Performance Evaluation of an Evolutionary Multiobjective Optimization Based Area Partitioning and Allocation Approach

Coverage analysis is essential for many coverage tasks (e.g., robotic grit-blasting, painting, and surface cleaning) performed by Autonomous Industrial Robots (AIRs). Coverage analysis enables (1) the performance evaluation (e.g., coverage rate and operation efficiency) of AIRs for a coverage task, and (2) the configuration design of a multi-AIR system (e.g., decision on the number of AIRs to be used). Multi-AIR coverage analysis of large and complex structures involves addressing various problems. Thus, a framework is presented in this paper that incorporates various modules (e.g., AIR reachability, AIR base placement, collision avoidance, and area partitioning and allocation) for appropriately addressing the associated problems. The modules within the framework provide the flexibility of utilizing different methods and algorithms, depending on the requirements of the target application. The framework is tested and validated by extensive analyses of 10 different scenarios with up to 10 AIRs.

So far, many performance indicators have been proposed to compare different evolutionary multiobjective optimization algorithms (MOEAs). Among them, the inverted generational distance (IGD) is one of the most commonly used, mainly because it can measure a population’s convergence, diversity, and evenness. However, the effectiveness of IGD highly depends on the quality of the reference set. That is to say, all the reference points should be as close to the Pareto front (PF) as possible and evenly distributed to become ready for a fair performance evaluation. Currently, it is still challenging to generate well-configured reference sets, even if the PF can be given analytically. Therefore, biased reference sets might be a significant source of systematic error. However, in most MOEA literature, biased reference sets are utilized in experiments without an error estimation, which may make the experimental results unconvincing. In this paper, we propose an exact IGD (eIGD) for continuous PF, which is derived from the original IGD under an additional assumption that the reference set is perfect, i.e., the PF itself is directly utilized as an infinite-sized reference set. Therefore, the IGD values produced by biased reference sets can be compared with eIGD so that systematic error can be quantitatively evaluated and analyzed.KeywordsMultiobjective optimizationEvolutionary computationPerformance indicatorReference setDifferential geometry

The hypervolume and the inverted generational distance (IGD) have been frequently used for the comparison of evolutionary multiobjective optimization algorithms. For the hypervolume, the relation between the location of a reference point and the optimal distribution of solutions has been studied in the literature. However, such a relation has not been studied for the IGD whereas IGD-based comparison results depend on the specification of reference points. Our intention is to clearly demonstrate the dependency of IGD-based comparison results on reference point specification. First, we explain difficulties of fair comparison in the following two cases: one is the use of all nondominated solutions among obtained solutions by compared algorithms as reference points, and the other is the use of a small number of uniformly sampled reference points. Discussions on these two cases show the necessity of a large number of uniformly sampled reference points on the entire Pareto front. Then, we show a bias of the IGD with such a reference point set through computational experiments. It is shown that the IGD tends to favor a solution set with much smaller diversity than a fully expanded solution set over the entire Pareto front. Finally, we propose a new specification method where reference points are uniformly sampled not only from the Pareto front but also from outside the Pareto front.

A fuzzy-guided adaptive algorithm with hierarchy mechanism for solving dynamic multi-objective optimization problems

The Hausdorff distance d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> is a widely used tool to measure the distance between different objects in several research fields. Possible reasons for this might be that it is a natural extension of the well-known and intuitive distance between points and/or the fact that d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> defines in certain cases a metric in the mathematical sense. In evolutionary multiobjective optimization (EMO) the task is typically to compute the entire solution set-the so-called Pareto set-respectively its image, the Pareto front. Hence, d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> should, at least at first sight, be a natural choice to measure the performance of the outcome set in particular since it is related to the terms spread and convergence as used in EMO literature. However, so far, d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> does not find the general approval in the EMO community. The main reason for this is that d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> penalizes single outliers of the candidate set which does not comply with the use of stochastic search algorithms such as evolutionary strategies. In this paper, we define a new performance indicator, Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> , which can be viewed as an “averaged Hausdorff distance” between the outcome set and the Pareto front and which is composed of (slight modifications of) the well-known indicators generational distance (GD) and inverted generational distance (IGD). We will discuss theoretical properties of Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (as well as for GD and IGD) such as the metric properties and the compliance with state-of-theart multiobjective evolutionary algorithms (MOEAs), and will further on demonstrate by empirical results the potential of Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> as a new performance indicator for the evaluation of MOEAs.

