Multi-objective Differential Evolution Algorithm Research Articles

A recursive framework for improving the performance of multi-objective differential evolution algorithms for gene selection

Gene selection is a pivotal process in machine-learning-driven medical diagnostics, where the goal is to identify a subset of genes from microarray expression profiles that can enhance the predictive accuracy of classifiers for disease diagnosis. The two key objectives of gene selection are to reduce the dimensionality of the data and to improve the accuracy of disease diagnosis, which is typically a multi-objective optimization problem. In recent years, multi-objective evolutionary algorithms (MOEAs) have gained wide attention in feature selection research, and several related algorithms have been produced. However, most algorithms tend to get stuck in local optimality when searching for solutions from a high-dimensional space. To solve the gene selection problem effectively, this study introduces a recursive multi-objective differential evolution algorithm with elite recursive strategy (RMODE-E) and a recursive multi-objective differential evolution algorithm with Pareto front recursive strategy (RMODE-P). RMODE-E amalgamates the features selected by the top E elite individuals, RMODE-P consolidates the features selected by the Pareto front set, and the combined features then serve as the foundation for subsequent recursive rounds of searching. The proposed feature subspace combination strategy not only reduces the recursive search space but also improves the capacity to find globally optimal feature subsets. Extensive experiments were conducted to compare our proposed algorithms with eight state-of-the-art evolutionary algorithms to validate their effectiveness. Experimental results demonstrate that RMODE-P has better global search capability as it achieves better best classification accuracy, mean classification accuracy, and minimal gene subset size.

Scheduling of Continuous Annealing With a Multi-Objective Differential Evolution Algorithm Based on Deep Reinforcement Learning

This paper studies a multi-objective scheduling problem in a continuous annealing operation of the steel industry, which is to simultaneously determine the production batch sizes for coils and their schedules so as to minimize the cost of setup, earliness, and tardiness. A large batch can reduce the setup costs but lead to a cost increase in earliness and tardiness. Thus, the conflict objectives of minimizing setup cost, earliness, and tardiness can be formulated separately. In this paper, we formulate a multi-objective optimization model for the problem and develop an adaptive multi-objective differential evolutionary based on deep reinforcement learning (AMODE-DRL) for effectively obtaining the Pareto solutions. In the proposed AMODE-RDL, DRL is integrated into the MODE algorithm as a controller, which can adaptively select mutation operators and parameters according to different search domains. Computational results on randomly generated instances and the practical problem instances show that DRL can effectively guide MODE to select mutation operators and parameters. The proposed algorithm can obtain better solutions compared to other powerful multi-objective evolutionary and adaptive MODE algorithms. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Setup usually leads to a decrease in production capacity and an increase in production costs. Similarly, earliness and tardiness costs are also crucial. These costs correspond to production, customer demand, and inventory, which are common objectives in production systems. However, the three objectives are generally conflicting, and it is very hard for practitioners to make appropriate decisions with manual experience. The multi-objective optimization methods can provide different scheduling for practitioners who may choose the suitable one according to current working conditions. Accordingly, the proposed model and algorithm can extend to other fields with similar characteristic problems.

Three-stage multi-modal multi-objective differential evolution algorithm for vehicle routing problem with time windows

In this paper, the mathematical model of Vehicle Routing Problem with Time Windows (VRPTW) is established based on the directed graph, and a 3-stage multi-modal multi-objective differential evolution algorithm (3S-MMDEA) is proposed. In the first stage, in order to expand the range of individuals to be selected, a generalized opposition-based learning (GOBL) strategy is used to generate a reverse population. In the second stage, a search strategy of reachable distribution area is proposed, which divides the population with the selected individual as the center point to improve the convergence of the solution set. In the third stage, an improved individual variation strategy is proposed to legalize the mutant individuals, so that the individual after variation still falls within the range of the population, further improving the diversity of individuals to ensure the diversity of the solution set. Based on the synergy of the above three stages of strategies, the diversity of individuals is ensured, so as to improve the diversity of solution sets, and multiple equivalent optimal paths are obtained to meet the planning needs of different decision-makers. Finally, the performance of the proposed method is evaluated on the standard benchmark datasets of the problem. The experimental results show that the proposed 3S-MMDEA can improve the efficiency of logistics distribution and obtain multiple equivalent optimal paths. The method achieves good performance, superior to the most advanced VRPTW solution methods, and has great potential in practical projects.

