Abstract
This paper compares the non-dominated sorting genetic algorithm (NSGA-II) and NSGA-III to solve multiobjective sectorization problems (MO-SPs). We focus on the effects of the parameters of the algorithms on their performance and we use statistical experimental design to find more effective parameters. For this purpose, the analysis of variance (ANOVA), Taguchi design and response surface method (RSM) are used. The criterion of the comparison is the number of obtained nondominated solutions by the algorithms. The aim of the problem is to divide a region that contains distribution centres (DCs) and customers into smaller and balanced regions in terms of demands and distances, for which we generate benchmarks. The results show that the performance of algorithms improves with appropriate parameter definition. With the parameters defined based on the experiments, NSGA-III outperforms NSGA-II. © 2020 IEEE.

Abstract
In this study, two novel stochastic models are introduced to solve the dynamic sectorization problem, in which sectors are created by assigning points to service centres. The objective function of the first model is defined based on the equilibration of the distance in the sectors, while in the second one, it is based on the equilibration of the demands of the sectors. Both models impose constraints on assignments and compactness of sectors. In the problem, the coordinates of the points and their demand change over time, hence it is called a dynamic problem. A new solution method is used to solve the models, in which expected values of the coordinates of the points and their demand are assessed by using the Monte Carlo simulation. Thus, the problem is converted into a deterministic one. The linear and deterministic type of the model, which is originally non-linear is implemented in Python's Pulp library and in this way the generated benchmarks are solved. Information about how benchmarks are derived and the obtained solutions are presented.

Abstract
This paper deals with multi-objective location-routing problems involving distribution centres and a set of customers. It proposes a new two-stage solution method that comprehends the concept of sectorization. Distribution centres are opened, and the corresponding opening cost is calculated. A subset of customers is assigned to each of them and, in this way, sectors are formed. The objective functions in assigning customers to distribution centres are the total deviation in demands of sectors and the total deviation in total distance of customers from centroid of sectors, which must be minimized Afterward, a route is determined for each sector to meet the demands of customers. At this stage, the objective function is the total distance on the routes in the sectors, that must be minimized Benchmarks are defined for the problem and the results acquired with the two-stage method are compared to those obtained with NSGA-II. It is observed that NSGA-II can achieve many non-dominated solutions.

Abstract
The process of sectorization aims at dividing a dataset into smaller sectors according to certain criteria, such as equilibrium and compactness. Sectorization problems appear in several different contexts, such as political districting, sales territory design, healthcare districting problems and waste collection, to name a few. Solution methods vary from application to application, either being exact, heuristics or a combination of both. In this paper, we propose two quadratic integer programming models to obtain a sectorization: one with compactness as the main criterion and equilibrium constraints, and the other considering equilibrium as the objective and compactness bounded in the constraints. These two models are also compared to ascertain the relationship between the criteria.

Abstract
In sectorization problems, a large district is split into small ones, usually meeting certain criteria. In this study, at first, two single-objective integer programming models for sectorization are presented. Models contain sector centers and customers, which are known beforehand. Sectors are established by assigning a subset of customers to each center, regarding objective functions like equilibrium and compactness. Pulp and Pyomo libraries available in Python are utilised to solve related benchmarks. The problems are then solved using a genetic algorithm available in Pymoo, which is a library in Python that contains evolutionary algorithms. Furthermore, the multi-objective versions of the models are solved with NSGA-II and RNSGA-II from Pymoo. A comparison is made among solution approaches. Between solvers, Gurobi performs better, while in the case of setting proper parameters and operators the evolutionary algorithm in Pymoo is better in terms of solution time, particularly for larger benchmarks. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.

Abstract
Sectorization refers to partitioning a large territory, network, or area into smaller parts or sectors considering one or more objectives. Sectorization problems appear in diverse realities and applications. For instance, political districting, waste collection, maintenance operations, forest planning, health or school districting are only some of the application fields. Commonly, sectorization problems respect a set of features necessary to be preserved to evaluate the solutions. These features change for different sectorization applications. Thus, it is important to conceive the needs and the preferences of the decision-makers about the solutions. In the current paper, we solve sectorization problems using the Genetic Algorithm by considering three objectives: equilibrium, compactness, and contiguity. These objectives are collected within a single composite objective function to evaluate the solutions over generations. Moreover, the Analytical Hierarchy Process, a powerful method to perceive the relative importance of several objectives regarding decision makers' preferences, is used to construct the weights. We observe the changes in the solutions by considering different sectorization problems that prioritize various objectives. The results show that the solutions' progress changed accurately to the given importance of each objective over generations.