Indicator-based Multi-objective Evolutionary Algorithms Research Articles

An indicator-based evolutionary algorithm with adaptive archive update cycle for multi-objective multi-robot task allocation

The multi-robot task allocation problem has garnered significant attention in research and development. This paper addresses the multi-robot task allocation problem by introducing a unified model that seamlessly transforms into four popular mathematical models through simple parameter adjustments. We propose an efficient indicator-based multi-objective evolutionary algorithm with a hybrid encoding scheme for task execution order and robot starting point information. The algorithm employs the hypervolume indicator for environmental selection to enhance convergence and the modified crowding distance for archive updates to promote diversity. Additionally, an adaptive archive update mechanism is designed for time efficiency. In experiments comparing our proposed algorithm with 8 state-of-the-art algorithms on 18 randomly generated instances of various sizes, along with 4 real-world problems from the TSPLIB benchmark, our algorithm consistently outperformed the comparison algorithms in five key indicators. These results underscore the effectiveness of our approach in addressing real-world multi-robot task allocation problems.

An adaptive parental guidance strategy and its derived indicator-based evolutionary algorithm for multi- and many-objective optimization

The indicator-based multi-objective evolutionary algorithms have demonstrated their superiority in handling diverse types of multi- and many-objective optimization problems. However, these evolutionary algorithms still face significant challenges in balancing convergence and diversity of the evolutionary population, despite numerous auxiliary mechanisms designed to improve their performance. To address this issue, an adaptive parental guidance strategy (APGS) is proposed in this paper. On the one hand, APGS leverages the current population to evaluate the quality of the newly generated offspring. On the other hand, it employs an adaptive threshold to select offspring individuals with better convergence and diversity. This approach enhances the convergence and diversity of the candidate solution set throughout the evolutionary process, thereby ensuring high-quality obtained solutions. By incorporating the APGS, this paper proposes a new indicator-based evolutionary algorithm with parental guidance (IEAPG). Simulation results on several test suites and real-world problem show that compared to PREA, SPEA/R, GrEA, TS-NSGA-II, HEA and MaOEA/IGD, the proposed IEAPG has better performance and robustness in dealing with different types of multi- and many-objective optimization problems. Furthermore, further investigation reveals that the incorporation of the APGS can significantly improve the performance of different categories of multi- and many-objective evolutionary algorithms.

Dual-Fuzzy-Classifier-Based Evolutionary Algorithm for Expensive Multiobjective Optimization

Multiobjective evolutionary algorithms (MOEAs) have been widely used to solve multiobjective optimization problems (MOPs). Conventional MOEAs usually require a large number of function evaluations (FEs) for evaluating the quality of solutions. However, only a limited number of FEs are affordable in many real-world optimization problems where the FEs are computationally or economically expensive. The use of a large number of FEs decreases the effectiveness of MOEAs in solving these problems. This article proposes a dual fuzzy classifier-based surrogate model (DFC) and a DFC-based multiobjective evolutionary algorithm framework (DFC-MOEA) for expensive optimization problems. The DFC model is used for offspring pre-selection to choose high-quality offspring solutions and for categorization to divide unevaluated solutions into three categories. In the proposed framework, two fuzzy classifiers are built to predict the quality of each unevaluated solution. To strike a balance between convergence and diversity, one classifier is designed for predicting the dominance relation of unevaluated solutions (i.e., convergence), and another classifier is designed for predicting the crowdedness of unevaluated solutions (i.e., diversity). The integration of the proposed framework in three different types of MOEAs (i.e., a dominance-based MOEA, an indicator-based MOEA, and a decomposition-based MOEA) demonstrates its usefulness in handling expensive optimization problems. Comprehensive experiments on four well-known test suites and several real-world optimization problems demonstrate that the proposed framework is able to improve the performance of MOEAs under a limited number of FEs.