Assessing the performance of Pareto front (PF) approximations is a key issue in the field of evolutionary multi/many-objective optimization. Inverted generational distance (IGD) has been widely accepted as a performance indicator for evaluating the comprehensive quality for a PF approximation. However, IGD usually becomes infeasible when facing a real-world optimization problem as it needs to know the true PF a priori. In addition, the time complexity of IGD grows quadratically with the size of the solution/reference set. To address the aforementioned issues, a grid-based IGD (Grid-IGD) is proposed to estimate both convergence and diversity of PF approximations for multi/many-objective optimization. In Grid-IGD, a set of reference points is generated by estimating PFs of the problem in question, based on the representative nondominated solutions of all the approximations in a grid environment. To reduce the time complexity, Grid-IGD only considers the closest solution within the grid neighborhood in the approximation for every reference point. Grid-IGD also possesses other desirable properties, such as Pareto compliance, immunity to dominated/duplicate solutions, and no need of normalization. In the experimental studies, Grid-IGD is verified on both the artificial and real PF approximations obtained by five many-objective optimizers. Effects of the grid specification on the behavior of Grid-IGD are also discussed in detail theoretically and experimentally.

Many real world problems are dynamic and multi-objective, which requires an optimisation algorithm to be able to continuously track the changing Pareto optimal set (POS) and Pareto optimal front (POF) over time. In this paper, a new variant of particle swarm optimisation (PSO) has been specifically designed by adaptively switching from competitive model to cooperative model to track for both POS and POF. In the proposed method, the competition is used to explore the search space, while the cooperation is applied to exploit the search space. The dynamic multi-objective functions are constructed to test the performance of the proposed algorithm. Both theoretical analysis and the numerical experiment have shown that the proposed algorithm is an excellent alternative for solving the dynamic multi-objective optimisation problems. Finally, the proposed method has been applied to the tuning of the parameters of PID controller for dynamic system in which a satisfactory control is obtained.

Testing and assessing the performance of proposed algorithms in a right way is a very important part in evolutionary optimization. If the number of optimization objectives is more than three, it's commonly referred as many-objective optimization problem(MaOP). Despite the fact there exists many test functions and assessment metrics, all of which are designed for testing and assessing the performance of evolutionary algorithms, the test functions DTLZ proposed by Deb(2005) and the assessment metrics Inverted Generational Distance(IGD) proposed by Bosman and Thierens(2003) are both widely recognized and used in most of evolutionary optimizations, especially in many-objective evolutionary optimization. When using IGD as a assessment tool, the vital task is to distribute reference points uniformly along the Pareto front of test functions. To overcome it, however, is really difficult with the number of objectives increase. For dealing with such difficulty, this paper introduce an approach called Inverse Transform Technique(ITT) suggested by Fang and Wang(1999) and successfully distributes reference points uniformly along the Pareto front of DTLZ test functions. In the final, three representative evolutionary algorithms: NSGA-II, MOEA/D and MOEA/D-M2M are selected to test their performance in dealing with many-objective optimization. As a result, extensive experiments and analysis are conducted on four DTLZ test functions from five to ten objectives.

Double-space environmental change detection and response strategy for dynamic multi-objective optimize problem

Selecting the best multi-objective algorithms for photovoltaic energy storage system (PV-ESS) design remains challenging due to limited benchmarking across renewable energy studies. This study addresses this gap through a systematic evaluation of four widely used multi-objective optimization algorithms: NSGA-II, Multi-Objective Particle Swarm Optimization (MOPSO), weighted-sum scalarization, and ε-constraint methods. Performance assessment utilized three Pareto front quality metrics: Inverted Generational Distance (IGD) for convergence quality, hypervolume (HV) for objective-space coverage, and spacing for solution distribution uniformity. The algorithms were tested on PV-ESS design problems in three developing economies (Nigeria, South Africa, India) under identical problem formulations and computational resources. NSGA-II achieved superior performance across all metrics in all three case studies. For convergence quality, NSGA-II attained a mean IGD of 0.0083, outperforming MOPSO by 29%, ε-constraint by 64%, and weighted-sum by 131%. For objective-space coverage, NSGA-II achieved a mean HV of 0. 700, representing 10–16% better coverage than other methods. For solution distribution, NSGA-II showed a mean spacing of 0.076, indicating 30–117% more uniform Pareto fronts. Computational efficiency analysis revealed that NSGA-II’s runtime is between 5.5 and 7.8 h per case, providing better quality–time ratios compared to ε-constraint methods (which are 18 times slower), while avoiding MOPSO’s premature convergence. Statistical validation confirmed NSGA-II’s superiority, with p &lt; 0.01 across all quality metrics. These results establish NSGA-II as the best algorithm for lifecycle-aware PV-ESS optimization, offering quantitative, evidence-based guidance for practitioners selecting optimization tools for renewable energy system design. The demonstrated performance leads to $ 45,000–$ 60,000 lifecycle cost savings per MW/MWh of system capacity through improved Pareto front identification.

