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Coello, Carlos A. Coello
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This chapter provides a short overview of multiobjective optimization using metaheuristics. The chapter includes a description of some of the main metaheuristics that have been used for multiobjective 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.
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Coello, Carlos A. Coello
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This chapter provides a short overview of multiobjective optimization using metaheuristics. The chapter includes a description of some of the main metaheuristics that have been used for multiobjective 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.
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Coello, Carlos A. Coello
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4 Citations
Summary
In this paper, we will briefly discuss the current state of the research on evolutionary multiobjective optimization, emphasizing the main achievements obtained to date. Achievements in algorithmic design are discussed from its early origins until the current approaches which are considered as the “second generation” in evolutionary multiobjective optimization. Some relevant applications are discussed as well, and we conclude with a list of future challenges for researchers working (or planning to work) in this area in the next few years.
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VillalobosArias, Mario; Coello, Carlos A. Coello; HernándezLerma, Onésimo
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1 Citations
This paper presents the asymptotic convergence analysis of Simulated Annealing, an Artificial Immune System and a General Evolutionary Algorithm for multiobjective optimization problems. In the case of a General Evolutionary Algorithm, we refer to any algorithm in which the transition probabilities use a uniform mutation rule. We prove that these algorithms converge if elitism is used.
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Durillo, Juan J.; GarcíaNieto, José; Nebro, Antonio J.; Coello, Carlos A. Coello; Luna, Francisco; Alba, Enrique
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48 Citations
Particle Swarm Optimization (PSO) has received increasing attention in the optimization research community since its first appearance in the mid1990s. Regarding multiobjective optimization, a considerable number of algorithms based on MultiObjective Particle Swarm Optimizers (MOPSOs) can be found in the specialized literature. Unfortunately, no experimental comparisons have been made in order to clarify which MOPSO version shows the best performance. In this paper, we use a benchmark composed of three wellknown problem families (ZDT, DTLZ, and WFG) with the aim of analyzing the search capabilities of six representative stateoftheart MOPSOs, namely, NSPSO, SigmaMOPSO, OMOPSO, AMOPSO, MOPSOpd, and CLMOPSO. We additionally propose a new MOPSO algorithm, called SMPSO, characterized by including a velocity constraint mechanism, obtaining promising results where the rest perform inadequately.
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López Jaimes, Antonio; Coello, Carlos A. Coello; Urías Barrientos, Jesús E.
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23 Citations
In this paper, we propose and analyze two schemes to integrate an objective reduction technique into a multiobjective evolutionary algorithm (moea) in order to cope with manyobjective problems. One scheme reduces periodically the number objectives during the search until the required objective subset size has been reached and, towards the end of the search, the original objective set is used again. The second approach is a more conservative scheme that alternately uses the reduced and the entire set of objectives to carry out the search. Besides improving computational efficiency by removing some objectives, the experimental results showed that both objective reduction schemes also considerably improve the convergence of a moea in manyobjective problems.
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Cabrera, Juan Carlos Fuentes; Coello, Carlos A. Coello
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7 Citations
In this chapter, we present a multiobjective evolutionary algorithm (MOEA) based on the heuristic called “particle swarm optimization” (PSO). This multiobjective particle swarm optimizer (MOPSO) is characterized for using a very small population size, which allows it to require a very low number of objective function evaluations (only 3000 per run) to produce reasonably good approximations of the Pareto front of problems of moderate dimensionality. The proposed approach first selects the leader and then selects the neighborhood for integrating the swarm. The leader selection scheme adopted is based on Pareto dominance and uses a neighbors density estimator. Additionally, the proposed approach performs a reinitialization process for preserving diversity and uses two external archives: one for storing the solutions that the algorithm finds during the search process and another for storing the final solutions obtained. Furthermore, a mutation operator is incorporated to improve the exploratory capabilities of the algorithm. The proposed approach is validated using standard test functions and performance measures reported in the specialized literature. Our results are compared with respect to those generated by the Nondominated Sorting Genetic Algorithm II (NSGAII), which is a MOEA representative of the stateoftheart in the area.