Improving multi-objective evolutionary algorithms using Grammatical Evolution

Multi-objective evolutionary algorithms (MOEAs) have become an effective choice to solve multi-objective optimization problems (MOPs). However, it is well known that Pareto dominance-based MOEAs struggle in MOPs with four or more objective functions due to a lack of selection pressure in high dimensional spaces. The main choices for dealing with such problems are decomposition-based and indicator-based MOEAs. In this work, we propose the use of Grammatical Evolution (an evolutionary computation search technique) to generate functions that can improve decomposition-based and indicator-based MOEAs. Namely, we propose a methodology to generate new scalarizing functions, which are known to have a great impact in the performance of decomposition-based MOEAs and in some indicator-based MOEAs. Additionally, we propose another methodology to generate hypervolume approximations, since the hypervolume is a popular performance indicator used not only in indicator-based MOEAs but also to assess performance of MOEAs. Using our first methodology, we generate two new scalarizing functions and provide their corresponding experimental validation to show that they exhibit a competitive behavior when compared against some well-known scalarizing functions such as ASF, PBI and the Tchebycheff scalarizing function. Using our second methodology, we produce 4 different hypervolume approximations and compare their performance against the Monte Carlo method and against two other state-of-the-art hypervolume approximations. The experimental results show that our functions exhibit a good compromise in terms of quality and execution time.

New Findings on Indicator-based Multi-Objective Evolutionary Algorithms: A Brief Summary

Quality indicators (QIs) are functions that assign a real value to a set that represents the Pareto front approximation of a multi-objective optimization problem. In the evolutionary multi-objective optimization community, QIs have been mainly employed in two ways: (1) for the performance assessment of multi-objective evolutionary algorithms (MOEAs), which produce Pareto front approximations, and (2) to be adopted as the backbone of selection mechanisms of MOEAs. Regardless of the continuing advances on QIs and their utilization in MOEAs, there are currently a vast number of open questions in this researcharea. In this doctoral thesis, we have focused on two main research directions: the design of new selection mechanisms based on the competition and cooperation of multiple QIs, aiming to compensate for the weaknesses (in terms of convergence and diversity properties) of individual QIs with the strengths of the others. The second research axis is the generation of new QIs that are compliant with the Pareto dominance relation extended to sets. Such QIs have a direct impact on the type of conclusions that can be drawn about the performance of MOEAs. Our experimental results have shown that the use of multiple QIs either to design new selection mechanisms or to construct new Pareto-compliant QIs is a promising research direction that can improve the capabilities of MOEAs and that allows for a performance assessment of MOEAs with a higher degree of confidence.

Preference incorporation into many-objective optimization: An Ant colony algorithm based on interval outranking

In this paper, we enriched Ant Colony Optimization (ACO) with interval outranking to develop a novel multi-objective ACO optimizer to approach problems with many objective functions. This proposal is suitable if the preferences of the Decision Maker (DM) can be modeled through outranking relations. The introduced algorithm (Interval Outranking-based ACO, IO-ACO) is the first ant-colony optimizer that embeds an outranking model to bear vagueness and ill-definition of the DM's preferences. This capacity is the most differentiating feature of IO-ACO because this issue is highly relevant in practice. IO-ACO biases the search towards the Region of Interest (RoI), the privileged zone of the Pareto frontier containing the solutions that better match the DM's preferences. Two widely studied benchmarks were utilized to measure the efficiency of IO-ACO, i.e., the DTLZ and WFG test suites. Accordingly, IO-ACO was compared with four competitive multi-objective optimizers: The Indicator-based Many-Objective ACO, the Multi-objective Evolutionary Algorithm Based on Decomposition, the Reference Vector-Guided Evolutionary Algorithm using Improved Growing Neural Gas, and the Indicator-based Multi-objective Evolutionary Algorithm with Reference Point Adaptation. The numerical results show that IO-ACO approximates the RoI better than leading metaheuristics based on approximating the Pareto frontier alone.

Meter Placement in Active Distribution System using Objective Discretization and Indicator-Based Multi-Objective Evolutionary Algorithm with Adaptive Reference Point Method

Meter Placement in Active Distribution System using Objective Discretization and Indicator-Based Multi-Objective Evolutionary Algorithm with Adaptive Reference Point Method

Weighted Indicator-Based Evolutionary Algorithm for Multimodal Multiobjective Optimization

