A Convergence Indicator for Multi-Objective Optimisation Algorithms

The algorithms of multi-objective optimisation had a relative growth in the last years. Thereby, it's requires some way of comparing the results of these. In this sense, performance measures play a key role. In general, it's considered some properties of these algorithms such as capacity, convergence, diversity or convergence-diversity. There are some known measures such as generational distance (GD), inverted generational distance (IGD), hypervolume (HV), Spread($\Delta$), Averaged Hausdorff distance ($\Delta_p$), R2-indicator, among others. In this paper, we focuses on proposing a new indicator to measure convergence based on the traditional formula for Shannon entropy. The main features about this measure are: 1) It does not require tho know the true Pareto set and 2) Medium computational cost when compared with Hypervolume.

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.

The problem of atmospheric duct inversion is usually solved as a single objective optimization problem. Based on ground-based Global Positioning System (GPS) phase delay and propagation loss, this paper develops a multi-objective method including the effect of source frequency and receiving antenna height. The diversity and convergence of solution sets are evaluated for seven multi-objective evolutionary algorithms with three performance metrics: Hypervolume (HV), Inverted Generational Distance (IGD), and the averaged Hausdorff distance (). The inversion results are compared with the simulation results, and the experimental comparison is conducted on three groups of test situations. The results demonstrate that the ranking of algorithm performance varies because of the different methods used to calculate performance metrics. Moreover, when the algorithms show overwhelming performance using performance metrics, the inversion result is not more close to the real value. In the comparison of computational experiments, it was found that, as the retrieved parameter dimension increases, the inversion result becomes more unstable. When the observed data are sufficient, the inversion result seems to be improved.

The multiobjective (MO) optimizers show great promise in solving constrained engineering structural problems. This paper introduces a MO version of the Brown Bear Optimization (BBO) algorithm, inspired by the foraging behavior of brown bears. The proposed Multiobjective Brown Bear Optimization (MOBBO) algorithm is applied to five structural optimization problems, including 10‐bar, 25‐bar, 60‐bar, 72‐bar, and 942‐bar trusses, aiming to minimize both mass and maximum nodal deflection simultaneously. Comparative evaluations against six benchmark algorithms demonstrate MOBBO’s superior convergence, solution diversity, and effectiveness in addressing highly constrained problems. The hypervolume (HV) and inverted generational distance (IGD) metrics place MOBBO in first rank according to the Friedman test, with an average standard deviation of 0.0002. Moreover, the spacing‐to‐extent (STE) and generational distance (GD) metrics rank MOBBO second. The final Friedman rank test highlights MOBBO’s overall dominance, achieving a first rank. Best Pareto plots, diversity graphs, and box plot analyses further suggest MOBBO’s superior performance and convergence compared to existing algorithms. Therefore, the MOBBO algorithm can be effectively applied to various MO optimization tasks in industry, offering refined global optimization solutions and contributing valuable insights to the field of MO algorithms.

The initial Moth-Flame Optimization (MFO) algorithm is appropriately changed to deal with multi-objective optimization problems described as Multi-objective Moth-Flame Optimization (MMFO). The primary idea of this MFO is the navigating technique of moths in nature which fly in night time by keeping a fixed angle relative to the moon. Nevertheless, these elegant bugs are entrapped in a useless/deadly spiral route around fabricated lights. Normally principles like using fixed-sized external archive make the recommended technique vary from the initial single-objective MFO. The efficiency of recommended MMFO is tested on eight multi-objective engineering design problems concerning remarkable precision and uniformity compared to Multi-objective Particle Swarm Optimization (MOPSO), Multi-objective Gray Wolf Optimizer (MOGWO), and Multi-objective Ant Lion Optimizer (MOALO). According to the results of different performance metrics, such as Generational Distance (GD), Inverted Generational Distance (IGD), and Maximum Spread (MS), the proposed algorithm can provide quality Pareto fronts with very competitive results.

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

In this paper, a new Multi-Objective Arithmetic Optimization Algorithm (MOAOA) is proposed for solving Real-World constrained Multi-objective Optimization Problems (RWMOPs). Such problems can be found in different fields, including mechanical engineering, chemical engineering, process and synthesis, and power electronics systems. MOAOA is inspired by the distribution behavior of the main arithmetic operators in mathematics. The proposed multi-objective version is formulated and developed from the recently introduced single-objective Arithmetic Optimization Algorithm (AOA) through an elitist non-dominance sorting and crowding distance-based mechanism. For the performance evaluation of MOAOA, a set of 35 constrained RWMOPs and five ZDT unconstrained problems are considered. For the fitness and efficiency evaluation of the proposed MOAOA, the results obtained from the MOAOA are compared with four other state-of-the-art multi-objective algorithms. In addition, five performance indicators, such as Hyper-Volume (HV), Spread (SD), Inverted Generational Distance (IGD), Runtime (RT), and Generational Distance (GD), are calculated for the rigorous evaluation of the performance and feasibility study of the MOAOA. The findings demonstrate the superiority of the MOAOA over other algorithms with high accuracy and coverage across all objectives. This paper also considers the Wilcoxon signed-rank test (WSRT) for the statistical investigation of the experimental study. The coverage, diversity, computational cost, and convergence behavior achieved by MOAOA show its high efficiency in solving ZDT and RWMOPs problems.

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.

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.

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.