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Leguizamón, Guillermo; Coello, Carlos A. Coello
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4 Citations
This paper presents a novel boundary approach which is included as a constrainthandling technique in an ant colony algorithm. The necessity of approaching the boundary between the feasible and infeasible search space for many constrained optimization problems is a paramount challenge for every constrainthandling technique. Our proposed technique precisely focuses the search on the boundary region and can be either used alone or in combination with other constrainthandling techniques depending on the type and number of problem constraints. For validation purposes, an ant algorithm is adopted as our search engine. We compare our proposed approach with respect to constrainthandling techniques that are representative of the stateoftheart in constrained evolutionary optimization using a set of standard test functions.
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L’opez, Antonio; Coello, Carlos A. Coello; Oyama, Akira; Fujii, Kozo
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7 Citations
In this paper, we use an alternative preference relation that couples an achievement function and the εindicator in order to improve the scalability of a MultiObjective Evolutionary Algorithm (moea) in manyobjective optimization problems. The resulting algorithm was assessed using the DebThieleLaumannsZitzler (dtlz) and the Walking FishGroup (wfg) test suites. Our experimental results indicate that our proposed approach has a good performance even when using a high number of objectives. Regarding the dtlz test problems, their main difficulty was found to lie on the presence of dominance resistant solutions. In contrast, the hardness of wfg problems was not found to be significantly increased by adding more objectives.
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MenchacaMendez, Adriana; Montero, Elizabeth; Riff, MaríaCristina; Coello, Carlos A. Coello
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1 Citations
In this paper, we study iSMSEMOA, a recently proposed approach that improves the wellknown S metric selection Evolutionary MultiObjective Algorithm (SMSEMOA). These two indicatorbased multiobjective evolutionary algorithms rely on hypervolume contributions to select individuals. Here, we propose to define a probability of using a randomly selected individual within the iSMSEMOA’s selection scheme. In order to calibrate the value of such probability, we use the EVOCA tuner. Our preliminary results indicate that we are able to save up to 33% of computations of the contribution to hypervolume with respect to the original iSMSEMOA, without any significant quality degradation in the solutions obtained. In fact, in some cases, the approach proposed here was even able to improve the quality of the solutions obtained by the original iSMSEMOA.
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Esquivel, Susana C.; Coello, Carlos A. Coello
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5 Citations
In this paper, we study the use of particle swarm optimization (PSO) for a class of nonstationary environments. The dynamic problems studied in this work are restricted to one of the possible types of changes that can be produced over the fitness landscape. We propose a hybrid PSO approach (called HPSO_dyn), which uses a dynamic macromutation operator whose aim is to maintain diversity. In order to validate our proposed approach, we adopted the test case generator proposed by Morrison & De Jong [1], which allows the creation of different types of dynamic environments with a varying degree of complexity. The main goal of this research was to determine the advantages and disadvantages of using PSO in nonstationary environments. As part of our study, we were interested in analyzing the ability of PSO for tracking an optimum that changes its location over time, as well as the behavior of the algorithm in the presence of high dimensionality and multimodality.
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Leguizamón, Guillermo; Coello, Carlos A. Coello
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4 Citations
The Ant Colony Optimization (ACO) metaheuristic embodies a large set of algorithms which have been successfully applied to a wide range of optimization problems. Although ACO practitioners have a long tradition in solving combinatorial optimization problems, many other researchers have recently developed a variety of ACO algorithms for dealing with continuous optimization problems. One of these algorithms is the socalled ACO
$_{\Bbb{R}}$
, which is one of the most relevant ACO algorithms currently available for continuous optimization problems. Although ACO
$_{\Bbb{R}}$
has been found to be successful, to the authors’ best knowledge its use in highdimensionality problems (i.e., with many decision variables) has not been documented yet. Such problems are important, because they tend to appear in realworld applications and because in them, diversity loss becomes a critical issue. In this paper, we propose an alternative ACO
$_{\Bbb{R}}$
algorithm (DACO
$_{\Bbb{R}}$
) which could be more appropriate for large scale unconstrained continuous optimization problems. We report the results of an experimental study by considering a recently proposed test suite. In addition, the parameters setting of the algorithms involved in the experimental study are tuned using an ad hoc tool. Our results indicate that our proposed DACO
$_{\Bbb{R}}$
is able to improve both, the quality of the results and the computational time required to achieve them.
