Pareto Optimal Solutions Research Articles

Multi-armed Bandit Experimental Design: Online Decision-Making and Adaptive Inference

Multi-armed bandit has been well known for its efficiency in online decision-making in terms of minimizing the loss of the participants’ welfare during experiments (i.e., the regret). In clinical trials and many other scenarios, the statistical power of inferring the treatment effects (i.e., the gaps between the mean outcomes of different arms) is also crucial. Nevertheless, minimizing the regret entails harming the statistical power of estimating the treatment effect because the observations from some arms can be limited. In this paper, we investigate the trade-off between efficiency and statistical power by casting the multi-armed bandit experimental design into a minimax multi-objective optimization problem. We introduce the concept of Pareto optimality to mathematically characterize the situation in which neither the statistical power nor the efficiency can be improved without degrading the other. We derive a useful sufficient and necessary condition for the Pareto optimal solutions to the minimax multi-objective optimization problem. Additionally, we design an effective Pareto optimal multi-armed bandit experiment that can be tailored to different levels of the trade-off between the two objectives. Moreover, we extend the design and analysis to the setting where the outcome of each arm consists of an adversarial baseline reward and a stochastic treatment effect, demonstrating the robustness of our design. Finally, motivated by clinical trials, we examine the setting where the employed experiment must split the experimental units into a small number of batches, and we propose a flexible Pareto optimal design. This paper was accepted by George Shanthikumar, data science. Funding: The authors thank the Massachusetts Institute of Technology (MIT)-IBM partnership in Artificial Intelligence and the MIT Data Science Laboratory for support. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.00492 .

A heuristic method for multi-objective hybrid flow shop scheduling problem with parent-child relationships and space constraints

ABSTRACT In modern ship manufacturing, a ship is divided into hundreds of blocks in the design stage, and the productivity of the block manufacturing process is vital to ship delivery. This paper investigates the production scheduling problem of ship blocks with the parent-child relationship; this relationship is characterised by the fact that sibling jobs at the same BOM layer must be processed on adjacent machines at their assembly stages. We formulate this problem as a multi-objective hybrid flow shop scheduling problem: the first objective is to minimise the total deviation from the target time of the jobs and the second one is to minimise the total processing time. To solve this problem, we propose a heuristic method based on the multi-objective genetic algorithm, in which an adjacent machine priority method is first proposed to generate feasible solutions that satisfy the space constraints. Then, we improve the individual quality in crossover and mutation of the algorithm via tabu search. We verify the performance of our algorithm by test instances generated from real manufacturing data of a shipbuilding company. Results show that the proposed method outperforms the existing algorithms when identifying the Pareto optimal solutions, especially for large-sized problems.

Novel statistical investigation on performance measures of WEDM: Optimization, microstructure and mechanical properties

A comparative performance analysis has been investigated on wire electrical discharge machining (WEDM) responses while machining a hybrid titanium matrix composite (TMC) varying the key input parameters like power (P), peak current (IP) and time-off (Toff). Two novel multi-objective optimization algorithms are developed namely desirable multi-objective genetic algorithm (DMOGA) and desirable multi-objective giant pacific octopus optimizer (DMOGPOO) for tackling various issues in many industries like automobile valve pins in crank and cam shafts, aerospace propeller and biomedical implants. The principal advantage of DMOGA to other algorithm is accuracy and robustness. The novelty fits in the iterative progression of growth of efficient grandee set, uttered as population congregating to a fitness function. Many techniques frequently encounter substandard solutions when evaluating MOO problems, as opposed to solving properly approximated functions of Pareto optimal solutions in targets. DMOGPOO is an enthralling statistical method which mimics the octopus’s predatory behavior, performs better than other multi-objective optimization (MOO) algorithms, where the desirable objective functions is fetched in python using MOGPOO. In a multi-objective foraging environment, the archive was utilized to imitate octopus predatory behavior and establish social hierarchies. DMOGPOO approach is designed with multi-objective formulations to preserve and guarantee enhanced coverage of optimum solutions across all goals. Experimental investigation is accepted on material removal rate (MRR), surface roughness (SR), kerf width (KW) and over cut (OC). Combined desirability in case of DMOGA is 0.716 which improved to 0.813 when DMOGPOO is proposed. MOO is improved with DMOGPOO of 13.547% when contrasted with DMOGA, with MRR of 3.81 mm3/min, SR of 0.79 µm, KW of 0.349 mm, OC of 0.099 mm, and combined desirability of 0.813. Improved optimality set is obtained when DMOGPOO is used. %improvement of MRR is 5.54%, SR is 75.95%, KW is 0.29%, and OC is 4.21%.

