The Algorithm Selection Problem (ASP) aims to identify the most suitable algorithm for a given problem instance. This paper introduces MODT-ASP, a multi-objective evolutionary framework designed to construct interpretable decision trees for ASP. The proposed approach overcomes the limitations of existing methods, such as scalability constraints and the challenge of balancing competing objectives, by employing a customized encoding scheme and specialized genetic operators. These components effectively explore trade-offs between predictive accuracy and model complexity via Pareto optimization. An enhanced version, EMODT-ASP, further integrates refined control mechanisms to improve generalization. Comprehensive experimental evaluations demonstrate the robustness and effectiveness of the proposed framework. In a large-scale linear programming benchmark comprising 1,004 problems and 532 algorithms, MODT-ASP consistently produces high-quality Pareto-optimal solutions. Furthermore, in the Open Algorithm Selection Challenge (OASC), evaluated across 8 heterogeneous scenarios, the proposed approach achieves third place overall, outperforming 6 of 8 OASC competitors and all 3 IP+VND variants reported in the recent literature. These results confirm its strong cross-domain applicability.

To ensure empirical evaluation of multi- and many-objective evolutionary algorithms, researchers perform benchmarking across test problems and algorithms. Due to the volume of performance data and the heterogeneity of problem characteristics, analyzing results becomes complex and prone to misinterpretation. Performance profiles have proven effective for visualizing and interpreting such results; however, they do not account for the relative difficulty or importance of individual problems and may overweight easy or less informative cases, potentially obscuring distinctions between algorithm performance. In this work, we address this limitation by extending the classical performance profile approach with a difficulty-aware weighting scheme that emphasizes more challenging problems. Weights can be assigned either a priori, based on problem characteristics such as the number of objectives or decision variables, or a posteriori, based on computational effort. We define and prove key mathematical properties of classical performance profiles, including local and global stability, and show that these properties extend to the proposed weighted formulation. By employing a difficulty-aware weighting scheme, the approach biases aggregation toward higher-dimensional instances, enabling a more discriminative assessment of scalability, robustness, and performance. The advantages of the weighted approach are demonstrated through experiments with algorithms applied to problem sets with numbers of objectives.

Abstract Effective control of Energy Storage Systems (ESS) is crucial for the secure and profitable operation of microgrids. In this context, ESSs are essential for enhancing the overall grid resilience, balancing supply, and mitigating voltage and frequency variations. This paper presents a novel neuroevolutionary method, coupling a modified version of the Multi-Objective Evolutionary Policy Search (MEPS) algorithm with the Cross-Entropy method, aimed at optimizing an ESS control problem. The modified MEPS, named Cascade-MEPS, employs a cascade weights mutation operator to refine policies by focusing on the most recent hidden node, ensuring localized and non-disruptive adjustments. The resulting algorithm, referred to as cross-entropy Cascade-MEPS (CE-CMEPS), utilizes the cross-entropy method as a depth initialization strategy, conducting an initial exploration of the weights space to initialize the population prior to Cascade-MEPS execution. Experimental validation on a newly proposed multi-objective ESS control problem demonstrates the efficacy of CE-CMEPS, showcasing performance improvements and reduced variation compared to standalone MEPS. Our results show that CE-CMEPS is an effective ESS discharge controller and a sustainable multi-objective reinforcement learning solution.

Evolutionary multiobjective and many-objective optimization (EMO and EMaO) algorithms are increasingly used to identify the true shape and location of the Pareto-optimal front using a few representative well-converged and well-distributed solutions. The reason for their popularity is due to their ability to provide a better understanding of objective relationships for optimal solutions, and also to facilitate the choice of a preferred solution using an interactive or post-optimal multicriterion decision analysis. However, since EMO and EMaO algorithms are stochastic, a single application may not provide a true representative set with a desired number of Pareto solutions reliably in repetitive runs and importantly with a well-distributed set of solutions. In this article, we propose a multistage framework involving reference-vector-based evolutionary multi- and many-objective algorithms (MuSt-EMO and MuSt-EMaO) that attempts to recursively rectify shortcomings of previous stages by careful executions of subsequent stages so that a prescribed number of well-distributed and well-converged solutions are achieved at the end. The proposed multistage approach is implemented to a number of popular reference vector-based EMO/EMaO algorithms and is applied on various multi- and many-objective test and real-world problems.

Abstract Optimal power flow (OPF) is a complex, highly nonlinear, NP-hard optimization problem, in which the goal is to determine the optimal operational parameters of a power-related system (in many cases a type of smart or micro grid) which guarantee an economic and effective power dispatch. In recent years, a number of approaches based on metaheuristics algorithms have been proposed to solve OPF problems. In this paper, we propose the use of the Cross-Entropy (CE) method as a first step depth search operator to assist population-based evolutionary methods in the framework of an OPF problem. Specifically, a new variant of the Coral Reefs Optimization with Substrate Layers algorithm boosted with CE method (CE+CRO-SL) is presented in this work. We have adopted the IEEE 57-Bus System as a test scenario which, by default, has seven thermal generators for power production for the grid. We have modified this system by replacing three thermal generators with renewable source generators, in order to consider a smart grid approach with renewable energy production. The performance of CE+CRO-SL in this particular case study scenario has been compared with that of well-known techniques such as population’s methods CMA-ES and EPSO (both boosted with CE). The results obtained indicate that CE+CRO-SL showed a superior performance than the alternative techniques in terms of efficiency and accuracy. This is justified by its greater exploration capacity, since it has internally operations coming from different heuristics, thus surpassing the performance of classic methods. Moreover, in a projection analysis, the CE+CRO-SL provides a profit of millions of dollars per month in all cases tested considering the modified version of the IEEE 57-Bus smart grid system.