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GarzaFabre, Mario; Pulido, Gregorio Toscano; Coello, Carlos A. Coello
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28 Citations
An important issue with Evolutionary Algorithms (EAs) is the way to identify the best solutions in order to guide the search process. Fitness comparisons among solutions in singleobjective optimization is straightforward, but when dealing with multiple objectives, it becomes a nontrivial task. Pareto dominance has been the most commonly adopted relation to compare solutions in a multiobjective optimization context. However, it has been shown that as the number of objectives increases, the convergence ability of approaches based on Pareto dominance decreases. In this paper, we propose three novel fitness assignment methods for manyobjective optimization. We also perform a comparative study in order to investigate how effective are the proposed approaches to guide the search in highdimensional objective spaces. Results indicate that our approaches behave better than six stateoftheart fitness assignment methods.
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Cagnina, Leticia C.; Esquivel, Susana C.; Coello, Carlos A. Coello
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8 Citations
This paper presents a particle swarm optimizer to solve constrained optimization problems. The proposed approach adopts a simple method to handle constraints of any type (linear, nonlinear, equality and inequality), and it also presents a novel mechanism to update the velocity and position of each particle. The approach is validated using standard test functions reported in the specialized literature and it’s compared with respect to algorithms representative of the stateoftheart in the area. Our results indicate that the proposed scheme is a promising alternative to solve constrained optimization problems using particle swarm optimization.
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Becerra, Ricardo Landa; Coello, Carlos A. Coello
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9 Citations
In this paper, we propose the use of a mathematical programming technique called the εconstraint method, hybridized with an evolutionary singleobjective optimizer: the cultured differential evolution. The εconstraint method uses the cultured differential evolution to produce one point of the Pareto front of a multiobjective optimization problem at each iteration. This approach is able to solve difficult multiobjective problems, relying on the efficiency of the singleobjective optimizer, and on the fact that none of the two approaches (the mathematical programming technique or the evolutionary algorithm) are required to generate the entire Pareto front at once. The proposed approach is validated using several difficult multiobjective test problems, and our results are compared with respect to a multiobjective evolutionary algorithm representative of the stateoftheart in the area: the NSGAII.
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VillalobosArias, Mario; Coello, Carlos A. Coello; HernándezLerma, Onésimo
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17 Citations
This paper presents a mathematical proof of convergence of a multiobjective artificial immune system algorithm (based on clonal selection theory). An specific algorithm (previously reported in the specialized literature) is adopted as a basis for the mathematical model presented herein. The proof is based on the use of Markov chains.
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López, Edgar Galván; Poli, Riccardo; Coello, Carlos A. Coello
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6 Citations
In this paper we propose an approach to Genetic Programming based on code reuse and we test it in the design of combinational logic circuits at the gatelevel. The circuits evolved by our algorithm are compared with circuits produced by human designers, by Particle Swarm Optimization, by an ncardinality GA and by Cartesian Genetic Programming.
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Schütze, Oliver; Laumanns, Marco; Coello, Carlos A. Coello
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10 Citations
In this paper we address the problem of approximating the ’knee’ of a biobjective optimization problem with stochastic search algorithms. Knees or entire kneeregions are of particular interest since such solutions are often preferred by the decision makers in many applications. Here we propose and investigate two update strategies which can be used in combination with stochastic multiobjective search algorithms (e.g., evolutionary algorithms) and aim for the computation of the knee and the kneeregion, respectively. Finally, we demonstrate the applicability of the approach on two examples.
