Zone design of specific sizes using adaptive additively weighted voronoi diagrams

Territory or zone design processes entail partitioning a geographic space, organized as a set of areal units, into different regions or zones according to a specific set of criteria that are dependent on the application context. In most cases, the aim is to create zones of approximately equal sizes (zones with equal numbers of inhabitants, same average sales, etc.). However, some of the new applications that have emerged, particularly in the context of sustainable development policies, are aimed at defining zones of a predetermined, though not necessarily similar, size. In addition, the zones should be built around a given set of seeds. This type of partitioning has not been sufficiently researched; therefore, there are no known approaches for automated zone delimitation. This study proposes a new method based on a discrete version of the adaptive additively weighted Voronoi diagram that makes it possible to partition a two-dimensional space into zones of specific sizes, taking both the position and the weight of each seed into account. The method consists of repeatedly solving a traditional additively weighted Voronoi diagram, so that each seed?s weight is updated at every iteration. The zones are geographically connected using a metric based on the shortest path. Tests conducted on the extensive farming system of three municipalities in Castile-La Mancha (Spain) have established that the proposed heuristic procedure is valid for solving this type of partitioning problem. Nevertheless, these tests confirmed that the given seed position determines the spatial configuration the method must solve and this may have a great impact on the resulting partition

Police districting problem: literature review and annotated bibliography

The police districting problem concerns the efficient and effective design of patrol sectors in terms of performance attributes. Effectiveness is particularly important as it directly influences the ability of police agencies to stop and prevent crime. However, in this problem, a homogeneous distribution of workload is also desirable to guarantee fairness to the police agents and an increase in their satisfaction. This chapter provides a systematic review of the literature related to the police districting problem, whose history dates back to almost 50 years ago. Contributions are categorized in terms of attributes and solution methodology adopted. Also, an annotated bibliography that presents the most relevant elements of each research is given

A generic geometric approach to territory design and districting

Territory design and districting may be viewed as the problem of grouping small geographic areas into larger geographic clusters called territories in such a way that the latter are acceptable according to relevant planning criteria. The availability of GIS on computers and the growing interest in Geo-Marketing leads to an increasing importance of this area. Despite the wide range of applications for territory design problems, when taking a closer look at the models proposed in the literature, a lot of similarities can be noticed. Indeed, the models are many times very similar and can often be, more or less directly, carried over to other applications. Therefore, our aim is to provide a generic application-independent model and present efficient solution techniques. We introduce a basic model that covers aspects common to most applications. Moreover, we present a method for solving the general model which is based on ideas from the field of computational geometry. Theoretical as well as computational results underlining the efficiency of the new approach will be given. Finally, we show how to extend the model and solution algorithm to make it applicable for a broader range of applications and how to integrate the presented techniques into a GIS

Towards a Unified Territory Design Approach – Applications, Algorithms and GIS Integration

Territory design may be viewed as the problem of grouping small geographic areas into larger geographic clusters called territories in such a way that the latter are acceptable according to relevant planning criteria. In this paper we review the existing literature for applications of territory design problems and solution approaches for solving these types of problems. After identifying features common to all applications we introduce a basic territory design model and present in detail two approaches for solving this model: a classical location–allocation approach combined with optimal split resolution techniques and a newly developed computational geometry based method. We present computational results indicating the efficiency and suitability of the latter method for solving large–scale practical problems in an interactive environment. Furthermore, we discuss extensions to the basic model and its integration into Geographic Information Systems

The ordered gradual covering location problem on a network

In this paper we develop a network location model that combines the characteristics of ordered median and gradual cover models resulting in the Ordered Gradual Covering Location Problem (OGCLP). The Gradual Cover Location Problem (GCLP) was specifically designed to extend the basic cover objective to capture sensitivity with respect to absolute travel distance. Ordered Median Location problems are a generalization of most of the classical locations problems like p-median or p-center problems. They can be modeled by using so-called ordered median functions. These functions multiply a weight to the cost of fulfilling the demand of a customer which depends on the position of that cost relative to the costs of fulfilling the demand of the other customers. We derive Finite Dominating Sets (FDS) for the one facility case of the OGCLP. Moreover, we present efficient algorithms for determining the FDS and also discuss the conditional case where a certain number of facilities are already assumed to exist and one new facility is to be added. For the multi-facility case we are able to identify a finite set of potential facility locations a priori, which essentially converts the network location model into its discrete counterpart. For the multi-facility discrete OGCLP we discuss several Integer Programming formulations and give computational results

