main-content

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 Adaptive Search Procedures) is an example of a Metaheuristic. To the layman, heuristics may be thought of as ‘rules of thumb’ but despite its imprecision, heuristics is a very rich field that refers to experience-based techniques for problem-solving, learning, and discovery. Any given solution/heuristic is not guaranteed to be optimal but heuristic methodologies are used to speed up the process of finding satisfactory solutions where optimal solutions are impractical. The introduction to this Handbook provides an overview of the history of Heuristics along with main issues regarding the methodologies covered. This is followed by Chapters containing various examples of local searches, search strategies and Metaheuristics, leading to an analyses of Heuristics and search algorithms. The reference concludes with numerous illustrations of the highly applicable nature and implementation of Heuristics in our daily life. Each chapter of this work includes an abstract/introduction with a short description of the methodology. Key words are also necessary as part of top-matter to each chapter to enable maximum search engine optimization. Next, chapters will include discussion of the adaptation of this methodology to solve a difficult optimization problem, and experiments on a set of representative problems.

### 1. Adaptive and Multilevel Metaheuristics

For the last decades, metaheuristics have become ever more popular as a tool to solve a large class of difficult optimization problems. However, determining the best configuration of a metaheuristic, which includes the program flow and the parameter settings, remains a difficult task. Adaptive metaheuristics (that change their configuration during the search) and multilevel metaheuristics (that change their configuration during the search by means of a metaheuristic) can be a solution for this. This chapter intends to make a quick review of the latest trends in adaptive metaheuristics and in multilevel metaheuristics.

Marc Sevaux, Kenneth Sörensen, Nelishia Pillay

### 2. Biased Random-Key Genetic Progamming

This chapter introduces biased random-key genetic programming, a new metaheuristic for evolving programs. Each solution program is encoded as a vector of random keys, where a random key is a real number randomly generated in the continuous interval [0, 1]. A decoder maps each vector of random keys to a solution program and assigns it a measure of quality. A Program-Expression is encoded in the chromosome using a head-tail representation which is later transformed into a syntax tree using a prefix notation rule. The artificial simulated evolution of the programs is accomplished with a biased random-key genetic algorithm. Examples of the application of this approach to symbolic regression are presented.

José Fernando Gonçalves, Mauricio G. C. Resende

### 3. Data Mining in Stochastic Local Search

This chapter explores some stochastic local search heuristics that incorporate a data mining procedure. The basic idea of using data mining inside a heuristic is to obtain knowledge from previous iterations performed by a heuristic to guide the search in next iterations. Patterns extracted from good quality solutions can be used to guide the search, leading to a more effective exploration of the solution space. This survey shows that memoryless heuristics may benefit from the use of data mining by obtaining better solutions in smaller computational times. Also, some results are revisited to demonstrate that even memory-based heuristics can benefit from using data mining by reducing the computational time to achieve good quality solutions.

Simone de Lima Martins, Isabel Rosseti, Alexandre Plastino

### 4. Evolution Strategies

Evolution strategies are classical variants of evolutionary algorithms which are frequently used to heuristically solve optimization problems, in particular in continuous domains. In this chapter, a description of classical and contemporary evolution strategies will be provided. The review includes remarks on the history of evolution strategies and how they relate to other evolutionary algorithms. Furthermore, developments of evolution strategies for nonstandard problems and search spaces will also be summarized, including multimodal, multi-criterion, and mixed-integer optimization. Finally, selected variants of evolution strategies are compared on a representative set of continuous benchmark functions, revealing strength and weaknesses of the different variants.

Michael Emmerich, Ofer M. Shir, Hao Wang

### 5. Matheuristics

As its name suggests, a matheuristic is the hybridization of mathematical programming with metaheuristics. The hallmark of matheuristics is the central role played by the mathematical programming model, around which the overall heuristic is built. As such, matheuristic is not a rigid paradigm but rather a concept framework for the design of mathematically sound heuristics. The aim of this chapter is to introduce the main matheuristic ideas. Three specific applications in the field of wind farm, packing, and vehicle routing optimization, respectively, are addressed and used to illustrate the main features of the method.

Martina Fischetti, Matteo Fischetti

### 6. Multi-start Methods

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, such as GRASP or tabu search, have been applied to this end. This survey briefly sketches historical developments that have motivated the field and then focuses on modern contributions that define the current state of the art. Two classical categories of multi-start methods are considered according to their domain of application: global optimization and combinatorial optimization. Additionally, several methods are reviewed to estimate the number of local optima in combinatorial problems. The estimation of this number can help to establish the complexity of a given instance, and also to choose the most convenient neighborhood, which is especially interesting in the context of multi-start methods. Experiments on three well-known combinatorial optimization problems are included to illustrate the local optima estimation techniques.

