Mathematical Problems in Engineering

Volume 2019, Article ID 2825854, 14 pages

https://doi.org/10.1155/2019/2825854

## Multiobjective Genetic Algorithms for Reinforcing Equal Population in Congressional Districts

^{1}Universidad Autónoma Metropolitana-Iztapalapa, Departamento de Ingeniería Eléctrica, Av. San Rafael Atlixco 186, Col. Vicentina, Del. Iztapalapa, México City 09340, Mexico^{2}Universidad Autónoma Metropolitana-Azcapotzalco, Departamento de Sistemas, Av. San Pablo 180, Colonia Reynosa Tamaulipas, México City 02200, Mexico

Correspondence should be addressed to Eric Alfredo Rincón-García; xm.mau.munax@nocnir

Received 17 May 2019; Revised 14 August 2019; Accepted 5 September 2019; Published 1 October 2019

Academic Editor: Benjamin Ivorra

Copyright © 2019 Alejandro Lara-Caballero et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Redistricting is the process of partitioning a set of basic units into a given number of larger groups for electoral purposes. These groups must follow federal and state requirements to enhance fairness and minimize the impact of manipulating boundaries for political gain. In redistricting tasks, one of the most important criteria is equal population. As a matter of fact, redistricting plans can be rejected when the population deviation exceeds predefined limits. In the literature, there are several methods to balance population among districts. However, further discussion is needed to assess the effectiveness of these strategies. In this paper, we considered two different strategies, mean deviation and overall range. Additionally, a compactness measure is included to design well-shaped districts. In order to provide a wide set of redistricting plans that achieve good trade-offs between mean deviation, overall range, and compactness, we propose four multiobjective metaheuristic algorithms based on NSGA-II and SPEA-II. The proposed strategies were applied in California, Texas, and New York. Numerical results show that the proposed multiobjective approach can be a very valuable tool in any real redistricting process.

#### 1. Introduction

The zone design problem consists in partitioning a given set of geographic units (GU’s) into *k* larger groups called zones in order to satisfy several criteria and constraints for a specific context. GU’s may represent diverse geographic spaces such as cities, counties, postal codes, or special geographic crafted areas of interest for a decision-maker (DM). The criteria can be the construction of zones with a specific shape and the same amount of clients or services among others. The zone design problem has a broad range of applications such as land use [1], commercial territory design [2], school districting [3], police districting [4], and service and maintenance zones [5]. Due to its complexity, which has shown to be NP-hard [6], there have been several heuristic approaches such as genetic algorithms [7], tabu search [8], and GRASP [9] to fulfill its objectives in reasonable time.

Most of the papers in the literature use an objective function that combines all objectives in a weighted sum, where the different weights represent their relative importance. The main problem of these methods is that nonconvex optimal solutions cannot be obtained by minimizing linear combination of the objectives; to find multiple solutions, the algorithm must be executed many times, and they are also sensitive to several characteristics of the search space such as discontinuity, multimodality, and nonuniformity [10]. The use of multiobjective algorithms and Pareto-based optimization techniques tries to avoid these pitfalls, but their use for the zone design problem is scarce, and their performance in this type of problems has not been fully investigated.

Regarding those few multiobjective approaches, we can find studies in GIS-based spatial zoning model [11], public service districting problem [12], commercial territory design [2], meter reading power distribution networks [13], planning of earthquake shelters [14], patrol sector [15], and health care system [16].

In this work, we focus on the political districting problem which is probably the most popular case of zone design due to its influence in democratic processes. It consists of grouping GU’s into a fixed number of zones according to several criteria such as contiguity, population equality, and compactness, to avoid political manipulation or gerrymandering. In political districting, there are several techniques reported to solve the problem. However, most of these techniques pose the political districting problem as a single-objective optimization technique [17, 18, 19]. There are scarce studies that deal with multiobjective approaches, for example, we can find the work of Guo et al. [20] where a multiobjective graph partitioning engine integrated with a Geographic Information System is proposed and tested in three cities of Australia. Ricca and Simeone [21] solve the political districting problem by a convex combination of the objectives and compare the behavior of four local search metaheuristics: descent, tabu search, simulated annealing, and old bachelor acceptance. Rincón-García et al. [22] present a multiobjective simulated annealing-based algorithm, where a set of weighting vectors are assigned to nondominated solutions, and the rejection of new solutions is based on the Metropolis criterion. Vanneschi et al. [23] provide a hybrid multiobjective method that combines a genetic algorithm with variable neighborhood and was tested on five US states. Caballero et al. [24] proposed a multiobjective algorithm based on an archived multiobjective simulated annealing (AMOSA) to solve the political redistricting problem, and the technique was applied on 12 datasets generated using electoral information from Mexico. It is important to mention that most of the aforementioned strategies considered a biobjective formulation of the problem where population equality and compactness are treated as objectives while contiguity is a side constraint.

