Pareto Front Research Articles

Multi-objective optimization of liquid metal bearing based on NSGA-II

Due to their exceptional static and dynamic properties along with high-temperature resistance, liquid metal bearings are extensively employed in challenging environments. This paper carried out a multi-objective optimization work for spiral groove journal liquid metal bearings (JLMB) by using a NSGA-II. The clearance of bearing C, spiral angle a, groove depth hg, groove ridge ratio β, groove number N, and groove spacing P were chosen as design variables to maximize the load-carrying capacity and the effective stiffness of liquid metal bearing. During optimization process, the Latin hypercube sampling was adopted to generate 41 sample points, and the Kriging surrogate model was used to establish the correlations between design variables and objective functions. The Pareto front with optimal structures was obtained after optimization. Subsequently, optimal parameters were determined from the Pareto front by using the TOPSIS decision method. Finally, a comparative analysis of bearing performance was conducted between the journal liquid metal bearings with optimal parameters and the original parameters. The results indicated that the optimal JLMB parameters are given as follows: C = 10 μm, a = 26°, hg = 5.2 μm, β = 0.7, P = 11.1 mm, N = 10. The static and dynamic performance of optimized JLMB has been significantly improved. Specifically, the load carrying capacity increased to 2781.2N, making a 1313% enhancement over the JLMB with original parameters. While the effective stiffness of the optimized JLMB surged to 1.86 × 108 N/m, a more than 20-fold increase.

An Improved Ant Colony Algorithm with Deep Reinforcement Learning for the Robust Multiobjective AGV Routing Problem in Assembly Workshops

Vehicle routing problems (VRPs) are challenging problems. Many variants of the VRP have been proposed. However, few studies on VRP have combined robustness and just-in-time (JIT) requirements with uncertainty. To solve the problem, this paper proposes the just-in-time-based robust multiobjective vehicle routing problem with time windows (JIT-RMOVRPTW) for the assembly workshop. Based on the conflict between uncertain time and JIT requirements, a JIT strategy was proposed. To measure the robustness of the solution, a metric was designed as the objective. Afterwards, a two-stage nondominated sorting ant colony algorithm with deep reinforcement learning (NSACOWDRL) was proposed. In stage I, ACO combines with NSGA-III to obtain the Pareto frontier. Based on the model, a pheromone update strategy and a transfer probability formula were designed. DDQN was introduced as a local search algorithm which trains networks through Pareto solutions to participate in probabilistic selection and nondominated sorting. In stage II, the Pareto frontier was quantified in feasibility by Monte Carlo simulation, and tested by diversity-robust selection based on uniformly distributed weights in the solution space to select robust Pareto solutions that take diversity into account. The effectiveness of NSACOWDRL was demonstrated through comparative experiments with other algorithms on instances. The impact of JIT strategy is analyzed and the effect of networks on the NSACOWDRL is further discussed.

A clustering-assisted adaptive evolutionary algorithm based on decomposition for multimodal multiobjective optimization

A multimodal multiobjective optimization problem can have multiple equivalent Pareto Sets (PSs). Since the number of PSs may vary in different problems, if the population is restricted to a fixed size, the number of solutions found for each PS will inevitably fluctuate widely, which is undesirable for decision makers. To address the issue, this paper proposes a clustering-assisted adaptive evolutionary algorithm based on decomposition (CA-MMEA/D), whose search process can be roughly divided into two stages. In the first stage, an initial exploration of decision space is carried out, and then solutions with good convergence are used for clustering to estimate the number and location of multiple PSs. In the second stage, new search strategies are developed on the basis of clustering, which can take advantage of unimodal search methods. Experimental studies show that the proposed algorithm outperforms some state-of-the-art algorithms, and CA-MMEA/D can keep the number of solutions found for each PS at a relatively stable level for different problems, thus making it easier for decision makers to choose the desired solutions. The research in this paper provides new ideas for the design of decomposition-based multimodal multiobjective algorithms.

