A data-driven hybrid multi-objective optimization framework for pressure swing adsorption systems

In multi-objective Bayesian optimization, an infill criterion is an important part, as it is the indicator to evaluate how much good a new set of solutions is, compared to a Pareto-front approximation set. This paper presents a deterministic algorithm for computing the Expected R2 Indicator for bi-objective problems and studies its use as an infill criterion in Bayesian Global Optimization. The R2-Indicator was introduced in 1998 by M. Hansen and A. Jaszkiewicz for performance assessment in multi-objective optimization and is more recently also used in indicator-based multi-criterion evolutionary algorithms (IBEAs). In Bayesian Global Optimization, we propose the Expected R2-indicator Improvement (ER2I) as an infill criterion. It is defined as the expected decrease of the R2 indicator by a point that is sampled from a predictive Gaussian distribution. The ER2I can also be used as a pre-selection criterion in surrogate-assisted IBEAs. It provides an alternative to the Expected Hypervolume-Indicator Improvement (EHVI) that requires a reference point, bounding the Pareto front from above. In contrast, the ER2I works with a utopian reference point that bounds the Pareto front from below. In addition, the ER2I supports preference modelling with utility functions and its computation time grows only linearly with the number of considered weight combinations. It is straightforward to approximate the ER2I by Monte Carlo Integration, but so far a deterministic algorithm to solve the non-linear integral remained unknown. We outline a deterministic algorithm for the computation of the bi-objective ER2I with Chebychev utility functions. Moreover, we study monotonicity properties of the ER2I w.r.t. parameters of the predictive distribution and numerical simulations demonstrate fast convergence to Pareto fronts of different shapes and the ability of the ER2I Bayesian optimization to fill gaps in the Pareto front approximation.

The transition from fossil fuels to renewable energy demands energy systems that are not only technically and economically sound but also environmentally and socially sustainable. This study proposed a novel multi-objective optimisation (MOO) framework for the design of a grid-connected hybrid renewable energy system (GC-HRES) that explicitly integrated Job Creation (JC) as a fourth objective along with traditional technical, economic, and environmental objectives, such as Loss of Power Supply Probability (LPSP), Cost of Energy (COE), and Renewable Energy Fraction (REF). Prior studies often treated JC as a post-optimisation metric, while this study incorporated JC into the MOO using employment technology-specific factors. Metaheuristic algorithms, Multi-Objective Particle Swarm Optimisation (MOPSO) and Non-Dominated Sorting Genetic Algorithm II (NSGA-II), were applied to find the optimal number of solar panels, wind turbines, and battery banks under scenarios which excluded and included JC. The results demonstrated that including JC reshaped the Pareto front, revealed new objective compromises and led to diverse configurations. MOPSO favoured solutions with higher JC and REF at the expense of cost and reliability, while NSGA-II achieved more balanced, cost-effective and reliable solutions with competitive JC values. Additional constraint sensitivity analysis further demonstrated the influence of battery bank constraints on solution feasibility. The results highlighted that integrating JC into the optimisation framework enriched design possibilities and fostered a more inclusive transition to renewable energy with a focus on GC-HRESs. This paper provided a flexible and replicable MOO framework that sets the foundation for socially attuned GC-HRES designs that align with broader sustainability goals.

Heuristic-based multi-objective optimization design for viscous dampers in RC frame structures considering soil-structure interaction

This work proposes a new method for approximating the Pareto front of a multi-objective simulation optimization problem (MOP) where the explicit forms of the objective functions are not available. The method iteratively approximates each objective function using a metamodeling scheme and employs a weighted sum method to convert the MOP into a set of single objective optimization problems. The weight on each single objective function is adaptively determined by accessing newly introduced points at the current iteration and the non-dominated points so far. A trust region algorithm is applied to the single objective problems to search for the points on the Pareto front. The numerical results show that the proposed algorithm efficiently generates evenly distributed points for various types of Pareto fronts. 1

