Multi-objective Test Functions Research Articles

Optimal design of electromagnetic repulsion mechanism based on hybrid infill criterion and kriging surrogate model

Abstract In the optimization design of the electromagnetic repulsion mechanism in finite element simulation, there are problems of high computational cost and long optimization time. To shorten the development cycle of the prototype and improve the efficiency of optimization, an optimization algorithm based on the Hybrid Infill Criterion and Kriging Surrogate Model (HIC-KSM) is proposed by organically combining the Expected Improvement (EI) infill criterion and the Pareto front infill criterion to make full use of the performance of parallel computation and to accelerate the convergence speed of the algorithm. The proposed algorithm is compared with the other optimization algorithms, the optimization results of five multi-objective test functions are compared to verify the efficiency of the proposed algorithm, and the algorithm is applied to the parameter optimization of the electromagnetic repulsion mechanism.

MBB-MOGWO: Modified Boltzmann-Based Multi-Objective Grey Wolf Optimizer.

The primary objective of multi-objective optimization techniques is to identify optimal solutions within the context of conflicting objective functions. While the multi-objective gray wolf optimization (MOGWO) algorithm has been widely adopted for its superior performance in solving multi-objective optimization problems, it tends to encounter challenges such as local optima and slow convergence in the later stages of optimization. To address these issues, we propose a Modified Boltzmann-Based MOGWO, referred to as MBB-MOGWO. The performance of the proposed algorithm is evaluated on multiple multi-objective test functions. Experimental results demonstrate that MBB-MOGWO exhibits rapid convergence and a reduced likelihood of being trapped in local optima. Furthermore, in the context of the Internet of Things (IoT), the quality of web service composition significantly impacts complexities related to sensor resource scheduling. To showcase the optimization capabilities of MBB-MOGWO in real-world scenarios, the algorithm is applied to address a Multi-Objective Problem (MOP) within the domain of web service composition, utilizing real data records from the QWS dataset. Comparative analyses with four representative algorithms reveal distinct advantages of our MBB-MOGWO-based method, particularly in terms of solution precision for web service composition. The solutions obtained through our method demonstrate higher fitness and improved service quality.

Design-space adaptation method for multiobjective and multidisciplinary optimization

This paper developed a new method that adaptively adjusts a design space by considering the actual solution distribution of a problem to overcome the conventional design-space adaptation method that assumes the solutions distribution to be a normal distribution because the distributions of solutions are rarely normal distributions for real-world problems. The developed method was applied to nineteen multiobjective test functions that are widely used to evaluate the characteristics and performance of optimization approaches. The results showed that this method adapted the design space to an appropriate design space where the solution existence probability was high. The optimization performance achieved using the developed method was higher than that of the conventional methods. Furthermore, the developed method was applied to the conceptual design of an unmanned spacecraft to confirm its validity in real-world design and multidisciplinary-optimization problems. The results showed that the Pareto solutions of the developed method were superior to those of conventional methods. Additionally, the optimization efficiency with the developed method was improved by more than 1.4 times over that of the conventional methods. In this regard, the developed method has the potential to be applied to complicated real-world optimization problems to achieve better performance and efficiency.

Multi-objective high-dimensional multi-fractional-order optimization algorithm for multi-objective high-dimensional multi-fractional-order optimization controller parameters of doubly-fed induction generator-based wind turbines

Aiming at the operation control of doubly-fed induction generator-wind energy systems, this study firstly introduces multi-dimensional information feedback and fractional-order theory and designs a high-dimensional multi-fractional-order controller in the rotor side converter. Secondly, for the tuning optimization of controller parameters of doubly-fed induction generators, this study proposes a high-dimensional multi-fractional-order optimization. Finally, this study proposes a multi-objective high-dimensional multi-fractional-order optimization algorithm for the multi-objective optimization problems of control parameters tuning and optimization. This study adopts an adaptive grid strategy to update non-dominated solutions for external archives with crowded conditions. Furthermore, this study adopts a roulette strategy to select the optimal global position in the iterative process. This study utilizes multi-objective test functions to verify the effectiveness of the proposed multi-objective optimization. The Pareto optimal solutions and five types of statistical data from two indexes show that the proposed multi-objective optimization has improved convergence, diversity, and operational stability. Compared with five algorithms, the proposed multi-objective optimization can obtain Pareto optimal solutions with uniform distribution and good diversity in the multi-objective tuning of control parameters of the designed multi-objective controllers for doubly-fed induction generators. Therefore, decision-makers can choose the control parameters according to the actual needs. The test results show that the designed multi-objective controller optimized by the proposed multi-objective optimization has more accurate power tracking ability and superior control performance.