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Sierra, Margarita Reyes; Coello, Carlos A. Coello
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We propose a new version of a multiobjective coevolutionary algorithm. The main idea of the proposed approach is to concentrate the search effort on promising regions that arise during the evolutionary process as a product of a clustering mechanism applied on the set of decision variables corresponding to the known Pareto front. The proposed approach is validated using several test functions taken from the specialized literature and it is compared with respect to its previous version and another approach that is representative of the stateoftheart in evolutionary multiobjective optimization.
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CruzCortés, Nareli; TrejoPérez, Daniel; Coello, Carlos A. Coello
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21 Citations
In this paper, we present a study of the use of an artificial immune system (CLONALG) for solving constrained global optimization problems. As part of this study, we evaluate the performance of the algorithm both with binary encoding and with realnumbers encoding. Additionally, we also evaluate the impact of the mutation operator in the performance of the approach by comparing Cauchy and Gaussian mutations. Finally, we propose a new mutation operator which significantly improves the performance of CLONALG in constrained optimization.
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MezuraMontes, Efrén; ReyesSierra, Margarita; Coello, Carlos A. Coello
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55 Citations
Summary
Differential Evolution is currently one of the most popular heuristics to solve singleobjective optimization problems in continuous search spaces. Due to this success, its use has been extended to other types of problems, such as multiobjective optimization. In this chapter, we present a survey of algorithms based on differential evolution which have been used to solve multiobjective optimization problems. Their main features are described and, based precisely on them, we propose a taxonomy of approaches. Some theoretical work found in the specialized literature is also provided. To conclude, based on our findings, we suggest some topics that we consider to be promising paths for future research in this area.
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Gómez, Juan Carlos; Hernández, Fernando; Coello, Carlos A. Coello; Ronquillo, Guillermo; Trejo, Antonio
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1 Citations
This paper introduces a Genetic Algorithm (GA) for training Artificial Neural Networks (ANNs) using the electromagnetic spectrum signal of a combustion process for flame pattern classification. Combustion requires identification systems that provide information about the state of the process in order to make combustion more efficient and clean. Combustion is complex to model using conventional deterministic methods thus motivate the use of heuristics in this domain. ANNs have been successfully applied to combustion classification systems; however, traditional ANN training methods get often trapped in local minima of the error function and are inefficient in multimodal and nondifferentiable functions. A GA is used here to overcome these problems. The proposed GA finds the weights of an ANN than best fits the training pattern with the highest classification rate.
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Becerra, Ricardo Landa; Coello, Carlos A. Coello
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10 Citations
A cultural algorithm is proposed in this paper. The main novel feature of this approach is the use of differential evolution as a population space. Differential evolution has been found to be very effective when dealing with real valued optimization problems. The knowledge sources contained in the belief space of the cultural algorithm are specifically designed according to the differential evolution population. Furthermore, we introduce an influence function that selects the source of knowledge to apply the evolutionary operators. Such influence function considerably improves the performance when compared to a previous version of the algorithm (developed by the same authors). We use a wellknown set of test functions to validate the approach, and compare the results with respect to the best constrainthandling technique known to date in evolutionary optimization.
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MenchacaMendez, Adriana; Coello, Carlos A. Coello
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4 Citations
We analyze here some properties of the maximin fitness function, which has been used by several researchers, as an alternative to Pareto optimality, for solving multiobjective optimization problems. As part of this analysis, we identify some disadvantages of the maximin fitness function and then propose mechanisms to overcome them. This leads to several selection operators for multiobjective evolutionary algorithms which are further analyzed. We incorporate them into an evolutionary algorithm, giving rise to the socalled MaximinClustering MultiObjective Evolutionary Algorithm (MCMOEA) approach. Our proposed approach is validated using standard test problems taken from the specialized literature, having from two to eight objectives. Our preliminary results indicate that our proposed approach is a good alternative to solve multiobjective optimization problems having both low dimensionality (two or three) and high dimensionality (more than three) in objective function space.
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Cortés, Nareli Cruz; Coello, Carlos A. Coello
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14 Citations
In this paper, we propose a new multiobjective optimization approach based on the clonal selection principle. Our approach is compared with respect to other evolutionary multiobjective optimization techniques that are representative of the stateoftheart in the area. In our study, several test functions and metrics commonly adopted in evolutionary multiobjective optimization are used. Our results indicate that the use of an artificial immune system for multiobjective optimization is a viable alternative.
