Non-dominated Solutions Research Articles

Multi-objective optimization of truss structure using multi-agent reinforcement learning and graph representation

This paper presents a novel method for multi-objective optimization of truss design utilizing multi-agent reinforcement learning and graph representation. The agents are trained to modify solutions to increase the hypervolume and spread the distribution of the non-dominated solutions. This iterative modification is modeled as a Markov game where multiple agents interact with the environment by changing the state of the environment, i.e., the solution and the current non-dominated solutions. Agents observe data representations of the state, modify each solution in the current non-dominated solutions, and receive their reward based on the improvement of the current non-dominated solutions. Each agent is modeled by the multi-agent deep deterministic gradient consisting of a policy function that predicts actions from its observation data and a value function that estimates rewards from the observation data, its action, and the actions of the other agents. The non-dominated solutions are represented as graph data and observed by the agents as the observation data through graph representation. The proposed method is applied to three multi-objective optimization problems: (1) a simple mathematical problem, (2) a 10-bar truss problem to minimize the structural volume and the displacement, and (3) trade-off designs of trusses to minimize the structural volume and bring the shape closer to the target shape. These numerical examples show the versatility of the trained agents to obtain superior solutions in practical structural optimization problems, compared to a conventional method.

Multipopulation-Based Differential Evolution for Large-Scale Many-Objective Optimization.

In recent years, numerous efficient many-objective optimization evolutionary algorithms have been proposed to find well-converged and well-distributed nondominated optimal solutions. However, their scalability performance may deteriorate drastically to solve large-scale many-objective optimization problems (LSMaOPs). Encountering high-dimensional solution space with more than 100 decision variables, some of them may lose diversity and trap into local optima, while others may achieve poor convergence performance. This article proposes a multipopulation-based differential evolution algorithm, called LSMaODE, which can solve LSMaOPs efficiently and effectively. In order to exploit and explore the exponential decision space, the proposed algorithm divides the population into two groups of subpopulations, which are optimized with different strategies. First, the randomized coordinate descent technique is applied to 10% of individuals to exploit the decision variables independently. This subpopulation maintains diversity in the decision space to avoid premature convergence into local optimum. Second, the remaining 90% of individuals are optimized with the nondominated guided random interpolation strategy, which interpolates individual among three nondominated solutions randomly. The strategy can guide the population convergent toward the nondominated solutions quickly, meanwhile, maintain good distribution in objective space. Finally, the proposed LSMaODE is evaluated on the LSMOP test suites from the scalability in both decision and objective dimensions. The performance is compared against five state-of-the-art large-scale many-objective evolutionary algorithms. The experimental results show that LSMaODE provides highly competitive performance.

Density Function-Based Trust Region Algorithm for Approximating Pareto Front of Black-Box Multiobjective Optimization Problems

In this paper, we consider a black-box multiobjective optimization problem, whose objective functions are computationally expensive. We propose a density function-based trust region algorithm for approximating the Pareto front of this problem. At every iteration, we determine a trust region and then in this trust region, select several sample points, at which are evaluated objective function values. In order to obtain non-dominated solutions in the trust region, we convert given objective functions into one function: scalarization. Then, we construct quadratic models of this function and the objective functions. In current trust region, we find optimal solutions of all single-objective optimization problems with these models as objectives. After that, we remove dominated points from the set of obtained solutions. In order to estimate the distribution of non-dominated solutions, we introduce a density function. By using this density function, we obtain the most “isolated” point among the non-dominated points. Then, we construct a new trust region around this point and repeat the algorithm. We prove convergence of proposed algorithm under the several assumptions. Numerical results show that even in case of tri-objective optimization problems, the points generated by proposed algorithm are uniformly distributed over the Pareto front.

