Rafael Marti

University of Valencia | UV · Department of Statistics and Operations Research

The dispersion problem consists of selecting a subset of elements from a data set in order to maximize its diversity, which has many applications in real-world scenarios. For the capacitated dispersion problem (CDP), it seeks for a subset such that the minimum distance among the selected elements is as large as possible while satisfying a demand co...

Discrete diversity optimization basically consists of selecting a subset of elements of a given set in such a way that the sum of their pairwise distances is maximized. Equity, on the other hand, refers to minimizing the difference between the maximum and the minimum distances in the subset of selected elements to balance their diversity. Both prob...

In an effort to balance the distribution of services across a given territory, dispersion and diversity models typically aim to maximize the minimum distance between any pair of facilities. Specifically, in the capacitated dispersion problem (CDP), each facility has an associated capacity or level of service, and the objective is to select a set of...

Tabu search is a meta-heuristic that guides a local heuristic search procedure to explore the solution space beyond local optimality. One of the main components of tabu search is its use of adaptive memory, which creates an intelligent search pattern based on strategic choices, as opposed to random selections that are widely applied in other method...

In spite of its practical application, solution representation has been scarcely used to study maximum diversity problems. In this chapter, we graphically represent the optimal solutions of some diversity models when solving Euclidean instances. These representations help us to understand and differentiate the models and their area of applicability...

Scatter search (SS) is a population-based metaheuristic introduced in the late 1970s as a heuristic for integer programming problems. The main concepts and principles are based on combining solution vectors, making limited use of randomization. Its flexibility in the development of the implementation has been shown very effective in the resolution...

Given an undirected graph, the minimum stretch spanning tree problem (MSSTP) deals with finding a spanning tree such that the maximum distance in the tree for adjacent nodes in the original graph, called the stretch, is minimum. This is an NP-hard problem with many applications in transportation and communication networks. We propose a general vari...

Given an undirected connected graph G, the Minimum Leaf Spanning Tree Problem (MLSTP) consists in finding a spanning tree T of G with minimum number of leaves. This is an NP-hard problem with applications in communications and water supply networks. In this paper, we propose a heuristic algorithm to provide high-quality solutions (spanning trees wi...

We now turn to the discussion of how to solve the linear ordering problem and the maximum diversity problem to (proven) optimality. In this chapter we start with the branch-and-bound method which is a general procedure for solving combinatorial optimization problems. In the subsequent chapters this approach will be realized for the LOP in a special...

This chapter focuses on the approach for solving the LOP to optimality which can currently be seen as the most successful one. It is a branch-and-bound algorithm, where the upper bounds are computed using linear programming relaxations.

In this chapter we elaborate on meta-heuristics for optimization from a beginner’s perspective. Basically, we start from scratch to describe the different methodologies and provide the reader with the elements and strategies to build and implement them successfully. Although we describe the adaptation of these methods to the linear ordering problem...

So far we developed a general integer programming approach for solving the LOP. It was based on the canonical IP formulation with equations and 3-dicycle inequalities which was then strengthened by generating mod-k-inequalities as cutting planes. In this chapter we will add further ingredients by looking for problemspecific inequalities. To this en...

Diversity and dispersion problems deal with selecting a subset of elements from a given set in such a way that their diversity is maximized. This study considers a practical location problem recently proposed in the context of max–min dispersion models. It is called the generalized dispersion problem, and it models realistic applications by introdu...

Since the linear ordering problem and the maximum diversity problem are both NP-hard, we cannot expect to be able to solve practical problem instances of arbitrary size to optimality. Depending on the size of an instance or depending on the available CPU time we will often have to be satisfied with computing approximate solutions. In addition, unde...

In this chapter we want to address some issues of interest for the LOP which we have not included in the previous chapters and point to some areas for possible further research. Additionally, in the last section, we consider some extensions of the MDP that include side constraints to model real problems in location and business.

The problem of maximizing diversity or dispersion deals with selecting a subset of elements from a given set in such a way that the distance among the selected elements is maximized. The definition of distance between elements is customized to specific applications, and the way that the overall diversity of the selected elements is computed results...

