An Immune-Inspired Approach with Interval Allocation in Solving Multimodal Multi-Objective Optimization Problems with Local Pareto Sets

In practical engineering,multi-objective optimization often encounters situations where multiple Pareto sets(PS)in the decision space correspond to the same Pareto front(PF)in the objective space,known as Multi-Modal Multi-Objective Optimization Problems(MMOP).Locating multiple equivalent global PSs poses a significant challenge in real-world applications,especially considering the existence of local PSs.Effectively identifying and locating both global and local PSs is a major challenge.To tackle this issue,we introduce an immune-inspired reproduction strategy designed to produce more offspring in less crowded,promising regions and regulate the number of offspring in areas that have been thoroughly explored.This approach achieves a balanced trade-off between exploration and exploitation.Furthermore,we present an interval allocation strategy that adaptively assigns fitness levels to each antibody.This strategy ensures a broader survival margin for solutions in their initial stages and progressively amplifies the differences in individual fitness values as the population matures,thus fostering better population convergence.Additionally,we incorporate a multi-population mechanism that precisely manages each subpopulation through the interval allocation strategy,ensuring the preservation of both global and local PSs.Experimental results on 21 test problems,encompassing both global and local PSs,are compared with eight state-of-the-art multimodal multi-objective optimization algorithms.The results demonstrate the effectiveness of our proposed algorithm in simultaneously identifying global Pareto sets and locally high-quality PSs.

Multi-Strategy Assisted Multi-Objective Whale Optimization Algorithm for Feature Selection

In classification problems,datasets often contain a large amount of features,but not all of them are relevant for accurate classification.In fact,irrelevant features may even hinder classification accuracy.Feature selection aims to alleviate this issue by minimizing the number of features in the subset while simultaneously minimizing the classification error rate.Single-objective optimization approaches employ an evaluation function designed as an aggregate function with a parameter,but the results obtained depend on the value of the parameter.To eliminate this parameter’s influence,the problem can be reformulated as a multi-objective optimization problem.The Whale Optimization Algorithm(WOA)is widely used in optimization problems because of its simplicity and easy implementation.In this paper,we propose a multi-strategy assisted multi-objective WOA(MSMOWOA)to address feature selection.To enhance the algorithm’s search ability,we integrate multiple strategies such as Levy flight,Grey Wolf Optimizer,and adaptive mutation into it.Additionally,we utilize an external repository to store non-dominant solution sets and grid technology is used to maintain diversity.Results on fourteen University of California Irvine(UCI)datasets demonstrate that our proposed method effectively removes redundant features and improves classification performance.The source code can be accessed from the website:https://github.com/zc0315/MSMOWOA.

Multi-objective optimization and evaluation of supercritical CO_(2) Brayton cycle for nuclear power generation

The supercritical CO_(2) Brayton cycle is considered a promising energy conversion system for Generation IV reactors for its simple layout,compact structure,and high cycle efficiency.Mathematical models of four Brayton cycle layouts are developed in this study for different reactors to reduce the cost and increase the thermohydraulic performance of nuclear power generation to promote the commercialization of nuclear energy.Parametric analysis,multi-objective optimizations,and four decision-making methods are applied to obtain each Brayton scheme’s optimal thermohydraulic and economic indexes.Results show that for the same design thermal power scale of reactors,the higher the core’s exit temperature,the better the Brayton cycle’s thermo-economic performance.Among the four-cycle layouts,the recompression cycle(RC)has the best overall performance,followed by the simple recuperation cycle(SR)and the intercooling cycle(IC),and the worst is the reheating cycle(RH).However,RH has the lowest total cost of investment(C_(tot))of$1619.85 million,and IC has the lowest levelized cost of energy(LCOE)of 0.012$/(kWh).The nuclear Brayton cycle system’s overall performance has been improved due to optimization.The performance of the molten salt reactor combined with the intercooling cycle(MSR-IC)scheme has the greatest improvement,with the net output power(W_(net)),thermal efficiencyη_(t),and exergy efficiency(η_(e))improved by 8.58%,8.58%,and 11.21%,respectively.The performance of the lead-cooled fast reactor combined with the simple recuperation cycle scheme was optimized to increase C_(tot) by 27.78%.In comparison,the internal rate of return(IRR)increased by only 7.8%,which is not friendly to investors with limited funds.For the nuclear Brayton cycle,the molten salt reactor combined with the recompression cycle scheme should receive priority,and the gas-cooled fast reactor combined with the reheating cycle scheme should be considered carefully.