Many-objective optimization of BEV design parameters based on gradient boosting decision tree models and the NSGA-III algorithm considering the ambient temperature

In this study, extensive efforts have been devoted to optimizing the energy distribution and performance of a battery electric vehicle (BEV). An integrated simulation model based on energy flow test data is built and validated, and a parallel framework to implement automatic batch simulations is developed. On this basis, eXtreme Gradient Boosting (XGBoost), Light Gradient Boosting Machine (LightGBM) and Categorical Boosting (CatBoost) models of the BEV are established and compared, and many-objective optimization is carried out based on the Non-dominated Sorting Genetic Algorithm-III (NSGA-III) algorithm. The results indicate that the developed parallel framework can efficiently perform automatic batch computations of the integrated simulation model for the BEV, and the CatBoost model demonstrates superior prediction performance on both the train and test sets. A uniformly distributed non-dominated set approximating the Pareto Front (PF) is obtained by the many-objective optimization, with the fourth non-dominated solution exhibiting a favorable optimization effect. The electricity consumption per 100 km, half-axis effective work, electricity recovered by battery, energy utilization and recovery efficiency are improved by 9.4 %, 12.2 %, 6.4 %, 96.3 % and 16.0 %, respectively. These findings can provide the theoretical basis, directional guidance and data support for the accurate modeling, batch simulation and many-objective optimization of BEVs.

MaOEA/D with adaptive external population guided weight vector adjustment

In order to make multiobjective evolutionary algorithm based on decomposition (MOEA/D) have better performance in dealing with many-objective optimization problems (MaOPs) with different Pareto fronts (PFs), this paper proposes a MOEA/D with adaptive external population guided weight vector adjustment (MaOEA/D-AEW). The algorithm modifies the MOEA/D method of constructing mating pools by calculating the utility function value of individuals in the population to produce high-quality offspring. For the maintenance of non-dominated solutions in the external population (EP), solutions exceeding the capacity of the EP are removed by the method of minimum Euclidean distance combined with the aggregation function value to ensure the convergence and diversity of non-dominated solutions in the EP. For the addition of weight vectors, a new sparsity evaluating method (SL+) is proposed to determine the sparsity of weight vectors by multiplying parallel distances between individuals. For the deletion of the weight vector, the minimum Euclidean distance combined with the aggregation function value is used to delete the weight vector corresponding to invalid or crowded individuals in the population. Comparing MaOEA/D-AEW with six state-of-the-art algorithms on regular PF and irregular PF test problems respectively, simulation results show that the proposed algorithm can adapt well to MaOPs with different PFs and significantly outperforms other comparative algorithms in terms of convergence and diversity of solution sets.

Enhancing the resilience of zero-carbon energy communities: Leveraging network reconfiguration and effective load carrying capability quantification

The integration of distributed energy resources (DERs) into contemporary energy communities (ECs) has revolutionized power systems, fostering sustainable and clean energy infrastructures. This paper focuses on the effective load-carrying capability (ELCC) to enhance grid resilience in the presence of DERs. We introduce an innovative network topology-based optimization framework that seamlessly integrates economic and resilience metrics within ECs while reducing carbon emissions. The Pareto front of non-dominated solutions for the proposed three-objective optimization problems is extracted, providing a comprehensive visualization of the trade-off between economic, resilience, and emission objectives, enabling informed decision-making. Analytical results, validated on the IEEE 33-bus test system, demonstrate the effectiveness of DER-based ELCC quantification in managing load supply during emergencies. Case studies show how the synergy between economic and resilience-based metrics significantly enhances grid resilience. The proposed framework has diverse applications, including enhancing grid adaptability to climate change, promoting sustainable energy integration, optimizing demand response strategies, and supporting the transition to a decarbonized energy community. This work addresses the challenges and opportunities in the evolving energy landscape, emphasizing the importance of our approach in achieving a cleaner and more resilient energy future.

A new decomposition-based multi-objective symbiotic organism search algorithm for solving truss optimization problems

This study presents a comprehensive study on the potential and efficacy of decomposition-based multi-objective symbiotic organism search (MOSOS/D) algorithm for truss structure optimization. The investigation is carried out on five benchmark truss structures, e.g., 37-bar, 60-bar, 72-bar, 120-bar, and 200-bar truss problems. The optimization performance of MOSOS/D is compared with other established algorithms, such as Multi-Objective Evolutionary Algorithm Based on Decomposition, Non-Dominated Sorting Genetic Algorithm-II, Multi-Objective Equilibrium Optimizer, Multi-Objective Marine Predator Algorithm, Decomposition-Based Multi-Objective Heat Transfer Search, Multi-Objective Passing Vehicle Search, Multi-Objective Evolutionary Algorithm Based on Decomposition Differential Evaluation, Multi-Objective Symbiotic Organisms Search, and Multi-Objective Multi-Verse Optimizer considering several metrics, including Generational Distance, Spacing, Spread, Inverted Generational Distance, Hypervolume and Runtime. The findings demonstrate MOSOS/D’s competitiveness in providing non-dominated solutions with less space between them. It also achieves a good balance between convergence and coverage and, thus, delivers diverse solutions with less computational complexity. However, the efficiency of the MOSOS/D algorithm varies depending on the complexity of the considered truss structure, with some algorithms occasionally outperforming it in one or two aspects for specific benchmarks. This study provides valuable insights to engineers and designers aiming to achieve optimal truss structure design.