In the past few years, multi-objective optimization algorithms have been extensively applied in several fields including engineering design problems. A major reason is the advancement of evolutionary multi-objective optimization (EMO) algorithms that are able to find a set of non-dominated points spread on the respective Pareto-optimal front in a single simulation. Besides just finding a set of Pareto-optimal solutions, one is often interested in capturing knowledge about the variation of variable values over the Pareto-optimal front. Recent innovization approaches for knowledge discovery from Pareto-optimal solutions remain as a major activity in this direction. In this article, a different data-fitting approach for continuous parameterization of the Pareto-optimal front is presented. Cubic B-spline basis functions are used for fitting the data returned by an EMO procedure in a continuous variable space. No prior knowledge about the order in the data is assumed. An automatic procedure for detecting gaps in the Pareto-optimal front is also implemented. The algorithm takes points returned by the EMO as input and returns the control points of the B-spline manifold representing the Pareto-optimal set. Results for several standard and engineering, bi-objective and tri-objective optimization problems demonstrate the usefulness of the proposed procedure.

© 2016 IEEE. Uncertainties in base placements of mobile, autonomous industrial robots can cause incomplete coverage in tasks such as grit-blasting and spray painting. Sensing and localization errors can cause such uncertainties in robot base placements. This paper addresses the problem of collaborative complete coverage under uncertainties through appropriate base placements of multiple mobile and autonomous industrial robots while aiming to optimize the performance of the robot team. A mathematical model for complete coverage under uncertainties is proposed and then solved using a stochastic multi-objective optimization algorithm. The approach aims to concurrently find an optimal number and sequence of base placements for each robot such that the robot team's objectives are optimized whilst uncertainties are accounted for. Several case studies based on a real-world application using a realworld object and a complex simulated object are provided to demonstrate the effectiveness of the approach for different conditions and scenarios, e.g. various levels of uncertainties, different numbers of robots, and robots with different capabilities.

In a wide range of engineering applications, complex constrained multi-objective optimization problems (CMOPs) present significant challenges, as the complexity of constraints often hampers algorithmic convergence and reduces population diversity. To address these challenges, we propose a novel algorithm named Constraint Intensity-Driven Evolutionary Multitasking (CIDEMT), which employs a two-stage, tri-task framework to dynamically integrates problem structure and knowledge transfer. In the first stage, three cooperative tasks are designed to explore the Constrained Pareto Front (CPF), the Unconstrained Pareto Front (UPF), and the -relaxed constraint boundary, respectively. A CPF-UPF relationship classifier is employed to construct a problem-type-aware evolutionary strategy pool. At the end of the first stage, each task selects strategies from this strategy pool based on the specific type of problem, thereby guiding the subsequent evolutionary process. In the second stage, while each task continues to evolve, a -driven knowledge transfer mechanism is introduced to selectively incorporate effective solutions across tasks. enhancing the convergence and feasibility of the main task. Extensive experiments conducted on 32 benchmark problems from three test suites (LIRCMOP, DASCMOP, and DOC) demonstrate that CIDEMT achieves the best Inverted Generational Distance (IGD) values on 24 problems and the best Hypervolume values (HV) on 22 problems. Furthermore, CIDEMT significantly outperforms six state-of-the-art constrained multi-objective evolutionary algorithms (CMOEAs). These results confirm CIDEMT’s superiority in promoting convergence, diversity, and robustness in solving complex CMOPs.

This paper examines the use of fitness sharing in evolutionary multiobjective optimization (EMO) algorithms to form a uniform distribution of niches along the non-dominated frontier. A long-standing, implicit assumption is that fitness sharing within an equivalence class, such as the Pareto optimal set, can form dynamically stable (under selection) subpopulations evenly spaced along the front. We show that this behavior can occur, but that it is highly unlikely. Rather, it is much more likely that a steady-state will be reached in which stable niches are maintained, but at inter-niche distances much less than the specified niche radius, with several times more niches than previously predicted, and with non-uniform sub-population sizes. These results might have implications for EMO population sizing, and perhaps even for EMO algorithm design itself.

Multi-objective SHADE with manta ray foraging optimizer for structural design problems