Rafael Martí, Jose A. Lozano, Alexander Mendiburu, Leticia Hernando

### 7. Multi-objective Optimization

This chapter provides a short overview of multi-objective optimization using metaheuristics. The chapter includes a description of some of the main metaheuristics that have been used for multi-objective optimization. Although special emphasis is made on evolutionary algorithms, other metaheuristics, such as particle swarm optimization, artificial immune systems, and ant colony optimization, are also briefly discussed. Other topics such as applications and recent algorithmic trends are also included. Finally, some of the main research trends that are worth exploring in this area are briefly discussed.

Carlos A. Coello Coello

### 8. Restart Strategies

This chapter is focused on restart strategies in optimization, which often provide a substantial algorithmic acceleration for randomized optimization procedures. Theoretical models that describe optimal restart strategies are presented alongside with their relations to parallel computing implementations.

Oleg V. Shylo, Oleg A. Prokopyev

### 9. Constraint-Based Local Search

Constraint-Based Local Search emerged in the last decade as a framework for declaratively expressing hard combinatorial optimization problems and solve them with local search techniques. It delivers tools to practitioners that enables them to quickly experiment with multiple models, heuristics, and meta-heuristics, focusing on their application rather than the delicate minutiae of producing a competitive implementation. At its heart, the declarative models are reminiscent of the modeling facilities familiar to constraint programming, while the underlying computational model heavily depends on incrementality. The net result is a platform capable of delivering competitive local search solutions at a fraction of the efforts needed with a conventional approach delivering model-and-run to local search users.

Laurent Michel, Pascal Van Hentenryck

### 10. Guided Local Search

Guided local search (GLS) is a meta-heuristic method proposed to solve combinatorial optimization problems. It is a high-level strategy that applies an efficient penalty-based approachpenalty-based approach to interact with the local improvement procedure. This interaction creates a process capable of escaping from local optima, which improves the efficiency and robustness of the underlying local search algorithms. Fast local search (FLS) is a way of reducing the size of the neighborhood to improve the efficiency of local search. GLS can be efficiently combined with FLS in the form of guided fast local search (GFLS). This chapter describes the principles of GLS and provides guidance for implementing and using GLS, FLS, and GFLS. It also surveys GLS extensions, hybrids, and applications to optimization, including multi-objective optimization.

Abdullah Alsheddy, Christos Voudouris, Edward P. K. Tsang, Ahmad Alhindi

### 11. Theory of Local Search

Local search is a widely used method to solve combinatorial optimization problems. As many relevant combinatorial optimization problems are NP-hard, we often may not expect to find an algorithm that is guaranteed to return an optimal solution in a reasonable amount of time, i.e., in polynomial time. Therefore, one often resorts to heuristic methods that return good, suboptimal solutions in reasonable running times. Local search is a heuristic method that is popular for its ability to trade solution quality against computation time. By spending more time, we will generally get better solutions. Well-known examples of local search approaches are iterative improvement, simulated annealing, and tabu search.The performance of local search, in terms of quality or running time, may be investigated empirically, probabilistically, and from a worst-case perspective. In this chapter we focus on the last option. That is, we give provable results on the worst-case performance of local search algorithms.Besides combinatorial optimization problems, the theory discussed in this chapter also finds its application in game theory and computational complexity.

W. Michiels, E. H. L. Aarts, J. Korst

### 12. Variable Neighborhood Descent

Local search heuristic that explores several neighborhood structures in a deterministic way is called variable neighborhood descent (VND). Its success is based on the simple fact that different neighborhood structures do not usually have the same local minimum. Thus, the local optima trap problem may be resolved by deterministic change of neighborhoods. VND may be seen as a local search routine and therefore could be used within other metaheuristics. In this chapter, we discuss typical problems that arise in developing VND heuristic: what neighborhood structures could be used, what would be their order, what rule of their change during the search would be used, etc. Comparative analysis of usual sequential VND variants is performed in solving traveling salesman problem.

### 13. Ant Colony Optimization: A Component-Wise Overview

The indirect communication and foraging behavior of certain species of ants have inspired a number of optimization algorithms for NP-hard problems. These algorithms are nowadays collectively known as the ant colony optimization (ACO) metaheuristic. This chapter gives an overview of the history of ACO, explains in detail its algorithmic components, and summarizes its key characteristics. In addition, the chapter introduces a software framework that unifies the implementation of these ACO algorithms for two example problems, the traveling salesman problem and the quadratic assignment problem. By configuring the parameters of the framework, one can combine features from various ACO algorithms in novel ways. Examples on how to find a good configuration automatically are given in the chapter. The chapter closes with a review of combinations of ACO with other techniques and extensions of the ACO metaheuristic to other problem classes.