Optimal active–reactive power dispatch problems (OARPD) are considered large scale optimization problems with a high nonlinear complexity. Usually, in OARPD the objective is to minimize the cost of the system operation. In 2018, the IEEE PES committee proposed a competition, the “Operational planning of sustainable power systems”, in which a test bed relating the OARPD and a renewable energy generation challenge within a smart grid was proposed. In this work we consider three test scenarios proposed in that competition. Specifically, we present a hybrid meta-heuristic optimization approach applied to the OARPD, the Canonical Differential Evolutionary Particle Swarm Optimization (C-DEEPSO), to tackle these test scenarios. Comparative results with other algorithms such as CMA-ES, EPSO, and CEEPSO indicate that C-DEEPSO shows a competitive performance when solving the OARPD problems.

Dominance move (DoM) is a binary quality indicator that can be used in multiobjective and many-objective optimization to compare two solution sets obtained from different simulations. The DoM indicator can differentiate the sets for certain important features, such asconvergence,spread,uniformity, andcardinality. DoM does not require any reference point or any representative Pareto solution set, and it has an intuitive and physical meaning, similar to the$\epsilon $-indicator. It calculates the minimum total move of members of one set so that all elements in another set are to be dominated or identical to at least one member of the first set. Despite the aforementioned desired properties, DoM is hard to calculate, particularly for higher dimensions. There is an efficient and exact method to calculate it in biobjective problems. This work proposes a novel approach to calculate DoM using a mixed-integer programming (MIP) approach, which can handle two sets with two or more objectives and is shown to overcome the issue of information loss associated with the$\epsilon $-indicator. Experiments in the biobjective space are done to verify the model’s correctness. Furthermore, other experiments, using 3-, 5-, 10-, 15-, 20-, 25-, and 30-objective problems, are performed to show how the model behaves in higher dimensional cases. Algorithms, such as IBEA, MOEA/D, NSGA-III, NSGA-II, and SPEA2, are used to generate the solution sets; however, any other algorithm can also be used with the proposed MIP-DoM indicator. Further extensions are discussed to handle certain idiosyncrasies with some solution sets and improve the quality indicator and its use for other scenarios.

In this work we present a pattern classification approach coupling the Neighbourhood Component Analysis (NCA) classifier with the Canonical Differential Evolutionary Particle Swarm Optimization (C-DEEPSO). The standard NCA uses the conjugate gradient method to minimize the classification error. Here we propose an approach using the C-DEEPSO instead. In the experimental design, the coupled approach is applied to 20 benchmark data sets, and its performance is compared with the standard NCA using the conjugate gradient. The experimental analysis shows the usage of an evolutionary approach to enhance the performance of a machine learning algorithm can be competitive when compared to well-known iterative optimization techniques, and even outperform them in some problems. A real-world problem classifying cyber-attacks to an industrial control system of gas pipelines is also solved by the proposed approach. The results obtained indicate the proposed approach can successfully identify possible cyber-attacks to the control system. In this way, the NCA coupled to C-DEEPSO can work as an Intrusion Detection Systems (IDS), being able to guarantee an acceptable security level.

The unit dispatch problem is defined as the attribution of operational values to each generation unit inside a hydro-power plant (HPP), given some criteria such as the total power to be generated, or the operational bounds of each unit. An optimal dispatch programming for hydroelectric units in HPP provides a larger production of electricity, with minimal water use. This paper presents an evolutionary approach to optimize the multi-criteria electric dispatch problem in a general HPP, based on a Multi-objective Evolutionary Swarm Hybridization (MESH) algorithm. The proposed approach integrates mathematical models and evolutionary swarm computation. The experimental analysis shows that the proposed MESH algorithm is able to reach competitive results when compared with classical evolutionary algorithms, the NGA-II and SPEA2 basing on ANOVA inference test. Results also show that the proposed MESH is able to save a large amount of water in the energy production process, supplying the requested load, and minimizing blackout risks and generating a profit around $275,000 monthly.

This paper discusses a dimensionality reduction procedure to tackle a many-objective formulation of a Vehicle Routing Problem with a Demand Responsive Transport (VRPDRT). The problem formulation presents eight objective functions that aim to reduce the operating costs while meeting passenger needs and providing a high-quality service. Two different dimensionality reduction-based approaches, aggregation and feature selection are employed to transform the many-objective formulation into a bi-objective one. The reduction, applied during the search evolution, follows a hierarchical clustering technique in which the objective functions' similarity and conflict are explored. The proposed approaches are compared with a classic version of MOEA/D that solves the problem in its original formulation. Moreover, different dimensionality reduction frequencies are tested to assess the impact on the algorithms' performance. When comparing the outcomes in the original objective space, the results show that the aggregation approach outperforms the feature selection method, regardless of the dimensionality reduction frequency. Furthermore, while there is no statistical difference between the MOEA/D and the aggregation approach and the MOEA/D outperforms the feature selection approaches.

