Dynamic Multi-objective Optimization Problems Research Articles

Scalable benchmarks and performance measures for dynamic multi-objective optimization

Dynamic multi-objective optimization problems (DMOPs) can be utilized to model certain real-world problems that have a dynamic nature. Algorithms for solving DMOPs can be evaluated and improved by comparing their performance on different benchmarks. However, some existing benchmarks for DMOPs have the limitation of non-uniform weights for decision variables. Additionally, dynamic many-objective optimization problems (DMaOPs) involve more than three objectives, but only a few existing benchmarks can be extended to accommodate DMaOPs. Furthermore, some existing performance measures for DMOPs may not effectively compare the relative performance differences between multiple algorithms or evaluate the search uniformity among different objectives. In this paper, we propose improvements to an existing benchmark for DMOPs by expanding the impact range of decision variables. Moreover, a benchmark framework that can be extended to accommodate DMaOPs is proposed, thus addressing a research gap between the optimization of DMOPs and DMaOPs. Additionally, a set of performance measures for DMOPs are proposed, which can evaluate the relative performance and search uniformity of multi-objective optimization algorithms. By comparing the performance of state-of-the-art and commonly used algorithms on test problems, we can gain a better understanding of the characteristics and strengths and weaknesses of the algorithms and test problems.

A Mahalanobis Distance-Based Approach for Dynamic Multiobjective Optimization With Stochastic Changes

In recent years, researchers have made significant progress in handling dynamic multi-objective optimization problems (DMOPs), particularly for environmental changes with predictable characteristics. However, little attention has been paid to DMOPs with stochastic changes. It may be difficult for existing dynamic multi-objective evolutionary algorithms (DMOEAs) to effectively handle this kind of DMOPs because most DMOEAs assume that environmental changes follow regular patterns and consecutive environments are similar. This paper presents a Mahalanobis Distance-based approach (MDA) to deal with DMOPs with stochastic changes. Specifically, we make an all-sided assessment of search environments via Mahalanobis distance on saved information to learn the relationship between the new environment and historical ones. Afterward, a change response strategy applies the learning to the new environment to accelerate the convergence and maintain the diversity of the population. Besides, the change degree is considered for all decision variables to alleviate the impact of stochastic changes on the evolving population. MDA has been tested on stochastic DMOPs with 2 to 4 objectives. The results show that MDA performs significantly better than the other latest algorithms in this paper, suggesting that MDA is effective for DMOPs with stochastic changes.

Dynamic Operation Optimization of Complex Industries Based on a Data-Driven Strategy

As industrial practices continue to evolve, complex process industries often exhibit characteristics such as multivariate correlation, dynamism, and nonlinearity, making traditional mechanism modeling inadequate in terms of addressing the intricacies of complex industrial problems. In recent years, with advancements in control theory and industrial practices, there has been a substantial increase in the volume of industrial data. Data-driven dynamic operation optimization techniques have emerged as effective solutions for handling complex industrial processes. By responding to dynamic environmental changes and utilizing advanced optimization algorithms, it is possible to achieve dynamic operational optimization in industrial processes, thereby reducing costs and emissions, improving efficiency, and increasing productivity. This correlates nicely with the goals set forth by conventional process operation optimization theories. Nowadays, this dynamic, data-driven strategy has shown significant potential in complex process industries characterized by multivariate correlations and nonlinear behavior. This paper approaches the subject from a data-driven perspective by establishing dynamic optimization models for complex industries and reviewing the state-of-the-art time series forecasting models to cope with changing objective functions over time. Meanwhile, aiming at the problem of concept drift in time series, this paper summarizes new concept drift detection methods and introduces model update methods to solve this challenge. In addressing the problem of solving dynamic multi-objective optimization problems, the paper reviews recent developments in dynamic change detection and response methods while summarizing commonly used as well as the latest performance measures for dynamic multi-objective optimization problems. In conclusion, a discussion of the research progress and challenges in the relevant domains is undertaken, followed by the proposal of potential directions for future research. This review will help to deeply understand the importance and application prospects of data-driven dynamic operation optimization in complex industrial fields.