Multiobjective Differential Evolution With Speciation for Constrained Multimodal Multiobjective Optimization

This paper proposes a novel differential evolution algorithm for solving constrained multimodal multiobjective optimization problems (CMMOPs), which may have multiple feasible Pareto optimal solutions with identical objective vectors. In CMMOPs, due to the coexistence of multimodality and constraints, it is difficult for current algorithms to perform well in both objective and decision spaces. The proposed algorithm uses speciation mechanism to induce niches preserving more feasible Pareto optimal solutions and adopts an improved environment selection criterion to enhance diversity. The algorithm can not only obtain feasible solutions but also retain more well-distributed feasible Pareto optimal solutions. Moreover, a set of constrained multimodal multiobjective test functions is developed. All these test functions have multimodal characteristics and contain multiple constraints. Meanwhile, this paper proposes a new indicator, which comprehensively considers the feasibility, convergence, and diversity of a solution set. The effectiveness of the proposed method is verified by comparing with the state-of-the-art algorithms on both test functions and real-world location-selection problem.

A novel hybrid wind speed prediction framework based on multi-strategy improved optimizer and new data pre-processing system with feedback mechanism

As a kind of renewable energy, wind energy has great potential for development and has been paid attention to by governments all over the world. However, due to the high uncertainty of wind speed, how to accurately predict wind speed and make use of wind energy has been recognized as a difficult problem. In order to solve this problem, a new hybrid wind speed prediction framework is proposed, which is composed of two subsystems, data preprocessing system and high-accuracy prediction system. In the system 1, the feedback mechanism is creatively added to the singular spectrum analysis (SSA) to find out the optimal decomposition-recombination strategy through the accuracy feedback. In the system 2, the unconstrained weighting mechanism is realized through the combination of combined neural network and multi-objective optimization algorithm to maximize the prediction accuracy on the premise of ensuring the stability of prediction. Besides, an improved meta-heuristic optimization algorithm based on cross-perturbation strategy (CP-JAYA) and its multi-objective form (MO-CPJAYA) are applied on two systems respectively to further improve the prediction ability of the framework. In 5 groups of experiments, the accuracy, advancement, generalization and sensitivity of the model are tested and compared with 13 other models. The proposed prediction framework has the best performance in all four sets of data. In 3 groups of discussions, we verify the advanced nature of CP-JAYA and MO-CPJAYA respectively through 13 single-objective test functions (CEC) and 4 multi-objective test functions (ZDT), and the speed advantage of the framework by recording the CPU running time.

An enhanced multiobjective bacterial foraging algorithm for the compaction parameter optimization of earth-rock dams

Compaction parameters can act as key decision variables. However, there is a lack of research on the optimization of compaction parameters considering quality, time, and costs. Existing algorithms have difficulty coping with this highly complex and nonlinear problem. To address this issue, an enhanced multiobjective bacterial foraging algorithm is proposed for searching the optimal compaction parameters of earth-rock dams. The cost and time are taken as multiple objectives, and the compaction quality is considered the main constraint because it must meet a certain standard. Then, the enhanced multiobjective bacteria foraging algorithm (EMOBFA) is proposed, which integrates multiple intelligent components into the fundamental single-objective bacterial foraging algorithm, including chaotic mapping, an adaptive step, a rotation learning strategy, and quantum computing. This further improves the speed and accuracy of the state-of-the-art multiobjective optimization algorithm. Finally, multiobjective test functions are used to compare the EMOBFA with commonly used multiobjective optimization algorithms to verify its generality, power, and scalability. A real-life large-scale water conservancy project in Southwest China shows that the proposed EMOBFA has realized the optimization of compaction parameters in the prior control of rolling construction.

Multi-objective pity beetle algorithm based optimal control of wastewater treatment process

Water is a critically scarce resource for industrial production and social life. The discharge of untreated wastewater into natural waters can pose a serious risk to human health. Most of urban wastewater treatment plants use biochemical methods, the most common of which is the biological reaction through activated sludge to degrade pollutants. Considering the public environmental protection and socio-economic needs, this paper establishes a wastewater treatment process (WWTP) optimization model considering the trade-off between effluent quality (EQ) and energy consumption (EC), and designs an dynamic optimal control framework based on a novel multi-objective evolutionary algorithm (MOEA) to track and control the key variables in the WWTP. The numerical experiments of multi-objective test functions show that the proposed MOEA has good convergence and distributivity. Simulation results based on the BSM1 platform show that the constructed framework can accurately adjust the set-points of controllers in time to improve the performance, which has broad application prospects in practical applications.