A multi-objective cooperation search algorithm for cascade reservoirs operation optimization considering power generation and ecological flows

With growing attention focused on energy generation and environmental conservation, the operation of cascade reservoirs that considers power generation benefits and ecological flow requirements plays an increasingly important role in water resource and power systems. However, the traditional optimization methods may fail to solve complex cascade reservoirs operation models with strong spatial-temporal coupling physical constraints. To effectively address this problem, this study proposes an efficient multi-objective cooperation search algorithm (MOCSA) that utilizes the modified team communication, reflective learning operator and internal competition operators to enhance global exploration and local exploitation. MOCSA is tested on a group of multi-objective benchmark functions and real-world constrained engineering problems. The results demonstrate that MOCSA can provide better results for most benchmark test functions in comparison with the competitive algorithms. Besides, MOCSA exhibits excellent search ability to find feasible solutions of constrained engineering problems. The proposed method is then applied to resolve a real-world reservoir system under different operation scenarios. Simulations indicate that MOCSA provides diversified decision options for the operation of cascade reservoir system and find a more diverse set of non-dominated solutions within the feasible solution space. Overall, this research presents MOCSA as an effective multi-objective optimization tool for complex cascade reservoir operation problems.

Multi-objective structural optimization for the automatic member grouping of truss structures using evolutionary algorithms

This paper aims to formulate the multi-objective structural optimization problem to find the best member grouping of truss structures. The weight of the structure and the different number of discrete cross-sectional areas are the conflicting objective functions to be minimized simultaneously, generating a Pareto front presenting the non-dominated solutions. Sixteen multi-objective evolutionary algorithms are adopted to solve the proposed optimization problems. Several benchmark problems and a new proposed optimization problem are analyzed. Even though original groupings based on symmetry are defined, it is possible to discover other competitive structural member configurations that can interest decision-makers regarding manufacturing, cutting, transportation, checking, and welding.

The multiobjective traveling salesman–repairman problem with profits: design and implementation of a variable neighborhood descent algorithm for a real scenario

AbstractThis paper introduces a problem that can be seen as a combination of the traveling salesman problem with profits and the traveling repairman problem with profits, coined as the multi‐objective traveling salesman–repairman problem with profits (Mo‐TSRPP). The objective of the Mo‐TSRPP is to simultaneously optimize three objectives: the total cost, total latency, and total profit. Indirectly, the number of nodes visited is also considered although not as an objective itself since it is determined by the size of every efficient solution in the Pareto front. The Mo‐TSRPP emerges as a real‐world problem in which a freelancer, which repairs appliances, wants to plan the daily route. Moreover, the daily plan does not require to visit all customers. To solve the problem, first, a greedy randomized adaptive procedure is designed to generate a set of high‐quality nondominated solutions and then, a variable neighborhood descent algorithm is applied for further improving the initial set. This procedure allows us to attain a good approximation of the Pareto front. To prove the performance of the proposal a comparison is done against three well‐known evolutionary algorithms: NSGA‐II, SPEA‐2, and MOEA/D. Finally, a realistic problem is shown and solved to illustrate the potential of the algorithm.

Collaborative Multiobjective Evolutionary Algorithms in the Search of Better Pareto Fronts: An Application to Trading Systems

Technical indicators use graphic representations of datasets by applying various mathematical formulas to financial time series of prices. These formulas comprise a set of rules and parameters whose values are not necessarily known and depend on many factors, such as the market in which they operate, the size of the time window, and so on. This paper focuses on the real-time optimization of the parameters applied for analyzing time series of data. In particular, we optimize the parameters of some technical financial indicators. We propose the combination of several Multiobjective Evolutionary Algorithms. Unlike other approaches, this paper applies a set of different Multiobjective Evolutionary Algorithms, collaborating to construct a global Pareto Set of solutions. Solutions for financial problems seek high returns with minimal risk. The optimization process is continuous and occurs at the same frequency as the investment time interval. This technique permits the application of the non-dominated solutions obtained with different MOEAs at the same time. Experimental results show that Collaborative Multiobjective Evolutionary Algorithms obtain up to 22% of profit and increase the returns of the commonly used Buy and Hold strategy and other multi-objective strategies, even for daily operations.