Tabu search is an optimization methodology that guides a local heuristic search procedure to explore the solution space beyond local optimality. It is substantiated by the hypothesis that an intelligent solving algorithm must incorporate memory to base its decisions on information collected during the search. The method creates in this way a learni...

In the context of simulation-based optimisation, this paper reviews recent work related to the role of metaheuristics, matheuristics (combinations of exact optimisation methods with metaheuristics), simheuristics (hybridisation of simulation with metaheuristics), biased-randomised heuristics for ‘agile’ optimisation via parallel computing, and lear...

The Time Capacitated Arc Routing Problem (TCARP) extends the classical Capacitated Arc Routing Problem by considering time-based capacities instead of traditional loading capacities. In the TCARP, the costs associated with traversing and servicing arcs, as well as the vehicle’s capacity, are measured in time units. The increasing use of electric ve...

The challenge of maximizing the diversity of a collection of points arises in a variety of settings, and the growing interest of dealing with diversity resulted in an effort to study these problems in the last few years. Generally speaking, maximizing diversity consists in selecting a subset of points from a given set in such a way that a measure o...

The problem of maximizing dispersion requires the selection of a specific number of elements from a given set, in such a way that the minimum distance between the pairs of selected elements is maximized. In recent years, this problem has received a lot of attention and has been solved with many complex heuristics. However, there is a recent variant...

Maximum diversity problems arise in many practical settings from facility location to social networks, and constitute an important class of NP-hard problems in combinatorial optimization. There has been a growing interest in these problems in recent years, and different mathematical programming models have been proposed to capture the notion of div...

The bipartite drawing problem is a well-known NP-hard combinatorial optimization problem with numerous applications. The aim is to minimize the number of edge crossings in a two-layer graph, in which the edges are drawn as straight lines. We consider the dynamic variant of this problem, called the dynamic bipartite drawing problem (DBDP), which con...

In this paper, we investigate the adaptation of the greedy randomized adaptive search procedure (GRASP) and variable neighborhood descent (VND) methodologies to the capacitated dispersion problem. Dispersion and diversity problems arise in the placement of undesirable facilities, workforce management, and social media, among others. Maximizing dive...

In the last few years, the application of decision making to logistic problems has become crucial for public and private organizations. Efficient decisions clearly contribute to improve operational aspects such as cost reduction or service improvement. The particular case of waste collection service considered in this paper involves a set of econom...

Graph drawing is a key issue in the field of data analysis, given the ever-growing amount of information available today that require the use of automatic tools to represent it. Graph Drawing Problems (GDP) are hard combinatorial problems whose applications have been widely relevant in fields such as social network analysis and project management....

In this work, we propose a heuristic procedure for a stochastic version of the uncapacitated r-allocation p-hub median problem with nonstop services. In particular, we assume that the number of hubs to which a terminal can be allocated is bounded from above by r. Additionally, we consider the possibility of shipping traffic directly between termina...

The Max-Sum diversity and the Max-Min diversity are two well-known optimization models to capture the notion of selecting a subset of diverse points from a given set. The resolution of their associated optimization problems provides solutions of different structures, in both cases with desirable characteristics. They have been extensively studied a...

Visualization of information is a relevant topic in Computer Science, where graphs have become a standard representation model, and graph drawing is now a well-established area. Within this context, edge crossing minimization is a widely studied problem given its importance in obtaining readable representations of graphs. In this paper, we focus on...

Heuristics are strategies using readily accessible, loosely applicable information to control problem solving. Algorithms, for example, are a type of heuristic. By contrast, Metaheuristics are methods used to design Heuristics and may coordinate the usage of several Heuristics toward the formulation of a single method. GRASP (Greedy Randomized Adap...

Multi-start procedures were originally conceived as a way to exploit a local or neighborhood search procedure, by simply applying it from multiple random initial solutions. Modern multi-start methods usually incorporate a powerful form of diversification in the generation of solutions to help overcome local optimality. Different metaheuristics, suc...

The challenge of maximizing the diversity of a collection of points arises in a variety of settings, and the growing interest of dealing with diversity resulted in an effort to study the management of equity. While the terms diversity and dispersion can be found in many optimization problems indistinguishable, we undertake to explore the different...

