Inverted Generational Distance Research Articles

Multi-objective scheduling for an energy-efficient flexible job shop problem with peak power constraint

Controlling electricity costs, which are closely related to the peak power reached, while simultaneously reducing completion time and energy consumption, is crucial for achieving sustainability in the manufacturing industry. However, most existing research on energy-efficient flexible job shop scheduling problems primarily focuses on optimizing completion time and energy consumption, often neglecting the critical aspect of controlling electricity costs. With decreasing resources and increasing energy prices, it is necessary to control electricity costs by limiting the peak power reached during production. To address this gap, this paper investigates an energy-efficient flexible job shop scheduling problem with peak power constraint considering setup and transportation time (EEFJSSP-PPST). We establish a mathematical model for EEFJSSP-PPST and propose an improved non-dominated sorting genetic algorithm II (INSGA-II) incorporating several key improvements. We first introduce three scheduling rules in decoding to optimize objectives and employ heuristic rules to generate a high-quality initial population. Then, we propose an innovative cluster crossover method to accelerate convergence and design an adaptive-based local optimization strategy to further enhance performance. Finally, our experiments explore the impacts of different degrees of the peak power constraint on objectives and evaluate the effectiveness of the three scheduling rules. Additionally, the performance of INSGA-II is tested using 135 benchmark instances. The results indicate that INSGA-II outperformed its variations without improvements, achieving the best hypervolume (HV) indicator in 80% of the instances and the best inverted generational distance (IGD) indicator in 68.89% of the instances. Compared to other state-of-the-art algorithms, INSGA-II achieves the best HV and IGD indicators in 82.2% and 81.5% of the instances, respectively, and demonstrates better convergence and a superior Pareto front. Therefore, we conclude that the improvements in INSGA-II are effective and enhance the algorithm’s performance, making it outstanding in solving EEFJSSP-PPST.

Modeling Approach of Cloud 4D Printing Service Composition Optimization Based on Non-Dominated Sorting Genetic Algorithm III

The manufacturing industry is currently experiencing a paradigm shift from traditional centralized systems to distributed, personalized, and cloud-based intelligent manufacturing ecosystems. The advent of 4-dimensional (4D) printing technology introduces dynamic characteristics to manufacturing design and functionality, necessitating the effective management of these emergent 4D printing services. This study aims to bridge the gap between the static nature of existing cloud manufacturing services and the dynamic requirements imposed by 4D printing technology. We propose a comprehensive multiobjective optimization model for cloud-based 4D printing service portfolios, incorporating the intricate complexities of 4D printing services and assessing the efficacy of the Non-Dominated Sorting Genetic Algorithm III (NSGA III) in optimizing these service portfolios to meet dynamic demands. In this research, the NSGA III algorithm is employed to develop a multiobjective optimization framework for 4D printing service portfolios, addressing critical issues such as service cost, time, quality, adaptability, and overall service optimization amidst fluctuating demand and service availability. The findings indicate that the NSGA III algorithm demonstrates superior performance in terms of generational distance (GD) and inverted generational distance (IGD), particularly excelling in convergence and diversity for high-dimensional optimization problems when compared to the comparison algorithms. The study concludes that the NSGA III algorithm exhibits significant potential in optimizing the orchestration of cloud-based 4D printing service portfolios, underscoring its effectiveness in managing the complexities associated with these services. This research provides valuable insights for the advancement of intelligent cloud-based 4D printing systems, paving the way for future developments in this field.

Multi-guided population co-evolutionary algorithm based on multiple similarity decomposition for large-scale flexible job shop scheduling problem