We propose a surrogate-assisted reference vector adaptation (SRVA) method to\nsolve expensive multi- and many-objective optimization problems with various\nPareto front shapes. SRVA is coupled with a multi-objective Bayesian\noptimization (MBO) algorithm using reference vectors for scalarization of\nobjective functions. The Kriging surrogate models for MBO is used to estimate\nthe Pareto front shape and generate adaptive reference vectors uniformly\ndistributed on the estimated Pareto front. We combine SRVA with expected\nimprovement of penalty-based boundary intersection as an infill criterion for\nMBO. The proposed algorithm is compared with two other MBO algorithms by\napplying them to benchmark problems with various Pareto front shapes.\nExperimental results show that the proposed algorithm outperforms the other two\nin the problems whose objective functions are reasonably approximated by the\nKriging models. SRVA improves diversity of non-dominated solutions for these\nproblems with continuous, discontinuous, and degenerated Pareto fronts.\nBesides, the proposed algorithm obtains much better solutions from early stages\nof optimization especially in many-objective problems.\n

A diversity indicator based on reference vectors for many-objective optimization

BACKGROUND Dual‐reflux pressure swing adsorption (DR‐PSA) is an advanced variant of conventional PSA systems, capable of simultaneously producing two high‐purity products with high recovery. However, inherent trade‐offs among purity, productivity, and cycle work necessitate multi‐objective optimization (MOO) to balance performance. This study investigates the MOO of the DR‐PSA process for biogas upgrading using artificial neural networks (ANNs) as surrogate models. Single‐hidden‐layer ANNs were developed with various structures and training algorithms and integrated into the non‐dominated sorting genetic algorithm‐II to solve MOO problems. RESULTS Different DR‐PSA performance metrics require distinct ANN architectures and algorithms, indicating no universal model selection. The Adam‐ selu‐ 446 model for CH 4 purity in light product (LP) and the Adam‐ relu ‐398 model for cycle work, both developed on a large dataset, exhibited high accuracy, with of 0.9975 and 0.9959, respectively. Meanwhile, the Adam‐ elu ‐212 model for bed productivity, developed using a small dataset, achieved an of 0.9988. The Pareto front for maximizing CH 4 purity in LP and bed productivity revealed a sharp knee region, indicating a significant trade‐off. Under constrained bed productivity and cycle work, distinct Pareto fronts emerged at different CH 4 purity levels (>91% and 91.5%), with fewer feasible solutions at higher purity. Ultimately, the optimized conditions resulted in an enhancement in CH 4 purity in LP by at least 17%, a reduction in bed productivity by as much as 34%, and a reduction in cycle work by at least 30%, compared to the base case. CONCLUSION ANN‐assisted MOO optimizes DR‐PSA performance, offering insights into ANN model development, Pareto front mapping, and identification of feasible operational regions. © 2025 Society of Chemical Industry (SCI).

A Tchebycheff-based multi-objective combined with a PSO–SQP dynamic real-time optimization framework for cycling energy systems

Radiation therapy treatment planning is inherently a multiobjective problem, aiming to obtain the best tradeoffs between irradiating the tumor with the prescribed dose and sparing as much as possible the surrounding healthy organs. Many different multiobjective approaches have been proposed for the optimization of radiation intensities for fixed beam irradiation directions. However, multiobjective beam angle optimization is seldom considered. The purpose of this paper is to introduce a new multiobjective optimization framework that explicitly and simultaneously considers the optimization of intensities and also beam directions. Whilst multiobjective optimization of radiation intensities considering a fixed set of beam directions gives rise to a single Pareto front, beam angle optimization gives rise to the appearance of multiple Pareto fronts, each one associated with a given beam angle set. Our framework proposes a beam angle set choice based on the evaluation of non-dominated solutions belonging to different Pareto fronts, using a tree-based approach and a performance indicator to assess the quality of each Pareto front. The proposed approach, illustrated by head-and-neck cancer cases, allows for more flexibility in the calculation of solutions and a better understanding of the existing compromises between different objectives.