A dynamic multi-objective evolutionary algorithm based on genetic engineering and improved particle swarm prediction strategy

Evolutionary and particle swarm optimisation (PSO) algorithms are widely used to solve various optimisation problems. However, both methods have limitations when applied to dynamic multi-objective optimisation problems. Therefore, we propose a dynamic multi-objective evolutionary algorithm based on genetic engineering and improved particle swarm prediction. Genetic engineering encompasses gene sequencing and editing, whereas improved particle swarm prediction involves enhancing information utilisation and dynamic performance. The improved PSO can be better combined with dynamic multi-objective optimization problems (DMOPs) for predicting changing populations, which we call improved particle swarm prediction. Gene sequencing clarifies the functions of different genes during the evolution process, and gene editing enables the algorithm to effectively avoid local optima. Expanding on this foundation, we redefined the notions of particle and global bests in the PSO strategy to improve the information utilisation. Additionally, we innovatively introduced the concept of acceleration, which greatly enhances the dynamic performance of the PSO. Finally, we conducted two experiments to verify the accuracy of our gene sequencing strategy and the performance of our optimisation algorithm. The results indicate that gene sequencing can effectively classify genes, and the developed optimisation algorithm significantly outperforms similar, state-of-the-art, and classical algorithms in terms of distribution and dynamics on 25 test problems.

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

Dynamic multi-objective optimization problems (DMOPs) which contain various Pareto-optimal front (PF) and Pareto-optimal set (PS) have gained much attention. Accurate environmental change detection reveals the change degree of DMOPs and contributes the algorithm to quickly respond to the environment changes. In order to fully detect environmental changes and efficiently track front, a double-space environmental change detection and response strategy (DSDRS) is proposed. It could detect whether the environment has changed while explore the change intensity of PF and PS, respectively. Moreover, different response strategies are implemented for PF and PS. For PF environmental changes, a multiple knee points-guided evolutionary strategy (MKGES) is proposed, which is driven by front shape information and adaptively responds to different PF change intensities. For PS environmental changes, a knowledge guided memory strategy (KGMS) is proposed, which guides population evolution based on environmental information. The effectiveness of DSDRS is confirmed by comparison with five evolutionary algorithms on 20 dynamic multiobjective benchmark functions. Simulation results demonstrate that the performance of proposed algorithm is outstanding on test functions with complex changing PF and PS.

A Cooperative Evolutionary Computation Algorithm for Dynamic Multiobjective Multi-AUV Path Planning

Multiple autonomous underwater vehicles (AUVs) are popular for executing submarine missions, which involve multiple targets distributed in a large and complex underwater environment. The path planning of multiple AUVs is a significant and challenging problem, which determines the location of surface points for AUV launch and plans the paths of AUVs for target traveling. Most existing works model the problem as a single-objective static optimization problem. However, the target missions may change over time, and multiple optimization objectives are usually expected for decision making. Thus, this paper models the problem as a dynamic multiobjective optimization problem and proposes a cooperative evolutionary computation algorithm to provide diverse and high-quality solutions for decision makers. In the proposed method, solutions are represented using a bi-layer encode scheme, in which the first layer indicates the surface location points and the second layer represents the traveling sequences of target missions. Multiple populations for multiple objectives (MPMO) framework is adopted to efficiently solve the dynamic multiobjective AUV optimization problem. In addition, a recombination-based sampling strategy is developed to improve convergence by fusing the information of multiple populations. Once a change occurs, an incremental response strategy is adopted to generate high-quality solutions for population evolution. Based on the dataset of New Zealand bathymetry, six complex underwater scenarios are constructed with a size of 50 km×50 km×10 km and 400 target missions for tests. Experimental results show that the proposed method outperforms the state-of-the-art algorithms in terms of solution diversity and optimality.

Learning to Guide Particle Search for Dynamic Multiobjective Optimization.