The performance assessment of multi-objective heuristic algorithms is one of the most significant contributions from the evolutionary optimization algorithms community. By contrast, performance assessment in the context of many-objective optimization is still a challenging, open research field. Recent advances have demonstrated disagreements between Pareto-compliant performance metrics, and indicated that reference fronts produced by benchmark generators of Pareto-optimal fronts could be further improved. In this work, we investigate these reference fronts with the help of multi-dimensional visualization techniques and Pareto-monotonic archivers. Interestingly, reference fronts produced by benchmark generators for DTLZ and WFG continuous optimization problems show significant issues, even when only three objectives are considered. Furthermore, given that input solution sets for five-objective problems are not high-quality, archivers are unable to output reasonable approximation fronts. We conclude that the performance assessment of EMO algorithms needs to urgently address reference front generation.

This paper tackles the short-term hydro-power unit commitment problem in a multi-reservoir system — a cascade-based operation scenario. For this, we propose a new mathematical modeling in which the goal is to maximize the total energy production of the hydro-power plant in a sub-daily operation, and, simultaneously, to maximize the total water content (volume) of reservoirs. For solving the problem, we discuss the Multi-objective Evolutionary Swarm Hybridization (MESH) algorithm, a recently proposed multi-objective swarm intelligence-based optimization method which has obtained very competitive results when compared to existing evolutionary algorithms in specific applications. The MESH approach has been applied to find the optimal water discharge and the power produced at the maximum reservoir volume for all possible combinations of turbines in a hydro-power plant. The performance of MESH has been compared with that of well-known evolutionary approaches such as NSGA-II, NSGA-III, SPEA2, and MOEA/D in a realistic problem considering data from a hydro-power energy system with two cascaded hydro-power plants in Brazil. Results indicate that MESH showed a superior performance than alternative multi-objective approaches in terms of efficiency and accuracy, providing a profit of $412,500 per month in a projection analysis carried out.

Dominance move (DoM) is a binary quality indicator to compare solution sets in multiobjective optimization. The indicator allows a more natural and intuitive relation when comparing solution sets. Like the ϵ-indicators, it is Pareto compliant and does not demand any parameters or reference sets. In spite of its advantages, the combinatorial calculation nature is a limitation. The original formulation presents an efficient method to calculate it in a bi-objective case only. This work presents an assignment formulation to calculate DoM in problems with three objectives or more. Some initial experiments, in the bi-objective space, were done to show that DoM has a similar interpretation as ϵ-indicators, and to show that our model formulation is correct. Next, other experiments, using three dimensions, were also done to show how DoM could be compared with other indicators: inverted generational distance (IGD) and hypervolume (HV). The assignment formulation for DoM is valid not only for three objectives but for more. Finally, there are some strengths and weaknesses, which are discussed and detailed.

In this paper, a hybrid single-objective metaheuristic, named as C-DEEPSO (Canonical Differential Evolutionary Particle Swarm Optimization), is proposed to solve large-scale optimization problems. C-DEEPSO can be viewed as an evolutionary algorithm with recombination rules borrowed from PSO or an swarm optimization method with selection and self-adaptiveness properties. To assess the algorithm performance, the algorithm is run over 15 benchmark continuous problems presented in CEC'2015. The algorithm is also applied over a real world large-scale problem. The results indicate that the proposed algorithm is an efficient and competitive method to handle such problems. The experimental results also show that the new approach reaches competitive results when compared to the reference algorithm DECC-G. An application of C-DEEPSO were perform to solve the electric dispatch in a large scale energy network, the IEEE 57 Bus-System. The results show that the algorithm is a good way to solve nonlinear problems respecting many constraints associated.

This work proposes an evolutionary approach to solve the Menu Planning Problem. Our work uses the Brazilian school context and our principal goal is to create menus that minimize the total cost of these menus. However, those menus must also satisfy requirements of the Brazilian government, such as: (i) student age group, (ii) school category, (iii) school duration time, (iv) school location, (v) variety of preparations, (vi) harmony of preparations, (vii) maximum amount to be paid for each meal and, (viii) lower and upper limits of macronutrients. The results demonstrate that the evolutionary approach is not only able to generate a set of inexpensive and healthy menus but also respect the required set of constraints. A constrained deterministic approach is performed to generate 5-day menu through a greedy-based function taking into account the normalized sum of all macronutrients and the monetary cost of the menu. A comparison between the 5-day menu obtained by the proposed approach and the constrained greedy-based approach menu is carried out. Despite the fact the obtained menu outperforms the greed-based menu taking into account the total cost, this difference is not so expressive. However, all macronutrients were outside the pre-defined range at least in one day of the week. The 5-day menu obtained by the proposed approach is evaluated by a nutritionist. The overall quality of the menu is outstanding and the time spent to generate it is 60 seconds.