As manufacturing shifts towards large-scale production, the size of the workshop increases, and its search space exponentially expands. It is difficult for existing algorithms to obtain an ideal scheduling solution in an acceptable time. For the large-scale flexible job shop scheduling problem (LSFJSP), a multi-guided population co-evolutionary algorithm based on multiple similarity decomposition (MPCSD) is designed. Faced with the problem of high dimensionality and complex solution space, a multiple similarity decomposition strategy is proposed. It proceeds to group based on similarity information at the dimension and population level. To obtain convergence-preferred and diversity-preferred dimension groupings, a training-set solution selection method is proposed. Inspired by the idea of divide-and-conquer, a multi-guided co-evolutionary strategy is proposed. It improves the exploration efficiency of the algorithm in the search space. To test effectiveness on more complex LSFJSP, a set of large-scale test problems including LS1-12 are designed. On LS1-12, MPCSD is compared with seven other algorithms to demonstrate its superiority. MPCSD performed best on 11, 7, and 10 of the 12 test problems on Inverted Generational Distance (IGD), Hypervolume (HV) and Schott’s Spacing Metric (SP), respectively. Meanwhile, the Relative Deviation (RD) results showed that MPCSD obtained the best fitness performance.

Trajectory Optimization to Enhance Observability for Bearing-Only Target Localization and Sensor Bias Calibration.

This study addresses the challenge of bearing-only target localization with sensor bias contamination. To enhance the system's observability, inspired by plant phototropism, we propose a control barrier function (CBF)-based method for UAV motion planning. The rank criterion provides only qualitative observability results. We employ the condition number for a quantitative analysis, identifying key influencing factors. After that, a multi-objective, nonlinear optimization problem for UAV trajectory planning is formulated and solved using the proposed Nonlinear Constrained Multi-Objective Gray Wolf Optimization Algorithm (NCMOGWOA). Simulations validate our approach, showing a threefold reduction in the condition number, significantly enhancing observability. The algorithm outperforms others in terms of localization accuracy and convergence, achieving the lowest Generational Distance (GD) (7.3442) and Inverted Generational Distance (IGD) (8.4577) metrics. Additionally, we explore the effects of the CBF attenuation rates and initial flight path angles.

The IGD-based prediction strategy for dynamic multi-objective optimization

In recent years, an increasing number of prediction-based strategies have shown promising results in handling dynamic multi-objective optimization problems (DMOPs), and prediction models are also considered to be very favorable. Nevertheless, some linear prediction models may not always be effective. In particular, when the motion direction trends of different individuals are not aligned, these models can yield inaccurate prediction results. Inverted generational distance (IGD) is a commonly used metric for evaluating the performance of algorithms. This paper proposes a prediction model based on the IGD metric. Specifically, we assume that the pareto optimal front (POF) of the population at the previous time step is the true POF, and the POF at the current time step is the approximate POF. We cluster the current population with reference to the euclidean distances from uniform points on the true POF to the current POF points, with slight overlap between adjacent clusters, enables a better tradeoff between convergence and diversity in the prediction process. We consider the movement directions of individuals within each cluster separately through different cluster distributions, while balancing the individual movement directions and the overall population movement direction by overlaying cluster coverage areas, thereby helping to avoid the clustered prediction population from getting trapped in local optima. Experimental results and comparisons with other algorithms demonstrate that this strategy exhibits strong competitiveness in handling DMOPs.

An OGSM-based multi-objective optimization model for partner selection in fresh produce supply chain considering carbon emissions

Incorporating partner selection with environmental commitment is recognized as a crucial aspect of green development in fresh produce supply chains. This paper presents a multi-objective programming (MOP) model for partner selection considering carbon emissions based on the ‘Objectives, Goals, Strategies, Measures’ (OGSM) framework. The OGSM serves as a strategic scheme to identify the long-term green development vision of a supply chain, ensuring that the activities of supply chain partners are aligned with common goals. Driven by the OGSM design, the MOP model is formulated with three conflicting objective functions to maximize profit, minimize carbon emission, and maximize the quality of fresh produce. The model helps decision-makers address decisions on partner selection with determined logistics volumes, transportation modes and packaging types under carbon emission constraints. To tackle the NP-hard problem, we develop an effective hybrid meta-heuristic algorithm called ɛ-NSGA, which complements the advantages of the ɛ-constraint method and NSGA-II (Non-dominated Sorting Genetic Algorithm-II). The experimental results based on parameter calibration of the Taguchi method and the Kruskal–Wallis test demonstrate that the proposed ɛ-NSGA outperforms other four heuristic algorithms in terms of the performance indices of GD (Generational Distance), spacing, IGD (Inverted Generational Distance), and HV (Hyper Volume). The implementation of the OGSM-MOP together with ɛ-NSGA in a real-world apple supply chain illustrates the effectiveness and practicality of the model. Finally, managerial insights into the utilization of the model are discussed and suggestions for decision-making to fulfill corporate environmental obligations are provided.