Dynamic multiobjective optimization problems (DMOPs) are characterized by multiple objectives that change over time in varying environments. More specifically, environmental changes can be described as various dynamics. However, it is difficult for existing dynamic multiobjective algorithms (DMOAs) to handle DMOPs due to their inability to learn in different environments to guide the search. Besides, solving DMOPs is typically an online task, requiring low computational cost of a DMOA. To address the above challenges, we propose a particle search guidance network (PSGN), capable of directing individuals' search actions, including learning target selection and acceleration coefficient control. PSGN can learn the actions that should be taken in each environment through rewarding or punishing the network by reinforcement learning. Thus, PSGN is capable of tackling DMOPs of various dynamics. Additionally, we efficiently adjust PSGN hidden nodes and update the output weights in an incremental learning way, enabling PSGN to direct particle search at a low computational cost. We compare the proposed PSGN with seven state-of-the-art algorithms, and the excellent performance of PSGN verifies that it can handle DMOPs of various dynamics in a computationally very efficient way.

A dynamic multi-objective evolutionary algorithm using center and multi-direction prediction strategies.

Dynamic multi-objective optimization problems have been popular because of its extensive application. The difficulty of solving the problem focuses on the moving PS as well as PF dynamically. A large number of efficient strategies have been put forward to deal with such problems by speeding up convergence and keeping diversity. Prediction strategy is a common method which is widely used in dynamic optimization environment. However, how to increase the efficiency of prediction is always a key but difficult issue. In this paper, a new prediction model is designed by using the rank sums of individuals, and the position difference of individuals in the previous two adjacent environments is defined to identify the present change type. The proposed prediction strategy depends on environment change types. In order to show the effectiveness of the proposed algorithm, the comparison is carried out with five state-of-the-art approaches on 20 benchmark instances of dynamic multi-objective problems. The experimental results indicate the proposed algorithm can get good convergence and distribution in dynamic environments.

A Novel Dynamic Multiobjective Optimization Algorithm With Non-Inductive Transfer Learning Based on Multi-Strategy Adaptive Selection.

In this article, a novel multi-strategy adaptive selection-based dynamic multiobjective optimization algorithm (MSAS-DMOA) is proposed, which adopts the non-inductive transfer learning (TL) paradigm to solve dynamic multiobjective optimization problems (DMOPs). In particular, based on a scoring system that evaluates environmental changes, the source domain is adaptively constructed with several optional groups to enrich the knowledge. Along with a group of guide solutions, the importance of historical experiences is estimated via the kernel mean matching (KMM) method, which avoids designing strategies to label individuals. The proposed MSAS-DMOA is comprehensively evaluated on 14 DMOPs, and the results show an overwhelming performance improvement in terms of both convergence and diversity as compared with other four popular DMOAs. In addition, ablation studies are also conducted to validate the superiority of the applied strategies in MSAS-DMOA, which can effectively alleviate the negative transfer phenomenon. Without the conventional labeling procedure, the proposed method also yields satisfactory results, which can provide valuable reference for designing other evolutionary transfer optimization (ETO) algorithms.

Dynamic Multi-Objective Evolutionary Algorithms: An Overview

Evolutionary algorithms have been widely explored and applied in optimization problems. The introduction of multi-objective evolutionary algorithms (MOEAs) has facilitated the adaptation and creation of new methods to handle more complex and realistic optimizations, such as dynamic multi-objective optimization problems (DMOPs). A dynamic MOEA (DMOEA) can be constructed by changing the MOEA structure and variation operators used to solve DMOPs. Furthermore, DMOEAs can implement change-detection strategies and mechanisms to handle the dynamics of the environment. DMOEAs are often designed to solve unconstrained DMOPs. However, several studies have been conducted to solve dynamic constrained MOPs and more recently, to solve dynamic many-objective optimization problems. There are many real-world DMOPs; however, few DMOEAs have been evaluated with respect to those problems. Evaluation of algorithm performance is an essential aspect of using DMOEAs. Several measures used to validate MOEAs have been adapted to assess DMOEAs and quantitatively compare them. This paper presents a broad review of DMOEAs. The information is summarized and organized according to current research branches. Furthermore, a more general classification scheme for DMOEAs is proposed, based on the method of incorporating diversity to solve DMOPs.