This paper presents a hybrid PSO algorithm (Particle Swarm Optimization) with an ILS (Iterated Local Search) operator for handling equality constraints problems in mono-objective optimization problems. The ILS can be used to locally search around the best solutions in some generations, exploring the attraction basins in small portions of the feasible set. This process can compensate the difficulty of the evolutionary algorithm to generate good solutions in zero-volume regions. The greatest advantage of the operator is the simple implementation. Experiments performed on benchmark problems shows improvement in accuracy, reducing the gap for the tested problems.

Metaheuristics for optimization based on the immune network theory are often highlighted by being able to maintain the diversity of candidate solutions present in the population, allowing a greater coverage of the search space. This work, however, shows that algorithms derived from the aiNET family for the solution of combinatorial problems may not present an adequate strategy for search space exploration, leading to premature convergence in local minimums. In order to solve this issue, a hybrid metaheuristic called VNS-aiNET is proposed, integrating aspects of the COPT-aiNET algorithm with characteristics of the trajectory metaheuristic Variable Neighborhood Search (VNS), as well as a new fitness function, which makes it possible to escape from local minima and enables it to a greater exploration of the search space. The proposed metaheuristic is evaluated using a scheduling problem widely studied in the literature. The performed experiments show that the proposed hybrid metaheuristic presents a convergence superior to two approaches of the aiNET family and to the reference algorithms of the literature. In contrast, the solutions present in the resulting immunological memory have less diversity when compared to the aiNET family approaches.

This article presents a non-deterministic approach to the Three-Dimensional Bin Packing Problem, using a genetic algorithm. To perform the packing, an algorithm was developed considering rotations, size constraints of objects and better utilization of previous free spaces (flexible width). Genetic operators have been implemented based on existing operators, but the highlight is the Real-Polarized crossover operator that produces new solutions with a certain disturbance near the best parent. The proposal presented here has been tested on instances already known in the literature and real instances. A visual comparison using boxplot was done and, in some situations, it was possible to say that the obtained results are statistically superior than the ones presented in the literature. In a given instance class, the presented Genetic Algorithm found solutions reaching up to 70% less bins.

In this work we propose a new algorithm, Delta-r-GRASP, for solving the allocation of Roadside Units (RSUs) in a Vehicular Network. Our goal is to find the minimum set of RSUs to meet a Deployment Δρ2ρ1. The Deployment Δρ2ρ1is a metric for specifying minimum communication guarantees from the infrastructure supporting the Vehicular Network. We compare our algorithm with a baseline algorithm, Delta-r. Moreover, we compare our results with the optimal value achieved by solver CPLEX. Our results demonstrate that our approach requires up to 85% fewer RSUs to achieve the same deployment efficiency, and our results differ no more than 15% from the optimal values.

In this work, we propose a GRASP+VNS algorithm for solving the allocation of Roadside Units (RSUs) in a Vehicular Network. Our main objective is to find the minimum set of RSUs to meet a Deployment Δρ2ρ1. The Deployment Δρ2ρ1is a metric for specifying minimal communication guarantees from the infrastructure supporting the Vehicular Network. We compare GRASP+VNS to some baseline algorithms: (i) Delta-g; (ii) Delta-r and, (iii) the optimal value. Our results demonstrate that our approach requires up to 90% less Roadside Units to meet the QoS required by Deployment Δρ2ρ1metric. Besides, different from the baseline algorithms, our approach find results that differ no more than 17% from the optimal values for all tested instances.

The problem of choosing an open-pit mining investment portfolio can be stated as, given a budget, picking among the possible projects the combination that will incur in the best increase for the mine's productivity when applied. Due to interaction between projects even a seemingly cheap and effective project may not be the most appropriate choice as, in the complete portfolio, it may interact badly with other projects. The objective of this paper is to produce an algorithm capable of finding an adequate solution to this kind of problem in a viable time frame. The proposed heuristic modifies an initial solution through a series of permutation operations with the objective of finding a better solution. Using data and projects from a real Brazilian mine, the algorithm is compared with the current adopted solutions. The algorithm is also used to solve problems of similar classes and its complexity order is estimated. For a collection of 15 projects applied to a medium port mining station, the algorithm is able to find the optimal solution with 93 evaluations of the objective function (in a 3-hour time frame) for the studied instance of the problem. The algorithm also indicates a linear complexity regarding to the number of projects.

This work presents extensions of the general methodology of employing quadratic approximations of the objective function and constraints for handling non-linear equality constraints in single-objective optimization problems. The methodology does not require any extra function evaluation since the quadratic approximations are constructed using only information that would be already obtained in the course of the optimization algorithms. The methodology is coupled with the Real Biased Genetic Algorithm to tackle non-linear single-objective optimization problems with two equality constraints. The modified algorithm is tested with a set of analytical problems. The results show the modified algorithm finds the constrained optima with enhanced precision and faster convergence. Considering that the new technique does not impose any additional cost to the algorithms, it can be stated that the technique is also suitable for costly black-box problems.