An improved differential evolution algorithm for multi-modal multi-objective optimization.

Multi-modal multi-objective problems (MMOPs) have gained much attention during the last decade. These problems have two or more global or local Pareto optimal sets (PSs), some of which map to the same Pareto front (PF). This article presents a new affinity propagation clustering (APC) method based on the Multi-modal multi-objective differential evolution (MMODE) algorithm, called MMODE_AP, for the suit of CEC'2020 benchmark functions. First, two adaptive mutation strategies are adopted to balance exploration and exploitation and improve the diversity in the evolution process. Then, the affinity propagation clustering method is adopted to define the crowding degree in decision space (DS) and objective space (OS). Meanwhile, the non-dominated sorting scheme incorporates a particular crowding distance to truncate the population during the environmental selection process, which can obtain well-distributed solutions in both DS and OS. Moreover, the local PF membership of the solution is defined, and a predefined parameter is introduced to maintain of the local PSs and solutions around the global PS. Finally, the proposed algorithm is implemented on the suit of CEC'2020 benchmark functions for comparison with some MMODE algorithms. According to the experimental study results, the proposed MMODE_AP algorithm has about 20 better performance results on benchmark functions compared to its competitors in terms of reciprocal of Pareto sets proximity (rPSP), inverted generational distances (IGD) in the decision (IGDX) and objective (IGDF). The proposed algorithm can efficiently achieve the two goals, i.e., the convergence to the true local and global Pareto fronts along with better distributed Pareto solutions on the Pareto fronts.

Manifold assistant multi-modal multi-objective differential evolution algorithm and its application in actual rolling bearing fault diagnosis

The fault diagnosis of rotating mechanical systems is crucial for ensuring the reliability of industrial equipment. In actual industrial scenarios, different combinations of fault features may yield the same diagnostic accuracy, but the difficulty and cost of obtaining different features generally vary. Therefore, in the feature extraction process, if multiple optimal feature combinations can be provided to decision-makers, it would be beneficial to select the feature subsets with lower computational cost or measurement difficulty that achieve the same fault diagnosis performance. Therefore, we model the fault feature selection task as a multi-modal multi-objective optimization problem (MMOP). To better solve MMOPs, a manifold assistant multi-modal multi-objective differential evolution (MA-MMODE) algorithm is proposed. It can establish a regularity model based on the distribution of the population to generate offspring and approximate the true Pareto set. Subsequently, this paper presents a preselection strategy that considers the distributions of both the decision space and objective space. The proposed preselection strategy can ensure that individuals with more promising solutions enter the mating pool to generate new offspring, thus exploring the search space more comprehensively. Using the CEC 2019 test set, the proposed MA-MMODE is compared with state-of-the-art algorithms, and the experimental results illustrate the effectiveness of the proposed algorithm. Moreover, based on two real rolling bearing testbeds, the proposed MA-MMODE is applied for feature selection in actual fault diagnosis. The experimental results show that the proposed MA-MMODE can provide multiple equivalent feature subsets to decision-makers while effectively improving the fault diagnosis accuracy for rolling bearings.

Reinforcement learning-based multi-objective differential evolution algorithm for feature selection

Feature Selection (FS) can be used to determine the optimal subset of features from a raw dataset by reducing dimensionality and improving accuracy. In this study, a reinforcement learning-based multi-objective differential evolution algorithm (RLMODE) for FS is proposed, which is modeled as a multi-objective optimization problem. First, a reinforcement learning-based offspring generation strategy is designed. The offspring generation strategy is based on the Q-learning framework, which considers each individual in the population as an agent. The dominance relationship between an agent and its predecessor is used to encode the state. A well-chosen action set containing three typical differential evolution mutation operators is available for each agent. The reward is used to update the exclusive Q-table. Moreover, a novel Pareto front (PF) relearning strategy is devised to allow adequate communication between individuals on the PF. The PF relearning strategy reevaluates the potential value of all individuals on the PF as a whole. This promotes the propagation of excellent solutions and improves PF diversity. The proposed RLMODE initially demonstrates its strength in benchmarks. The excellent results for 17 datasets demonstrate RLMODE’s merits in terms of dimensionality reduction and accuracy improvement. Therefore, the proposed RLMODE method is a promising FS technique.