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SantanaQuintero, Luis V.; RamírezSantiago, Noel; Coello, Carlos A. Coello; Luque, Julián Molina; HernándezDíaz, Alfredo García
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10 Citations
This paper presents a new multiobjective evolutionary algorithm which consists of a hybrid between a particle swarm optimization approach and some concepts from rough sets theory. The main idea of the approach is to combine the high convergence rate of the particle swarm optimization algorithm with a local search approach based on rough sets that is able to spread the nondominated solutions found, so that a good distribution along the Pareto front is achieved. Our proposed approach is able to converge in several test functions of 10 to 30 decision variables with only 4,000 fitness function evaluations. This is a very low number of evaluations if compared with today’s standards in the specialized literature. Our proposed approach was validated using nine standard test functions commonly adopted in the specialized literature. Our results were compared with respect to a multiobjective evolutionary algorithm that is representative of the stateoftheart in the area: the NSGAII.
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MezuraMontes, Efrén; Coello, Carlos A. Coello
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24 Citations
We propose an evolutionarybased approach to solve engineering design problems without using penalty functions. The aim is to identify and maintain infeasible solutions close to the feasible region located in promising areas. In this way, using the genetic operators, more solutions will be generated inside the feasible region and also near its boundaries. As a result, the feasible region will be sampled wellenough as to reach better feasible solutions. The proposed approach, which is simple to implement, is tested with respect to typical penalty function techniques as well as against stateoftheart approaches using four mechanical design problems. The results obtained are discussed and some conclusions are provided.
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Coello, Carlos A. Coello; Cortés, Nareli Cruz
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285 Citations
In this paper, we propose an algorithm based on the clonal selection principle to solve multiobjective optimization problems (either constrained or unconstrained). The proposed approach uses Pareto dominance and feasibility to identify solutions that deserve to be cloned, and uses two types of mutation: uniform mutation is applied to the clones produced and nonuniform mutation is applied to the “not so good” antibodies (which are represented by binary strings that encode the decision variables of the problem to be solved). We also use a secondary (or external) population that stores the nondominated solutions found along the search process. Such secondary population constitutes the elitist mechanism of our approach and it allows it to move towards the true Pareto front of a problem over time. Our approach is compared with three other algorithms that are representative of the stateoftheart in evolutionary multiobjective optimization. For our comparative study, three metrics are adopted and graphical comparisons with respect to the true Pareto front of each problem are also included. Results indicate that the proposed approach is a viable alternative to solve multiobjective optimization problems.
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MezuraMontes, Efrén; MuñozDávila, Lucía; Coello, Carlos A. Coello
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5 Citations
This document presents a proposal to incorporate a fitness inheritance mechanism into an Evolution Strategy used to solve the general nonlinear programming problem. The aim is to find a tradeoff between a lower number of evaluations of each solution and a good performance of the approach. A set of test problems taken from the specialized literature was used to test the capabilities of the proposed approach to save evaluations and to maintain a competitive performance.
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Coello, Carlos A. Coello; Rivera, Daniel Cortés; Cortés, Nareli Cruz
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33 Citations
In this paper, we propose an algorithm based on an artificial immune system to solve job shop scheduling problems. The approach uses clonal selection, hypermutations and a library of antibodies to construct solutions. It also uses a local selection mechanism that tries to eliminate gaps between jobs in order to improve solutions produced by the search mechanism of the algorithm. The proposed approach is compared with respect to GRASP (an enumerative approach) in several test problems taken from the specialized literature. Our results indicate that the proposed algorithm is highly competitive, being able to produce better solutions than GRASP in several cases, at a fraction of its computational cost.