The Best Compromise Solution for Multi-objective Programming Problems

This paper focuses on multi-objective optimization problems that are an important part of operations research. This part is concerned with mathematical optimization problems involving two or more interdependent objectives to be optimized simultaneously. Thus, there is not a single optimal solution for multi-objective problems, but a set of solutions that represents the compromise (Pareto-optimal, efficient, non-dominated, trade-off, or non-inferior) solutions and can be visualized as Pareto front in the objective space. The best solution of this set has the shortest distance to the ideal (utopian) solution, whereas the ideal solution optimizes all objective functions, which often cannot be found. The main contribution of this paper is to introduce some methods to find the best compromise solution. These methods depend on new calculations for the normal of objectives. They can help to reduce the overall computational distance of the searching process. Therefore, they are flexible and stable. Besides, some numerical examples are presented to demonstrate the effectiveness of proposed methods with discussing their similarities and differences. The experimental results show that the proposed methods are effective and efficient for many different multi-objective (convex and non-convex) problems.

Greenhouse environmental control target constrained by discrete surface curvature and multi-objective optimization algorithm

Protected greenhouse provides a suitable micro-environment for crop growth, which enhances the quality and yield of agricultural products. How to optimize the greenhouse environment is currently a pressing issue in protected agriculture. In this research, we propose a new environmental control method from the perspectives of crop photosynthesis rate and environmental control cost. The photosynthesis rate data are collected from a nested experiment of tomatoes, and these data are combined with the support vector regression algorithm to establish a photosynthesis rate model. The cost function of greenhouse environmental control is established by theoretical calculation. The non-dominated solution sets of photosynthesis rate and control cost are obtained through an improved non-dominated sorting genetic algorithm Ⅱ. To determine the optimal solution among the numerous Pareto solutions, the geometric characteristics of the photosynthetic response surface are considered to constrain the control target. Then, a discrete surface curvature method is proposed to computer the constraint region. The intersection between the Pareto solutions and the constraint region is regarded as the environmental control target. The validation experiment demonstrates that the proposed method can effectively enhance photosynthetic accumulation and reduce control cost. Compared to the traditional saturation method, this method achieves a 53% cost reduction while still obtaining 89% photosynthetic accumulation. Compared to the U-chord method, this method increases photosynthetic accumulation by 12% with only a 3% increase in cost. The research has important implications for protected agriculture production.

A Pareto-based hybrid genetic simulated annealing algorithm for multi-objective hybrid production line balancing problem considering disassembly and assembly

ABSTRACT Most existing studies about line balancing problems mainly focus on disassembly and assembly separately, which rarely integrate these two modes into a system. However, as critical activities in the remanufacturing field, assembly and disassembly share many similarities, such as working tools and processing sequence. Thus, this paper proposes a multi-objective hybrid production line balancing problem with a fixed number of workstations (HPLBP-FNW) considering disassembly and assembly to optimise cycle time, total cost, and workload smoothness simultaneously. And a novel Pareto-based hybrid genetic simulated annealing algorithm (PB-HGSA) is designed to solve it. In PB-HGSA, the two-point crossover and hybrid mutation operator are proposed to produce potential non-dominated solutions (NDSs). Then, a local search method based on a parallel simulated annealing algorithm is designed for providing a depth search around the NDSs to balance the global and local search ability. Numerical results by comparing PB-HGSA with the well-known algorithms verify the effectiveness of PB-HGSA in solving HPLBP-FNW. Moreover, the managerial insights based on a case study are given to inspire enterprise companies to consider hybrid production line in the remanufacturing process, which is beneficial to reduce the cycle time and total cost and improve the service life of the equipment.