In this paper, a novel hybrid single-objective metaheuristic, the so called C-DEEPSO (Canonical Differential Evolutionary Particle Swarm Optimization), is proposed and tested. C-DEEPSO can be viewed as an evolutionary algorithm with recombination rules borrowed from PSO, or a swarm optimization method with selection and self-adaptiveness properties proper from DE. A case study on the problem of optimal control for reactive sources in energy production by Wind Power Plants (WPP), solved by means of Optimal Power Flow (OPF-like), is used to test the new hybrid algorithm and to evaluate its performance. C-DEEPSO is compared to the baseline algorithm, DEEPSO, and to a reference algorithm, Mean-Variance Mapping Optimization (MVMO). The experiments indicate that the proposed algorithm is efficient and competitive, capable to tackle this large-scale problem. The results also show that the new approach exhibits better results, when compared to MVMO.

The Vehicle Routing Problem (VRP) has been largely studied over the last years, since problems involving the transport of persons and/or goods have great practical application. This paper addresses the Vehicles Routing Problem with Demand Responsive Transport (VRPDRT), a type of transport which enables customers to be taken to your destination like a taxi or minibus in order to reduce operating costs and to meet customer needs. A multiobjective approach is proposed to VRPDRT in which five different objective functions are used. Using an iterative methodology, known as aggregation tree, the objective functions are used to construct a bi-objective version for the problem. The proposed bi-objective optimization problem is solved via NSGA-II and SPEA2 and the algorithm performances are compared using S-Metric. Through a statistical test, the results shows with 95% of confidence that the NSGA-II presents better convergence when compared with SPEA2.

In this work we tackle the bus stop selection step for the School Bus Routing Problem (SBRP). Our goal is to minimize the number of bus stops in order to assign all students to a bus stop respecting a home-to-bus-stop walking distance constraint. Our strategy creates a large number of possible bus stops points in a road network and uses a pseudo-random constructive heuristic algorithm to assign students to a bus stops. Our approach is tested on a real georeferenced data of a Brazilian city and is compared with a different methodology. Results demonstrate that the proposed approach is able to find good solutions for this optimization problem. Besides, the higher the number of possible points to install bus stops, the smaller is the number of bus stops required to attend all students.

In this work the allocation of Roadside Units (RSUs) in a V2I network is modeled as a Maximum Coverage Problem. The main objective is to maximize the number of distinct vehicles contacting the infrastructure. Two different approaches are presented to solve the problem. The first one is an ILP model that can found optimal solutions or give sharp upper and lower bounds for the problem. The second one is a GRASP-based heuristic that can found close-to-optimal solutions. The GRASP-based heuristic is compared with a previous work achieving better results. Furthermore, a new metric to measure the efficiency of a Deployment strategy is presented.

Recent works raised the hypothesis that the assignment of a geometry to the decision variable space of a combinatorial problem could be useful both for providing meaningful descriptions of the fitness landscape and for supporting the systematic construction of evolutionary operators (the geometric operators) that make a consistent usage of the space geometric properties in the search for problem optima. This paper introduces some new geometric operators that constitute the realization of searches along the combinatorial space versions of the geometric entities descent directions and subspaces. The new geometric operators are stated in the specific context of the wireless sensor network dynamic coverage and connectivity problem (WSN-DCCP). A genetic algorithm (GA) is developed for the WSN-DCCP using the proposed operators, being compared with a formulation based on integer linear programming (ILP) which is solved with exact methods. That ILP formulation adopts a proxy objective function based on the minimization of energy consumption in the network, in order to approximate the objective of network lifetime maximization, and a greedy approach for dealing with the system's dynamics. To the authors' knowledge, the proposed GA is the first algorithm to outperform the lifetime of networks as synthesized by the ILP formulation, also running in much smaller computational times for large instances.

Nowadays, the population growth and economic development causes the need for electricity power to increase every year. An unit dispatch problem is defined as the attribution of operational values to each generation unit inside a power plant, given some criteria to be obeyed like the total power to be generated, operational bounds of these units etc. In this context, an optimal dispatch programming for hydroelectric units in energy plants provides a bigger production of electricity to be generated with a minimal water amount. This paper presents an optimization solution for hydroelectric generating system of a plant, using Differential Evolution algorithms. The novel mathematical model proposed and validation of the obtained algorithms will be performed with practical simulation experiments. Throughout the text, the equations and models for the system simulation will be fully described, and the experiments and results will be objectively analysed through statistical inference. Simulation results indicate savings of 6.5 million litres of water for each month of operation using the proposed solution.

This paper presents a method based on Neural Networks and Evolutionary Algorithms to solve the Hydroelectric Unit Commitment Problem. A Neural Network is used to model the production function and a novel approach based on movable partitions is proposed, which makes it easier to model the desired power output equality constraint in the optimization modeling. Three evolutionary algorithms are tested in order to find optimized operation points: differential evolution DE/best/1/bin, a balanced version of DE and Particle Swarm Optimization algorithm (PSO). The results show that the proposed method is effective in terms of water consumption, reaching in some cases more than 1% of economy whether compared to the traditional commitment strategy.