Multi-objective generalized normal distribution optimization: a novel algorithm for multi-objective problems

This study introduces the Multi-objective Generalized Normal Distribution Optimization (MOGNDO) algorithm, an advancement of the Generalized Normal Distribution Optimization (GNDO) algorithm, now adapted for multi-objective optimization tasks. The GNDO algorithm, previously known for its effectiveness in single-objective optimization, has been enhanced with two key features for multi-objective optimization. The first is the addition of an archival mechanism to store non-dominated Pareto optimal solutions, ensuring a detailed record of the best outcomes. The second enhancement is a new leader selection mechanism, designed to strategically identify and select the best solutions from the archive to guide the optimization process. This enhancement positions MOGNDO as a cutting-edge solution in multi-objective optimization, setting a new benchmark for evaluating its performance against leading algorithms in the field. The algorithm's effectiveness is rigorously tested across 35 varied case studies, encompassing both mathematical and engineering challenges, and benchmarked against prominent algorithms like MOPSO, MOGWO, MOHHO, MSSA, MOALO, MOMVO, and MOAOS. Utilizing metrics such as Generational Distance (GD), Inverted Generational Distance (IGD), and Maximum Spread (MS), the study underscores MOGNDO's ability to produce Pareto fronts of high quality, marked by exceptional precision and diversity. The results affirm MOGNDO's superior performance and versatility, not only in theoretical tests but also in addressing complex real-world engineering problems, showcasing its high convergence and coverage capabilities. The source codes of the MOGNDO algorithm are publicly available at https://nimakhodadadi.com/algorithms-%2B-codes.

Multi-objective optimization of dual resource integrated scheduling problem of production equipment and RGVs considering conflict-free routing.

In flexible job shop scheduling problem (FJSP), the collision of bidirectional rail guided vehicles (RGVs) directly affects RGVs scheduling, and it is closely coupled with the allocation of production equipment, which directly affects the production efficiency. In this problem, taking minimizing the maximum completion time of RGVs and minimizing the maximum completion time of products as multi-objectives a dual-resource integrated scheduling model of production equipment and RGVs considering conflict-free routing problem (CFRP) is proposed. To solve the model, a multi-objective improved discrete grey wolf optimizer (MOID-GWO) is designed. Further, the performance of popular multi-objective evolutionary algorithms (MOEAs) such as NSGA-Ⅱ, SPEA2 and MOPSO are selected for comparative test. The results show that, among 42 instances of different scales designed, 37, 34 and 28 instances in MOID-GWO are superior to the comparison algorithms in metrics of generational distance (GD), inverted GD (IGD) and Spread, respectively. Moreover, in metric of Convergence and Diversity (CD), the Pareto frontier (PF) obtained by MOID-GWO is closer to the optimal solution. Finally, taking the production process of a construction machinery equipment component as an example, the validity and feasibility of the model and algorithm are verified.

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).

Improved multi objective particle swarm optimization based reactive power optimization for ensuring voltage security of power systems

In this study an improved multi objective particle swarm optimization (IMOPSO) algorithm is proposed for power system reactive power optimization with the objective of ensuring voltage security. The multi objective particle swarm optimization (MOPSO) is improved by introducing an adapted binary crossover (ABX) to the new positions obtained by the basic particle swarm optimization (PSO) algorithm. Additionally, diversity maintenance strategy is added to the algorithm by employing crowding distance (CD) calculation. The developed algorithm is tested and compared with standard MOPSO and non dominated sorting genetic algorithm (NASGA II). The comparison is made based on the degree of closeness to the true pareto front, as measured by the inverted generational distance (IGD), and based on diversity, as measured by the CDs . The test is made using ZDT1, ZDT2, and ZDT3 test functions. The IMOPSO showed improved performance over MOPSO and NASGA II algorithms in terms of convergence to the true pareto front (PF) and in terms of the speed of convergence as well as in maintaining diversity. The algorithm is then implemented to reactive power optimization of IEEE 14 bus test system. For the implementation purpose, the voltage stability and voltage deviation components of voltage security are formulated as a multi objective functions. The implementation has resulted diverse options of optimal settings of reactive power controlling parameters. The optimal settings proved to produce an improved voltage security as measured in terms of voltage deviation and voltage stability.