Scatter search (SS) is a population-based metaheuristic that has been shown to yield high-quality outcomes for hard combinatorial optimization problems. It uses strategies for combining solution vectors, making limited use of randomization, that have proved effective in a variety of problem settings. The fundamental concepts and principles were fir...

In this work we propose a heuristic procedure for a stochastic version of the Unca-pacitated r-Allocation p-Hub Median Problem with non-stop services. In particular, we assume that the number of hubs to which a terminal can be allocated is bounded from above by r. Additionally, we consider the possibility of shipping traffic directly between termin...

In this paper, we study the visualization of complex structures in the context of automatic graph drawing. Constructing geometric representations of combinatorial structures, such as networks or graphs, is a difficult task that requires an expert system. The automatic generation of drawings of graphs finds many applications from software engineerin...

The obnoxious p-median problem consists of selecting p locations, considered facilities, in a way that the sum of the distances from each nonfacility location, called customers, to its nearest facility is maximized. This is an NP-hard problem that can be formulated as an integer linear program. In this paper, we propose the application of a variabl...

Metaheuristics have become a very popular family of solution methods for optimization problems because they are capable of finding “acceptable” solutions in a “reasonable” amount of time. Most optimization problems in practice are too complex to be approached by exact methods that can guarantee finding global optimal solutions. The time required to...

Linear programming is perhaps the best-known tool for optimization. Linear programming is a general-purpose framework that allows a real system to be abstracted as a model with a linear objective function subject to a set of linear constraints.

In this chapter, we describe the process of designing heuristic procedures to solve combinatorial optimization problems.

Models, as a simplified representation of reality, are used daily in an attempt to control or understand some aspects of a real system. Simplification of reality is the accepted view of the modeling process, which assumes that reality represents the absolute truth. Without getting too deep into a philosophical discourse, it is worth mentioning the...

In this chapter we describe one of the most successful methodologies for obtaining high quality solutions to difficult optimization problems. It was proposed by Fred Glover (Decision Sciences 8:371–392, 1997) [3] under the term Tabu Search, which refers to the way in which the method explores the solution region of a given problem. We will use Clus...

This essential metaheuristics tutorial provides descriptions and practical applications in the area of business analytics. It addresses key problems in predictive and prescriptive analysis, while also illustrating how problems that arise in business analytics can be modelled and how metaheuristics can be used to find high-quality solutions. Readers...

Automated graph-drawing systems utilize procedures to place vertices and arcs in order to produce graphs with desired properties. Incremental or dynamic procedures are those that preserve key characteristics when updating an existing drawing. These methods are particularly useful in areas such as planning and logistics, where updates are frequent....

Drawings of graphs have many applications and they are nowadays well-established tools in computer science in general, and optimization in particular. Project scheduling is one of the many areas in which representation of graphs constitutes an important instrument. The experience shows that the main quality desired for drawings of graphs is readabi...

The bi-criteria p-median p-dispersion is a challenging optimization problem that belongs to the family of location problems. To our knowledge, no metaheuristic has been proposed to this problem, where a multi-objective approach has to be considered. In this paper, we propose a multi-objective Scatter Search implementation in which three different i...

Hub networks are commonly used in telecommunications and logistics to connect origins to destinations in situations where a direct connection between each origin–destination (o-d) pair is impractical or too costly. Hubs serve as switching points to consolidate and route traffic in order to realize economies of scale. The main decisions associated w...

In this work, we investigate the adaptation of the Greedy Randomized Adaptive Search Procedure (GRASP) and Iterated Greedy methodologies to the Capacitated Clustering Problem (CCP). In particular, we focus on the effect of the balance between randomization and greediness on the performance of these multi-start heuristic search methods when solving...

In this paper, we investigate the adaptation of the Greedy Randomized Adaptive Search Procedure (GRASP) and Iterated Greedy methodologies to the Capacitated Clustering Problem (CCP). In particular, we focus on the effect of the balance between randomization and greediness on the performance of these multi-start heuristic search methods when solving...