Multiobjective enterprise development algorithm for optimizing structural design by weight and displacement

This study presents a novel metaheuristic optimization algorithm for complex multiobjective engineering problems. By integrating advanced population and nondominated sorting techniques into the existing single-objective enterprise development algorithm, this new multiobjective approach effectively identifies Pareto-optimal solutions. The algorithm leverages these techniques to explore engineering solutions within multiobjective search spaces. We evaluated its performance using 29 multiobjective mathematical benchmark problems and conducted a comparative analysis against ten established metaheuristic optimization algorithms. The results demonstrate that the algorithm produces highly accurate approximations of Pareto-optimal fronts while maintaining a uniform distribution of solutions. Additionally, the algorithm was applied to real-world engineering challenges, including the optimization of structures such as the 942-bar tower, the 1536-bar double-layer barrel vault, and the 1410-bar double-layer dome truss, with the primary objectives of minimizing both structural weight and maximum nodal deflection. The findings highlight the algorithm's effectiveness in solving practical engineering problems and consistently achieving optimal Pareto fronts.

Crashworthiness and optimization design of additive manufacturing double gradient lattice enhanced thin-walled tubes under dynamic impact loading

It is of paramount importance to ensure the protection of both infrastructure and personnel in scenarios characterized by a high strain rate dynamic impact. In order to develop lighter weight and higher crashworthiness energy-absorbing structures, this paper incorporates the bionic design of lattice reinforced thin-walled tube structures in the form of double gradient, as a means of achieving the aforementioned objectives. Material mechanical properties and the designed structures at high strain rate were researched using Split Hopkinson Pressure Bar (SHPB) experiment. The effects of gradient form and gradient parameters on the deformation pattern and crashworthiness of lattice enhanced thin-walled tubes are researched by means of the validated finite element model and the experimentally fitted Johnson-Cook material model. The results indicate that the specific energy absorption of the double gradient lattice enhanced thin-walled tube (T3L3) is increased by 36.19 % and the peak crush force is reduced by 21.19 % compared to the lattice enhanced thin-walled tube without gradient. Finally, the multi-objective optimization analysis of the T3L3 with the objective of achieving the maximum SEA and minimum PCF is carried out by Response Surface Methodology (RSM) and Non-dominated Sorting Genetic Algorithm-II (NSGA-II) to find the Pareto optimal solution set. Double-gradient bionic structure offers a new idea for designing energy-absorbing structures under high-speed impact conditions.

Intelligent mix design of steel fiber reinforced concrete using a particle swarm algorithm based on a multi-objective optimization model

Steel fiber reinforced concrete (SFRC) is widely used in construction and is important for concrete-based applications. Nevertheless, compared with conventional concrete, it is difficult to efficiently and accurately design SFRC with a given mix ratio, because many factors affect the SFRC performance. This study proposes a multi-objective optimization model using numerical simulations and artificial intelligence to effectively derive the optimal SFRC mix proportion. For a quick and accurate relationship between the SFRC mix ratio and compressive strength, Latin hypercube sampling was used for sampling the random variables determining the mix ratio, yielding a set of random numbers. Subsequently, a finite-element simulation dataset was constructed, and a backpropagation neural network (BPNN) was used for predicting the complex nonlinear relationship between the raw material mix proportions and uniaxial compressive strength (UCS). The BPNN model was then utilized as the objective function in the multi-objective particle swarm optimization model, with the mix ratio parameters as the input variables and the compressive strength, unit production cost, and carbon dioxide emission as objective functions, which facilitated the search for the optimal mix proportion, yielding a Pareto-optimal solution set. Finally, based on the engineering preferences, the best solution was determined to be the recommended mix proportion.

Dynamic Event-Triggered Safe Control for Nonlinear Game Systems With Asymmetric Input Saturation.

This article focuses on the Pareto optimal issues of nonlinear game systems with asymmetric input saturation under dynamic event-triggered mechanism (DETM). First, the safe control is guaranteed by transforming the system with safety constraints into the one without state constraints utilizing barrier function. The united cost function integrating nonquadratic utility function is constructed to provide the foundation to achieve the Pareto optimal solutions. Then, the adaptive dynamic programming method with concurrent learning is proposed to approximate the Pareto optimal strategies wherein both current and historical data are utilized. To further lessen the consumptions of computation/communication resources, the DETM is integrated into the adaptive algorithm framework which can avoid Zeno phenomena. All the signals of the closed-loop system are proved to be uniformly ultimately bounded. Finally, the simulation results are given to validate the effectiveness of the proposed method from several aspects.

An optimization framework for achieving optimal hydrocyclone's performance aligning with decision-makers' preferences

Previous hydrocyclone optimization often neglected interactions among key performance objectives, which limits hydrocyclone wide applications to meet increasingly diverse industry demands. This study proposes an optimization framework to identify the most suitable hydrocyclone design and operating conditions with conflicting key performance objectives. An advanced multi-objective evolutionary algorithm (PICEA-g) is employed to generate Pareto-optimal solutions that capture trade-offs among multiple conflicting objectives. A novel data-driven predictive algorithm, INFO-ELM, is introduced to establish nonlinear relationships between key variables and performance objectives, thereby accelerating the search for Pareto-optimal solutions by PICEA-g. Furthermore, a multi-criteria decision-making method (TOPSIS) is utilized to determine the optimal solution based on decision-makers' preferences, ensuring consistency between preference information and decision outcomes. The framework's effectiveness is validated across various separation scenarios using two decision-making strategies. This study offers a comprehensive approach to address trade-offs in hydrocyclone optimization, widening its application in diverse separation scenarios.