Identifying vulnerabilities of industrial control systems using evolutionary multiobjective optimisation

In this paper, we propose a novel methodology to assist in identifying vulnerabilities in real-world complex heterogeneous industrial control systems (ICS) using two Evolutionary Multiobjective Optimisation (EMO) algorithms, NSGA-II and SPEA2. Our approach is evaluated on a well-known benchmark chemical plant simulator, the Tennessee Eastman (TE) process model. We identified vulnerabilities in individual components of the TE model and then made use of these vulnerabilities to generate combinatorial attacks. The generated attacks were aimed at compromising the safety of the system and inflicting economic loss. Results were compared against random attacks, and the performance of the EMO algorithms was evaluated using hypervolume, spread, and inverted generational distance (IGD) metrics. A defence against these attacks in the form of a novel intrusion detection system was developed, using machine learning algorithms. The designed approach was further tested against the developed detection methods. The obtained results demonstrate that the developed EMO approach is a promising tool in the identification of the vulnerable components of ICS, and weaknesses of any existing detection systems in place to protect the system. The proposed approach can serve as a proactive defense tool for control and security engineers to identify and prioritise vulnerabilities in the system. The approach can be employed to design resilient control strategies and test the effectiveness of security mechanisms, both in the design stage and during the operational phase of the system.

English Speech Emotion Classification Based on Multi-Objective Differential Evolution

Speech signals involve speakers’ emotional states and language information, which is very important for human–computer interaction that recognizes speakers’ emotions. Feature selection is a common method for improving recognition accuracy. In this paper, we propose a multi-objective optimization method based on differential evolution (MODE-NSF) that maximizes recognition accuracy and minimizes the number of selected features (NSF). First, the Mel-frequency cepstral coefficient (MFCC) features and pitch features are extracted from speech signals. Then, the proposed algorithm implements feature selection where the NSF guides the initialization, crossover, and mutation of the algorithm. We used four English speech emotion datasets, and K-nearest neighbor (KNN) and random forest (RF) classifiers to validate the performance of the proposed algorithm. The results illustrate that MODE-NSF is superior to other multi-objective algorithms in terms of the hypervolume (HV), inverted generational distance (IGD), Pareto optimal solutions, and running time. MODE-NSF achieved an accuracy of 49% using eNTERFACE05, 53% using the Ryerson audio-visual database of emotional speech and song (RAVDESS), 76% using Surrey audio-visual expressed emotion (SAVEE) database, and 98% using the Toronto emotional speech set (TESS). MODE-NSF obtained good recognition results, which provides a basis for the establishment of emotional models.

Blind Source Separation with Strength Pareto Evolutionary Algorithm 2 (SPEA2) Using Discrete Wavelet Transform

This paper presents a new method for separating the mixed audio signals of simultaneous speakers using Blind Source Separation (BSS). The separation of mixed signals is an important issue today. In order to obtain more efficient and superior source estimation performance, a new algorithm that solves the BSS problem with Multi-Objective Optimization (MOO) methods was developed in this study. In this direction, we tested the application of two methods. Firstly, the Discrete Wavelet Transform (DWT) was used to eliminate the limited aspects of the traditional methods used in BSS and the small coefficients in the signals. Afterwards, the BSS process was optimized with the multi-purpose Strength Pareto Evolutionary Algorithm 2 (SPEA2). Secondly, the Minkowski distance method was proposed for distance measurement by using density information in the discrimination of individuals with raw fitness values for the concept of Pareto dominance. With this proposed method, the originals (original source signals) were estimated by separating the randomly mixed male and two female speech signals. Simulation and experimental results proved that the efficiency and performance of the proposed method can effectively solve BSS problems. In addition, the Pareto front approximation performance of this method also confirmed that it is superior in the Inverted Generational Distance (IGD) indicator.