Reliability-based design of high-performance hydrocyclones: Multi-objective optimization, fabrication using 3D-printing and experimental analysis

In the literature, several works aim to improve the performance of hydrocyclones using geometry optimization. However, these approaches do not consider the uncertainties in the models, design variables, and parameters, which can lead to solutions that do not correspond to the expected results when applied to real systems. Considering that, the present work proposed a reliability-based optimization of the hydrocyclone dimensions using the Advanced Mean Value (AMV) technique associated with the Multi-Objective Differential Evolution (MODE) algorithm. For that purpose, the maximization of total efficiency and minimization of underflow-to-throughput ratio and Euler number were taken as objectives and constraints were set to ensure an acceptable performance of the hydrocyclone in the main aspects of the process. Some of the obtained devices were selected to be fabricated through an additive manufacturing technique (3D printing) and, after that, be experimentally tested. The optimization results showed that the reliability-based optimization affected the Pareto curves in two different ways: a shortening or a displacement of the curve in relation to the nominal one, depending on which constraints are being activated. The experimental results demonstrated that the hydrocyclones obtained by the reliable approach had a greater tendency to meet the set constraints in practice. Finally, hydrocyclones with great experimental performance were obtained considering separation efficiency, concentration capacity, and energy consumption.

Reinforcement learning-based differential evolution algorithm for constrained multi-objective optimization problems

Many real-world problems can be established as Constrained Multi-objective Optimization Problems (CMOPs). It is still challenging to automatically set efficient parameters for Constrained Multi-Objective Evolutionary Algorithms (CMOEAs) to solve these CMOPs. A Reinforcement Learning-based Multi-Objective Differential Evolution (RLMODE) algorithm is proposed, in which the main parameters are dynamically adjusted. During the evolution process, the offspring generated is evaluated and compared with its corresponding parents, the relationship between the offspring and parent can adjust the parameters of RLMODE by the Reinforcement Learning (RL) technique. The feedback mechanism can produce the most appropriate parameters for RLMODE, which pushes the population towards feasible regions. The proposed RLMODE is evaluated on thirty functions and compared with some popular CMOEAs. The performance indicator IGD has revealed that the proposed RLMODE is competitive. Then, they are applied to solve the UAV path planning problem with three objectives and a constraint. The real application has further demonstrated the superiority of the proposed RLMODE.

An [formula omitted]-constrained multiobjective differential evolution with adaptive gradient-based repair method for real-world constrained optimization problems

Over the past decade, incorporating information from the objective function into the constraint-handling process has garnered considerable attention in evolutionary algorithm research. Stemming from this, multiobjective optimization has emerged as a promising approach that simultaneously optimizes the objective function and constraints. However, the challenges associated with optimizing objective functions and satisfying constraints exhibit significant variability. Some constraints and/or objective functions can be exceptionally challenging, necessitating specific methods to identify the optimal solution within a limited feasible region. This study proposes an adaptive gradient descent-based repair method to enhance the search capability for both objective function optimization and constraint satisfaction. This method leverages objective function information to rectify infeasible solutions using gradient descent, thereby reducing the limitations of a purely constraint-based approach and automating the application of the repair method. Furthermore, an enhanced variant of the ɛ-constrained multiobjective differential evolution algorithm is developed for solving constrained optimization problems. The efficacy of the proposed approach is assessed using 57 benchmark test functions derived from real-world applications. Empirical results demonstrate that our approach is capable of locating high-quality solutions, outperforming several selected state-of-the-art algorithms.