Performance optimization of a printed circuit heat exchanger for the recuperated gas turbine

To further improve the thermal efficiency of gas turbines, the recuperator needs to be employed and the printed circuit heat exchanger with airfoil fins meets the recuperator's requirements of high heat transfer capacity, low flow resistance, and high reliability. However, the effect of airfoil fin pitches is not clear and the structure optimization method should be developed. In this study, the heat transfer quantity and pressure drop per length of the flue gas in the airfoil channels are numerically investigated under different Reynolds numbers and airfoil fin pitches. The results show that the effect of the airfoil transverse pitch has a great effect on the performance of the flue gas and the heat transfer quantity is increased by 82.39 %–96.08 % when the transverse pitch is decreased from 6.0 mm to 1.2 mm. At the same time, the pressure drop per length is increased to 189.39–842.47 Pa/m. The heat transfer performance is varied not obviously when the airfoil longitudinal pitch is varied from 4 mm to 15 mm. Based on the numerical data, the multiple-objective genetic algorithm combined with the backpropagation neural network is proposed for the performance predictions and optimizations. The corresponding Pareto front is obtained and the technique for order preference by similarity to the ideal solution method is employed to obtain the optimal point. Under the current conditions, the heat transfer quantity of the optimal point is 521.25 kW/m2, which is decreased by 34.5 % compared to the highest value, and the pressure drop per length of the optimal point is 108.45 Pa/m, which is decreased by 85.2 % compared to the highest value.

Utilizing neural networks and genetic algorithms in AI-assisted CFD for optimizing PCM-based thermal energy storage units with extended surfaces

Phase Change Materials (PCMs) offer significant benefits for applications such as electronic cooling and Thermal Energy Storage (TES) due to their high energy storage capacity and stability. However, their limited thermal conductivity restricts widespread utilization. To address this limitation, the use of fins has become a common technique to enhance heat transfer. Optimizing fin lengths and locations is challenging, as they affect natural convection. This study aims to develop a model using a feed-forward back-propagation network, trained on simulation data, for designing a TES unit. A design matrix, incorporating factors such as fin lengths and heights, is obtained through response surface methodology. Single- and multi-objective Genetic Algorithms (GAs) are employed to seek optimal solutions, considering Complete Melting Time (CMT) and total fin length. The results demonstrate the excellent performance of the network model, with an error of less than 2.98%. The single-objective GA yields a minimum CMT. In a finless enclosure, the CMT is 338.5% higher compared to the single-objective configuration. The multi-objective GA, using Euclidean distance specified from the Pareto front to balance weight and material costs, results in a CMT that is 192% higher compared to the single-objective optimization but with a total fin length that is 286% shorter. While these improvements are significant, it is important to note that fins can increase material and manufacturing costs, as well as the overall unit weight. This artificial intelligence-computational fluid dynamics method not only predicts TES unit performance but also reduces computational costs in designing an optimal TES unit.

Grouping and scheduling multiple sports leagues: an integrated approach

This paper introduces the multi-league grouping and scheduling problem, which integrates the grouping of teams into leagues and the scheduling of each league. This involves two possibly conflicting objectives: minimizing travel distance and minimizing capacity violations of venues shared by teams. We formulate this problem as a bi-objective mixed-integer programming model. Given the NP-hardness of the grouping problem, the integrated problem is particularly challenging. Hence, we design a two-layer constructive heuristic to efficiently approximate the Pareto set, using simulated annealing on the outer layer and an integer programming model on the inner layer. We further develop a speed-up version where the inner layer is solved heuristically. We develop a series of large-scale problem instances, including one based on data from the Royal Belgian Football Association. In a computational study, we compare our algorithms with an epsilon-constraint method and evaluate their results using various multi-objective solution quality metrics.