Dynamic niching particle swarm optimization with an external archive-guided mechanism for multimodal multi-objective optimization

Multimodal multi-objective optimization problems (MMOPs) contain multiple equivalent Pareto optimal sets (PSs) corresponding to the same Pareto front (PF). However, simultaneously locating well-distributed and well-converged multiple equivalent global PSs and PF remains challenging. Therefore, this paper proposes dynamic niching particle swarm optimization (PSO) with an external archive-guided (AG) mechanism, termed DNPSO-AG, for solving MMOPs. In DNPSO-AG, a clustering-based dynamic niching technique is integrated with PSO to divide the population into multiple niches. In addition, a leader updating method controls the updating of the leaders. Furthermore, a novel external archive-guided mechanism guides the evolution of multiple niches and enhances the distribution of solutions, which comprises two strategies: the adaptive division of the external archive strategy, which adaptively divides the external archive into multiple sub-archives, and the distance-based sub-archive and niche matching strategy, which assigns sub-archives to multiple niches for maintenance. The experimental results demonstrate that the proposed DNPSO-AG outperforms seven other state-of-the-art competitors on the CEC 2019 MMOP test suite in terms of the inverted generational distance (IGD) and IGD in the decision space (IGDX) metrics, with improvements of 21.3% and 9.1% over the best-performing competitor, respectively.

Linear LSA-NSGAII optimization: A case study in optimal switch placement in distribution network

In this paper, a new hybrid of lightning search algorithm (LSA) and non-dominated sorting genetic algorithm II (NSGAII) is proposed in order to increase the accuracy and efficiency of the conventional multi-objective NSGAII. In the proposed method, the important parameters of the NSGAII are optimized using LSA. To verify the effectiveness of the proposed algorithm, it is applied to 19 well-known standard test functions, including unconstrained functions, UF1–UF7, constrained functions, CF1–CF7, Schaffer1, Schaffer2, ZDT1, ZDT2, and ZDT3. The inverted generational distance (IGD) is used to check the efficiency of the proposed algorithm. Accordingly, the proposed algorithm is compared with the most used multi-objective algorithms. Besides, the proposed algorithm is applied to a practical case study of optimal switch, including reclosers and disconnectors, placement in the presence of distributed generation (DG) sources. The distribution networks are the Roy-Billinton test system (RBTS) and a part of the real network in Mazandaran province, in Iran. The simulation results confirm the superiority of the proposed method. For example, in practical case studies, the proposed algorithm for optimal recloser placement in the real network shows 47.09% and 4.691% cost improvement compared with multi-objective particle swarm optimization (MOPSO) and NSGAII, respectively. These improvements for RBTS-BUS2 are 3.725% and 3.502%, respectively. Also, for the RBTS-BUS2, the proposed algorithm result shows 1% and 0.732% reliability improvement compared with MOPSO and NSGAII, respectively.

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.

Adaptive stochastic fractal search algorithm for multi-objective optimization

Stochastic fractal search algorithm, a relatively efficient algorithm has certain advantages in solving multi-objective optimization problems. Decomposition-based strategy is effective for solving multi-objective optimization, which guarantees the diversity of solutions. On the other hand, a challenge for decomposition-based strategy is to enhance the algorithm’s convergence while maintaining the algorithm’s distribution. This paper proposes a reference vector-guided stochastic fractal search algorithm for multi-objective optimization(RVSFS). In the proposed algorithm, the leader solution of each subspace is selected by reference vectors to assist the generation of new solutions and ensure the convergence of the algorithm. A new selection criterion length–angle distance (LAD) is proposed. Furthermore, the Gaussian walk and Levy flight are conducted in different searching stages to balance the exploration and exploitation ability. To speed up convergence and maintain the diversity of solutions, an adaptive dynamic proportion mechanism is designed when using the random fractals and Gaussian walks to generate offspring. Since information is shared and exchanged among different solutions in each iteration, the searching process is accelerated and the exploration capability can be enhanced contemporary. On top of this, a vector-guided enhanced selection operator is designed in the elite selection step to provide desirable distribution. Our experimental results on 45 benchmarks from UF and WFG test suits show prominent competitiveness compared with five state-of-the-art algorithms. RVSFS is very competitive on all 9 UF benchmarks in terms of the metric inverted generational distance and also performs well on 14 of 36 benchmarks from WFG in terms of the metric inverted generational distance and the metric hypervolume.