Interpretable self-organizing map assisted interactive multi-criteria decision-making following Pareto-Race

The problem-solving task of the multi-criteria decision-making (MCDM) approach involves decision makers’ (DMs’) interaction by incorporating their preferences to arrive at one or more preferred near Pareto-optimal solution(s). The Pareto-Race is an interactive MCDM approach that relies on finding preferred solutions as the DM navigates on the Pareto surface by selecting the preferred reference direction and uniform/varying speed using the concept of Achievement Scalarization Function (ASF) or its augmented version (AASF). Selecting a reference direction or speed is possible, algorithmically, but the effectiveness of the Pareto-Race concept relies on suitable visualization methods that present the series of past solutions and convey the current objective trade-off information to assist the DM’s in decision-making task. Traditional multi-dimensional Pareto-optimal front visualization methods like parallel coordinate plots, radial visualization, and heat maps are deemed insufficient for this purpose. Therefore, the paper proposes integrating the concept of the Pareto-Race MCDM approach with a recently developed interpretable self-organizing map (iSOM) based visualization method. The iSOM maps high-dimensional data into lower-dimensional space, facilitating visual convenience for the DM. The proposed method requires a finite-size representation of non-dominated solutions near the complete Pareto Front to generate iSOM plots of objectives. We also propose visualizing the metrics such as closeness to constraint boundaries, trade-off value, and robustness using iSOM to check the quality of the preferred solutions, which is rarely considered in existing MCDM approaches. iSOM plots of objective functions, closeness to constraint, trade-off, and robustness metrics assist the DM in effectively choosing new reference direction and step size in each iteration to arrive at the most preferred solution. The proposed iSOM-enabled Pareto-Race approach is demonstrated on benchmark analytical and real-world engineering examples with 3 to 10 objectives.

Enhanced Genetic Method for Optimizing Multiple Sequence Alignment

In the realm of bioinformatics, Multiple Sequence Alignment (MSA) is a pivotal technique used to optimize the alignment of multiple biological sequences, guided by specific scoring criteria. Existing approaches addressing the MSA challenge tend to specialize in distinct biological features, leading to variability in alignment outcomes for the same set of sequences. Consequently, this paper proposes an enhanced evolutionary-based approach that simplifies the sequence alignment problem without considering the sequences in the non-dominated solution. Our method employs a multi-objective optimization technique that uniquely excludes non-dominated solution sets, effectively mitigating computational complexities. Utilizing the Sum of Pairs and the Total Conserved Column as primary objective functions, our approach offers a novel perspective. We adopt an integer coding approach to enhance the computational efficiency, representing chromosomes with sets of integers during the alignment process. Using the SABmark and BAliBASE datasets, extensive experimentation is conducted to compare our method with existing ones. The results affirm the superior solution quality achieved by our approach compared to its predecessors. Furthermore, via the Wilcoxon signed-rank test, a statistical analysis underscores the statistical significance of our model’s improvement (p < 0.05). This comprehensive approach holds promise for advancing Multiple Sequence Alignment in bioinformatics.

An Approach for Predicting Social, Environmental, and Economic Product Impacts and Characterizing the Associated Sustainability Tradespace in Engineering Design

Abstract There is a growing demand for sustainable products and systems. Sustainability encompasses environmental, social, and economic aspects, often referred to as the three pillars of sustainability. To make more sustainable design decisions, engineers need tools to predict the environmental, social, and economic impacts of products and characterize potential sustainability trade-offs. To predict the total impact of a product, the quantity of functional units of the product in society and impact of each product needs to be estimated. This article uses agent-based modeling (ABM), combined with tools such as life cycle assessment (LCA), to predict impacts across all three pillars of sustainability. Using the product impact results the multidimensional sustainability tradespace can be characterized. The approach described in this article is based on three main components for the predictive modeling of product impacts and the characterization of the sustainability trade space, i) ABM of product adoption, ii) the assessment of product impacts, and iii) an approach for the characterization of product sustainability trade-offs at the population level. The tradespace characterization uses a Pareto-based method presented visually to find the non-dominated solutions in the product impact space. To illustrate and describe how to use the method, a case study is presented that predicts the impact of residential solar panels in a region of the United States under various scenarios. The findings of the case study can help policy makers understand suitable implementation strategies for residential solar panels while considering the impact trade-offs involved.