A modified back analysis method for deep excavation with multi-objective optimization procedure

Real-time prediction of excavation-induced displacement of retaining pile during the deep excavation process is crucial for construction safety.This paper proposes a modified back analysis method with multi-objective optimization procedure,which enables a real-time prediction of horizontal displacement of retaining pile during construction.As opposed to the traditional stage-by-stage back analysis,time series monitoring data till the current excavation stage are utilized to form a multi-objective function.Then,the multi-objective particle swarm optimization (MOPSO) algorithm is applied for parameter identification.The optimized model parameters are immediately adopted to predict the excavation-induced pile deformation in the continuous construction stages.To achieve efficient parameter optimization and real-time prediction of system behavior,the back propagation neural network (BPNN) is established to substitute the finite element model,which is further implemented together with MOPSO for automatic operation.The proposed approach is applied in the Taihu tunnel excavation project,where the effectiveness of the method is demonstrated via the comparisons with the site monitoring data.The method is reliable with a prediction accuracy of more than 90%.Moreover,different optimization algorithms,including non-dominated sorting genetic algorithm (NSGA-II),Pareto Envelope-based Selection Algorithm II (PESA-II) and MOPSO,are compared,and their influences on the prediction accuracy at different excavation stages are studied.The results show that MOPSO has the best performance for high dimensional optimization task.

Crashworthiness Design and Multi-Objective Optimization of Bionic Thin-Walled Hybrid Tube Structures

Thin-walled structures are widely used in cars due to their lightweight construction and energy-absorbing properties.However,issues such as high initial stress and lowenergy-absorbing efficiency arise.This study proposes a novel energy-absorbing structure inwhich a straight tube is combinedwith a conical tube and a bamboo-inspired bulkhead structure is introduced.This configuration allows the conical tube to flip outward first and then fold together with the straight tube.This deformation mode absorbs more energy and less peak force than the conical tube sinking and flipping inward.Through finite element numerical simulation,the specific energy absorption capacity of the structure is increased by 26%compared to that of a regular circular cross-section tube.Finally,the impact resistance of the bionic straight tapered tube structure is further improved through multi-objective optimization,promoting the engineering application and lightweight design of hybrid cross-section tubes.

Multi-objective global optimization approach predicted quasi-layered ternary TiOS crystals with promising photocatalytic properties

Titanium dioxide(TiO_(2))has attracted considerable research attentions for its promising applications in solar cells and photocatalytic devices.However,the intrinsic challenge lies in the relatively low energy conversion efficiency of TiO_(2),primarily attributed to the substantial band gaps(exceeding 3.0 eV)associated with its rutile and anatase phases.Leveraging multi-objective global optimization,we have identified two quasi-layered ternary Ti-O-S crystals,composed of titanium,oxygen,and sulfur.The calculations of formation energy,phonon dispersions,and thermal stability confirm the chemical,dynamical and thermal stability of these newly discovered phases.Employing the state-of-art hybrid density functional approach and many-body perturbation theory(quasiparticle GW approach and Bethe-Salpeter equation),we calculate the optical properties of both the TiOS phases.Significantly,both phases show favorable photocatalytic characteristics,featuring band gaps suitable for visible optical absorption and appropriate band alignments with water for effective charge carrier separation.Therefore,ternary compound TiOS holds the potential for achieving high-efficiency photochemical conversion,showing our multi-objective global optimization provides a new approach for novel environmental and energy materials design with multicomponent compounds.

Prediction Model-based Multi-objective Optimization for Mix-ratio Design of Recycled Aggregate Concrete

The prediction model for mechanical properties of RAC was established through the Bayesian optimization-based Gaussian process regression(BO-GPR)method,where the input variables in BO-GPR model depend on the mix ratio of concrete.Then the compressive strength prediction model,the material cost,and environmental factors were simultaneously considered as objectives,while a multi-objective gray wolf optimization algorithm was developed for finding the optimal mix ratio.A total of 730 RAC datasets were used for training and testing the predication model,while the optimal design method for mix ratio was verified through RAC experiments.The experimental results show that the predicted,testing,and expected compressive strengths are nearly consistent,illustrating the effectiveness of the proposed method.