Manuel López-Ibáñez, Thomas Stützle, Marco Dorigo

### 14. Evolutionary Algorithms

Evolutionary algorithms (EAs) are population-based metaheuristics, originally inspired by aspects of natural evolution. Modern varieties incorporate a broad mixture of search mechanisms, and tend to blend inspiration from nature with pragmatic engineering concerns; however, all EAs essentially operate by maintaining a population of potential solutions and in some way artificially ‘evolving’ that population over time. Particularly well-known categories of EAs include genetic algorithms (GAs), Genetic Programming (GP), and Evolution Strategies (ES). EAs have proven very successful in practical applications, particularly those requiring solutions to combinatorial problems. EAs are highly flexible and can be configured to address any optimization task, without the requirements for reformulation and/or simplification that would be needed for other techniques. However, this flexibility goes hand in hand with a cost: the tailoring of an EA’s configuration and parameters, so as to provide robust performance for a given class of tasks, is often a complex and time-consuming process. This tailoring process is one of the many ongoing research areas associated with EAs.

David Corne, Michael A. Lones

### 15. Genetic Algorithms

This chapter presents the fundamental concepts of genetic algorithms (GAs) that have become an essential tool for solving optimization problems in a wide variety of fields. The first part of this chapter is devoted to the revision of the basic components for the design of GAs. We illustrate this construction process through its application for solving three widely known optimization problems as knapsack problem, traveling salesman problem, and real-parameter optimization. The second part of the chapter focuses on the study of diversification techniques that represent a fundamental issue in order to achieve an effective search in GAs. In fact, analyzing its diversity has led to the presentation of numerous GA models in the literature. Similarly, the hybridization with other metaheuristics and optimization methods has become a very fruitful research area. The third part of the chapter is dedicated to the study of these hybrid methods. In closing, in the fourth part, we outline the wide spectrum of application areas that shows the level of maturity and the wide research community of the GA field.

Carlos García-Martínez, Francisco J. Rodriguez, Manuel Lozano

### 16. GRASP

GRASP (greedy randomized adaptive search procedure) is a multistart metaheuristic for computing good-quality solutions of combinatorial optimization problems. Each GRASP iteration is usually made up of a construction phase, where a feasible solution is constructed, and a local search phase which starts at the constructed solution and applies iterative improvement until a locally optimal solution is found. Typically, the construction phase of GRASP is a randomized greedy algorithm, but other types of construction procedures have been also proposed. Repeated applications of a construction procedure yields diverse starting solutions for the local search. This chapter gives an overview of GRASP describing its basic components and enhancements to the basic procedure, including reactive GRASP and intensification strategies.

Paola Festa, Mauricio G. C. Resende

### 17. Hyper-heuristics

This chapter presents a literature review of the main advances in the field of hyper-heuristics, since the publication of a survey paper in 2013. The chapter demonstrates the most recent advances in hyper-heuristic foundations, methodologies, theory, and application areas. In addition, a simple illustrative selection hyper-heuristic framework is developed as a case study. This is based on the well-known Iterated Local Search algorithm and is presented to provide a tutorial style introduction to some of the key basic issues. A brief discussion about the implementation process in addition to the decisions that had to be made during the implementation is presented. The framework implements an action selection model that operates on the perturbation stage of the Iterated Local Search algorithm to adaptively select among various low-level perturbation heuristics. The performance and efficiency of the developed framework is evaluated across six well-known real-world problem domains.

Michael G. Epitropakis, Edmund K. Burke

### 18. Iterated Greedy

Iterated greedy is a search method that iterates through applications of construction heuristics using the repeated execution of two main phases, the partial destruction of a complete candidate solution and a subsequent reconstruction of a complete candidate solution. Iterated greedy is based on a simple principle, and methods based on this principle have been proposed and published several times in the literature under different names such as simulated annealing, iterative flattening, ruin-and-recreate, large neighborhood search, and others. Despite its simplicity, iterated greedy has led to rather high-performing algorithms. In combination with other heuristic optimization techniques such as a local search, it has given place to state-of-the-art algorithms for various problems. This paper reviews the main principles of iterated greedy algorithms, relates the basic technique to the various proposals based on this principle, discusses its relationship with other optimization techniques, and gives an overview of problems to which iterated greedy has been successfully applied.