This paper presents a specialized multiobjective evolutionary algorithm SPEA2 (Strength Pareto Evolutionary Algorithm 2) coupled, separetely, with four placement heuristics for solving the 2D Guillotine Strip Packing Problem. In this study, the problem requires minimization of both the amount of wasted material and the number of independent cuts required by a packing. With the goal of solving this multiobjective version of the problem, the construction phase of the GRASP algorithm (Greedy Randomized Adaptive Search Procedure) is used to generate a portion of the initial population of SPEA2. Four different placement heuristics, Next-Fit, a variation of Next-Fit, Best-Fit and First-Fit, were coupled with SPEA2 and were tested on a set of test data. The results show that the presented methodology is able to generate a good set of candidate solutions for each test problem. A statistical comparison methodology, based on multiobjective principles, was used to compare the four algorithm variants.

Local search techniques based on Convex Quadratic Approximation (CQA) of functions are studied here, in order to speed up the convergence and the quality of solutions in evolutionary multiobjective algorithms. The hybrid methods studied here pick up points from the nondominated population and determine a CQA for each objective function. Since the CQA of the functions and the respective weighted sums are convex, fast deterministic methods can be used in order to generate approximated Pareto-optimal solutions from the approximated functions. A new scheme is proposed in this paper, using a CQA model that represents a lower bound for the function points, which can be solved via linear programming. This scheme and also another one using the methodology of linear matrix inequality (LMI) for CQA are coupled with a canonical implementation of the NSGA-II. Comparison tests are performed, using Monte Carlo simulations, considering the S-metric with an equivalent final number of evaluated objective functions and the algorithm execution time. The results indicate that the proposed scheme is promising.

This paper presents a statistical based comparison methodology for performing evolutionary algorithm comparison under multiple merit criteria. The analysis of each criterion is based on the progressive construction of a ranking of the algorithms under analysis, with the determination of significance levels for each ranking step. The multicriteria analysis is based on the aggregation of the different criteria rankings via a non-dominance analysis which indicates the algorithms which constitute the efficient set. In order to avoid correlation effects, a principal component analysis pre-processing is performed. Bootstrapping techniques allow the evaluation of merit criteria data with arbitrary probability distribution functions. The algorithm ranking in each criterion is built progressively, using either ANOVA or first order stochastic dominance. The resulting ranking is checked using a permutation test which detects possible inconsistencies in the ranking—leading to the execution of more algorithm runs which refine the ranking confidence. As a by-product, the permutation test also delivers$p$-values for the ordering between each two algorithms which have adjacent rank positions. A comparison of the proposed method with other methodologies has been performed using reference probability distribution functions (PDFs). The proposed methodology has always reached the correct ranking with less samples and, in the case of non-Gaussian PDFs, the proposed methodology has worked well, while the other methods have not been able even to detect some PDF differences. The application of the proposed method is illustrated in benchmark problems.

The evolution in the microelectronics and embedded systems has expanded the employment of Wireless Sensor Networks (WSNs). The energy limitation of the nodes is a very important restriction of those structures and should be always considered during the network design. The search for energy-efficient WSNs must take into account aspects which are essential for the proper operation of the network, such as area coverage and network connectivity. This paper proposes an evolutionary approach for performing the design of WSNs, considering the dynamic nature of the problem. A genetic algorithm, which aims to maximize the lifetime of the network, is employed for establishing the sequence in which the sensor nodes are activated, ensuring that the minimum coverage (established a priori) and the connectivity constraints are met. Results achieved by the proposed algorithm in a 81-sensor node instance are compared with a former work, in order to validate the approach which is presented here.

This work presents a Linear Matrix Inequality (LMI) formulation for training Radial Basis Function (RBF) neural networks, considering the context of multiobjective learning. The multiobjective learning approach treats the bias-variance dilemma in neural network modeling as a bi-objective optimization problem: the minimization of the empirical risk measured by the sum of squared error over the training data, and the minimization of the structure complexity measured by the norm of the weight vector. We transform the multiobjective problem into a constrained mono-objective one, using the ϵ-constraint method. This mono-objective problem can be efficiently solved using an LMI formulation. A procedure for choosing the width parameter of the radial basis functions is also presented. The results show that the proposed methodology provides generalization control and high quality solutions.

This paper presents the hierarchical cluster counting (HCC), a new quality metric for nondominated sets generated by multi-objective optimizers that is based on hierarchical clustering techniques. In the computation of the HCC, the samples in the estimate set are sequentially grouped into clusters. The nearest clusters in a given iteration are joined together until all the data is grouped in only one class. The distances of fusion used at each iteration of the hierarchical agglomerative clustering process are integrated into one value, which is the value of the HCC for that estimate set. The examples show that the HCC metric is able to evaluate both the extension and uniformity of the samples in the estimate set, making it suitable as a unary diversity metric for multiobjective optimization.

The increase in the demand for wireless sensor networks (WSNs) has intensified studies which aim to obtain energy-efficient solutions, since the energy storage limitation is critical in those systems. However, there are other aspects which usually must be ensured in order to provide an efficient design of WSNs, such as area coverage and network connectivity. This paper proposes a multiobjective hybrid approach for solving the dynamic coverage and connectivity problem (DCCP) in flat WSN subjected to node failures. It combines a multiobjective global on-demand algorithm (MGoDA), which improves the current DCCP solution using a genetic algorithm, with a local online algorithm (LoA), which is intended to restore the network coverage when one or more failures occur. The proposed approach is compared with an integer linear programming (ILP) based approach and a similar mono-objective approach with regard to coverage, energy consumption and residual energy of the solution provided by each method. Results achieved for a test instance show that the hybrid approach presented can obtain good solutions with a considerably smaller computational cost than ILP. The multiobjective approach still provides a feasible method for extending WSNs lifetime with slight decreasing in the network mean coverage.