Even Search in a Promising Region for Constrained Multi-Objective Optimization

In recent years, a large number of approaches to constrained multi-objective optimization problems(CMOPs) have been proposed, focusing on developing tweaked strategies and techniques for handling constraints. However, an overly finetuned strategy or technique might overfit some problem types,resulting in a lack of versatility. In this article, we propose a generic search strategy that performs an even search in a promising region. The promising region, determined by obtained feasible non-dominated solutions, possesses two general properties.First, the constrained Pareto front(CPF) is included in the promising region. Second, as the number of feasible solutions increases or the convergence performance(i.e., approximation to the CPF) of these solutions improves, the promising region shrinks. Then we develop a new strategy named even search,which utilizes the non-dominated solutions to accelerate convergence and escape from local optima, and the feasible solutions under a constraint relaxation condition to exploit and detect feasible regions. Finally, a diversity measure is adopted to make sure that the individuals in the population evenly cover the valuable areas in the promising region. Experimental results on 45 instances from four benchmark test suites and 14 real-world CMOPs have demonstrated that searching evenly in the promising region can achieve competitive performance and excellent versatility compared to 11 most state-of-the-art methods tailored for CMOPs.

Multi-objective Design of Blending Fuel by Intelligent Optimization Algorithms

Fuel design is a complex multi-objective optimization problem in which facile and robust methods are urgently demanded.Herein,a complete workflow for designing a fuel blending scheme is presented,which is theoretically supported,efficient,and reliable.Based on the data distribution of the composition and properties of the blending fuels,a model of polynomial regression with appropriate hypothesis space was established.The parameters of the model were further optimized by different intelligence algorithms to achieve high-precision regression.Then,the design of a blending fuel was described as a multi-objective optimization problem,which was solved using a Nelder–Mead algorithm based on the concept of Pareto domination.Finally,the design of a target fuel was fully validated by experiments.This study provides new avenues for designing various blending fuels to meet the needs of next-generation engines.

Multi-Objective Optimization of VBHF in Deep Drawing Based on the Improved QO-Jaya Algorithm

Blank holder force(BHF)is a crucial parameter in deep drawing,having close relation with the forming quality of sheet metal.However,there are different BHFs maintaining the best forming effect in different stages of deep drawing.The variable blank holder force(VBHF)varying with the drawing stage can overcome this problem at an extent.The optimization of VBHF is to determine the optimal BHF in every deep drawing stage.In this paper,a new heuristic optimization algorithm named Jaya is introduced to solve the optimization efficiently.An improved“Quasi-oppositional”strategy is added to Jaya algorithm for improving population diversity.Meanwhile,an innovated stop criterion is added for better convergence.Firstly,the quality evaluation criteria for wrinkling and tearing are built.Secondly,the Kriging models are developed to approximate and quantify the relation between VBHF and forming defects under random sampling.Finally,the optimization models are established and solved by the improved QO-Jaya algorithm.A VBHF optimization example of component with complicated shape and thin wall is studied to prove the effectiveness of the improved Jaya algorithm.The optimization results are compared with that obtained by other algorithms based on the TOPSIS method.

Multi-Objective Optimization of the Ultrasonic Scalpel Rod and Tip with Improved Performance:Vibration Frequency,Amplitude,and Service Life

Ultrasonic scalpel design for minimally invasive surgical procedures is mainly focused on optimizing cutting performance.However,an important issue is the low fatigue life of traditional ultrasonic scalpels,which affects their long-term reliability and effectiveness and creates hidden dangers for surgery.In this study,a multi-objective optimal design for the cutting performance and fatigue life of ultrasonic scalpels was proposed using finite element analysis and fatigue simulation.The optimal design parameters of resonance frequency and amplitude were determined.By setting the transition fillet and keeping the gain structure away from the node position to enable the scalpel to have a high service life with excellent cutting performance.The frequency modulation method of setting the vibration node bosses at the node position and setting the vibration antinode grooves at the antinode position was compared.Then,the mechanism of the influence of various design elements,such as tip,shank,node position,and antinode position,on the resonance frequency,amplitude,and fatigue life of the ultrasonic scalpel was analyzed,and the optimal design principles of the ultrasonic scalpel were obtained.The proposed ultrasonic scalpel design was confirmed by simulations,impedance measurements,and liver tissue cutting experiments,demonstrating its feasibility and enhanced performance.This research introduces innovative design strategies to improve the fatigue life and performance of ultrasonic scalpels to address an important issue in minimally invasive surgery.