Thomas Stützle, Rubén Ruiz

### 19. Iterated Local Search

Iterated local search is a metaheuristic that embeds an improvement heuristic within an iterative process generating a chain of solutions. Often, the improvement method is some kind of local search algorithm and, hence, the name of the metaheuristic. The iterative process in iterated local search consists in a perturbation of the current solution, leading to some intermediate solution that is used as a new starting solution for the improvement method. An additional acceptance criterion decides which of the solutions to keep for continuing this process. This simple idea has led to some very powerful algorithms that have been successfully used to tackle hard combinatorial optimization problems. In this chapter, we review the main ideas of iterated local search, exemplify its application to combinatorial problems, discuss historical aspects of the development of the method, and give an overview of some successful applications.

Thomas Stützle, Rubén Ruiz

### 20. Memetic Algorithms

Memetic algorithms provide one of the most effective and flexible metaheuristic approaches for tackling hard optimization problems. Memetic algorithms address the difficulty of developing high-performance universal heuristics by encouraging the exploitation of multiple heuristics acting in concert, making use of all available sources of information for a problem. This approach has resulted in a rich arsenal of heuristic algorithms and metaheuristic frameworks for many problems. This chapter discusses the philosophy of the memetic paradigm, lays out the structure of a memetic algorithm, develops several example algorithms, surveys recent work in the field, and discusses the possible future directions of memetic algorithms.

Carlos Cotta, Luke Mathieson, Pablo Moscato

### 21. Particle Swarm Methods

Particle swarm optimization has gained increasing popularity in the past 15 years. Its effectiveness and efficiency has rendered it a valuable metaheuristic approach in various scientific fields where complex optimization problems appear. Its simplicity has made it accessible to the non-expert researchers, while the potential for easy adaptation of operators and integration of new procedures allows its application on a wide variety of problems with diverse characteristics. Additionally, its inherent decentralized nature allows easy parallelization, taking advantage of modern high-performance computer systems. The present work exposes the basic concepts of particle swarm optimization and presents a number of popular variants that opened new research directions by introducing novel ideas in the original model of the algorithm. The focus is placed on presenting the essential information of the algorithms rather than covering all the details. Also, a large number of references and sources is provided for further inquiry. Thus, the present text can serve as a starting point for researchers interested in the development and application of particle swarm optimization and its variants.

Konstantinos E. Parsopoulos

### 22. POPMUSIC

This chapter presents POPMUSIC, a general decomposition-based framework within the realm of metaheuristics and matheuristics that has been successfully applied to various combinatorial optimization problems. POPMUSIC is especially useful for designing heuristic methods for large combinatorial problems that can be partially optimized. The basic idea is to optimize subparts of solutions until a local optimum is reached. Implementations of the technique to various problems show its broad applicability and efficiency for tackling especially large-size instances.

Éric D. Taillard, Stefan Voß

### 23. Random-Key Genetic Algorithms

A random-key genetic algorithm is an evolutionary metaheuristic for discrete and global optimization. Each solution is encoded as an array of n random keys, where a random key is a real number, randomly generated, in the continuous interval [0, 1). A decoder maps each array of random keys to a solution of the optimization problem being solved and computes its cost. The algorithm starts with a population of p arrays of random keys. At each iteration, the arrays are partitioned into two sets, a smaller set of high-valued elite solutions and the remaining nonelite solutions. All elite elements are copied, without change, to the next population. A small number of random-key arrays (the mutants) are added to the population of the next iteration. The remaining elements of the population of the next iteration are generated by combining, with the parametrized uniform crossover of Spears and DeJong (On the virtues of parameterized uniform crossover. In: Proceedings of the fourth international conference on genetic algorithms, San Mateo, pp 230–236, 1991), pairs of arrays. This chapter reviews random-key genetic algorithms and describes an effective variant called biased random-key genetic algorithms.

José Fernando Gonçalves, Mauricio G. C. Resende

### 24. Scatter Search

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 first proposed in the 1960s and 1970s as an extension of mathematical relaxation techniques for combinatorial optimization problems. Its framework is flexible, allowing the development of implementations with varying degrees of sophistication.This chapter provides a grounding in the scatter search methodology that will allow readers to create successful applications of their own. To illustrate this, we present a scatter search implementation for a NP $$\mathcal {NP}$$ -hard variant of the classic p-hub median problem, for which we describe search elements, mechanisms, and strategies to generate, combine, and improve solutions.