Application of high-dimensional multi-objective differential evolutionary algorithm based on global ordering to heat supply pipe network resistance identification

The distribution of resistance coefficients of heat supply pipe networks is the key data guiding the hydraulic balance adjustment of heat networks. Since the heat supply pipe network is composed of many pipe segments and the resistance coefficient of each pipe segment is different during heat supply operation, the identification of the resistance coefficient of the heat supply pipe network is an optimization problem with multiple objective functions. In this paper, a high-dimensional multi-objective differential evolutionary algorithm based on global sorting is developed as a method to identify the resistance coefficients of the heat network and the multi-objective algorithm is applied to the resistance identification of the heat network, and the computational process of resistance identification is improved. The fuzzy mathematical method is applied to the process of resistance identification, and a set of optimal solution sets are generated through the identification of each pipe segment and the optimal solutions are selected from the optimal solution sets based on the fuzzy degree of subordination to solve the problem of determining the optimal solutions. The problem of determining the optimal solution is solved. The results show that compared with the single-objective algorithm, the high-dimensional multi-objective differential evolutionary algorithm based on global sorting produces a uniform and concentrated optimal solution set, and the optimal solution accuracy is higher.

Multiobjective Differential Evolution Algorithm Balancing Multiple Stakeholders for Low-Carbon Order Scheduling in E-Waste Recycling

Order scheduling is an important part of the e-waste recycling process, which can influence the quantity and efficiency of the recycling. With the sustainable development of e-waste recycling, low-carbon order scheduling becomes a significant and challenging reverse logistics scheduling problem. However, it is difficult to obtain an effective low-carbon order schedule considering the conflicting interests of the multiple stakeholders, including enterprises, drivers, customers, and governments. To address this issue, a multi-objective order scheduling model (MOOSM) and a multi-objective differential evolution algorithm balancing multiple stakeholders (MODE-MS) are proposed in this paper. First, to embody the interests of different stakeholders, three time-dependent key variables are calculated by the road congestion and vehicle load, including the velocity, travelling time and carbon emission. Second, with the above key variables, a five-objective order scheduling model is formulated to describe the low-carbon order scheduling problem in e-waste recycling. Third, for solving the MOOSM, a multi-objective differential evolution algorithm based on an adaptive evolutionary search strategy is developed to obtain the low-carbon and stakeholders satisfied scheduling schemes. The experimental results validate the feasibility of MOOSM and the effectiveness of MODE-MS. By comparing with four state-to-art algorithms, the advantages of the proposed MODE-MS are further demonstrated in solving the low-carbon order scheduling.

A fuzzy multi-objective optimization model for sustainable healthcare supply chain network design

Healthcare supply chains play a crucial role, which enables the implementation of optimization strategies that have rapidly emerged as highly effective means for improving the overall structure of pharmaceutical and healthcare supply chains.. In the healthcare industry, parameters such as increasing the quality of service, as well as optimizing costs, environmental, and social factors play a unique role in supply chain management. To improve the healthcare supply chain network, this study proposed a novel optimization model to optimize multiple objectives, including minimizing the total costs and environmental impacts, while maximizing the social factors by creating jobs simultaneously. To address the effects of uncertain parameters, a fuzzy optimization method alongside the multi-objective gray wolf optimizer (MOGWO), non-dominated sorting genetic algorithm II (NSGA-II), multi-objective differential evolution algorithm (MODEA), and ε-constraint are applied to optimize the model. Also, a case study of the pharmaceutical industry demonstrates the model's efficacy in a real-life context. The numerical results show the MOGWO manages to create high-quality Pareto solutions with a good spread at the Pareto boundary within a short time compared to the ε-constraint approach. Further, it has shown a more robust performance compared to MODEA and NSGA-II, indicating the efficiency of MOGWO, among other solution methods and other objective indicators.

A hybrid electricity load prediction system based on weighted fuzzy time series and multi-objective differential evolution

With advances in science and technology, the demand for electricity is increasing dramatically. Consequently, reliable short-term power load prediction is critical to ensure the safety, efficiency, and cost-effectiveness of electricity grids. However, owing to the volatility and randomness of power load time series, traditional forecasting models have struggled to meet the current power system requirements for load prediction precision and stability. In light of this, this study proposes a new hybrid short-term electricity load prediction system that combines the variational modal decomposition (VMD) technique, multi-objective differential evolutionary algorithm (MODE), and weighted fuzzy time series (WFTS) prediction model to increase load prediction precision and stability. Four Belgian datasets (spring, summer, autumn, and winter) were used for short-term power load prediction experiments and were compared with prevalent prediction methods. The experimental results indicate that the designed system obtains MAPESpring=0.6670%, MAPESummer=0.5835%, MAPEAutumn=0.7049%, MAPEWinter=0.7123% and STDSpring=60.2575, STDSpring=62.6573, STDSpring=72.7393, STDSpring=71.5549, which are obviously superior to the comparison models in terms of prediction accuracy and stability.

