Abstract
Collecting and transporting solid waste is a constant problem for municipalities and populations in general. Waste management should take into account the preservation of the environment and the reduction of costs. The goal with this paper is to address a real-life solid waste problem. The case reveals some general and specific characteristics which are not rare, but are not widely addressed in the literature. Furthermore, new methods and models to deal with sectorization and routing are introduced, which can be extended to other applications. Sectorization and routing are tackled following a two-phase approach. In the first phase, a new method is described for sectorization based on electromagnetism and Coulomb's Law. The second phase addresses the routing problems in each sector. The paper addresses not only territorial division, but also the frequency with which waste is collected, which is a critical issue in these types of applications. Special characteristics related to the number and type of deposition points were also a motivation for this work. A new model for a Mixed Capacitated Arc Routing Problem with Limited Multi-Landfills is proposed and tested in real instances. The computational results achieved confirm the effectiveness of the entire approach.

Abstract
This article deals with a real-life waste collection routing problem. To efficiently plan waste collection, large municipalities may be partitioned into convenient sectors and only then can routing problems be solved in each sector. Three diverse situations are described, resulting in three different new models. In the first situation, there is a single point of waste disposal from where the vehicles depart and to where they return. The vehicle fleet comprises three types of collection vehicles. In the second, the garage does not match any of the points of disposal. The vehicle is unique and the points of disposal (landfills or transfer stations) may have limitations in terms of the number of visits per day. In the third situation, disposal points are multiple (they do not coincide with the garage), they are limited in the number of visits, and the fleet is composed of two types of vehicles. Computational results based not only on instances adapted from the literature but also on real cases are presented and analyzed. In particular, the results also show the effectiveness of combining sectorization and routing to solve waste collection problems. (c) 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(2), 155-165 2015

Abstract
Sectorization means dividing a whole into parts (sectors), a procedure that occurs in many contexts and applications, usually to achieve some goal or to facilitate an activity. The objective may be a better organization or simplification of a large problem into smaller sub-problems. Examples of applications are political districting and sales territory division. When designing/comparing sectors some characteristics such as contiguity, equilibrium and compactness are usually considered. This paper presents and describes new generic measures and proposes a new measure, desirability, connected with the idea of preference. © 2015, Springer International Publishing Switzerland.

Abstract
Logistic decisions involving the location of facilities in connection with vehicle routing appear in many contexts and applications. Given a set of potential distribution centers (DC) and a group of clients, the choice of which DC to open together with the design of a number of vehicle routes, satisfying clients' demand, may define Location-Routing Problems (LRP). This paper contributes with a new method, the 4-Phase Method (4-PhM), to deal with Capacitated LRP. Relevant advantages of 4-PhM are its generality, the possibilities of handling Multiple-Criteria and of facing large dimension problems. This last aptitude is a consequence of the sectorization phases, which permit a simplification of the solution space. Sectors are constructed by two Simulated Annealing based procedures, and they follow SectorEl, a sectorization approach inspired by electrostatics. In the last phase, the results obtained are evaluated using multicriteria analysis. Here, decision makers play an important role by reflecting preferences in a pairwise comparison matrix of the Analytic Hierarchy Process. Computational results, based on randomly generated instances, confirm the expectations about 4-PhM and its potentiality to deal with LRP.

Abstract
This paper deals with a multi-objective location-routing problem (MO-LRP) and follows the idea of sectorization to simplify the solution approaches. The MO-LRP consists of sectorization, sub-sectorization, and routing sub-problems. In the sectorization sub-problem, a subset of potential distribution centres (DCs) is opened and a subset of customers is assigned to each of them. Each DC and the customers assigned to it form a sector. Afterward, in the sub-sectorization stage customers of each DC are divided into different sub-sector. Then, in the routing sub-problem, a route is determined and a vehicle is assigned to meet demands. To solve the problem, we design two approaches, which adapt the sectorization, sub-sectorization and routing sub-problems with the non-dominated sorting genetic algorithm (NSGA-II) in two different manners. In the first approach, NSGA-II is used to find non-dominated solutions for all sub-problems, simultaneously. The second one is similar to the first one but it has a hierarchical structure, such that the routing sub-problem is solved with a solver for binary integer programming in MATLAB optimization toolbox after solving sectorization and sub-sectorization sub-problem with NSGA-II. Four benchmarks are used and based on a comparison between the obtained results it is shown that the first approach finds more non-dominated solutions. Therefore, it is concluded that the simultaneous approach is more effective than the hierarchical approach for the defined problem in terms of finding more non-dominated solutions. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021.

Abstract
This paper deals with multi-objective location-routing problems (MO-LRPs) and follows a sectorizationapproach, which means customers are divided into different sectors, and a distribution centre is opened for eachsector. The literature has considered objectives such as minimizing the number of opened distribution centres,the variances of compactness, distances and demands in sectors. However, the achievement of these objectivescannot guarantee the geographical separation of sectors. In this sense, and as the geographical separation ofsectors can have significant practical relevance, we propose a new objective function and solve a benchmarkof problems with the non-dominated sorting genetic algorithm (NSGA-II), which finds multiple non-dominatedsolutions. A comparison of the results shows the effectiveness of the introduced objective function, since, in thenon-dominated solutions obtained, the sectors are more geographically separated when the values of the objectivefunction improve.