Rafael Martí, Ángel Corberán, Juanjo Peiró

### 25. Tabu Search

Tabu search (TS) is a solution methodology within the area of metaheuristics. While the methodology applies to optimization problems in general, most TS applications have been and continue to be in discrete optimization. A key and distinguishing feature of tabu search is the use of special strategies based on adaptive memory. The underlying philosophy is that an effective search for optimal solutions should involve a flexible process that responds to the objective function landscape in a manner that allows it to learn appropriate directions to exploit specific areas of the solution space and useful departures to explore new terrain. The adaptive memory structures of tabu search enable the implementation of procedures that are capable of searching effectively and produce solutions of suitable quality within reasonable computational effort.

Manuel Laguna

### 26. Variable Neighborhood Search

Variable neighborhood search (VNS) is a metaheuristic for solving combinatorial and global optimization problems. Its basic idea is systematic change of neighborhood both within a descent phase to find a local optimum and in a perturbation phase to get out of the corresponding valley. In this chapter we present the basic schemes of variable neighborhood search and some of its extensions. We next present four families of applications of VNS in which it has proved to be very successful: (i) finding feasible solutions to large mixed-integer linear programs, by hybridization of VNS and local branching, (ii) finding good feasible solutions to continuous nonlinear programs, (iii) finding programs in automatic fashion (artificial intelligence field) by building variable neighborhood programming methodology, and (iv) exploring graph theory in order to find conjectures, refutations, and proofs or ideas of proofs.

### 27. A History of Metaheuristics

This chapter describes the history of metaheuristics in five distinct periods, starting long before the first use of the term and ending a long time in the future.The field of metaheuristics has undergone several paradigm shifts that have changed the way researchers look upon the development of heuristic methods. Most notably, there has been a shift from the method-centric period, in which metaheuristics were thought of as algorithms, to the framework-centric period, in which researchers think of metaheuristics as more general high-level frameworks, i.e., consistent collections of concepts and ideas that offer guidelines on how to go about solving an optimization problem heuristically.Tremendous progress has been made in the development of heuristics over the years. Optimization problems that seemed intractable only a few decades ago can now be efficiently solved. Nevertheless, there is still much room for evolution in the research field, an evolution that will allow it to move into the scientific period. In this period, we will see more structured knowledge generation that will benefit both researchers and practitioners.Unfortunately, a significant fraction of the research community has deluded itself into thinking that scientific progress can be made by resorting to ever more outlandish metaphors as the basis for so-called “novel” methods. Even though considerable damage to the research field will have been inflicted by the time these ideas have been stamped out, there is no doubt that science will ultimately prevail.

Kenneth Sörensen, Marc Sevaux, Fred Glover

### 28. Parallel Metaheuristic Search

The chapter presents a general picture of parallel meta-heuristic search for optimization. It recalls the main concepts and strategies in designing parallel meta-heuristics, pointing to a number of contributions that instantiated them for neighborhood- and population-based meta-heuristics, and identifies trends and promising research directions. The focus is on cooperation-based strategies, which display remarkable performances, in particular strategies based on asynchronous exchanges and the creation of new information out of exchanged data to enhance the global guidance of the search.

Teodor Gabriel Crainic

### 29. Theoretical Analysis of Stochastic Search Algorithms

Theoretical analyses of stochastic search algorithms, albeit few, have always existed since these algorithms became popular. Starting in the 1990s, a systematic approach to analyze the performance of stochastic search heuristics has been put in place. This quickly increasing basis of results allows, nowadays, the analysis of sophisticated algorithms such as population-based evolutionary algorithms, ant colony optimization, and artificial immune systems. Results are available concerning problems from various domains including classical combinatorial and continuous optimization, single and multi-objective optimization, and noisy and dynamic optimization. This chapter introduces the mathematical techniques that are most commonly used in the runtime analysis of stochastic search heuristics in finite, discrete spaces. Careful attention is given to the very popular artificial fitness levels and drift analyses techniques for which several variants are presented. To aid the reader’s comprehension of the presented mathematical methods, these are illustrated by analysis of simple evolutionary algorithms for artificial example functions. The chapter is concluded by providing references to more complex applications and further extensions of the techniques for the obtainment of advanced results.

Per Kristian Lehre, Pietro S. Oliveto

### 30. City Logistics

This chapter provides an introductory overview of city logistics systems, highlighting the specific characteristics that make them different from general logistics problems. It analyzes the types of decisions involved in managing city logistics applications, from strategic, tactical, and operational, and identifies the key models to address them. This analysis identifies types of problems, location, location routing, and variants of routing problems with time windows, all those with ad hoc formulations, derived from the constraints imposed by policy and operational regulations, technological conditions, or other specificities of urban scenarios, which result in variants of the classical models that, for its size and complexity, become a fertile field for metaheuristic approaches to define algorithms to solve the problems. Some of the more relevant cases are studied in this chapter, and guidelines for further and deeper insights on other cases are provided to the reader through a rich set of bibliographical references.