Building retrofit optimization considering future climate and decision-making under various mindsets

Building retrofit is effective in reducing building energy use and improving comfort levels for existing buildings. However, conducting multi-objective optimization for individual buildings can be challenging due to the laborious computational cost of using white box models and the difficulty in visualizing and understanding the decision-making process. Additionally, the impact of climate change has not been fully considered for the post-retrofit lifecycle. This research proposes a pragmatic automated scheme that integrates a feature selection method based on marginal abatement cost analysis and variance-based sensitivity analysis, multi-objective optimization supported by non-dominated sorting differential evolution (NSDE) algorithm, tailor-made decision-making support under different mindsets, and tree-based retrospection scheme of decision-making pathways. The simulation engine used in this study is a low-order white box modeling tool developed by the research team. The proposed scheme was applied to two educational buildings with different thermal characteristics, and the results showed that a certain number of sampling sizes were needed to achieve reliable feature selection results. The hierarchical clustering based decision-making support scheme has demonstrated robustness in visualizing and supporting decision-making for Pareto front. Two retrofit mindsets - aggressive and balanced - were assumed in the decision-making process, and the proposed method produced distinct final solutions accoridng to the two mindsets. This framework can support informed decision-making, helping stakeholders implement sustainable practices and transition to a low-carbon built environment.

Dynamic-multi-task-assisted evolutionary algorithm for constrained multi-objective optimization

Compared with common multi-objective optimization problems, constrained multi-objective optimization problems demand additional consideration of the treatment of constraints. Recently, many constrained multi-objective evolutionary algorithms have been presented to reconcile the relationship between constraint satisfaction and objective optimization. Notably, evolutionary multi-task mechanisms have also been exploited in solving constrained multi-objective problems frequently with remarkable outcomes. However, previous methods are not fully applicable to solving problems possessing all types of constraint landscapes and are only superior for a certain type of problem. Thus, in this paper, a novel dynamic-multi-task-assisted constrained multi-objective optimization algorithm, termed DTCMO, is proposed, and three dynamic tasks are involved. The main task approaches the constrained Pareto front by adding new constraints dynamically. Two auxiliary tasks are devoted to exploring the unconstrained Pareto front and the constrained Pareto front with dynamically changing constraint boundaries, respectively. In addition, the first auxiliary task stops the evolution automatically after reaching the unconstrained Pareto front, avoiding the waste of subsequent computational resources. A series of experiments are conducted with eight mainstream algorithms on five benchmark problems, and the results confirm the generality and superiority of DTCMO.

Multi-objective resistance-capacitance optimization algorithm: An effective multi-objective algorithm for engineering design problems

Focusing on practical engineering applications, this study introduces the Multi-Objective Resistance-Capacitance Optimization Algorithm (MORCOA), a new approach for multi-objective optimization problems. MORCOA uses the transient response behaviour of resistance-capacitance circuits to navigate complex optimization landscapes and identify global optima when faced with many competing objectives. The core approach of MORCOA combines a dynamic elimination-based crowding distance mechanism with non-dominated sorting to generate an ideal and evenly distributed Pareto front. The algorithm's effectiveness is evaluated through a structured, three-phase analysis. Initially, MORCOA is applied to five benchmark problems from the ZDT test suite, with performance assessed using various metrics and compared against state-of-the-art multi-objective optimization techniques. The study then expands to include seven problems from the DTLZ benchmark collection, further validating MORCOA's effectiveness. The final phase involves applying MORCOA to six real-world constrained engineering design problems. Notably, the optimization of a honeycomb heat sink, which is crucial in thermal management systems, is a significant part of this study. This phase uses a range of performance measures to assess MORCOA's practical application and efficiency in engineering design. The results highlight MORCOA's robustness and efficiency in both real-world engineering applications and benchmark problems, demonstrating its superior capabilities compared to existing algorithms. The effective use of MORCOA in real-world engineering design problems indicates its potential as an adaptable and powerful tool for complex multi-objective optimization tasks.