Multimodal multiobjective problems (MMOPs) arise frequently in the real world, in which multiple Pareto-optimal solution (PS) sets correspond to the same point on the Pareto front. Traditional multiobjective evolutionary algorithms (MOEAs) show poor performance in solving MMOPs due to a lack of diversity maintenance in the decision space. Thus, recently, many multimodal MOEAs (MMEAs) have been proposed. However, for most existing MMEAs, the convergence performance in the objective space does not meet expectations. In addition, many of them cannot always obtain all equivalent Pareto solution sets. To address these issues, this study proposes an MMEA based on a weighted indicator, termed MMEA-WI. The algorithm integrates the diversity information of solutions in the decision space into an objective space performance indicator to maintain the diversity in the decision space and introduces a convergence archive to ensure a more effective approximation of the Pareto-optimal front (PF). These strategies can readily be applied to other indicator-based MOEAs. The experimental results show that MMEA-WI outperforms some state-of-the-art MMEAs on the chosen benchmark problems in terms of the inverted generational distance (IGD) and IGD in the decision space (IGDX) metrics.

On the Effect of the Cooperation of Indicator-Based Multiobjective Evolutionary Algorithms

For almost 20 years, quality indicators (QIs) have promoted the design of new selection mechanisms of multiobjective evolutionary algorithms (MOEAs). Each indicator-based MOEA (IB-MOEA) has specific search preferences related to its baseline QI, producing Pareto front approximations with different properties. In consequence, an IB-MOEA based on a single QI has a limited scope of multiobjective optimization problems (MOPs) in which it is expected to have a good performance. This issue is emphasized when the associated Pareto front geometries are highly irregular. In order to overcome these issues, we propose here an island-based multiindicator algorithm (IMIA) that takes advantage of the search biases of multiple IB-MOEAs through a cooperative scheme. Our experimental results show that the cooperation of multiple IB-MOEAs allows IMIA to perform more robustly (considering several QIs) than the panmictic versions of its baseline IB-MOEAs as well as several state-of-the-art MOEAs. Additionally, IMIA shows a Pareto-front-shape invariance property, which makes it a remarkable optimizer when tackling MOPs with complex Pareto front geometries.

An integrated multi-objective optimization method with application to train crashworthiness design

The collision performance of crashworthy structures can be improved using the optimization design. However, conflicting objective functions and the non-uniqueness of solutions hinder the Pareto optimum selection. This study constructs a hybrid optimization approach (TA-FBGV) integrating the multi-objective optimization and the multiple criteria decision-making (MCDM). TA-FBGV combines the two-phase differential evolution (ToPDE) method, indicator-based multi-objective evolutionary algorithm (AR-MOEA), fuzzy best worst (F-BW) strategy, grey relational analysis (GRA), and VIsekriterijumsko KOmpromisno Rangiranje (VIKOR) integrating method (G-VIKOR) to address difficulties of the Pareto optimum selection. The ToPDE method is employed to generate uniform samples. AR-MOEA is used to produce Pareto optimal solutions. Weighting values of competing objectives are calculated through the F-BW strategy. G-VIKOR is applied to choose the final Pareto optimum from the Pareto front. Subsequently, the multi-objective optimization of the train underframe structure is performed using TA-FBGV. The results show that the Pareto optimum obtained by G-VIKOR is a good compromising solution which locates near the knee point provided by the minimum distance selection method (TMDSM). This implies that the G-VIKOR approach can achieve a good trade-off among conflictive objectives. Compared with the initial model, although the energy absorption of the Pareto optimum is decreased, the initial peak crushing force and structural mass are reduced. Optimization results indicate that the TA-FBGV approach is efficient to select the Pareto optimum for the structure optimization design.

Indicator-based Multi-objective Evolutionary Algorithms

For over 25 years, most multi-objective evolutionary algorithms (MOEAs) have adopted selection criteria based on Pareto dominance. However, the performance of Pareto-based MOEAs quickly degrades when solving multi-objective optimization problems (MOPs) having four or more objective functions (the so-called many-objective optimization problems), mainly because of the loss of selection pressure. Consequently, in recent years, MOEAs have been coupled with indicator-based selection mechanisms in furtherance of increasing the selection pressure so that they can properly solve many-objective optimization problems. Several research efforts have been conducted since 2003 regarding the design of the so-called indicator-based (IB) MOEAs. In this article, we present a comprehensive survey of IB-MOEAs for continuous search spaces since their origins up to the current state-of-the-art approaches. We propose a taxonomy that classifies IB-mechanisms into two main categories: (1) IB-Selection (which is divided into IB-Environmental Selection, IB-Density Estimation, and IB-Archiving) and (2) IB-Mating Selection. Each of these classes is discussed in detail in this article, emphasizing the advantages and drawbacks of the selection mechanisms. In the final part, we provide some possible paths for future research.