In this paper we study the hub location problem, where the goal is to identify an optimal subset of facilities (hubs) to minimize the transportation cost while satisfying certain capacity constraints. In particular, we target the single assignment version, in which each node in the transportation network is assigned to only one hub to route its tra...

Graphs are used to represent reality in several areas of knowledge. Drawings of graphs have many applications, from project scheduling to software diagrams. The main quality desired for drawings of graphs is readability, and crossing reduction is a fundamental aesthetic criterion for a good representation of a graph. In this paper we target the edg...

The Maximum Minsum Dispersion Problem (Max-Minsum DP) is a strongly NP-Hard problem that belongs to the family of equitable dispersion problems. When dealing with dispersion, the operations research literature has focused on optimizing efficiency-based objectives while neglecting, for the most part, measures of equity. The most common efficiency-ba...

We tackle a combinatorial problem that consists of finding the optimal configuration of a binary matrix. The configuration is determined by the ordering of the rows in the matrix and the objective function value is associated with a value B, the so-called bandpass number. In the basic version of the problem, the objective is to maximize the number...

Embedded systems have become an essential part of our lives, mainly due to the evolution of technology in the last years. However, the power consumption of these devices is one of their most important drawbacks. It has been proven that an efficient use of the memory of the device also improves its energy performance. This work efficiently solves th...

The term layout problem comes from the context of Very Large-Scale Integration (VLSI) circuit design. Graph layouts are optimization problems where the main objective is to project an original graph into a predefined host graph, usually a horizontal line. In this paper, some of the most relevant linear layout problems are reviewed, where the purpos...

The challenge of maximizing the diversity of a collection of points arises in a variety of settings, and the growing interest of dealing with diversity resulted in an effort to study the management of equity. While the terms diversity and dispersion can be found in many optimization problems indistinguishable, we undertake to explore the different...

Location problems have been extensively studied in the optimization literature, being probably the p-median one of the most tackled models. The Obnoxious p-median is an interesting variant that appears in the context of hazardous location. The aim of this paper is to formally introduce a bi-objective optimization model for this problem, in which a...

Given a set of positive integers S, the minimum generating set problem consists in finding a set of positive integers T with a minimum cardinality such that every element of S can be expressed as the sum of a subset of elements in T. It constitutes a natural problem in combinatorial number theory and is related to some real-world problems, such as...

In this paper we study the hub location problem. The goal is to identify an optimal subset of facilities (hubs) to minimize the transportation cost while satisfying certain capacity constraints. In particular, we target the single assignment version, in which each node in the transportation network is assigned to only one hub to route its traffic....

In this work we propose a heuristic procedure for a stochastic version of the Uncapacitated r-Allocation p-Hub Median Problem with non-stop services. In particular, we assume that the number of hubs to which a terminal can be allocated, is bounded from above by r. Additionally we consider the possibility of shipping traffic directly between termina...

The capacitated single assignment hub location problem with modular link capacities is a variant of the classical hub location problem in which the cost of using edges is not linear but stepwise, and the hubs are restricted in terms of transit capacity rather than in the incoming traffic. We propose a metaheuristic algorithm based on strategic osci...

The Obnoxious p-Median problem consists in selecting a subset of p facilities from a given set of possible locations, in such a way that the sum of the distances between each customer and its nearest facility is maximized. The problem is -hard and can be formulated as an integer linear program. It was introduced in the 1990s, and a branch and cut m...

The structure and properties of sexual-contacts networks in human populations is a topic of key interest in connection with the spread of sexually transmitted infections (STI). However, this problem has received scarce attention, and the modelling of STI epidemiology is usually based on theoretical proposals in terms of the network structure usuall...

Black box optimizers have a long tradition in the field of operations research. These procedures treat the objective function evaluation as a black box and therefore do not take advantage of its specific structure. Black-box optimization refers to the process in which there is a complete separation between the evaluation of the objective function —...

Scatter search is a population-based method that has been shown to yield high-quality outcomes for combinatorial optimization problems. It uses strategies for combining solution vectors that have proved effective in a variety of problem settings. In this paper, we present a scatter search implementation for an -hard variant of the classic p-hub med...