A novel territory design model arising in the implementation of the WEEE-Directive

The problem discussed in this paper is motivated by the new recycling directiveWEEE of the EC. The core of this law is, that each company which sells electrical or electronic equipment in a European country has the obligation to recollect and recycle an amount of returned items which is proportional to its market share. To assign collection stations to companies, in Germany for one product type a territory design approach is planned. However, in contrast to classical territory design, the territories should be geographically as dispersed as possible to avoid that a company, resp. its logistics provider responsible for the recollection, gains a monopoly in some region. First, we identify an appropriate measure for the dispersion of a territory. Afterwards, we present a first mathematical programming model for this new problem as well as a solution method based on the GRASP methodology. Extensive computational results illustrate the suitability of the model and assess the effectiveness of the heuristic

A generic geometric approach to territory design and districting

Territory design and districting may be viewed as the problem of grouping small geographic areas into larger geographic clusters called territories in such a way that the latter are acceptable according to relevant planning criteria. The availability of GIS on computers and the growing interest in Geo-Marketing leads to an increasing importance of this area. Despite the wide range of applications for territory design problems, when taking a closer look at the models proposed in the literature, a lot of similarities can be noticed. Indeed, the models are many times very similar and can often be, more or less directly, carried over to other applications. Therefore, our aim is to provide a generic application-independent model and present efficient solution techniques. We introduce a basic model that covers aspects common to most applications. Moreover, we present a method for solving the general model which is based on ideas from the field of computational geometry. Theoretical as well as computational results underlining the efficiency of the new approach will be given. Finally, we show how to extend the model and solution algorithm to make it applicable for a broader range of applications and how to integrate the presented techniques into a GIS

Towards a Unified Territory Design Approach – Applications, Algorithms and GIS Integration

Territory design may be viewed as the problem of grouping small geographic areas into larger geographic clusters called territories in such a way that the latter are acceptable according to relevant planning criteria. In this paper we review the existing literature for applications of territory design problems and solution approaches for solving these types of problems. After identifying features common to all applications we introduce a basic territory design model and present in detail two approaches for solving this model: a classical location–allocation approach combined with optimal split resolution techniques and a newly developed computational geometry based method. We present computational results indicating the efficiency and suitability of the latter method for solving large–scale practical problems in an interactive environment. Furthermore, we discuss extensions to the basic model and its integration into Geographic Information Systems

The maximum dispersion problem

In the maximum dispersion problem, a given set of objects has to be partitioned into a number of groups. Each object has a non-negative weight and each group has a target weight, which may be different for each group. In addition to meeting the target weight of each group, all objects assigned to the same group should be as dispersed as possible with respect to some distance measure between pairs of objects. Potential applications for this problem come from such diverse fields as the problem of creating study groups or the design of waste collection systems. We develop and compare two different (mixed-) integer linear programming formulations for the problem. We also study a specific relaxation that enables us to derive tight bounds that improve the effectiveness of the formulations. Thereby, we obtain an upper bound by finding in an auxiliary graph subsets of given size with minimal diameter. A lower bound is derived based on the relation of the optimal solution of the relaxation to the chromatic number of a series of auxiliary graphs. Finally, we propose an exact solution scheme for the maximum dispersion problem and present extensive computational experiments to assess its efficiency

GRAPH Reinforcement Learning for Operator Selection in the ALNS Metaheuristic

ALNS is a popular metaheuristic with renowned efficiency in solving combinatorial optimisation problems. However, despite 16 years of intensive research into ALNS, whether the embedded adaptive layer can efficiently select operators to improve the incumbent remains an open question. In this work, we formulate the choice of operators as a Markov Decision Process, and propose a practical approach based on Deep Reinforcement Learning and Graph Neural Networks. The results show that our proposed method achieves better performance than the classic ALNS adaptive layer due to the choice of operator being conditioned on the current solution. We also discuss important considerations such as the size of the operator portfolio and the impact of the choice of operator scales. Notably, our approach can also save significant time and labour costs for handcrafting problem-specific operator portfolios