Since its introduction in 1964 by the US Supreme Court, the equal population criterion has widely been studied and played a paramount role in American politics. In different real scenarios, population equality is considered essential and even small deviations from the ideal had been challenged in court. Much of the redistricting jurisprudence devotes to this feature, and there are several tribunal cases that set precedence within other criteria [25]. Additionally, the legislative body standards are quite strict, requiring equal population “as nearly as is practicable,” so districts must have approximately the same number of inhabitants or the redistricting plan could be rejected. There are several methods for measuring population equality. Generally, it is calculated by the sum of the absolute deviations of all the districts divided by the total number of districts. However, in some states, it is determined by the difference in population between the largest and the smallest district divided by the ideal number of inhabitants. Nevertheless, neither of these two measures on its own provide a full picture of the degree of population equality. For example, if we fixate (do not modify) the largest and the smallest districts, very different redistricting plans can be arranged, even though the best solution should be a redistricting plan where all the remaining districts are closely clustered around the ideal population. This analysis can only be done if both measures are examined simultaneously. Therefore, we propose a multiobjective approach where both measures, and compactness, are considered as objectives to optimize, whereas contiguity is incorporated as a constraint. We designed two multiobjective algorithms based on the nondominated sorting genetic algorithm II (NSGA-II) and the strength Pareto evolutionary algorithm II (SPEA-II) and evaluated their performance in three of the most populated states in the United States: California, Texas, and New York. The remainder of this paper is organized as follows: in Section 2, some relevant multiobjective optimization concepts are defined; the problem definition is provided in Section 3; Section 4 includes a description of the proposed multiobjective heuristic algorithms; in Section 6, the computational experiments are explained and discussed. Finally, conclusions and future lines of research are included in Section 7.

#### 2. Multiobjective Optimization

Many real-world scenarios involve simultaneously optimization of several objectives in order to solve a certain problem [26, 27, 28]. Formally, a general multiobjective optimization problem (MOP) can be formulated as [29]where is the feasible region in the decision space and is a vector of decision variables. consists of *m* objective functions, , where is the objective space.

The objectives in (1) are often in conflict with each other. Improvement of one objective may lead to the deterioration of another. Therefore, a single solution, which can optimize all objectives simultaneously, does not exist. Instead, in an MOP, the goal is to find the best trade-off solutions, called the Pareto optimal solutions, that are important to a DM. To define the concept of optimality for a multiobjective problem, the following definitions are provided.

*Definition 1. *Let *x* and *y* be vectors of decision variables such that , we say that *x* dominates *y*, denoted as , if and only if, for all .

*Definition 2. *A feasible solution of problem (1) is called a Pareto optimal solution, if and only if there is no other solution such that . The set of all efficient solutions or Pareto optimal solutions is called the Pareto set (PS), which is denoted as .

*Definition 3. *If is Pareto optimal, then is called as a nondominated point or an efficient point. The set of all nondominated points is referred to the nondominated frontier or the Pareto front (PF). The Pareto front is the image of the PS in the objective space, which is denoted as .

The main goal when solving an MOP is to provide the DM the so-called Pareto optimal set of Pareto optimal (or nondominated) solutions.

#### 3. Problem Definition

Political redistricting can be characterized as a multiobjective combinatorial optimization problem with criteria and constraints that fulfill democratic ideals. There are several criteria based on geographic, socioeconomic, and cultural attributes suggested in the literature to avoid political interference or gerrymandering, and some are usually imposed by law in different countries. However, there is a general consensus that population equality, contiguity, and compactness are fundamental in any electoral democratic process. In this work, we use a multiobjective model that considers population equality and compactness as objectives and contiguity as a constraint.