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

Dynamic multi-objective optimization problems (DMOPs) require an algorithm to track the true Pareto-optimal front (POF) or Pareto-optimal set (POS) quickly and accurately when the environment suffers changes. However, many existing prediction-based methods ignore the association of individual movement directions, resulting in biased predictions and misleading the subsequent search process. To address this problem, a fuzzy-guided adaptive algorithm with hierarchy mechanism for solving DMOPs (FGAHM) is proposed, where a membership function is used to map movement vectors of the center point and individuals to the fuzzy sets for measuring the association. Thus, entire population is adaptively divided into three hierarchies, high, medium, and low based on the fuzzy relative entropy (FRE). Highly associated subpopulation is committed to leading the movement direction of the population through the linear prediction model. Medially associated subpopulation is dedicated to correcting the movement direction of the population and estimating changes in POS and POF by the multiview prediction. Lowly associated subpopulation is discarded, replaced by the t-test sampling to promote the population convergence and diversity. Although three subpopulations carry out different optimization purposes, they also synergistically contribute to the evolution of the entire population. Systematic experiments demonstrate that the performance of the proposed FGAHM outperforms seven state-of-the-art algorithms on 27 benchmark problems with 10 decision variables and 2 or 3 objectives, as evaluated by inverted generational distance (IGD), mean inverted generational distance (MIGD), DMIGD, mean hypervolume (MHV), and mean Schott's spacing (MSP).

A framework based on generational and environmental response strategies for dynamic multi-objective optimization

Due to the dynamics and uncertainty of the dynamic multi-objective optimization problems (DMOPs), it is difficult for algorithms to find a satisfactory solution set before the next environmental change, especially for some complex environments. One reason may be that the information in the environmental static stage cannot be used well in the traditional framework. In this paper, a novel framework based on generational and environmental response strategies (FGERS) is proposed, in which response strategies are run both in the environmental change stage and the environmental static stage to obtain population evolution information of those both stages. Unlike in the traditional framework, response strategies are only run in the environmental change stage. For simplicity, the feed-forward center point strategy was chosen to be the response strategy in the novel dynamic framework (FGERS-CPS). FGERS-CPS is not only to predict change trend of the optimum solution set in the environmental change stage, but to predict the evolution trend of the population after several generations in the environmental static stage. Together with the feed-forward center point strategy, a simple memory strategy and adaptive diversity maintenance strategy were used to form the complete FGERS-CPS. On 13 DMOPs with various characteristics, FGERS-CPS was compared with four classical response strategies in the traditional framework. Experimental results show that FGERS-CPS is effective for DMOPs.

A Knowledge Guided Transfer Strategy for Evolutionary Dynamic Multiobjective Optimization

The key task in dynamic multiobjective optimization problems (DMOPs) is to find Pareto-optima closer to the true one as soon as possible once a new environment occurs. Previous dynamic multiobjective evolutionary algorithms (DMOEAs) normally focus on DMOPs with regular environmental changes, but neglect widespread random one, limiting their applications in real-world fields. To address this issue, a knowledge guided transfer strategy based DMOEA is proposed in the paper. First, knowledge described as a two-tuple is extracted under each historical environment and preserved to a knowledge pool. Redundant knowledge is recognized and adaptively removed so as to guarantee the diversity of the pool. Second, a knowledge matching strategy is developed to re-evaluate the representative of each stored knowledge under a new environment, with the purpose of finding the most valuable one to promote positive knowledge transfer. Third, an improved knowledge transfer mechanism based on subspace alignment is introduced. By integrating it with knowledge reuse mechanism, a hybrid transfer strategy is constructed to adaptively select the most suitable one in terms of the similarity degree of selected knowledge on the current environment, and then generate a new initial population. Experiments on 20 benchmark problems demonstrate that the knowledge guided transfer strategy outperforms five state-of-the-art algorithms, achieving good versatility in solving DMOPs with both regular and random changes.

Interindividual Correlation and Dimension-Based Dual Learning for Dynamic Multiobjective Optimization

Dynamic multiobjective optimization problems (DMOPs) are characterized by their multiple objectives, constraints, and parameters that may change over time. The challenge in solving DMOPs is how to track the varying Pareto optimal solution sets quickly and accurately. Therefore, an inter-individual correlation and dimension-based dual learning method is proposed in this paper. Two learning strategies, decomposition-based inter-individual correlation transfer learning (DICTL) and dimension-wise learning (DL), are developed to respectively generate one-half of the initial population in the new environment. More specifically, DICTL learns the inter-individual correlation from the final population of the adjacent environment and then transfers it to the new environment, aiming to maintain the diversity and distribution of the predicted population. While DL extracts the changing pattern of dynamic environments from the high-quality solutions of historical environments in the perspective of variable dimension, trying to improve the quality of the population and accelerate the convergence. The designed two learning strategies (DICTL&DL) work complementarily and collaboratively to make the algorithm adapt to dynamic environments better and faster. Comprehensive experiments have been conducted by comparing the proposed method with four state-of-the-art algorithms on 14 benchmark problems. The results demonstrate the superiority of the proposed method.