The pickup and delivery problem (PDP) is a very common and important problem, which has a large number of real-world applications in logistics and transportation. In PDP, customers send transportation requests to pick up an object from one place and deliver it to another place. This problem is under the focus of researchers since the last two decades with multiple variations. In the literature, different variations of PDP with different number of objectives and constraints have been considered. Depending on the number of objectives, multi and many-objective evolutionary algorithms have been applied to solve the problem and to study the conflicts between objectives. In this paper, PDP is formulated as a many-objective pickup and delivery problem (MaOPDP) with delay time of vehicle having six criteria to be optimized. To the best of our knowledge, this variation of PDP has not been considered in the literature. To solve the problem, this paper proposes a memetic I-DBEA (Improved Decomposition Based Evolutionary Algorithm), which is basically the modification of an existing many-objective evolutionary algorithm called I-DBEA. To demonstrate the superiority of our approach, a set of experiments have been conducted on a variety of small, medium and large-scale problems. The quality of the results obtained by the proposed approach is compared with five existing multi and many-objective evolutionary algorithms using three different multi-objective evaluation measures such as hypervolume (HV), inverted generational distance (IGD) and generational distance (GD). The experimental results demonstrate that the proposed algorithm has significant advantages over several state-of-the-art algorithms in terms of the quality of the obtained solutions.

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

In this paper, we evaluate the effects of adaptive discretization of design variables in real-coded evolutionary computations (RCECs). While the appropriate granularity of design variables can improve convergence in RCECs, it is difficult to decide the appropriate one in advance in most of the practical optimization problems. Besides, when the granularity is too coarse, the diversity may be lost. To address these difficulties, we propose two adaptive discretization techniques that discretize each design variable using granularity determined according to the indicator of solution distribution state in design space. In this study, standard deviation(SD) or estimated probability density function(ePDF) is used as an indicator for determining granularities of design variables. We use NSGA-II as an RCEC and thirteen benchmark problems including engineering problems. The generational distance (GD) and inverted generational distance (IGD) metrics are used for investigating the performance of convergence and diversity, respectively. To make sure the statistical difference of results, the Wilcoxon rank-sum test and Welch's t-test are applied in each problem. The results of experiments show that both of the proposed methods can automatically improve convergence in many problems. In addition, it is confirmed that the diversity is also maintained.

This paper introduces the Improved Evolution Strategy Reinforcement Learning (I-ES) algorithm.The I-ES algorithm is designed to minimize the makespan and total energy consumption (TEC) in a multi-objective dynamic scheduling problem within a hybrid flow shop.It addresses key challenges such as flexible preventive maintenance for machines, random job arrivals, uncertain processing times, and the setup time.An experimental comparison of the I-ES-based approach with Evolution Strategy Reinforcement Learning (ES) algorithms and scheduling rules, which are a combination of job selection rules and machine selection rules, has been carried out by designing problem examples.The Generational Distance (GD) and Inverted Generational Distance (IGD) metrics of the I-ES algorithm are obtained by averaging all the instances as 389.14 and 1476.25,respectively, which are smaller than the other compared algorithms, so the results show that the I-ES algorithm can obtain solutions with superior convergence compared to that of the ES algorithm and dispatching rules.

Supply-chain scheduling in modern production systems faces significant challenges due to the complexity of balancing cost efficiency, delivery lead times, and order variability—especially under stochastic demand and uncertain transportation conditions. Conventional metaheuristic approaches, such as the standard Non-dominated Sorting Genetic Algorithm II (NSGA-II), often struggle with premature convergence, limited population diversity, and rigid parameter settings that hinder adaptability in dynamic environments. To address these limitations, this study proposes a novel Hybrid Adaptive NSGA-II (HA-NSGA-II) framework, enhanced by two key innovations: (1) a Global-Intensity Mutation Operator (GIMO) that strengthens population diversity by dynamically adjusting mutation intensity across the entire solution space; and (2) an Adaptive Parameter Self-Tuning (APST) module based on reinforcement learning, which intelligently regulates crossover and mutation rates in response to evolutionary progress and environmental changes. A tri-objective optimization model is formulated to simultaneously minimize total supply chain cost, order quantity variance (mitigating the bullwhip effect), and customer delivery delays. The proposed HA-NSGA-II demonstrates superior convergence and diversity performance compared to traditional NSGA-II, offering a more robust and adaptive solution for complex, real-world supply chain scheduling under uncertainty. The model is tested on a simulated supply chain with 5 factories, 15 warehouses, and 50 customers facing random demand and Gaussian-distributed lead times. The new HA-NSGA-II is compared with traditional NSGA-II, SPEA2, and MOEA/D based on Hypervolume (HV), Inverted Generational Distance (IGD), Spread (Δ), Total Inventory Cost, Bullwhip Index (BI), and Robustness Index (RI). Results of 50 simulation runs demonstrate that HA-NSGA-II surpasses all the baseline algorithms with 12.9% enhancement in HV, 41.1% improvement in IGD, and 13.7% reduction in convergence time. In addition, it delivers a 14.6% supply chain cost reduction, a 20.4% bullwhip effect suppression, and 25% accelerated times of fulfillment. The research concludes that the HA-NSGA-II provides a scalable and effective scheduling approach applicable for real-time Industry 4.0 supply chains. Work in the future involves expanding the framework towards multi-period dynamic environments using IoT and blockchain-based visibility.