A [Math Processing Error]-indicator-based shuffled frog leaping algorithm for many-objective optimization problems

Many-objective optimization problems take challenges to multi-objective evolutionary algorithms. A number of nondominated solutions in population cause a difficult selection towards the Pareto front. To tackle this issue, a series of indicatorbased multi-objective evolutionary algorithms (MOEAs) have been proposed to guide the evolution progress and shown promising performance. This paper proposes an indicator-based manyobjective evolutionary algorithm called e-indicator-based shuffled frog leaping algorithm (e-MaOSFLA), which adopts the shuffled frog leaping algorithm as an evolutionary strategy and a simple and effective e-indicator as a fitness assignment scheme to press the population towards the Pareto front. Compared with four stateof-the-art MOEAs on several standard test problems with up to 50 objectives, the experimental results show that e-MaOSFLA outperforms the competitors.

An Indicator-Based Multiobjective Evolutionary Algorithm With Reference Point Adaptation for Better Versatility

During the past two decades, a variety of multiobjective evolutionary algorithms (MOEAs) have been proposed in the literature. As pointed out in some recent studies, however, the performance of an MOEA can strongly depend on the Pareto front shape of the problem to be solved, whereas most existing MOEAs show poor versatility on problems with different shapes of Pareto fronts. To address this issue, we propose an MOEA based on an enhanced inverted generational distance indicator, in which an adaptation method is suggested to adjust a set of reference points based on the indicator contributions of candidate solutions in an external archive. Our experimental results demonstrate that the proposed algorithm is versatile for solving problems with various types of Pareto fronts, outperforming several state-of-the-art evolutionary algorithms for multiobjective and many-objective optimization.

A Large-Scale Experimental Evaluation of High-Performing Multi- and Many-Objective Evolutionary Algorithms.

Research on multi-objective evolutionary algorithms (MOEAs) has produced over the past decades a large number of algorithms and a rich literature on performance assessment tools to evaluate and compare them. Yet, newly proposed MOEAs are typically compared against very few, often a decade older MOEAs. One reason for this apparent contradiction is the lack of a common baseline for comparison, with each subsequent study often devising its own experimental scenario, slightly different from other studies. As a result, the state of the art in MOEAs is a disputed topic. This article reports a systematic, comprehensive evaluation of a large number of MOEAs that covers a wide range of experimental scenarios. A novelty of this study is the separation between the higher-level algorithmic components related to multi-objective optimization (MO), which characterize each particular MOEA, and the underlying parameters-such as evolutionary operators, population size, etc.-whose configuration may be tuned for each scenario. Instead of relying on a common or "default" parameter configuration that may be low-performing for particular MOEAs or scenarios and unintentionally biased, we tune the parameters of each MOEA for each scenario using automatic algorithm configuration methods. Our results confirm some of the assumed knowledge in the field, while at the same time they provide new insights on the relative performance of MOEAs for many-objective problems. For example, under certain conditions, indicator-based MOEAs are more competitive for such problems than previously assumed. We also analyze problem-specific features affecting performance, the agreement between performance metrics, and the improvement of tuned configurations over the default configurations used in the literature. Finally, the data produced is made publicly available to motivate further analysis and a baseline for future comparisons.

Hypervolume-based multi-objective local search

This paper presents a multi-objective local search, where the selection is realized according to the hypervolume contribution of solutions. The HBMOLS algorithm proposed is inspired from the IBEA algorithm, an indicator-based multi-objective evolutionary algorithm proposed by Zitzler and Kunzli in 2004, where the opti- mization goal is defined in terms of a binary indicator defining the selection operator. In this paper, we use the indicator optimization principle, and we apply it to an iterated local search algorithm, using hypervolume contri- bution indicator as selection mechanism. The methodology proposed here has been defined in order to be easily adaptable and to be as parameter-independent as possible. We carry out a range of experiments on the multi-objective flow shop problem and the multi-objective quadratic assignment problem, using the hypervolume contribution selection as well as two different binary indicators which were initially proposed in the IBEA algorithm. Experi- mental results indicate that the HBMOLS algorithm is highly effective in comparison with the algorithms based on binary indicators.