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.

Abstract
Sectorization problems have significant challenges arising from the many objectives that must be optimised simultaneously. Several methods exist to deal with these many-objective optimisation problems, but each has its limitations. This paper analyses an application of Preference Inspired Co-Evolutionary Algorithms, with goal vectors (PICEA-g) to sectorization problems. The method is tested on instances of different size difficulty levels and various configurations for mutation rate and population number. The main purpose is to find the best configuration for PICEA-g to solve sectorization problems. Performance metrics are used to evaluate these configurations regarding the solutions’ spread, convergence, and diversity in the solution space. Several test trials showed that big and medium-sized instances perform better with low mutation rates and large population sizes. The opposite is valid for the small size instances. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.

Abstract
Sectorization consists of grouping the basic units of a large territory to deal with a complex problem involving different criteria. Resectorization rearranges a current sectorization avoiding substantial changes, given a set of conditions. The paper considers the case of the distribution of geographic areas of fire brigades in the north of Portugal so that they can protect and rescue the population surrounding the fire stations. Starting from a current sectorization, assuming the geographic and population characteristics of the areas and the fire brigades’ response capacity, we provide an optimized resectorization considering two objectives: to reduce the rescue time by maximizing the compactness criterion, and to avoid overload situations by maximizing the equilibrium criterion. The solution method is based on the Non-dominated Sorting Genetic Algorithm (NSGA-II). Finally, computational results are presented and discussed. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.

Abstract

Abstract
Sectorization is the division of a large area, territory or network into smaller parts considering one or more objectives. Dynamic sectorization deals with situations where it is convenient to discretize the time horizon in a certain number of periods. The decisions will not be isolated, and they will consider the past. The application areas are diverse and increasing due to uncertain times. This work proposes a conceptualization of dynamic sectorization and applies it to a distribution problem with variable demand. Furthermore, Genetic Algorithm is used to obtain solutions for the problem since it has several criteria; Analytical Hierarchy Process is used for the weighting procedure. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.

Abstract
Sectorization is concerned with dividing a large territory into smaller areas, also known as sectors. This process usually simplifies a complex problem, leading to easier solution approaches to solving the resulting subproblems. Sectors are built with several criteria in mind, such as equilibrium, compactness, contiguity, and desirability, which vary with the applications. Sectorization appears in different contexts: sales territory design, political districting, healthcare logistics, and vehicle routing problems (agrifood distribution, winter road maintenance, parcel delivery). Environmental problems can also be tackled with a sectorization approach; for example, in municipal waste collection, water distribution networks, and even in finding more sustainable transportation routes. This work focuses on sectorization concerning the location of the area's centers and allocating basic units to each sector. Integer programming models address the location-allocation problems, and various formulations implementing different criteria are compared. Methods to deal with multiobjective optimization problems, such as the e-constraint, the lexicographic, and the weighted sum methods, are applied and compared. Computational results obtained for a set of benchmarking instances of sectorization problems are also presented.