An application of adaptive normalization evolutionary optimization ANMOGA for missile fin design based on trajectory parameters

System design is a complicated and iterative process. Accordingly, a robust optimization procedure should be involved during the design process to minimize complications and time consumed for the whole process. The present research proposes a new methodology to design missile fins that can lessen the gap between the conceptual and final phases during the missile design process. The proposed methodology is based on a new dynamic adaptive multi-objective optimization algorithm accompanied by a selection scheme to choose a single optimum design amongst a Pareto set of solutions. The adaptive normalization multi-objective evolutionary algorithm can be used in any system design. This algorithm is capable of predicting Pareto fronts for multi-objective optimization problems at very good dispersion. In addition, a selection scheme is provided to help decision-makers choose a single optimum amongst Pareto front points. The proposed algorithm is tested against several multi-objective test functions as well as the application of a selection scheme to the obtained Pareto fronts. The results showed the robustness of the proposed algorithm in the prediction of convex fronts. However, it still needs more improvement to deal with non-convex ones. An application for the considered algorithm is also provided as a case study to design tail fins for a conventional missile configuration. The output configuration of the designed fins has very reasonable dimensions when compared to similar configurations.

Improved whale optimization algorithm and its application in vehicle structural crashworthiness

Whale optimization algorithm (WOA) is a novel-innovative swarm-based meta-heuristic algorithm with excellent performance, but it may still be trapped into local extremum for troublesome problems. To this end, an improved multi-objective whale optimization algorithm (IMOWOA) is proposed to cover the shortages. Firstly, in the search stage of WOA, individual difference is considered to strengthen the exploration ability, and evolution operators are introduced to regenerate the stagnated population to prevent premature convergence. Next, the performance of IMOWOA is compared with MOWOA and other classical optimization algorithms, and a series of multi-objective test functions are used. The results on the convergence and diversity of Pareto front confirm that IMOWOA has better feasibility and competitiveness. Finally, integrated with the least squares support vector regression (LSSVR) model, IMOWOA is applied to the deterministic optimization of vehicle structural crashworthiness. The conclusion testified the efficiency of IMOWOA in the field of vehicle crashworthiness.

An adaptive disturbance multi-objective evolutionary algorithm based on decomposition

In solving multi-objective optimisation problems, the uniformly distributed weight vector of decomposition based multi-objective evolutionary algorithm (MOEA/D) is not completely suitable for the non-uniformly distributed Pareto front (PF). In order to solve the situation above, this paper proposes an adaptive disturbance multi-objective evolutionary algorithm based on decomposition (AD-MOEA/D), which introduces the disturbance individuals and disturbance weight vectors during the evolution. The disturbance individuals maintain the population diversity and improve convergence accuracy. The disturbance weight vectors assist the weight vectors to adjust adaptively and improve the distribution of PF. Besides, both disturbance individuals and disturbance weight vectors are produced according to the actual evolution, which will not participate in evolution when it is not necessary. The experimental results on multi-objective test functions show that the PF optimised by AD-MOEA/D has better convergence and distribution.

A novel grey prediction evolution algorithm for multimodal multiobjective optimization

In practical applications, solving multimodal multiobjective optimization problems (MMOPs), which have multiple Pareto optimal sets (PSs) in the decision space mapping to the same Pareto front (PF) in the objective space, has great significance for decision makers. The issue of how to maintain diversity in both decision space and objective space remains a key problem for existing multimodal multiobjective evolutionary algorithms. To address this issue, a novel grey prediction evolution algorithm for multimodal multiobjective optimization, termed MMGPE, is proposed in this paper. This is the first time that the grey prediction evolution algorithm (GPE), which is a recently proposed competitive optimization algorithm with strong exploration capability, is improved for MMOPs. These improvements are conducted in the following four aspects: (1) an initialization operator based on particle swarm optimization, (2) an adaptive parameter setting strategy depending on the domain of decision variables of MMOPs, (3) an accelerating convergence mechanism inspired by the niche principle, and (4) an environmental selection operator based on a nondominated sorting mechanism and a special crowding distance approach. The performance of MMGPE is compared with two state-of-the-art multiobjective evolutionary algorithms and four multimodal multiobjective evolutionary algorithms on 11 multimodal multiobjective test functions. MMGPE is also applied to solve a practical problem. The results show the MMGPE’s effectiveness and superiority in achieving the goal of finding multiple PSs while obtaining a well-distributed PF.