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HernandezDiaz, Alfredo G.; Coello, Carlos A. Coello; SantanaQuintero, Luis V.; Perez, Fatima; Molina, Julian; Caballero, Rafael
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2 Citations
Recent works have shown how hybrid variants of gradientbased methods and evolutionary algorithms perform better than a pure evolutionary method both for singleobjective and multiobjective optimization. This same idea has been used with Evolutionary Multiobjective Optimization (EMO), obtaining also very promising results. In most cases, gradient information is used as part of the mutation operator (and only for unconstrained MOPs), in order to move every generated point to the exact Pareto front. In our approach, we use the KarushKuhnTucker optimality condition for constrained optimization problems to combine the information provided by the gradient vector of each objective function and the gradient vectors of constraint functions to obtain a feasible movement direction in those points near the border. In our approach, gradients of the objective functions will be approximated using quadratic regressions, trying to avoid local optima. The proposed algorithm is able to converge on several nonlinear constrained multiobjective optimization problems obtained from a benchmark, consuming few objective function evaluations (between 150 and 1000). Our results indicate that our proposed scheme may produce a significant reduction in the computational cost, while producing results of good quality, when it is incorporated into a hybrid MOEA or when it is used to seed an EMO algorithm.
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MezuraMontes, Efrén; Coello, Carlos A. Coello; TunMorales, Edy I.
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In this paper, we propose a differential evolution algorithm to solve constrained optimization problems. Our approach uses three simple selection criteria based on feasibility to guide the search to the feasible region. The proposed approach does not require any extra parameters other than those normally adopted by the Differential Evolution algorithm. The present approach was validated using test functions from a wellknown benchmark commonly adopted to validate constrainthandling techniques used with evolutionary algorithms. The results obtained by the proposed approach are very competitive with respect to other constrainthandling techniques that are representative of the stateoftheart in the area.
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Coello, Carlos A. Coello; Lamont, Gary B.; Veldhuizen, David A. Van
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Successfully engineering Multiobjective Evolutionary Algorithms (MOEAs) involves thoroughly addressing many different issues. However, the performance concepts of efficiency and effectiveness are paramount. MOEAs are stochastic, populationbased computational procedures mimicking evolutionary concepts and operations in attempts to find satisfactory, if not optimal, solutions of problems with multiple objectives. Evolutionary Algorithms (EAs) and MOEAs are adaptive stochastic search techniques classified under the umbrella of soft computing; generic EAs such as Genetic Algorithms, Evolution Strategies, Evolutionary Programming, and Genetic Programming are all successfully used in MOEA implementations
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Coello, Carlos A. Coello; Lamont, Gary B.; Veldhuizen, David A. Van
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Problems with multiple objectives arise in a natural fashion in most disciplines and their solution has been a challenge to researchers for a long time. Despite the considerable variety of techniques developed in Operations Research (OR) and other disciplines to tackle these problems, the complexities of their solution calls for alternative approaches.
The use of evolutionary algorithms (EAs) to solve problems of this nature has been motivated mainly because of the populationbased nature of EAs which allows the generation of several elements of the Pareto optimal set in a single run. Additionally, the complexity of some multiobjective optimization problems (MOPs) (e.g., very large search spaces, uncertainty, noise, disjoint Pareto curves, etc.) may prevent use (or application) of traditional OR MOPsolution techniques.
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Coello, Carlos A. Coello; Lamont, Gary B.; Veldhuizen, David A. Van
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Why test multiobjective evolutionary algorithms (MOEAs)? To evaluate, compare, classify, and improve algorithm performance (effectiveness and effi ciency). What is a MOEA test? Should we use a multiobjective optimization problem (MOP) test function, a MOP test suite, pedagogical functions, or a realworld problem? How to find an appropriate MOEA test?
Should we rely on the MOEA literature, on historical use, on test generators, or on well known realworld applications? When to test? Should we adopt and incremental algorithm and test development methodology or should we wait until the final stage of algorithm development to test it?
How should we design a MOEA test? Evidently, several important issues must be taken into consideration. For example: basic assumptions, computational platform selection, statistical tools, performance measures selection, experimental plan, among others. Thus, considerable effort must be spent not only in defining proper MOP tests and in generating the proper design of MOEA experiments, but also in employing the appropriate performance measures and experiment conditions, as well as the proper statistical tools that allow a fair algorithmic comparison. In this chapter, the development of various MOP test suites is discussed in detail.