Deep reinforcement learning as multiobjective optimization benchmarks: Problem formulation and performance assessment

The successful deployment of Deep learning in several challenging tasks has been translated into complex control problems from different domains through Deep Reinforcement Learning (DRL). Although DRL has been extensively formulated and solved as single-objective problems, nearly all real-world RL problems often feature two or more conflicting objectives, where the goal is to obtain a high-quality and diverse set of optimal policies for different objective preferences. Consequently, the development of Multi-Objective Deep Reinforcement Learning (MODRL) algorithms has gained a lot of traction in the literature. Generally, Evolutionary Algorithms (EAs) have been demonstrated to be scalable alternatives to the classical DRL paradigms when formulated as an optimization problem. Hence it is reasonable to employ Multi-objective Evolutionary Algorithms (MOEAs) to handle MODRL tasks. However, there are several factors constraining the progress of research along this line: first, there is a lack of a general problem formulation of MODRL tasks from an optimization perspective; second, there exist several challenges in performing benchmark assessments of MOEAs for MODRL problems. To overcome these limitations: (i) we present a formulation of MODRL tasks as general multi-objective optimization problems and analyze their complex characteristics from an optimization perspective; (ii) we present an end-to-end framework, termed DRLXBench, to generate MODRL benchmark test problems for seamless running of MOEAs (iii) we propose a test suite comprising of 12 MODRL problems with different characteristics such as many-objectives, degenerated Pareto fronts, concave and convex optimization problems, etc. (iv) Finally, we present and discuss baseline results on the proposed test problems using seven representative MOEAs.

Optimization of Operational Parameters of Plant Protection UAV.

The operational parameters of plant protection unmanned aerial vehicles (UAVs) significantly impact spraying effectiveness, but the underlying mechanism remains unclear. This paper conducted a full factorial experiment with varying flight speeds, heights, and nozzle flow rates to collect parameter space data. Using the Kriging surrogate model, we characterized this parameter space and subsequently optimized the average deposition rate and coefficient of variation by employing a variable crossover (mutation) probability multi-objective genetic algorithm. In the obtained Pareto front, the average sedimentation rate is no less than 46%, with a maximum of 56.08%, and the CV coefficient is no more than 13.91%, with a minimum of only 8.42%. These optimized parameters enhance both the average deposition rate and spraying uniformity compared to experimental data. By employing these optimized parameters in practical applications, a balance between the maximum average deposition rate and minimum coefficient of variation can be achieved during UAV spraying, thereby reducing pesticide usage, promoting sustainable agriculture, and mitigating instances of missed spraying and re-spraying.

Design and multiobjective optimization of a two‐point contact ladder‐climbing robot using a genetic algorithm

AbstractThis paper presents the design and optimization of a climbing robot. The design of a ladder‐climbing robot is done with fundamental mathematical considerations. The designed robot is robust enough to manage all environmental calamities, and at the same time, it is optimized for lightweight to reduce the actuator's cost and ease of transportation. An analytical evaluation is carried out for both static and dynamic conditions to determine strength and motion characteristics. The multiobjective optimization of the design parameters of a ladder‐climbing robot is done to obtain optimized values of design parameters. The formulation of an optimization problem that considers the minimization of weight and natural frequency is performed. Using an evolutionary genetic algorithm (GA) for the multicriteria optimization problem is solved, and a Pareto front solution is obtained. The optimal values of the parameters are decided based on the knee selection technique. As both objective functions are contradictory, the optimum results significantly improve the robot's performance. Controlling the proportional–integral–derivative (PID) parameters is crucial as the robot climbs with a two‐point contact gait pattern. The controlling parameters impart stability to the robot. PID parameters like proportional, integral and derivative gain are tunned using the GA. Finally, the developed prototype is tested on the ladders of the tower, and satisfactory climbing motion is achieved.

Multi-objective parameter matching of elastic ring for rotor system by using particle swarm optimization method

PurposeElastic rings served as the elastic supporting elements which have been extensively used in the aeroengines for maneuverable planes with high overloading. However, under extreme conditions, the elastic ring contacts the bearing seat, causing elastic ring failure. Therefore, it is necessary to optimize the matching parameters of the elastic ring in order to suppress the occurrence of elastic ring failure under harsh working conditions.Design/methodology/approachIn this paper, a rotor system supported by elastic rings is researched and a multi-objective parameter matching method of elastic ring is proposed, considering the elastic ring failure, rotor system’s frequency forbidden zone and rotor system’s dynamic response. Then, the particle swarm optimization algorithm is used to dynamically constrain the parameter matching space and obtain the ideal solution for the elastic ring parameter matching.FindingsBy analyzing the elastic ring’s matching results (different unbalanced forces and disk masses), the relationship between the trend of Pareto front changes and rotor system parameters is studied. In addition, the rotor system’s dynamic characteristics before and after parameter matching are analyzed.Originality/valueThis article provides guidance for the design of elastic rings by matching the parameters of elastic rings.