Design optimization of metamaterial units using a genetic algorithm based optimization methodology

Purpose The purpose of this paper is to develop a pertinent design optimization methodology for symmetric designs of a metamaterial (MM) unit. Design/methodology/approach A cell division mechanism is introduced and used to design a new selecting mechanism in the proposed algorithm, a non-dominated sorting cellular genetic algorithm (NSCGA). Findings The numerical results on solving standard multi-objective test functions and a prototype MM unit positively demonstrate the advantages of the proposed NSCGA. Originality/value A new NSGAII-based optimization algorithm, NSCGA, for multi-objective optimization designs of a MM unit is proposed.

A novel Whale Optimization Algorithm integrated with Nelder–Mead simplex for multi-objective optimization problems

Recently, several meta-heuristics and evolutionary algorithms have been proposed for tackling optimization problems. Such methods tend to suffer from degraded performance when solving multi-objective optimization problems due to addressing the conflicting goals of finding accurate estimation of Pareto optimal solutions and increasing their distribution across all objectives. In this paper, the Whale Optimization Algorithm (WOA) is improved and extended to solve such multi-objective optimization problems with the purpose of alleviating these drawbacks. The improvements include: (1) modifying the distance control factor of the standard WOA to contain values generated dynamically instead of a fixed one, (2) the trade-off between moving toward the opposite of the best solution and its original values based on a certain probability to prevent stuck into local minima, and (3) accelerating the convergence and coverage using Nelder–Mead method and the Pareto Archived Evolution Strategy (PAES). The proposed algorithm is tested on three benchmark multi-objective test functions (DTLZ, CEC 2009, and GLT), including 25 test functions, to verify its effectiveness by comparing with nine robust multi-objective algorithms. The experiments demonstrate the superiority of the proposed algorithm compared to some of the existing multi-objective algorithms in the literature.

Rethinking the differential evolution algorithm

Selection operation plays a significant role in differential evolution algorithm. A new differential evolution algorithm based on an improved selection process is presented in this work. It was studied that there was neither a practical method to maintain the distribution of population nor a correction to the variables out of bounds in mutation process in a standard differential evolution algorithm. The fast non-dominated sorting approach and the spatial distance algorithm which were applied to the beginning of the selection process, as well as a method to fix the transboundary variables in the mutation process, were adopted to optimize the differential evolution algorithm. The reformative algorithm could obtain a uniformly distributed and effective Pareto-optimal sets when applied to the classical multi-objective test functions; it performed prominently in the experiment of optimizing the quality, the cost and the time in a construction project compared with the previous work.

Multi-Verse Algorithm based Approach for Multi-criteria Path Planning of Unmanned Aerial Vehicles

In this paper, a method based on a Multiobjective Multi-Verse Optimizer (MOMVO) is proposed and successfully implemented to solve the unmanned aerial vehicles’ path planning problem. The generation of each coordinate of the aircraft is reformulated as a multiobjective optimization problem under operational constraints. The shortest and smoothest path by avoiding all obstacles and threats is the solution of such a hard optimization problem. A set of competitive metaheuristics such as Multiobjective Salp Swarm Algorithm (MSSA), Grey Wolf Optimizer (MOGWO), Particle Swarm Optimization (MOPSO) and Non-dominated Sorting Genetic Algorithm II (NSGA-II) are retained as comparison tools for the problem’s resolution. To assess the performance of the reported algorithms and conclude about their effectiveness, an empirical study is firstly performed for solving different multiobjective test functions from the literature. These algorithms are then used to obtain a set of optimal Pareto solutions for the multi-criteria path planning problem. An efficient Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) of Multi Criteria Decision-Making (MCDM) model is investigated to find the optimal solution from the non-dominant ones. Demonstrative results and statistical analysis are presented and compared in order to show the effectiveness of the proposed MOMVO-based path planning technique.

A Multi-objective Improved Squirrel Search Algorithm based on Decomposition with External Population and Adaptive Weight Vectors Adjustment

