target would require large-scale cross-county busing.  However, we are encouraged by the improvement achieved in our plan with minimal student moves compared to other current options. In terms of demographic distribution, our plan reduces the variation across groups at all three levels and across all demographic groups. The result is a more diverse student base distribution.”

We respect the work that the AAC committee undertook using inaccurate data and without expertise in statistics, mathematical modeling, or demographic analysis. The values in standard deviation they report are across all levels of schools, and our example is for high school only, but we believe similar results could be found across all levels. 

Standard deviation of FARM percentage across high schools was 16.3 for the pre-BoE action districts, and this would have decreased to 15.1 for the second AAC plan, which amounts to a 7.3% reduction in FARM SD. 

Our optimized plan decreased SD in FARM percentage for high schools from 16.3 to 11.9 for a 27.0% reduction. Thus, the target of 25% reduction was in fact feasible. 

Our plan also balanced capacity without moving walkers, and increased average distance per student by only 0.1 miles or 0.3 minutes driving time compared to current districts, which is less than all other plans. 

A future staff, with a strong background in planning, demographics, and statistics, and access to accurate data could develop district plans that balance capacity, and decrease SD in FARM %, without increasing travel time to schools.

The process that is being used to develop school districts in Howard County is inherently inequitable and unfair. Those who have the resources, time, and connections to organize a response to those plans are able to ensure that their neighborhoods are not adversely affected while those without resources cannot do the same.  In contrast, using quantitative statistical measures would enable development of optimal plans for the county in general.