StoSS - Sectorization to Simplify and Solve

D3S - Decision Support System for Sectorization: A Tutorial

This paper explores the problem of sectorization of a parcel delivery service that wants to assign an action region to each of its teams, regarding the number of deliveries scheduled for each zone, so that there is a balanced service amongst sectors, covering contiguous zones, and considering limited capacities for the teams. Besides being relatively easy to model, the available optimization tools and software provide poor results when dimension increases in these types of problems, with computational capacity exceeding. In this paper an integer programming model, combined with an heuristic to return a faster solution, was implemented to solve a sectorization problem in two different situations. The main advantage of the strategy proposed, compared to previous ones, is its simplicity and easy implementation while still returning an optimal solution.

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.KeywordsSectorizationDynamic sectorizationGenetic algorithmAnalytical hierarchy process

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.

One of the most widely used methods in multi-objective optimization problems is the weighted sum method. However, in this method, defining the weights of objectives is always a challenge. Various methods have been suggested to achieve the weights, one of which is Shannon’s entropy method. In this study, a bi-objective model is introduced to solve the sectorization problem. As a solution method, the model is transformed into two single-objective ones. Also, the bi-objective model is solved for the case where the weights are equal to one. The gained three results from a benchmark are supposed as alternatives in a decision matrix. After the limitation of this approach appears, solutions from different benchmarks are added to the matrix. With Shannon’s entropy method, the weights of the objective functions are got from the decision matrix. The limitations of the approach and possible causes are discussed.

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.

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 centers. 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.

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.

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.

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.

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. Keywords—Sectorization, NSGA-II, NSGA-III, Statistically Parameter Tuning, Analysis of Variance, Design of Experiments, Taguchi Method, Response Surface Method

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.

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.