Jaume Barceló, Hanna Grzybowska, Jesús Arturo Orozco

### 31. Cutting and Packing

Cutting and Packing (C&P) problems arise in many industrial and logistics applications, whenever a set of small items, with different shapes, has to be assigned to large objects with specific shapes so as to optimize some objective function. Besides some characteristics common to combinatorial optimization problems, the distinctive feature of this field is the existence of a geometric subproblem, to ensure that the items do not overlap and are completely contained in the large objects. The geometric tools required to deal with this subproblem depend on the shapes (rectangles, circles, irregular) and on the specific conditions of the problem being solved. In this chapter, after an introduction that describes and classifies Cutting and Packing problems, we review the basic strategies that have appeared in the literature for designing constructive algorithms, local search procedures, and metaheuristics for problems with regular and irregular shapes.

Ramón Alvarez-Valdes, Maria Antónia Carravilla, José Fernando Oliveira

### 32. Diversity and Equity Models

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 models behind them.In particular, this chapter describes the mathematical models for two diversity and three equity models. Additionally, we also review two related models that have recently received special attention. This chapter also reviews heuristics and metaheuristics for finding near-optimal solutions for these problems, where constructive and local search-based methods, such as greedy randomized adaptive search procedure (GRASP) and tabu search, play an important role.

Fernando Sandoya, Anna Martínez-Gavara, Ricardo Aceves, Abraham Duarte, Rafael Martí

### 33. Evolutionary Algorithms for the Inverse Protein Folding Problem

Protein structure prediction is an essential step in understanding the molecular mechanisms of living cells with widespread application in biotechnology and health. The inverse folding problem (IFP) of finding sequences that fold into a defined structure is in itself an important research problem at the heart of rational protein design. In this chapter, a multi-objective genetic algorithm (MOGA) using the diversity-as-objective (DAO) variant of multi-objectivization is presented, which optimizes the secondary structure similarity and the sequence diversity at the same time and hence searches deeper in the sequence solution space. To validate the final optimization results, a subset of the best sequences was selected for tertiary structure prediction. Comparing secondary structure annotation and tertiary structure of the predicted model to the original protein structure demonstrates that relying on fast approximation during the optimization process permits to obtain meaningful sequences.

Sune S. Nielsen, Grégoire Danoy, Wiktor Jurkowski, Roland Krause, Reinhard Schneider, El-Ghazali Talbi, Pascal Bouvry

### 34. Linear Layout Problems

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 purpose is to minimize the objective function: the Cutwidth, the Minimum Linear Arrangement, the Vertex Separation, the SumCut, and the Bandwidth. Each problem is presented with its formal definition and it is illustrated with a detailed example. Additionally, the most relevant heuristic methods in the associated literature are reviewed together with the instances used in their evaluation. Since linear layouts represent a challenge for optimization methods in general and, for heuristics in particular, this review pays special attention to the strategies and methodologies which provide high-quality solutions.

Eduardo G. Pardo, Rafael Martí, Abraham Duarte

### 35. Maritime Container Terminal Problems

Maritime container terminals are essential infrastructures in global supply chains. Their high management complexity and heterogeneous processes make them an interesting field to apply heuristics. A brief overview of the main optimization problems found at maritime container terminals and a review of the way they are related to each other are firstly introduced. In order to solve these problems, several heuristics are presented and analyzed. The computational results reveal that they are suitable to be applied in practical scenarios due to the fact that they provide high-quality solutions in short computational times.

Christopher Expósito-Izquierdo, Eduardo Lalla-Ruiz, Jesica de Armas, Belén Melián-Batista, J. Marcos Moreno-Vega

### 36. Metaheuristics for Medical Image Registration

In the last few decades, image registration (IR) has been a very active research area in computer vision. Applications of IR cover a broad range of real-world problems, including remote sensing, medical imaging, artificial vision, and computer-aided design. In particular, medical IR is a mature research field with theoretical support and two decades of practical experience. Formulated as either a continuous or combinatorial optimization problem, medical IR has been traditionally tackled by iterative numerical optimization methods, which are likely to get stuck in local optima and deliver suboptimal solutions. Recently, a large number of medical IR methods based on different metaheuristics, mostly belonging to evolutionary computation, have been proposed. In this chapter, we review the most recognized of these algorithms and develop an experimental comparison over real-world IR scenarios.