Constraints Separation Based Evolutionary Multitasking for Constrained Multi-Objective Optimization Problems

Constrained multi-objective optimization problems(CMOPs)generally contain multiple constraints,which not only form multiple discrete feasible regions but also reduce the size of optimal feasible regions,thus they propose serious challenges for solvers.Among all constraints,some constraints are highly correlated with optimal feasible regions;thus they can provide effective help to find feasible Pareto front.However,most of the existing constrained multi-objective evolutionary algorithms tackle constraints by regarding all constraints as a whole or directly ignoring all constraints,and do not consider judging the relations among constraints and do not utilize the information from promising single constraints.Therefore,this paper attempts to identify promising single constraints and utilize them to help solve CMOPs.To be specific,a CMOP is transformed into a multitasking optimization problem,where multiple auxiliary tasks are created to search for the Pareto fronts that only consider a single constraint respectively.Besides,an auxiliary task priority method is designed to identify and retain some high-related auxiliary tasks according to the information of relative positions and dominance relationships.Moreover,an improved tentative method is designed to find and transfer useful knowledge among tasks.Experimental results on three benchmark test suites and 11 realworld problems with different numbers of constraints show better or competitive performance of the proposed method when compared with eight state-of-the-art peer methods.

Multi-Objective Optimization of Aluminum Alloy Electric Bus Frame Connectors for Enhanced Durability

The widespread adoption of aluminumalloy electric buses,known for their energy efficiency and eco-friendliness,faces a challenge due to the aluminum frame’s susceptibility to deformation compared to steel.This issue is further exacerbated by the stringent requirements imposed by the flammability and explosiveness of batteries,necessitating robust frame protection.Our study aims to optimize the connectors of aluminum alloy bus frames,emphasizing durability,energy efficiency,and safety.This research delves into Multi-Objective Coordinated Optimization(MCO)techniques for lightweight design in aluminum alloy bus body connectors.Our goal is to enhance lightweighting,reinforce energy absorption,and improve deformation resistance in connector components.Three typical aluminum alloy connectors were selected and a design optimization platform was built for their MCO using a variety of software and methods.Firstly,through three-point bending experiments and finite element analysis on three types of connector components,we identified optimized design parameters based on deformation patterns.Then,employing Optimal Latin hypercube design(OLHD),parametric modeling,and neural network approximation,we developed high-precision approximate models for the design parameters of each connector component,targeting energy absorption,mass,and logarithmic strain.Lastly,utilizing the Archive-based Micro Genetic Algorithm(AMGA),Multi-Objective Particle Swarm Optimization(MOPSO),and Non-dominated SortingGenetic Algorithm(NSGA2),we explored optimized design solutions for these joint components.Subsequently,we simulated joint assembly buckling during bus rollover crash scenarios to verify and analyze the optimized solutions in three-point bending simulations.Each joint component showcased a remarkable 30%–40%mass reduction while boosting energy absorption.Our design optimization method exhibits high efficiency and costeffectiveness.Leveraging contemporary automation technology,the design optimization platform developed in this study is poised to facilitate intelligent optimization of lightweight metal components in future applications.

Enhancing Evolutionary Algorithms With Pattern Mining for Sparse Large-Scale Multi-Objective Optimization Problems

Sparse large-scale multi-objective optimization problems(SLMOPs)are common in science and engineering.However,the large-scale problem represents the high dimensionality of the decision space,requiring algorithms to traverse vast expanse with limited computational resources.Furthermore,in the context of sparse,most variables in Pareto optimal solutions are zero,making it difficult for algorithms to identify non-zero variables efficiently.This paper is dedicated to addressing the challenges posed by SLMOPs.To start,we introduce innovative objective functions customized to mine maximum and minimum candidate sets.This substantial enhancement dramatically improves the efficacy of frequent pattern mining.In this way,selecting candidate sets is no longer based on the quantity of nonzero variables they contain but on a higher proportion of nonzero variables within specific dimensions.Additionally,we unveil a novel approach to association rule mining,which delves into the intricate relationships between non-zero variables.This novel methodology aids in identifying sparse distributions that can potentially expedite reductions in the objective function value.We extensively tested our algorithm across eight benchmark problems and four real-world SLMOPs.The results demonstrate that our approach achieves competitive solutions across various challenges.