Machine Learning Enabled Design and Optimization for 3D‐Printing of High‐Fidelity Presurgical Organ Models

AbstractThe development of a general‐purpose machine learning algorithm capable of quickly identifying optimal 3D‐printing settings can save manufacturing time and cost, reduce labor intensity, and improve the quality of 3D‐printed objects. Existing methods have limitations which focus on overall performance or one specific aspect of 3D‐printing quality. Here, for addressing the limitations, a multi‐objective Bayesian Optimization (BO) approach which uses a general‐purpose algorithm to optimize the black‐box functions is demonstrated and identifies the optimal input parameters of direct ink writing for 3D‐printing different presurgical organ models with intricate geometry. The BO approach enhances the 3D‐printing efficiency to achieve the best possible printed object quality while simultaneously addressing the inherent trade‐offs from the process of pursuing ideal outcomes relevant to requirements from practitioners. The BO approach also enables us to effectively explore 3D‐printing inputs inclusive of layer height, nozzle travel speed, and dispensing pressure, as well as visualize the trade‐offs between each set of 3D‐printing inputs in terms of the output objectives which consist of time, porosity, and geometry precisions through the Pareto front.

Target Material Property‐Dependent Cluster Analysis of Inorganic Compounds

The cluster analysis of materials categorizes them according to similarities based on the features of materials, providing insight into the relationship between the materials. Conventional cluster analyses typically use basic features derived from the chemical composition and crystal structure without considering target material properties such as the bandgap and dielectric constant. However, such approaches do not meet demands for grading materials according to properties of interest simultaneously with chemical and structural similarities. Herein, a clustering method grouping similar materials in terms of both the target properties and basic features is proposed. The clustering is compared considering the cohesive energy with that considering the bandgap of metal oxides, showing that their categorizations are clearly different. Further, several clusters classified by the bandgap are analyzed, and coordination environments related to each range of the bandgap are revealed. The clustering for the electronic static dielectric constant identifies a cluster involving several perovskite‐type oxides and balancing with the bandgap near the Pareto front. The method enables analyses with different viewpoints from those of the conventional clustering and feature importance analyses by taking the relationship between the target property and the basic features into account.

Multimodal multiobjective differential evolution algorithm based on enhanced decision space search

Multimodal multiobjective optimization problems (MMOPs) have attracted extensive research interest. These problems are characterized by the presence of multiple equivalent optimal solutions in the decision space, all corresponding to the same optimal values in the objective space. However, effectively finding a high-quality and evenly distributed Pareto sets (PSs) remains a challenge for researchers. This paper introduces a multimodal multiobjective differential evolution algorithm based on enhanced decision space search (MMODE_EDSS). By adopting two types of strategies to enhance the decision space search capability, the algorithm generates multiple high-quality non-dominated solutions. In the early stages of evolution, neighborhood information is used to enhance search capabilities, while in the later stages, data interpolation methods following clustering are employed for searching. Moreover, to improve the overall population distribution, an environmental selection mechanism based on dual-space crowding distance is adopted. The effectiveness of the proposed algorithm, MMODE_EDSS, is evaluated by comparing it with eight state-of-the-art multimodal multiobjective evolutionary algorithms (MMOEAs). Experimental results confirm the significant advantages of MMODE_EDSS.

Decentralized pure exchange processes on networks

AbstractWe define a class of pure exchange Edgeworth trading processes that under minimal assumptions converge to a stable set in the space of allocations, and characterise the Pareto set of these processes. Choosing a specific process belonging to this class, that we define fair trading, we analyse the trade dynamics between agents located on a weighted network. We determine the conditions under which there always exists a one-to-one map between the set of networks and the set of limit points of the dynamics, and derive an analog of the Second Welfare Theorem for networks. This result is used to explore what is the effect of the network topology on the trade dynamics and on the final allocation.