A multiobjective differential evolution algorithm with subpopulation region solution selection for global and local Pareto optimal sets

Current multimodal multiobjective optimization (MMO) has mostly focused on locating multiple equivalent global Pareto optimal sets (PSs) or local PSs in the decision space, rarely considering finding both simultaneously. To address this issue, this paper explores a crowded individual activation multiobjective differential evolution algorithm based on subpopulation region solution selection, named CAMODE_SR. In CAMODE_SR, a subpopulation region solution selection mechanism is developed to search for more uniformly distributed global and local Pareto optimal solutions. Specifically, two types of specific neighborhood radii are designed depending on the location information of the Pareto optimal solutions discovered during evolution. The two types of radii construct specific neighborhoods for the global and local optimal solutions to maintain the solutions with better diversity. To improve the exploitation capacity of the population, a crowded individual activation mechanism is proposed by developing an activation function associated with iteration. The activation function activates the crowded individuals in local regions to exploit more superior solutions near global optimal solutions. The proposed CAMODE_SR achieves a good tradeoff between the performance of locating both global and local PSs. Extensive experiments on the CEC 2020 MMO benchmark functions demonstrate that the proposed CAMODE_SR is significantly superior to nine state-of-the-art MMEAs in tackling multimodal multiobjective optimization problems with global and local PSs.

Gas system scheduling strategy for steel metallurgical process based on multi-objective differential evolution

Effective gas system scheduling is vital for ensuring the safe operation and energy efficiency of steel production. However, the dynamic and cyclical nature of gas generation and consumption leads to fluctuations in gas network pressure and gas holder levels. To address this issue, this paper proposes a novel strategy for gas system scheduling based on multi-objective differential evolution. The originality lies in fully considering the workload of operators, effectively reducing the workload while ensuring gas network stability. We analyze the composition of the gas system and design a scheduling scheme. Subsequently, a novel gas holder level model based on mechanism analysis is established, which calculates the gas holder level by analyzing the volume and pressure of the gas. Then, a gas system scheduling model with a new objective function based on a multi-objective differential evolution algorithm is provided. By comparing with the two recent methods, this method reduced the number of adjustments by 13.9% and 9.75%, respectively, effectively reducing the workload while ensuring gas system stability. Through statistical analysis for eight consecutive months, the results show that this method can reduce the gas emission rate from 1.34% to 0.88%, demonstrating its correctness and effectiveness.

Reinforcement learning-based multi-objective differential evolution for wind farm layout optimization

Wind farm layout optimization is a challenging issue which demands to discover some trade-off solutions considering various criteria, such as the power generated and the cost of the farm. Due to the complexity of the problem, we developed a reinforcement learning-based multi-objective differential evolution (RLMODE) algorithm to address the issue. In the developed algorithm, RL technique is applied to coordinate the parameter of DE algorithm, which can balance the local and global search. A tournament-based mutation operator is used to accelerate the convergence of the RLMODE algorithm. We tested the performance of the proposed RLMODE in two wind scenarios. The spread and spacing indicators of the algorithm are the best; the power generated by the solution from the RLMODE algorithm is the most when compared with some representative optimization algorithms and existing methods.