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Coello, Carlos A. Coello; Lamont, Gary B.; Veldhuizen, David A. Van
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Although the application of classical multiobjective optimization techniques to solve problems in different areas (e.g., management, engineering and science) started as early as 1951 (see Section 1.6.2 from Chapter 1), MultiObjective Evolutionary Algorithms (MOEAs) were applied for the first time until the mid1980s. However, since the late 1990s, there has been a considerable increase in the number of applications of MOEAs. This has been mainly originated by the success of MOEAs in solving realworld problems. MOEAs have generated either competitive or better results than those produced using other search techniques. This has made the task of classifying MOEA applications difficult and subjective. Trying to deal with this problem, it was decided to use a rather simple and general classification in this chapter, trying to fit each paper reviewed within the closest category according to the focus of the work. For example, a paper that is related to scheduling and naval engineering but is more focused on the second subject, is classified under “environmental, naval and hydraulic engineering”. This avoids overlapping to a certain extent, but can be confusing for some people. Therefore, it was decided to add as many entries as possible to the analytical index provided at the end of this book to facilitate the search. Additionally, italics characters are used throughout this chapter to indicate the specific name of an application, in an attempt to facilitate the search of specific information.
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Aguirre, Arturo; Equihua, Edgar C. González; Coello, Carlos A. Coello
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1 Citations
In this paper,we propose the use of Information Theory as the basis of the fitness function for Boolean circuit design.Boolean functions are implemented by means of multiplexers and genetic programming. Entropy based measures such as Mutual Information and Conditional Entropy are investigated as tools for similarity measures between circuits.A comparison of synthesized (through evolution)and minimized circuits through other methods denotes the advantages of the InformationTheoretical approach.
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Aguirre, Arturo Hernández; Coello, Carlos A. Coello
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2 Citations
In this paper, we propose the use of Information Theory as the basis for designing a fitness function for Boolean circuit design using Genetic Programming. Boolean functions are implemented by replicating binary multiplexers. Entropybased measures, such as Mutual Information and Normalized Mutual Information are investigated as tools for similarity measures between the target and evolving circuit. Three fitness functions are built over a primitive one. We show that the landscape of Normalized Mutual Information is more amenable for being used as a fitness function than simple Mutual Information. The evolutionary synthesized circuits are compared to the known optimum size. A discussion of the potential of the InformationTheoretical approach is given.
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Coello, Carlos A. Coello; Lamont, Gary B.; Veldhuizen, David A. Van
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Evolutionary Algorithms (EAs) are not the only search techniques that have been used to deal with multiobjective optimization problems. In fact, as other search techniques (e.g., Tabu search and simulated annealing) have proved to have very good performance in many combinatorial (as well as other types of) optimization problems, it is only natural to think of extensions of such approaches to deal with multiple objectives.
The Operations Research (OR) and EA communities have shown a clear interest in pursuing these extensions. Since the multiobjective formulation of combinatorial optimization problems (e.g., the quadratic assignment problem) are known to be NPcomplete, they present real challenges to researchers. Additionally, many realworld problems (e.g., scheduling) require efficient approaches that can at least approximate P_{true} and PF_{true} in a reasonable amount of time.
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Coello, Carlos A. Coello; Lamont, Gary B.; Veldhuizen, David A. Van
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beneficial to realworld applications, local search structures have been proposed to drive the search towards the Pareto front more effectively and efficiently. A number of generic local search techniques have been proposed along with problem domain specific methods. These approaches are discussed in this chapter with thoughts on integrating new innovative local search with MOEAs. Another emerging area of MOEA research is applying coevolutionary techniques. Relatively few researchers have explored the idea of combining coevolution with MOEAs. This chapter presents various researchers’ algorithmic processes for Coevolutionary MOEAs (CMOEA) with each researcher’s efforts summarized, categorized, and analyzed. Some potential concept and future applications of MOEA coevolution are also suggested. Exercises, discussion questions, and possible research directions for MOEA local search and coevolution are presented at the end of the chapter.