Andrea Valsecchi, Enrique Bermejo, Sergio Damas, Oscar Cordón

### 37. Metaheuristics for Natural Gas Pipeline Networks

In this chapter an overview of metaheuristic algorithms that have been very successful in tackling a particular class of natural gas pipeline network optimization problems is presented. In particular, the problem of minimizing fuel consumption incurred by the compressor stations driving natural gas in pipeline networks is addressed. This problem has been studied from different angles over the past few years by virtue of its tremendous economical impact. First, a general mathematical framework for this class of problems is presented. After establishing the most relevant model properties and fundamental network topologies, which are key factors for choosing an appropriate solution technique, current state-of-the-art metaheuristics are presented for handling different versions of this problem. This work concludes by highlighting the most relevant and important challenges of this very exciting area of research in natural gas transportation networks.

### 38. Network Optimization

Several real-world problems can be modeled as graph problems. Graph algorithms and theories that have evolved for decades can be applied to solve the problem on hand. Interestingly, many of these graph problems can be solved polynomially, while small changes in a problem definition turn the problem difficult. In this chapter, we explore this path from polynomial network problems to NP-hard ones. Along the chapter, we visit several problems, dedicating more extended discussions to two real-world problems: the weight setting problem, originated from telecommunication networks, and the virtual network embedded problem, a recent stated optimization problem from the computer network area. For these two problems, we discuss their heuristic solution, since only small instances can be solved exactly within a reasonable amount of time.

Luciana S. Buriol

### 39. Optimization Problems, Models, and Heuristics in Wireless Sensor Networks

This chapter provides an overview and a comprehensive discussion of problems, models, algorithms, and applications in a vast and growing literature of wireless sensor networks. Being a particular kind of ad hoc network, many power management and communication protocols may be designed specifically for those networks. The critical issues considered in these protocols are the objectives, the quality of communication, the energy consumption, and the network lifetime. Moreover, due to the large-scale aspect inherent in some applications, traditional exact solution approaches are not practical, so heuristics may be adopted instead. The chapter starts by introducing the main concepts in the design of WSN and a wide range of problems and applications. Basic formulations and algorithms are also discussed, together with their benefits and drawbacks.

Vinicius Morais, Fernanda S. H. Souza, Geraldo R. Mateus

### 40. Particle Swarm Optimization for the Vehicle Routing Problem: A Survey and a Comparative Analysis

In the last few years, a number of books and survey papers devoted to the vehicle routing problem (VRP) or to its variants or to the methods used for the solution of one or more variants of the VRP have been published. Also, in these years, the field of swarm intelligence algorithms has had a significant growth. One of the most important swarm intelligence algorithms is the particle swarm optimization (PSO). Although the particle swarm optimization was first published in 1995, it took around 10 years in order researchers to publish papers using a PSO algorithm for the solution of variants of the VRP. However, in the last 10 years, many journal papers, conference papers, and book chapters have been published where a variant of VRP is solved using a PSO algorithm. Thus, it is significant to present a survey paper where a review and brief analysis of the most important of these papers will be given. This is the main focus of this chapter.

Yannis Marinakis, Magdalene Marinaki, Athanasios Migdalas

### 41. Scheduling Heuristics

The scheduling of operations over resources is a relevant theoretical and practical problem with applications in many fields and disciplines, including the manufacturing industry. Scheduling problems are as varied as the reality they model. Additionally, some scheduling settings are among the hardest combinatorial problems there are. This is a perfect scenario for heuristic methods where high-quality robust solutions can be obtained in a short amount of time. This chapter concentrates on heuristics for production scheduling problems and summarizes the main results that range from simple rules to advanced metaheuristics. The importance of proper scheduling in practice is first highlighted, along with its difficulty and relevance. A summary of the scheduling notation is also given. Basic scheduling techniques, dispatching rules, combined rules, advanced heuristics, and an introduction to metaheuristics are also summarized in the chapter. While necessarily brief and incomplete, this chapter serves as an introductory point to those interested readers seeking to delve in the vast and rich world of scheduling heuristics. Some pointers to fruitful future research avenues are also provided. A large list of journal articles and monographs are provided as a reference for additional details and study.

Rubén Ruiz

### 42. Selected String Problems

This chapter overviews some string selection and comparison problems, with special emphasis on the optimization and operational research perspective. It also proposes a simple and efficient ILP-based heuristic that can be used for any of the considered problems.

