Abstract
In this paper, we consider three Relaxation Adaptive Memory Programming (RAMP) approaches for solving the Uncapacitated Facility Location Problem (UFLP), whose objective is to locate a set of facilities and allocate these facilities to all clients at minimum cost. Different levels of sophistication were implemented to measure the performance of the RAMP approach. In the simpler level, (Dual-) RAMP explores more intensively the dual side of the problem, incorporating a Lagrangean Relaxation and Subgradient Optimization with a simple Improvement Method on the primal side. In the most sophisticated level, RAMP combines a Dual-Ascent procedure on the dual side with a Scatter Search (SS) procedure on primal side, forming the Primal-Dual RAMP (PD-RAMP). The Dual-RAMP algorithm starts with (dual side) the dualization of the initial problem, and then a projection method projects the dual solutions into the primal solutions space. Next, (primal side) the projected solutions are improved through an improvement method. In the PD-RAMP algorithm, the SS procedure is incorporated in the primal side to carry out a more intensive exploration. The algorithm alternates between the dual and the primal side until a fixed number of iterations is achieved. Computational experiments on a standard testbed for the UFLP were conducted to assess the performance of all the RAMP algorithms.

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 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 is the process of grouping a set of previously defined basic units (points or small areas) into a fixed number of sectors. Sectorization is also known in the literature as districting or territory design, and is usually performed to optimize one or more criteria regarding the geographic characteristics of the territory and the planning purposes of sectors. The most common criteria are equilibrium, compactness and contiguity, which can be measured in many ways. Sectorization is similar to clustering but with a different motivation. Both aggregate smaller units into groups. But, while clustering strives for inner similarity of data, sectorization aims at outer homogeneity [1]. In clustering, groups should be very different from each other, and similar points are classified in the same cluster. In sectorization, groups should be very similar to each other, and therefore very different points can be grouped in the same sector. We classify sectorization problems into four types: basic sectorization, sectorization with service centers, resectorization, and dynamic sectorization. A Decision Support System for Sectorization, D3S, is being developed to deal with these four types of problems. Multi-objective genetic algorithms were implemented in D3S using Python, and a user-friendly web interface was developed using Django. Several applications can be solved with D3S, such as political districting, sales territory design, delivery service zones, and assignment of fire stations and health services to the population.