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Coello, Carlos A. Coello; Lamont, Gary B.; Veldhuizen, David A. Van
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Many MOEA development efforts acknowledge various facets of underlying MOEA theory, but make limited contributions when simply citing relevant issues raised by others. Some authors, however, exhibit significant theoretical detail. Their work provides basic MOEA models and associated theories. Table 6.1 lists contemporary efforts reflecting MOEA theory development. In essence, a MOEA is searching for optimal elements in a partially ordered set or in the Pareto optimal set. Thus, the concept of convergence to P_{true} and PF_{true} is integral to the MOEA search process.
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Coello, Carlos A. Coello; Lamont, Gary B.; Veldhuizen, David A. Van
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2 Citations
One aspect that most of the current research on evolutionary multiobjective optimization (EMO) often disregards is the fact that the solution of a multiobjective optimization problem (MOP) really involves three stages: measurement, search, and decision making.
Being able to find P_{true} does not completely solve an MOP. The decision maker (DM) still has to choose a single solution out of this set. The process of selecting a single solution is not trivial. In fact, there is a set of methodologies regarding how and when to incorporate decisions from the DM into the search process.
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Coello, Carlos A. Coello; Lamont, Gary B.; Veldhuizen, David A. Van
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Both researchers and practitioners in science, engineering, government, and industry certainly have a strong interest in knowing stateoftheart multiobjective optimization techniques. For researchers, this is the normal procedure to trigger new and original algorithmic contributions. For practitioners, this knowledge allows them to choose the most appropriate algorithm(s) for their specific multiobjective problem (MOP) domain application. From the decision maker’s (DM) perspective, it is desired that only a “few” solutions are available for ease of decision. Thus, as presented in Chapter 1, one is attempting to optimize a vector objective function possibly with constraints resulting in tradeoffs between the multiple objectives. This chapter employs the various generic mathematical definitions defined in Chapter 1 for discussing multiobjective evolutionary algorithm (MOEA) design. It is desired that an MOEA generates MOP solutions in P_{true} which provide a tradeoff of performance (efficiency, effectiveness) for specific system model objectives (cost/profit, constraints, etc.) that may mutually conflict. For example, the classical multiobjective knapsack problem (profit and weight) and drug development (cost vs. effectiveness) represent vectors of two objectives. Maximizing one objective such as profit usually does not optimize another such as reliability.
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Coello, Carlos A. Coello; Lamont, Gary B.; Veldhuizen, David A. Van
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Regarding the scientific method of experimentation, it is desirable to construct an accurate, reliable, consistent and nonarbitrary representation of multiobjective evolutionary algorithm (MOEA) architectures and performance over a variety of multiobjective optimization problems (MOPs). In particular, through the use of standard procedures and criteria, one should attempt to minimize the influence of bias or prejudice of the experimenter when testing a MOEA hypothesis. The design of each experiment must conform then to an accepted “standard” approach as reflected in any generic scientific method. When employing the scientific method, the detailed design of MOEA experiments can draw heavily from outlines presented by Barr et al. [93] and Jackson et al. [765]. These generic articles discuss computational experiment design for heuristic methods, providing guidelines for reporting results and ensuring their reproducibility. Specifically, they suggest that a welldesigned experiment follows the following steps: 1. Define experimental goals; 2. Choose measures of performance  metrics; 3. Design and execute the experiment; 4. Analyze data and draw conclusions; 5. Report experimental results.
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MezuraMontes, Efrén; Coello, Carlos A. Coello
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Summary
In this chapter, we present a survey of constrainthandling techniques based on evolutionary multiobjective optimization concepts. We present some basic definitions required to make this chapter selfcontained, and then introduce the way in which a global (singleobjective) nonlinear optimization problem is transformed into an unconstrained multiobjective optimization problem. A taxonomy of methods is also proposed and each of them is briefly described. Some interesting findings regarding common features of the approaches analyzed are also discussed.
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