Space division and adaptive selection strategy based differential evolution algorithm for multi-objective satellite range scheduling problem

Satellite range scheduling always plays a crucial role in tracking, telemetry, and control of the spacecraft. With the significant increase in the number and type of satellites in orbit, the demands of users for satellite range scheduling are becoming more and more diverse. However, existing studies for the satellite range scheduling problem (SRSP) rarely consider multiple optimization objectives, which is quite significant in practice. Hence, this paper investigates a multi-objective satellite range scheduling problem (MO-SRSP) that optimizes three objectives simultaneously: the overall profits of tasks, load-balance of antennas, and completion timeliness of tasks. To address MO-SRSP, this paper establishes a mathematical model on the basis of analysis of MO-SRSP and proposes a multi-objective evolutionary algorithm, which is called multi-objective differential evolution algorithm based on space division and adaptive selection strategy (MODE-SDAS). The space division strategy will uniformly divide the objective space into a set of subspaces and preserve a set of non-dominated solutions in each subspace during the environmental selection even if some of these solutions are dominated by other solutions in other subspaces, so as to maintain the diversity of the algorithm. The adaptive selection strategy will adaptively allocate computational resources to different subspaces to improve the convergence of the population. Besides, problem-specific designs such as coding and encoding methods, the discrete differential evolution operator, as well as objective-specific individual variation operators are incorporated into the algorithm to enhance the search capability. Finally, extensive experiments based on simulation test cases are carried out to verify the effectiveness and efficiency of MODE-SDAS. The comparison results show that MODE-SDAS significantly outperforms its competitors in terms of three metrics. Meanwhile, knee point analysis, sensitivity analysis, and application insights are presented.

Two-Layered Hierarchical Optimization Strategy With Distributed Potential Game for Interconnected Hybrid Energy Systems.

Due to the existence of different stakeholders, it makes competitive game characteristic in hybrid energy systems (HESs). Combined with the high-dimensional complexity and output uncertainty of distributed energy resources, the optimal operation of HESs can be a more challenging problem. Here, this article proposes a potential game-based two-layered hierarchical optimization strategy to deal with this problem. With consideration of its high-dimensional complexity, a two-layered hierarchical HES model is created, consisting of an upper-level and a lower-level model. For properly solving competitive relationships among different stakeholders in the upper-level model, a multiagent system for stakeholders is created and a potential game is employed with a distributed primal-dual perturbed algorithm, and its convergence and optimality have been both proved. Moreover, an uncertainty and robustness analysis is done with coordination between lower and upper models, which deduces a feasible robust uncertainty interval in the lower-level model. For better dealing with the lower-level model, a gradient descent-based multiobjective differential evolution (GD-MODE) algorithm is utilized to optimize the economic cost and emission issue simultaneously, producing a set of Pareto-optimal schemes. Combined with simulation results, it is proven that the proposed method can reduce computational complexity as well as properly deal with uncertainty problems for the optimal operation of HESs.

Multiobjective Optimized Deployment of Edge-Enabled Wireless Visual Sensor Networks for Target Coverage

In wireless visual sensor networks, the generation and transmission of huge amounts of image data consume much energy of sensor nodes, and their routing and processing take quite a long time. It is of great importance to shorten event reporting delay and prolong network lifetime, which can be achieved by the appropriate deployment of edge nodes that can not only collect but also process data. This work investigates how to jointly optimize sensor node deployment, edge node deployment, data routing, and data offloading to minimize the number of deployed sensor nodes, the number of deployed edge nodes, and event reporting delay and maximize network lifetime. We formulate this problem as a mixed integer nonlinear program and propose a multiobjective differential evolution algorithm to solve it. A large number of simulation results demonstrate that it can deliver a more accurate Pareto set than the nondominated sorting genetic algorithm III.

A constrained multiobjective differential evolution algorithm based on the fusion of two rankings

The tradeoff between objective functions and constraints is a key issue that needs to be addressed by constrained multiobjective optimization algorithms, and constraint handling techniques (CHTs) are an important technique for balancing objective functions and constraints. In this paper, a novel CHT that fuses two rankings is proposed. Specifically, each individual is assigned two rankings: one ranking calculated based on Pareto dominance (regardless of constraints) and another calculated based on the constrained dominance principle (CDP). The fitness value of an individual is the weighted sum of these two rankings, and the weight is related to the generation number and the proportion of feasible solutions in the current generation. Based on the proposed CHT, a constrained multiobjective differential evolution algorithm is proposed. To generate high-quality offspring, the proposed constrained multiobjective differential evolution algorithm combines four mutation operations as core components of the search algorithm. The proposed algorithm is compared with eight state-of-the-art algorithms in experiments with five test suites, and the experimental results show that the proposed algorithm performs significantly better than the eight state-of-the-art algorithms.