Domain Generalization-Based Dynamic Multiobjective Optimization: A Case Study on Disassembly Line Balancing

The objective of disassembly lines is to disassemble End-of-Life products in a remanufacturing field. The disassembly line balancing problem (DLBP) considers how to allocate disassembly operations to operators on the disassembly line to optimize predetermined goals, such as cycle time. In practice, various environmental uncertainties (e.g., uncertain product quality) exist in the disassembly line. These uncertainties entail DLBP essentially a dynamic multi-objective optimization problem. This study presents a dynamic DLBP (D-DLB) to model the effect of environmental uncertainties on the assignment of disassembly operations. Furthermore, a prediction-based dynamic optimization algorithm, termed domain generalization-based dynamic multi-objective evolutionary algorithm (DG-DMOEA), combining meta-learning with multi-objective optimization, is proposed to solve D-DLB. In DG-DMOEA, a meta-learning algorithm is employed to learn the parameters of a solution-generative model from the Pareto-optimal sets in all historical environments. Subsequently, the solution-generative model is applied to generate a high-quality initial population that can assist multi-objective optimization algorithms in finding the Pareto-optimal set in the new environment faster. Since no information in the new environment is required, learning can begin before the new environment arrives, significantly reducing computational time. Moreover, different solution-generative models can be designed for different DMOPs. Therefore, DG-DMOEA can thoroughly combine real-world problem properties to represent knowledge. The experimental results show that, compared with state-of-the-art methods, DG-DMOEA can considerably improve the quality of solutions and significantly enhance the ability to react quickly to environmental changes.

Dynamic adaptive multi-objective optimization algorithm based on type detection

Dynamic multi-objective optimization problems (DMOPs) are multi-objective problems that are influenced by dynamically changing environmental parameters. Most current algorithms for solving DMOPs only respond to dynamic changes in the decision space or objective space and also ignore the impact of the type of DMOPs on the algorithm. The changes in the Pareto-optimal solution (POS) and Pareto-optimal front (POF) may affect the type of change in DMOPs. Therefore, this paper proposed an adaptive dynamic multi-objective evolutionary algorithm for type detection (TDA-DMOEA). First, the dynamic detection operator is designed to identify the types of dynamic problems. The Wilcoxon signed-rank test and Hyper Volume (HV) are used to detect the difference of POS and POF in two adjacent environments respectively. Then, different response strategies are designed to cope with different types of changes in DMOP. In particular, a multi-angle-based transfer learning method (MA-TL) with a closed kernel function is derived when faced with simultaneous changes in POS and POF. Finally, a comprehensive study of the commonly used benchmark set of DMOPs is presented, and the proposed algorithm achieves better performance in optimizing DMOPs.

Model based optimization of fertilization with treated wastewater reuse

Treated wastewater contains vital crop nutrients and offers the possibility for not only irrigation but also fertilization from a renewable source. The feasibility of replacing conventional chemical fertilizers with wastewater reuse is investigated, focusing on the implications of coupling irrigation and fertilization. We consider a multi-objective dynamic optimization problem of crop irrigation and fertigation with treated wastewater, with the objectives of maximizing crop production and minimizing environmental costs. A double modeling method is proposed that relies on both a modern complex crop model – the simulation model – and a low-order dynamical systems model – the control model – to solve efficiently the optimization problem, whilst maintaining a detailed representation of the system of study. Through a case study, we show that it is possible to achieve high levels of crop production. The maximum nutrient concentration of the reclaimed waters CNmax is a key parameter of the system performance: when CNmax is high enough, it is possible to substitute entirely conventional fertilization for irrigation with wastewater. However, when CNmax is small, our results highlight that meeting crop needs would require excessive irrigation, with the added risk of dilution of the nutrients already present in soil. We find that there is a maximum amount of nutrients that can be delivered efficiently to the soil-crop system and identify the conditions when wastewater irrigation is only be capable of providing a portion of the crop needs and should be complimented with other sources of fertilization. The double modeling method and the control model provide a means to estimate the key quantities to determine the best strategy to adopt for wastewater fertilization, as well as to obtain efficient and simple controls, that could be applied in practice.