Multi-objective optimization of comprehensive performance enhancement for proton exchange membrane fuel cell based on machine learning

The comprehensive performance of proton exchange membrane fuel cells depends on operating conditions. This paper innovatively uses the Pearson correlation coefficient to screen the optimization objectives (uniformity index of oxygen, standard deviation of temperature, net power density), and obtains the optimal operating conditions of the proton exchange membrane fuel cell through a multi-objective optimization method. The optimized dataset comes from the simulation results of the three-dimensional numerical model, and the regression model is established through the response surface method. Moreover, the non-dominated sorting genetic algorithm-II is used for processing to obtain the Pareto front solution set, and the optimal operating conditions are obtained from it through the Technique for order preference by similarity to an ideal solution. The analysis of variance result shows that the influence of cathode operating conditions on the comprehensive performance is greater than that of anode, especially the influence of cathode stoichiometry ratio is the most significant. The optimal solution obtained 1.0981 %, 10.5845 %, and 1.0376 % enhancement compared to the optimal values in the simulation results. The differences between the three optimization objectives are only 0.8190 %, 1.0315 %, and 0.8789 % as verified by numerical simulation, thus the machine learning results are reliable and accurate.

Multi-objective optimization of heat transfer performance of H-type fin with vortex generator based on NSGA-II

To further enhance the performance of heat exchangers and bolster waste heat recovery, vortex generators (VGs) of various shapes are added to the H-type fins. This study explores the impact of VGs’ attack angles and configurations on the thermal performance of H-type finned tubes under different Reynolds (Re) numbers. Additionally, multiple dimensionless evaluation metrics are introduced to assess the flow and heat transfer characteristics with VGs, and correlations for predicting the Nusselt (Nu) number and pressure drop at varying attack angles are developed. The results reveal that at low Re numbers, trapezoidal winglets yield the highest comprehensive performance factor (R-factor), while at high Re numbers, rectangular winglets prevail. Compared to trapezoidal winglet configurations, rectangular winglets enhance the Nu by 6.49%-6.67%. Among the various attack angle configurations, the 45° rectangular winglets exhibit the most favorable overall heat transfer performance, with a Nu increase of 12.07% compared to standard H-type fins. It is observed that installing more than two VGs can result in an R-value less than 1, suggesting that excessive VG placement in H-type fin heat exchangers is not advisable. Finally, the Non-dominated Sorting Genetic Algorithm II (NSGA-II) is employed to identify Pareto optimal points, which exhibit lower resistance and enhanced convective heat transfer. The Pareto optimal set, in comparison to different attack angle CFD data, shows a 31.27% reduction in resistance and a 15.72% increase in Nu.

A Multiobjective Genetic Algorithm to Evolving Local Interpretable Model-Agnostic Explanations for Deep Neural Networks in Image Classification

Deep convolutional neural networks have become a dominant solution for numerous image classification tasks. However, a main criticism is the poor explainability due to the black-box characteristic, which hurdles the extensive usage of deep convolutional neural networks. To address this issue, this paper proposes a new evolutionary multi-objective based method, which aims to explain the behaviours of deep convolutional neural networks by evolving local explanations on specific images. To the best of our knowledge, this is the first evolutionary multi-objective method to evolve local explanations. The proposed method is model-agnostic, i.e. it is applicable to explain any deep convolutional neural networks. ImageNet is used to examine the effectiveness of the proposed method. Three well-known deep convolutional neural networks -VGGNet, ResNet, and MobileNet, are chosen to demonstrate the modelagnostic characteristic. Based on the experimental results, it can be observed that the local explanations are understandable to end-users, who need to check the sensibility of the evolved explanations to decide whether to trust the predictions made by the deep convolutional neural networks. Furthermore, the local explanations evolved by the proposed method improves the confidence of deep convolutional neural networks making the predictions. Lastly, the pareto front and convergence analyses indicate that the proposed method can form a good set of nondominated solutions.