Multi-Objective Gray Wolf Optimizer with Cost-Sensitive Feature Selection for Predicting Students’ Academic Performance in College English

Feature selection is a widely utilized technique in educational data mining that aims to simplify and reduce the computational burden associated with data analysis. However, previous studies have overlooked the high costs involved in acquiring certain types of educational data. In this study, we investigate the application of a multi-objective gray wolf optimizer (GWO) with cost-sensitive feature selection to predict students’ academic performance in college English, while minimizing both prediction error and feature cost. To improve the performance of the multi-objective binary GWO, a novel position update method and a selection mechanism for a, b, and d are proposed. Additionally, the adaptive mutation of Pareto optimal solutions improves convergence and avoids falling into local traps. The repairing technique of duplicate solutions expands population diversity and reduces feature cost. Experiments using UCI datasets demonstrate that the proposed algorithm outperforms existing state-of-the-art algorithms in hypervolume (HV), inverted generational distance (IGD), and Pareto optimal solutions. Finally, when predicting the academic performance of students in college English, the superiority of the proposed algorithm is again confirmed, as well as its acquisition of key features that impact cost-sensitive feature selection.

A hybrid multi-objective optimization method for nuclear essential service water system design

Evolutionary algorithms have proven to be very successful in solving multi-objective optimization problems (MOPs). However, their performance often deteriorates when constraints are introduced. This paper proposes a hybrid heuristic intelligence (HHI) approach to remedy this issue. First, a classification operator is designed to sort individuals by their relative constraint values, which enlarges the size of feasible solutions in the early process. Then, by combining constraints with the non-dominated sorting method, we define a new selection operator to increase the probability of selecting individuals who have low constraint values but high objective values. Finally, we use hybrid optimized operations to simultaneously enhance the convergence and diversity performance by performing different operators in a certain domain. The proposed method achieves state-of-the-art performance on test functions ZDT1, Binh2, and OSY, with a remarkable decrease of 24% in the inverted generational distance (IGD) on OSY. For further applications, our method is also tested with an engineering model established for the design of the nuclear essential service water system (SEC). The final result shows that without taking hundreds of hours, the final solution of HHI has met all constraints and reduced the total cost by 8.9%© YEAR The Authors. Published by Elsevier Ltd.

A Hybrid Chaotic-Based Multiobjective Differential Evolution Technique for Economic Emission Dispatch Problem

The economic emission dispatch problem (EEDP) is a nonconvex and nonsmooth multiobjective optimization problem in the power system field. Generally, fuel cost and total emissions of harmful gases are the problem objective functions. The EEDP decision variables are output powers of thermal generating units (TGUs). To make the EEDP problem more practical, valve point loading effects (VPLEs), prohibited operation zones (POZs), and power balance constraints should be included in the problem constraints. In order to solve this complex and constrained EEDP, a new multiobjective optimization technique combining the differential evolution (DE) algorithm and chaos theory is proposed in this study. In this new multiobjective optimization technique, a nondomination sorting principle and a crowding distance calculation are employed to extract an accurate Pareto front. To avoid being trapped in local optima and enhance the conventional DE algorithm, two different chaotic maps are used in its initialization, crossover, and mutation phases instead of random numbers. To overcome difficulties caused by the equality constraint describing the power balance constraint, a slack TGU is defined to compensate for the gap between the total generation and the sum of the system load and total power losses. Then, the optimal power outputs of all thermal units except the slack unit are determined by the suggested optimization technique. To assess the effectiveness and applicability of the proposed method for solving the EEDP, the six-unit and ten-unit systems are used. Moreover, obtained results are compared with other new optimization techniques already developed and tested for the same purpose. The superior performance of the ChMODE is also evaluated by using various metrics such as inverted generational distance (IGD), hyper-volume (HV), spacing metric (SM), and the average satisfactory degree (ASD).