A Correlation-Guided Layered Prediction Approach for Evolutionary Dynamic Multiobjective Optimization

When solving dynamic multiobjective optimization problems by evolutionary algorithms, the historical moving directions of some special points along the Pareto front, such as the center and knee points, are widely employed to predict the Pareto-optimal solutions (POS). However, special points may be impacted by certain individuals with a large direction deviation, and thus mislead the tracking of dynamic POS. To solve this issue, a correlation-guided layered prediction approach for solving dynamic multiobjective optimization problems is proposed in this paper, where multiple prediction models are integrated by considering the correlation of individuals’ moving directions. To be specific, the population is clustered into three subpopulations (i.e., high, mid, and low correlation) by correlation analysis to perform different prediction behaviors. The high correlation subpopulation aims to predict the moving direction via a linear prediction model. The mid correlation subpopulation is devoted to predicting the manifold change of POS by self-adaptively using the direction and length correction models. The diversity preservation is considered by the low correlation subpopulation. While the three subpopulations focus on different optimization tasks, they also cooperate to track the dynamic POS. The comprehensive experimental results on a variety of benchmark test problems demonstrate the superiority of the proposed approach, as compared with some state-of-the-art prediction-based dynamic multiobjective algorithms.

A novel knee-guided algorithm based on frequency analysis for non-cyclic dynamic multiobjective optimization problems

In real life, many dynamic multiobjective optimization problems (DMOPs) have both time-varying and non-cyclic properties. Their non-cyclic nature differentiates the dynamics of the problem from all previous moments, and the changes become increasingly complex over time. Therefore, geometric changes in the Pareto-optimal front (PF) are more difficult to track. A novel knee-guided algorithm based on frequency analysis (NKGFA) is designed to solve non-cyclic DMOPs. To identify sufficient feature points to describe the change in PF, a knee-guided prediction strategy based on frequency analysis is proposed. From the perspective of frequency-domain analysis, almost all concave and convex knee points that contain critical feature information on the PF are identified. By predicting the frequency-domain information of the knees, the population’s evolutionary trend is obtained in the frequency-domain space. An interpolation strategy based on approximate frontier manifold construction is proposed to ensure that the population is uniformly distributed on the PF. This strategy established an approximate PF between concave and convex knee points. Moreover, it adaptively interpolated the new individuals in the approximate PF. A non-cyclic dynamic (NCD) benchmark test suite is designed, including PFs with increasingly complex geometric features. The effectiveness of the NKGFA is verified by comparing it with four algorithms in the NCD test suite. The NKGFA performed best on 26, 23, and 29 of the 39 test problems on Inverted Generational Distance (IGD), Schott’s Spacing Metric (SP), and Hypervolume Difference (HVD), respectively.

Dynamic multi-objective workflow scheduling for combined resources in cloud

Cloud resource providers offer idle resources to users as spot instances. The price of the instances changes with market supply and demand, and the dynamic price can have a significant impact on workflow scheduling. In this work, we use a combination of spot and on-demand instances as the foundation cloud resource and characterize the dynamic workflow scheduling problem as a dynamic multi-objective optimization problem (DMOP), where the dynamics originate from the dynamic price of spot instances. The scheduling solution is found by considering three objectives: maximizing the reliability of the instances while minimizing the makespan and cost. In addition, we provide an enhanced MOEAD algorithm called MOEA/D-URDI that combines diversity introduction and uniform random sampling, where the uniform random sampling paradigm is used to generate the initial weight vector. The dynamic multi-objective optimization evolutionary algorithm DMOEA/D-URDI is then created by combining the method with a dynamic optimization framework. Our technique beats existing algorithms, according to experimental data based on dynamic benchmark sets and three well-known scientific procedures in terms of metrics on dynamic benchmark sets and better ensures reliability in scheduling scientific workflows while reducing makespan and cost.