Christian Blum, Paola Festa

### 43. Supply Chain Management

Supply chain management (SCM) is related to the management of all activities along a network of organizations to provide a good or a service to final customers. The efficiency of these activities can have a great impact on customer‘s satisfaction and cost reduction. However, SCM is not just the sum of activities along the supply chain but, instead, it must consider the organization, supervision, and control of all activities in the chain from an integrated and collaborative perspective aiming to provide a competitive advantage. From this point of view, an increase in the dimension and complexity of the decision problems involved is expected, as several actors with different goals must be considered to administrate efficiently the activities within the supply chain.This chapter briefly reviews the main concepts of SCM, identifying relevant decision and optimization problems and discussing possible solution approaches. Heuristics and metaheuristics are two of the best optimization tools to be used in solving and providing business insights for the SCM problems. This chapter also describes some successful metaheuristic approaches to SCM and it examines future research trends. A large number of applications of metaheuristics to SCM integrating new subjects, such as open and big data, smart cities, and online decision-making, just to mention a few, are foreseen.

Helena Ramalhinho Lourenço, Martín Gómez Ravetti

### 44. The Maximum Clique and Vertex Coloring

In this chapter we review heuristic approaches for two classical and closely related problems of finding a maximum clique and an optimal vertex coloringColoringVertex coloring. Both problems have a wide variety of practical applications, and due to their computational intractability, a significant effort has been focused on developing heuristic methods. This chapter discusses construction heuristics, local search strategies, and metaheuristics designed and/or adapted for the maximum cliqueClique and vertex coloringColoringVertex coloring problems.

Oleksandra Yezerska, Sergiy Butenko

### 45. The Multi-plant Lot Sizing Problem with Multiple Periods and Items

The lot sizing problem consists in determining lot sizes and their scheduling to meet the required demands. This chapter focuses on a multi-plant lot sizing problem with planning for multiple items, each plant with their own demands and multiple periods. In spite of not been widely investigated, their theoretical and practical importance are supported by applications from private and public sectors. This chapter pays special attention to the multi-plant uncapacitated lot sizing problem (MPULSP). In particular, a novel network flow formulation is introduced, and some computational experiments were carried out to assess the performance of commercial solvers in solving a number of large instance problems. Moreover, a comparative analysis is performed by considering two other formulations for the MPULSP found in the literature.

Mariá C. V. Nascimento, Horacio H. Yanasse, Desiree M. Carvalho

### 46. Trees and Forests

Trees and forests have been a fascinating research topic in Operations Research (OR)/Management Science (MS) throughout the years because they are involved in numerous difficult problems, have interesting theoretical properties, and cover a large number of practical applications. A tree is a finite undirected connected simple graph with no cycles, while a set of independent trees is called a forest. A spanning tree is a tree covering all nodes of a graph. In this chapter, key components for solving difficult tree and forest problems, as well as insights to develop efficient heuristics relying on such structures, are surveyed. They are usually combined to obtain very efficient metaheuristic algorithms, hybrid methods, and matheuristics. Some emerging topics and trends in trees and forests are pointed out. This is followed by two case studies: a Lagrangian-based heuristic for the minimum degree-constrained spanning tree problem and an evolutionary algorithm for a generalization of the bounded-diameter minimum spanning tree problem. Both problems find applications in network design, telecommunication, and transportation fields, among others.

Andréa Cynthia Santos, Christophe Duhamel, Rafael Andrade

### 47. World’s Best Universities and Personalized Rankings

This chapter presents a heuristic for a multi-objective ranking problem using a dataset of international interest as an example of its application, namely, the ranking of the world’s top educational institutions. The problem of ranking academic institutions is a subject of keen interest for administrators, consumers, and research policy makers. From a mathematical perspective, the proposed heuristic addresses the need for more transparent models and associated methods related to the problem of identifying sound relative rankings of objects with multiple attributes. The low complexity of the method allows software implementations that scale well for thousands of objects as well as permitting reasonable visualization. It is shown that a simple and multi-objective-aware ranking system can easily be implemented, which naturally leads to intuitive research policies resulting from varying scenarios presented within. The only assumption that this method relies on is the ability to sort the candidate objects according to each given attribute. Thus the attributes could be numerical or ordinal in nature. This helps to avoid the selection of an ad hoc single score based on an arbitrary assignment of attributes’ weights as other heuristics do. To illustrate the use of this proposed methodology, results are presented and obtained using the dataset on the ranking of world universities (of the years 2007–2012), by academic performance, published annually by ARWU.

Mario Inostroza-Ponta, Natalie Jane de Vries, Pablo Moscato