In order to further improve the performance of the Multi-objective Evolutionary Algorithm Based on Decomposition (MOEA/D) in solving multi-objective optimization problems, this paper constructs a multi-objective optimization algorithm by taking MOEA/D as the multi-objective framework and the Squirrel Search Algorithm (SSA) as the core evolutionary strategy. Besides, both of them are modified and the Multi-objective Improved Squirrel Search Algorithm based on Decomposition with External Population and Adaptive Weight Vectors Adjustment (MOEA/D-EWA-ISSA) is proposed. MOEA/D-EWA-ISSA establishes an external population for every individual to retain the evolutionary information and maintain the population diversity, external individuals participate in evolution and produce better offspring, the convergence and distribution of Pareto Front (PF) are improved. As for SSA, the jumping search method and the progressive search method are introduced to it, different evolutionary strategies are provided to solve subproblems, which further improves the ability of core evolutionary strategy to solve subproblems and the convergence of the obtained PF. Furthermore, MOEA/D-EWA-ISSA adjusts every weight vector adaptively according to the population’s actual evolutionary direction and the representative neighbor weight vectors, the distribution of the obtained PF is improved as well. The experimental results on multi-objective test functions show that the convergence and distribution of PF have obvious improvement when the improved SSA, the improved MOEA/D and the whole MOEA/D-EWA-ISSA are used to solve multi-objective optimization problems.

A Pareto-Based Hybrid Whale Optimization Algorithm with Tabu Search for Multi-Objective Optimization

Multi-Objective Problems (MOPs) are common real-life problems that can be found in different fields, such as bioinformatics and scheduling. Pareto Optimization (PO) is a popular method for solving MOPs, which optimizes all objectives simultaneously. It provides an effective way to evaluate the quality of multi-objective solutions. Swarm Intelligence (SI) methods are population-based methods that generate multiple solutions to the problem, providing SI methods suitable for MOP solutions. SI methods have certain drawbacks when applied to MOPs, such as swarm leader selection and obtaining evenly distributed solutions over solution space. Whale Optimization Algorithm (WOA) is a recent SI method. In this paper, we propose combining WOA with Tabu Search (TS) for MOPs (MOWOATS). MOWOATS uses TS to store non-dominated solutions in elite lists to guide swarm members, which overcomes the swarm leader selection problem. MOWOATS employs crossover in both intensification and diversification phases to improve diversity of the population. MOWOATS proposes a new diversification step to eliminate the need for local search methods. MOWOATS has been tested over different benchmark multi-objective test functions, such as CEC2009, ZDT, and DTLZ. Results present the efficiency of MOWOATS in finding solutions near Pareto front and evenly distributed over solution space.

Epsilon-multiobjective particle swarm optimization-based tuning of sensitivity functions for polynomial control design

In this paper, a novel systematic method for the polynomial controllers (denoted as RST controllers) design and tuning is proposed and successfully implemented based on an epsilon-multiobjective particle swarm optimization (ε-MOPSO) algorithm. The ε-domination concept is used to further improve both properties of the non-premature convergence towards Pareto-optimal sets and the diversity among the found solutions. The RST polynomials coefficients’ computation is formulated as a constrained multiobjective optimization problem under operational frequency-domain constraints. The proposed methodology aims to optimize a desired output sensitivity function satisfying nominal performance and H∞ robustness specifications of the closed-loop system. The use of a suitable fixed parts of the optimized output sensitivity function will provide partial poles placement of the closed-loop dynamics. The inverse of such an optimized desired sensitivity function defines the performance and robustness H∞ weighting filter. Such a proposed ε-MOPSO algorithm is firstly compared with several homologous MOPSO, non-dominated sorting genetic algorithm II (NSGA-II) and multiobjective differential evolution (MODE) algorithms for a benchmark of multiobjective test functions. The algorithms’ performances are evaluated in terms of several metrics in order to show the superiority and the effectiveness of the proposed method. An application to the axis position control of a flexible transmission system with varying loads is then performed. Demonstrative simulations are carried out to show the remarkable superiority and effectiveness of the proposed ε-MOPSO-tuned digital RST controllers compared with several well-known methods available in the literature.

Multimodal multiobjective optimization with differential evolution

Abstract This paper proposes a multimodal multiobjective Differential Evolution optimization algorithm (MMODE). The technique is conceived for deployment on problems with a Pareto multimodality, where the Pareto set comprises multiple disjoint subsets, all of which map to the same Pareto front. A new contribution is the formulation of a decision-variable preselection scheme that promotes diversity of solutions in both the decision and objective space. A new mutation-bound process is also introduced as a supplement to a classical mutation scheme in Differential Evolution methods, where offspring that lie outside the search bounds are given a second opportunity to mutate, hence reducing the density of individuals on the boundaries of the search space. New multimodal multiobjective test functions are designed, along with analytical expressions for their Pareto sets and fronts. Some test functions introduce more complicated Pareto-front shapes and allow for decision-space dimensions greater than two. The performance of the MMODE algorithm is compared with five other state-of-the-art methods. The results show that MMODE realizes superior performance by finding more and better distributed Pareto solutions.