This work presents an evolutionary approach for solving a difficult problem of combinatorial optimization, the DCMST (degree-constrained minimum spanning tree problem). Three genetic algorithms which embed candidate solutions in the continuous space are proposed here for solving the DCMST. The results achieved by these three algorithms have been compared with four other existing algorithms according to three merit criteria: i) quality of the best solution found; ii) computational effort spent by the algorithm, and; iii) convergence tendency of the population. The three proposed algorithms have provided better results for both solution quality and population convergence, with reasonable computational cost, in tests performed for 25-node and 50-node test instances. The results suggest that the proposed algorithms are well suited for dealing with the problem under study.

In this paper, we propose a multiobjective evolutionary approach to design a microwave heating device. The goal is to heat the maximum amount of water, above certain temperature, and spending the minimum energy. The device is modeled as a loss multilayer dielectric irradiated by microwave power. The resulting bi-objective problem is then solved using SPEA2 and a set of solutions is obtained. The results show that SPEA2 finds a higher number of non-dominated solution when compared with the traditional approaches used in this problem, within lower computational cost.

Support Vector Machines (SVMs) are considered state-of-the-art learning machines techniques for classification problems. This paper studies the training of SVMs in the special case of problems in which the raw data to be used for training purposes is composed of both labeled and unlabeled data - the semi-supervised learning problem. This paper proposes the definition of an intermediate problem of attributing labels to the unlabeled data as a multiobjective optimization problem, with the conflicting objectives of minimizing the classification error over the training data set and maximizing the regularity of the resulting classifier. This intermediate problem is solved using an evolutionary multiobjective algorithm, the SPEA2. Simulation results are presented in order to illustrate the suitability of the proposed technique.

The increasing demand for Wireless Sensor Networks (WSN) has intensified studies which aim to obtain energy-efficient solutions, since the energy storage limitation is critical in those systems. However, there are other aspects which usually must be ensured in order to get an acceptable performance of WSNs, such as area coverage and network connectivity. This paper proposes a procedure for network performance enhancement: a multiobjective hybrid approach for solving the Dynamic Coverage and Connectivity Problem in flat WSN subjected to node failures.Results achieved for a test instance show that the hybrid approach can improve the performance of the WSN obtaining good solutions with a considerably smaller computational cost than ILP.

This paper presents an enhanced approach for comparing evolutionary algorithm. This approach is based on three statistical techniques: (a) Principal Component Analysis, which is used to make the data uncorrelated; (b) Bootstrapping, which is employed to build the probability distribution function of the merit functions; and (c) Stochastic Dominance Analysis, that is employed to make possible the comparison between two or more probability distribution functions. Since the approach proposed here is not based on parametric properties, it can be applied to compare any kind of quantity, regardless the probability distribution function. The results achieved by the proposed approach have provided more supported decisions than former approaches, when applied to the same problems.

This work studies the mGA operator (Micro Genetic Algorithm), that has been proposed in literature as a "local search" operator for optimization with Genetic Algorithm. A new interpretation for this operator behavior is proposed, showing the role that this operator can have in a "global search". Such interpretation will possibly allow the definition of some directives for this operator parameter tuning, leading to more efficient GA that reach the optima with greater probability, spending less objective function evaluations. Some preliminary tests, conducted over problems of nonlinear functions with continuous variables, are presented, leading to some specific conjectures about what should be such directives.

This paper studies the issue of space coordinate change in genetic algorithms, based on two methods: convex quadratic approximations, and principal component analysis. In both methods, the procedure employs only the objective function samples that have already been obtained through the usual genetic algorithm operations, without the need of any additional function evaluation. The two procedures have been tested over a set of benchmark problems, and the data has been analyzed via a stochastic dominance analysis procedure. In both cases, the results suggest that in the transformed coordinates the genetic algorithm can able to deal with ill-conditioned problems in less iterations and with greater proportion of successful attempts, in comparison to the genetic algorithm without coordinate transformation.

This paper proposes a local search optimizer that, employed as an additional operator in multiobjective evolutionary techniques, can help to find more precise estimates of the Pareto-optimal surface with a smaller cost of function evaluation. The new operator employs quadratic approximations of the objective functions and constraints, which are built using only the function samples already produced by the usual evolutionary algorithm function evaluations. The local search phase consists of solving the auxiliary multiobjective quadratic optimization problem defined from the quadratic approximations, scalarized via a goal attainment formulation using an LMI solver. As the determination of the new approximated solutions is performed without the need of any additional function evaluation, the proposed methodology is suitable for costly black-box optimization problems.