Multi-Objective Optimization Algorithm for Grouping Decision Variables Based on Extreme Point Pareto Frontier

The existing algorithms for solving multi-objective optimization problems fall into three main categories:Decomposition-based,dominance-based,and indicator-based.Traditional multi-objective optimization problemsmainly focus on objectives,treating decision variables as a total variable to solve the problem without consideringthe critical role of decision variables in objective optimization.As seen,a variety of decision variable groupingalgorithms have been proposed.However,these algorithms are relatively broad for the changes of most decisionvariables in the evolution process and are time-consuming in the process of finding the Pareto frontier.To solvethese problems,a multi-objective optimization algorithm for grouping decision variables based on extreme pointPareto frontier(MOEA-DV/EPF)is proposed.This algorithm adopts a preprocessing rule to solve the Paretooptimal solution set of extreme points generated by simultaneous evolution in various target directions,obtainsthe basic Pareto front surface to determine the convergence effect,and analyzes the convergence and distributioneffects of decision variables.In the later stages of algorithm optimization,different mutation strategies are adoptedaccording to the nature of the decision variables to speed up the rate of evolution to obtain excellent individuals,thusenhancing the performance of the algorithm.Evaluation validation of the test functions shows that this algorithmcan solve the multi-objective optimization problem more efficiently.

Large-Scale Multi-Objective Optimization Algorithm Based on Weighted Overlapping Grouping of Decision Variables

The large-scale multi-objective optimization algorithm(LSMOA),based on the grouping of decision variables,is an advanced method for handling high-dimensional decision variables.However,in practical problems,the interaction among decision variables is intricate,leading to large group sizes and suboptimal optimization effects;hence a large-scale multi-objective optimization algorithm based on weighted overlapping grouping of decision variables(MOEAWOD)is proposed in this paper.Initially,the decision variables are perturbed and categorized into convergence and diversity variables;subsequently,the convergence variables are subdivided into groups based on the interactions among different decision variables.If the size of a group surpasses the set threshold,that group undergoes a process of weighting and overlapping grouping.Specifically,the interaction strength is evaluated based on the interaction frequency and number of objectives among various decision variables.The decision variable with the highest interaction in the group is identified and disregarded,and the remaining variables are then reclassified into subgroups.Finally,the decision variable with the strongest interaction is added to each subgroup.MOEAWOD minimizes the interactivity between different groups and maximizes the interactivity of decision variables within groups,which contributed to the optimized direction of convergence and diversity exploration with different groups.MOEAWOD was subjected to testing on 18 benchmark large-scale optimization problems,and the experimental results demonstrate the effectiveness of our methods.Compared with the other algorithms,our method is still at an advantage.

Enhancing Hyper-Spectral Image Classification with Reinforcement Learning and Advanced Multi-Objective Binary Grey Wolf Optimization

Hyperspectral(HS)image classification plays a crucial role in numerous areas including remote sensing(RS),agriculture,and the monitoring of the environment.Optimal band selection in HS images is crucial for improving the efficiency and accuracy of image classification.This process involves selecting the most informative spectral bands,which leads to a reduction in data volume.Focusing on these key bands also enhances the accuracy of classification algorithms,as redundant or irrelevant bands,which can introduce noise and lower model performance,are excluded.In this paper,we propose an approach for HS image classification using deep Q learning(DQL)and a novel multi-objective binary grey wolf optimizer(MOBGWO).We investigate the MOBGWO for optimal band selection to further enhance the accuracy of HS image classification.In the suggested MOBGWO,a new sigmoid function is introduced as a transfer function to modify the wolves’position.The primary objective of this classification is to reduce the number of bands while maximizing classification accuracy.To evaluate the effectiveness of our approach,we conducted experiments on publicly available HS image datasets,including Pavia University,Washington Mall,and Indian Pines datasets.We compared the performance of our proposed method with several state-of-the-art deep learning(DL)and machine learning(ML)algorithms,including long short-term memory(LSTM),deep neural network(DNN),recurrent neural network(RNN),support vector machine(SVM),and random forest(RF).Our experimental results demonstrate that the Hybrid MOBGWO-DQL significantly improves classification accuracy compared to traditional optimization and DL techniques.MOBGWO-DQL shows greater accuracy in classifying most categories in both datasets used.For the Indian Pine dataset,the MOBGWO-DQL architecture achieved a kappa coefficient(KC)of 97.68%and an overall accuracy(OA)of 94.32%.This was accompanied by the lowest root mean square error(RMSE)of 0.94,indicating very precise predictions with minimal error.In the case of the Pavia University dataset,the MOBGWO-DQL model demonstrated outstanding performance with the highest KC of 98.72%and an impressive OA of 96.01%.It also recorded the lowest RMSE at 0.63,reinforcing its accuracy in predictions.The results clearly demonstrate that the proposed MOBGWO-DQL architecture not only reaches a highly accurate model more quickly but also maintains superior performance throughout the training process.