For a high-speed and high-acceleration planar parallel mechanism, a design for control (DFC) approach is proposed for the optimization of the mechanism structure. Several evaluation indexes of kinematics and dynamics performance are established for the object functions and constraints, and then a standard multi-objective optimization problem can be formed. The dimension synthesis is carried out by solving the multi-objective optimization with NSGA-II (non-dominated sorting genetic algorithm-II) algorithm. The final result shows that the mechanism with the optimized parameters can eliminate the systematic coupling and enhance the dynamic performance greatly, which provides a solid mechanism hardware platform for designing the high-speed control system.

The performance of a plate heat exchanger (PHE) using water as the working fluid with zigzag flow channels was optimized in the present study. The optimal operating conditions of the PHE are explored experimentally by the Taguchi method, with effectiveness as the objective function. The results are further verified by the analysis of variance (ANOVA). In addition, the zigzag flow channel geometry is optimized by the non-dominated sorting genetic algorithm-II (NSGA-II), in which the effectiveness and pressure drop of the PHE are considered the two objective functions in the multi-objective optimization process. The experimental results show that the ratio of flow rates is the most important factor affecting the effectiveness of the PHE. The optimal operating conditions are the temperatures of 95 °C and 10 °C at the inlets of hot and cold water flows, respectively, with a cold/hot flow rate ratio of 0.25. The resultant effectiveness is 0.945. Three geometric parameters of the zigzag flow channel are considered, including the entrance length, the bending angle, and the fillet radius. The sensitivity analysis of the parameters reveals that a conflict exists between the two objective functions, and multi-objective optimization is necessary for the zigzag flow channel geometry. The numerical simulations successfully obtain the Pareto optimal front for the two objective functions, which benefits the determination of the geometric design for the zigzag flow channel.

Multi-objective optimization for limiting tunnel-induced damages considering uncertainties

The Hierarchical Voltage Regulation (HVR) is an efficient alternative to the traditional solutions used in the power system voltage control. It consists of three levels: primary, secondary and tertiary, among which the Secondary Voltage Regulation (SVR) level is the core of the HVR ensuring the coordination of a network area reactive power resources by controlling the voltage of the pilot bus, the area most representative bus. Moreover, in the recent years, the shunt FACTS devices, especially the Static var Compensators (SVC), have shown very good performances in local voltage control at a reasonable cost. Therefore, including these devices in the HVR scheme has become an important task. In this paper, the application of a multi-objective optimization to find the optimal placement of SVCs in the SVR environment is proposed; the objective functions to be minimized are the total power losses and the reactive power generated in the power system. The Non-dominated Sorting Genetic Algorithms II (NSGA-II) is used to generate the Pareto front which contains several optimal solutions and will be used by the Decision Maker (DM) to choose the best compromise solution. In this work, a fuzzy-based mechanism is proposed to select the best compromise solution from the Pareto set. Tests and simulations are performed on a model of the Italian power system.

A surrogate-assisted multi-objective evolutionary algorithm with dimension-reduction for production optimization

Minimizing carbon emissions in power system operations has become a critical challenge as the energy sector transitions toward sustainability. This paper presents a high-fidelity simulation framework for CO<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf>-optimized energy dispatch, employing an enhanced Non-Dominated Sorting Genetic Algorithm II (NSGA-II) tailored for emission-oriented optimization. Unlike traditional economic dispatch approaches that primarily focus on cost, the proposed framework explicitly minimizes total CO<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> emissions while ensuring demand–supply balance, operational limits, and ramp-rate feasibility. Key enhancements to the baseline NSGA-II include adaptive crossover–mutation operators, hybrid population initialization, and constraint-repair mechanisms, which improve convergence speed, robustness, and solution diversity. The framework is validated through multiple case studies incorporating stochastic load demand and renewable variability. Results show that the enhanced NSGA-II achieves nearly 20% emission reduction compared to baseline economic dispatch, with faster convergence and a broader Pareto-optimal solution set than standard NSGA-II. Furthermore, dispatch outcomes demonstrate increased reliance on renewable and storage units, reducing the share of coal and gas generation. These findings highlight the potential of the proposed approach to support low-carbon, policy-aligned energy system operations and provide a foundation for future extensions in multi-objective optimization, real-time scheduling, and carbon pricing integration.