##### 3.1. Constraints

Let us define as a set of GU’s that must be grouped into *k* zones or districts. Let be the set of all the GU’s that belong to the *i*-th zone, and let *S* be a districting plan, . Then, each district can be defined through a set of binary variables such that if the *l*-th GU belongs to district and otherwise. In order to consider *S* a feasible districting plan, all zones must attain a contiguity property, meaning that each GU in a district can be connected to every other GU in the zone via GU’s that are also in the district. In addition, the following constraints must be satisfied:

Constraint (2) implies that each district must be nonempty; in other words, a district must contain at least one GU. Constraints (3) and (4) assure that each GU is assigned to exactly one zone. Finally, constraint (4) indicates that a districting plan must be complete in terms that all units are assigned to a district. Thus, a redistricting plan can be considered feasible if (**C1**) each district is connected, (**C2**) the number of districts is equal to *k*, and (**C3**) each GU is assigned to exactly one district.

##### 3.2. Population Equality

Population equality pursues the one man, one vote principle, and it seeks that districts should have about the same population. In this paper, we use two different population equality measures, a mean deviation and an overall range.

The mean deviation is equal to the absolute deviation of a district divided by the product of the total number of districts and the ideal population:where represents the population of the district *i*, is the average district population, and *k* is the number of districts to be designed.

The overall range is given by the difference in population between the largest and the smallest districts divided by the ideal population:where and represent the population of the largest and smallest districts, respectively, and is the average district population.

Equations (5) and (6) measure the level of population deviation among the districts. The lower the value for , or , the better population equality. In an ideal case, the number of inhabitants in each district is equal to the average district population, and in this case, equations (5) and (6) reach their lowest possible value of zero.

##### 3.3. Compactness

Compactness deals with the promotion of regular shapes among the districts. It is included to avoid the creation of irregular zones for political purposes. Although compactness is considered essential to prevent the creation of unfair political boundaries, there is no consensus to define it. There are different measures, or shape indices, proposed in the literature that can be used to quantify the compactness of a district, see [30, 31, 32] for surveys on compactness metrics. We decided to use a simple and widely used compactness measure that compares the area and the perimeter of each district as follows:where represents the perimeter and the area of district .

From equation (7), we can observe that the more compact all the districts are, the closer the cost is to 0.

#### 4. Heuristic Algorithms

In this section, we include a description of the proposed multiobjective algorithms for the redistricting problem. These algorithms are able to produce a set of nondominated redistricting plans that satisfy constraints **C1**–**C3**, whereas objectives , mean deviation, , overall range, and , compactness are minimized.

In order to obtain a good balanced design of the districts, we use two popular multiobjective evolutionary algorithms, namely, NSGA-II and SPEA-II. We first present the main framework for these multiobjective techniques. We then describe the tailored components for these genetic algorithms such as the solution encoding and genetic operators.

##### 4.1. NSGA-II

The nondominated sorting genetic algorithm II (NSGA-II) is a multiobjective metaheuristic originally proposed by Deb et al. [10]. NSGA-II blends the main characteristics of a genetic algorithm and the concept of Pareto dominance.

Firstly, NSGA-II creates a random parent population, , of size *N*. At each generation, solutions are ranked into several classes or fronts according to its nondomination level. All nondominated solutions are included in front level 1, or front 1. This front represents the best efficient set and is temporarily disregarded from the population. Iteratively, nondominated solutions are determined and assigned to front level 2, or front 2. This new front represents the second best efficient set. The process is repeated until the population is empty. This procedure is called fast nondominated sort.

In order to maintain population diversity, a second value called crowding distance is calculated for solutions that belong to the same nondominated front. This measure estimates population density around a solution in the objective space. The extreme points of each front are assigned with an infinite distance, so they are preserved and can introduce more dispersion in the population. As a consequence, every chromosome will have two attributes, the nondomination rank and a crowding distance.

Next, a binary tournament is applied. Two solutions are picked randomly from the population, and the winner is the lowest ranked individual; if the rank is the same for both, the winner will be the one with the highest crowding distance. This strategy is applied to select *N* pairs of parents, subsequently the crossover and mutation operators are applied to obtain a new population, . Finally, the fast nondominated sort and the crowding distance are applied to all solutions in , and the *N* best-ranked solutions are retained to the next population. This process is repeated a predefined number of generations. The pseudocode of NSGA-II is shown in Algorithm 1.