Development of an assessment-based planting structure optimization model for mitigating agricultural greenhouse gas emissions

Optimization of crop structure is an efficient way to reduce greenhouse gas (GHGs) from agriculture production. However, carbon footprint have rarely been incorporated into previous planting structure optimization models due to the challenges of assessing the spatial and temporal distribution of agricultural carbon footprint for multiple crops in irrigated districts. In addition, previous planting structure models suffered from strong subjectivity in objective function determination, and the obtained non-dominated solution set offered difficulties to decision-makers in selecting specific implementation options. To fill such gaps, an integrated accounting-assessment-optimization-decision making (AAODM) approach was proposed, which remedies the shortcomings of previous crop planting structure optimization models in carbon footprint mitigation, and overcomes the subjectivity of objective function determination and the difficulty in selecting specific implementation options. Firstly, life cycle assessment (LCA) method was used to account for the multi-year agricultural carbon footprints of multiple crops in the irrigation district. The optimization objective functions of planting structure optimization models can then be determined based on the assessment method of carbon footprint influencing factors. Next, the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) was used to generate a non-dominated solution set of the optimization model. The optimal planting structure can be finally obtained based on decision making methods by determining the maximum harmonic mean (HM) and knee points (KPs) of the non-dominated solution set. The developed AAODM approach was then applied to a case study of agricultural crop management in Bayan Nur City, China. The results showed that the level of economic development was a key factor influencing the increase in carbon footprint in Bayan Nur City over the past 20 years. The regulation of the level of economic development would significantly influence the agricultural carbon footprint in Bayan Nur City. Moreover, two optimal crop cultivation patterns were provided for decision-makers by selecting solutions from the Pareto front with decision making methods. The comparison results with other methods showed that the solutions obtained by NSGA-II were superior to MOPSO in terms of carbon reduction. The developed AAODM approach for agricultural GHG mitigation could help agricultural production systems in achieving low carbon emissions and high efficiency.

MULTI-RESOURCE ALLOCATION AND LEVELING IN MULTI-PROJECT SCHEDULING PROBLEM WITH HYBRID-CHROMOSOME NON-DOMINATED SORTING GENETIC ALGORITHM II

Multi-resource allocation and leveling in multi-project (MR-AL-MP) scheduling refers to the attempt of producing a project schedule with minimum project duration and maximum resource utilization while complying with all precedence and resource availability constraints in a multi-project environment involving multiple resources. This study proposes a model that integrates both resource allocation and leveling models into a unified framework. This study develops a modified version of the Non-Dominated Sorting Genetic Algorithm II (NSGA-II), called Hybrid-Chromosome NSGA-II, as the optimization algorithm. For validation purposes, the performance of Hybrid-Chromosome NSGA-II is compared with two benchmark metaheuristic algorithms which are Multi-Objective Particle Swarm Optimization (MOPSO) and Multi-Objective Symbiotic Organisms Search (MOSOS) in optimizing a case study. It is shown that the proposed model and algorithm are able to produce a set of non-dominated solutions that represent the feasible trade-off relationships between the objectives. Furthermore, the Hybrid-Chromosome NSGA-II is superior to MOPSO and MOSOS in terms of the quality, spread, and diversity of the solutions.

Co-evolutionary and Elite learning-based bi-objective Poor and Rich Optimization algorithm for scheduling multiple workflows in the cloud

Cloud computing is a cost-effective environment for deploying large-scale scientific applications. However, multi-workflow scheduling has great challenge since users may request a series of applications with different Quality of Service (QoS) at the same time. In this paper, a Co-evolutionary and Elite learning-based bi-objective Poor and Rich Optimization algorithm (CE-PRO) is proposed for scheduling applications to minimize the makespan and cost of each workflow. First, an MPMO framework is combined with PRO to optimize two objectives by two populations, respectively for better balancing the search diversity and convergence speed, where each population is updated by an improved PRO, which adopts the middle-class sub-population and re-defines the update mechanism for rich individuals to enhance search diversity and reduce the possibility of falling into local optima. Second, to restrain each population focusing overly on its respective objective, a global information exchange pool is innovatively designed to save the non-dominated solutions ever found, which will be used back as the shared guiding solutions to foster inter-population communication and co-evolution during an evolutionary process. Third, a hybrid mutation-based Elite Enhancement Strategy (EES) is developed by introducing multiple scales of mutation operations into elite solutions alternatively and iteratively to exploit excellent individuals and explore more trade-off solutions. Extensive experiments are conducted on real world scientific workflows with different types and scales, and the experimental results demonstrate that in most cases, our proposed CE-PRO outperforms its peers in the number of obtained non-dominated solutions, and the solution diversity and quality as well. In particular, the dominance of CE-PRO is superior to its peers by at least 25.62%.