Constrained Multi-Objective Optimization With Deep Reinforcement Learning Assisted Operator Selection

Solving constrained multi-objective optimization problems with evolutionary algorithms has attracted considerable attention.Various constrained multi-objective optimization evolutionary algorithms(CMOEAs)have been developed with the use of different algorithmic strategies,evolutionary operators,and constraint-handling techniques.The performance of CMOEAs may be heavily dependent on the operators used,however,it is usually difficult to select suitable operators for the problem at hand.Hence,improving operator selection is promising and necessary for CMOEAs.This work proposes an online operator selection framework assisted by Deep Reinforcement Learning.The dynamics of the population,including convergence,diversity,and feasibility,are regarded as the state;the candidate operators are considered as actions;and the improvement of the population state is treated as the reward.By using a Q-network to learn a policy to estimate the Q-values of all actions,the proposed approach can adaptively select an operator that maximizes the improvement of the population according to the current state and thereby improve the algorithmic performance.The framework is embedded into four popular CMOEAs and assessed on 42 benchmark problems.The experimental results reveal that the proposed Deep Reinforcement Learning-assisted operator selection significantly improves the performance of these CMOEAs and the resulting algorithm obtains better versatility compared to nine state-of-the-art CMOEAs.

A Multi-Objective Optimization for Locating Maintenance Stations and Operator Dispatching of Corrective Maintenance

In this study,we introduce a novel multi-objective optimization model tailored for modern manufacturing,aiming to mitigate the cost impacts of operational disruptions through optimized corrective maintenance.Central to our approach is the strategic placement of maintenance stations and the efficient allocation of personnel,addressing a crucial gap in the integration of maintenance personnel dispatching and station selection.Our model uniquely combines the spatial distribution of machinery with the expertise of operators to achieve a harmonious balance between maintenance efficiency and cost-effectiveness.The core of our methodology is the NSGA Ⅲ+Dispatch,an advanced adaptation of the Non-Dominated Sorting Genetic Algorithm Ⅲ(NSGA-Ⅲ),meticulously designed for the selection of maintenance stations and effective operator dispatching.This method integrates a comprehensive coding process,crossover operator,and mutation operator to efficiently manage multiple objectives.Rigorous empirical testing,including a detailed analysis from a taiwan region electronic equipment manufacturer,validated the effectiveness of our approach across various scenarios of machine failure frequencies and operator configurations.The findings reveal that the proposed model significantly outperforms current practices by reducing response times by up to 23%in low-frequency and 28.23%in high-frequency machine failure scenarios,leading to notable improvements in efficiency and cost reduction.Additionally,it demonstrates significant improvements in oper-ational efficiency,particularly in selective high-frequency failure contexts,while ensuring substantial manpower cost savings without compromising on operational effectiveness.This research significantly advances maintenance strategies in production environments,providing the manufacturing industry with practical,optimized solutions for diverse machine malfunction situations.Furthermore,the methodologies and principles developed in this study have potential applications in various other sectors,including healthcare,transportation,and energy,where maintenance efficiency and resource optimization are equally critical.

Energy-Saving Distributed Flexible Job Shop Scheduling Optimization with Dual Resource Constraints Based on Integrated Q-Learning Multi-Objective Grey Wolf Optimizer

The distributed flexible job shop scheduling problem(DFJSP)has attracted great attention with the growth of the global manufacturing industry.General DFJSP research only considers machine constraints and ignores worker constraints.As one critical factor of production,effective utilization of worker resources can increase productivity.Meanwhile,energy consumption is a growing concern due to the increasingly serious environmental issues.Therefore,the distributed flexible job shop scheduling problem with dual resource constraints(DFJSP-DRC)for minimizing makespan and total energy consumption is studied in this paper.To solve the problem,we present a multi-objective mathematical model for DFJSP-DRC and propose a Q-learning-based multi-objective grey wolf optimizer(Q-MOGWO).In Q-MOGWO,high-quality initial solutions are generated by a hybrid initialization strategy,and an improved active decoding strategy is designed to obtain the scheduling schemes.To further enhance the local search capability and expand the solution space,two wolf predation strategies and three critical factory neighborhood structures based on Q-learning are proposed.These strategies and structures enable Q-MOGWO to explore the solution space more efficiently and thus find better Pareto solutions.The effectiveness of Q-MOGWO in addressing DFJSP-DRC is verified through comparison with four algorithms using 45 instances.The results reveal that Q-MOGWO outperforms comparison algorithms in terms of solution quality.