The biological control of plagues in agriculture, a practice that has been growing around the world, is performed by leaving a suitable quantity of natural enemies of the plague in the farm during the finite time horizon of the farming cycle. This work proposes a multi-objective mathematical solution for the problem of optimal biological plague control for soy farmings, considering the control cost and the cost of farming damage due to plague. The system model is non-linear with impulsive control dynamics, in order to cope with the real-problem feature of control action, that should be performed in a finite number of discrete time instants. The dynamic optimization problem is solved using the NSGA-II, a fast and elitist multiobjective genetic algorithm. The results suggest a dual plague control policy, in which the relative price of control action versus the associated additional harvesting determine the usage of either a low control action or a higher well-defined one.

This paper presents a new operator for genetic algorithms that enhances convergence in the case of multiple nonlinear equality constraints. The proposed operator, named CQA-MEC (Constraint Quadratic Approximation for Multiple Equality Constraints), performs the steps: (i) the approximation of the non-linear constraints via quadratic functions; (ii) the determination of exact equality-constrained projections of some points onto the approximated constraint surface, via an iterative projection algorithm; and (iii) the re-insertion of the constraint- satisfying points in the genetic algorithm population. This operator can be interpreted both as a local search engine (that employs local approximations of constraint functions for correcting the feasibility) and a kind of elitism operator for equality constrained problems that plays the role of "fixing" the best estimates of the feasible set. The proposed operator has the advantage of not requiring any additional function evaluation per algorithm iteration, solely making usage of the information that is already obtained in the course of the usual genetic algorithm iterations. The test cases that were performed suggest that the new operator can enhance both the convergence speed (in terms of the number of function evaluations) and the accuracy of the final result.

A large number of evolutionary algorithms for solving multiobjective optimization problems has been already developed. Several merit factors for comparing the outcomes of these algorithms have also been proposed. However, evaluating Pareto-surface sample sets is still considered an open problem, since the result of a multiobjective evolutionary algorithm is a collection of vectors forming a nondominated set, that can be viewed under rather different merit criteria. In this paper, we present a new performance metric: the Integrated Sphere Counting. This metric is motivated on two reasoning principles: (i) the Pareto-surface is an object that is to be described via sample sets, in a sense that is similar to the sampled function description in signal processing; and (ii) the resolution that is to be employed in the Pareto-surface sample set depends on the decision-making procedure resolution, instead of the surface structure itself. We test this metric with two benchmark problems: the 0/1 Knapsack Problem and ZDT number 6 test suite.

In this paper, we propose a local search methodology to be coupled with a Genetic Algorithm to solve optimization problems with non-linear constraints. This methodology uses quadratic approximations for both objective function and constraints. In the local search phase, these quadratic approximations define an associated problem that is solved using a linear matrix inequality (LMI) formulation. The number of function evaluations needed for finding the point of optimum is significantly reduced with this procedure, what makes the proposed methodology suitable for dealing with costly black-box optimization problems. A case study is presented: the well- known TEAM 22 benchmark problem, an expensive problem of electromagnetic design. The results show that the hybrid algorithm has a better performance when compared to the same Genetic Algorithm without the proposed local search operator.

The use of local search in evolutionary techniques is believed to enhance the performance of the algorithms, giving rise to memetic or hybrid algorithms. However, in many continuous optimization problems the additional cost required by local search may be prohibitive. Thus we propose the local learning of the objective and constraint functions prior to the local search phase of memetic algorithms, based on the samples gathered by the population through the evolutionary process. The local search operator is then applied over this approximated model. We perform some experiments by combining our approach with a real-coded genetic algorithm. The results demonstrate the benefit of the proposed methodology for costly black-box functions.

We devise in this paper a local search procedure for multiobjective genetic algorithms (GAs). The proposed local search process employs quadratic approximations for all objective functions involved in the optimization problem. The samples gathered by the algorithm along the evolutionary process are used to fit these quadratic approximations around the point selected to local search, therefore no extra cost of function evaluation is required. After that, a locally improved solution is easily estimated from the quadratic associated problem. We demonstrate the hybridization of our proposed procedure with SPEA 2.

This paper proposes a procedure for space coordinate change, inside genetic algorithms, based on convex quadratic approximations of the general nonlinear objective function. It is shown that in the transformed coordinates the genetic algorithm is able to And the problem optimum in less iterations and with greater proportion of successful attempts. The proposed procedure employs only the objective function samples that have already been obtained through the usual genetic algorithm operations. It means that there is no need of any additional function evaluation. The proposed procedure was tested with a set of benchmark problems. In all cases, the proposed algorithm has been able to repeatedly find solutions closer to the true solution than those found by the same genetic algorithm without coordinate change. The results suggest that the modification can enhance the convergence rate and accuracy of genetic algorithms.

This paper presents a new operator for genetic algorithms that enhances their convergence in the case of nonlinear problems with nonlinear equality constraints. The proposed operator, named CQA (constraint quadratic approximation), can be interpreted as both a local search engine (that employs quadratic approximations of both objective and constraint functions for guessing a solution estimate) and a kind of elitism operator that plays the role of 'fixing" the best estimate of the feasible set. The proposed operator has the advantage of not requiring any additional function evaluation per algorithm iteration, solely making use of the information that would be already obtained in the course of the usual genetic algorithm iterations. The test cases that were performed suggest that the new operator can enhance both the convergence speed (in terms of the number of function evaluations) and the accuracy of the final result.