The capacitated clustering problem (CCP) consists of forming a specified number of clusters or groups from a set of elements in such a way that the sum of the weights of the elements in each cluster is within some capacity limits, and the sum of the benefits between the pairs of elements in the same cluster is maximized. This problem—which has been...

The goal of this work is to develop an improved procedure for the solution of the lexicographic bottleneck variant of the assembly line balancing problem (LB-ALBP). The objective of the LB-ALBP is to minimize the workload of the most heavily loaded workstation, followed by the workload of the second most heavily loaded workstation and so on. This p...

We study the problem of minimizing the profile of a graph and develop a solution method by following the tenets of scatter search. Our procedure exploits the network structure of the problem and includes strategies that produce a computationally efficient and agile search. Among several mechanisms, our search includes path relinking as the basis fo...

In this paper we review and propose different adaptations of the GRASP metaheuristic to solve multiobjective combinatorial optimization problems. In particular, we describe several alternatives to specialize the construction and improvement components of GRASP when two or more objectives are considered. GRASP has been successfully coupled with Path...

Multi-start procedures were originally conceived as a way to exploit a local or neighborhood search procedure, by simply applying it from multiple random initial solutions. Modern multi-start methods usually incorporate a powerful form of diversification in the generation of solutions to help overcome local optimality. Different metaheuristics, suc...

Scatter search (SS) is a population-based metaheuristic that has been shown to yield high-quality outcomes for hard combinatorial optimization problems. It uses strategies for combining solution vectors, making limited use of randomization, that have proved effective in a variety of problem settings. The fundamental concepts and principles were fir...

In this paper, we address an optimization problem resulting from the combination of the well-known travelling salesman and knapsack problems. In particular, we target the orienteering problem, originated in the context of sport, which consists of maximizing the total score associated with the vertices visited in a path within the available time. Th...

We propose several new hybrid heuristics for the differential dispersion problem, the best of which consists of a GRASP with sampled greedy construction with variable neighborhood search for local improvement. The heuristic maintains an elite set of high-quality solutions throughout the search. After a fixed number of GRASP iterations, exterior pat...

Embedded systems have become an essential part of our lives, thanks to their evo-lution in the recent years, but the main drawback is their power consumption. This paper is focused on improving the memory allocation of embedded systems to reduce their power consumption. We propose a parallel variable neighborhood search algorithm for the dynamic me...

The quadratic multiple knapsack problem (QMKP) consists in assigning a set of objects, which interact through paired profit values, exclusively to different capacity-constrained knapsacks with the aim of maximising total profit. Its many applications include the assignment of workmen to different tasks when their ability to cooperate may affect the...

Telecommunication systems use optical signals to transmit information. The strength of a signal in an optical network deteriorates and loses power as it goes farther from the source, mainly due to attenuation. Therefore, to enable the signal to arrive its intended destination with good quality, it is necessary to regenerate the signal periodically...

The quadratic minimum spanning tree problem consists of determining a spanning tree that minimizes the sum of costs of the edges and pairs of edges in the tree. Many algorithms and methods have been proposed for this hard combinatorial problem, including several highly sophisticated metaheuristics. This article presents a simple Tabu Search (TS) fo...

In this paper we propose a heuristic for the Uncapacitated rr-Allocation pp-Hub Median Problem. In the classical pp-hub location problem, given a set of nodes with pairwise traffic demands, we must select pp of them as hub locations and route all traffics through them at a minimum cost. We target here an extension, called the rr-allocation pp-hub m...

Resumen En este trabajo se propone un algoritmo basado en Scat-ter Search para resolver el problema de la minimización del Profile. Se utiliza la estrategia Path Relinking como método base para la genera-ción de nuevas soluciones mediante combinación. El problema del Profile (PMP) es NP-duro y tiene aplicaciones relevantes en técnicas de análi-sis...

In this paper, we propose a hybrid metaheuristic algorithm to solve the cyclic antibandwidth problem. This hard optimization problem consists of embedding an n-vertex graph into the cycle Cn, such that the minimum distance (measured in the cycle) of adjacent vertices is maximized. It constitutes a natural extension of the well-known antibandwidth p...