A grid self-adaptive exploration-based algorithm for multimodal multiobjective optimization

In multimodal multiobjective optimization, the key is to find as many equivalent Pareto optimal solutions as possible through broad exploration in the decision space. The grid search strategy can achieve quick convergence by guiding the evolution with historical information in each grid while enabling broad exploration. However, inaccurate utilization of information in grids may lead to losing numerous potential solutions, especially on imbalanced problems. In order to resolve this issue, a grid self-adaptive exploration-based algorithm (GSEA) is proposed in this paper. In GSEA, the historical information in the grid is accurately utilized through grid-based self-adaptive exploration and niche clearing methods, which retain a large number of solutions with potential and effectively handle multimodal multiobjective optimization problems (MMOPs). Experimental results show that the proposed algorithm outperforms seven other state-of-the-art multimodal multiobjective evolutionary algorithms (MMEAs) on two types of MMOPs, and the approach can effectively deal with the MMOPs with middle-scale decision variables as memory allows.

Multi-objective optimization of L-shaped fins in rectangular phase change energy storage units based on genetic algorithm

Phase change energy storage technology holds immense potential in the field of energy storage, and enhancing the efficiency of energy storage systems has long been a focal point of industry attention. This study aims to optimize the design of an L-shaped fin structure to improve the melting rate of the phase change material (PCM) within a rectangular phase change energy storage unit and enhance the overall system performance. Through the utilization of numerical simulations and the response surface method (RSM), the influence of fin design parameters - specifically, the length and thickness of the main segment and the length and thickness of the branch segment - on the energy storage per unit mass (Em) and energy storage efficiency (Pt) of the energy storage unit is evaluated. The results reveal that the length of both the main and branch segments of the L-shaped fin significantly impacts the melting rate of the PCM, whereas the thickness of these segments has a comparatively lesser effect. Additionally, when the length of the main segment reaches a sufficient value, increasing the length of the branch segment effectively addresses the challenge of PCM melting difficulties at the bottom of the rectangular phase change cavity caused by natural convection. The Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) is employed for the multi-objective optimization of the L-shaped fin shape, and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) with entropy weighting is applied for decision-making to determine the optimal solution. Notably, when the weights of Em and Pt are set at 44.9 % and 55.1 % respectively, the overall performance of the energy storage unit is optimized. The Pareto-optimal decision solution exhibits a significant improvement of nearly threefold in Pt compared to the case without fins, while the reduction in Em is only 5.24 %. This design approach presents a novel perspective for determining fin parameters in phase change energy storage devices and is expected to drive advancements in phase change energy storage technology.

Multi-Objective Robust Design Optimization for Crashworthiness Enhancement of Hybrid 2D Triaxially Braided Composite Tube Using Evolutionary Algorithms.

An innovative optimal design framework is developed aiming at enhancing the crashworthiness while ensuring the lightweight design of a hybrid two-dimensional triaxial braided composite (2DTBC) tube, drawing insights from the mesostructure of the composite material. To achieve these goals, we first compile the essential mechanical properties of the 2DTBC using a concentric cylinder model (CCM) and an analytical laminate model. Subsequently, a kriging surrogate model to elucidate the intricate relationship between design variables and macroscopic crashworthiness is developed and validated. Finally, employing multi-objective evolutionary optimization, we identify Pareto optimal solutions, highlighting that reducing the total fiber volume and increasing the glass fiber content in the total fiber volume are crucial for optimal crashworthiness and the lightweight design of the hybrid 2DTBC tube. By integrating advanced predictive modeling techniques with multi-objective evolutionary optimization, the proposed approach not only sheds light on the fundamental principles governing the crashworthiness of hybrid 2DTBC but also provides valuable insights for the design of robust and lightweight composite structures.

Interval-Valued Multiobjective Programming Problems Based on Convex Cones

The new solution concepts of interval-valued multiobjective optimization problems using ordering cone are proposed in this paper. An equivalence relation is introduced to divide the collection of all bounded closed intervals into the equivalence classes. The family of all equivalence classes is also called a quotient set. In this case, this quotient set can turn into a vector space under some suitable settings for vector addition and scalar multiplication. The notions of ordering cone and partial ordering on a vector space are essentially equivalent. It means that an ordering in the quotient set can be defined to study the Pareto optimal solution in multiobjective optimization problems. In this paper, we consider the multiobjective optimization problem such that its coefficients are taken to be the bounded closed intervals. With the help of the convex cone, we can study the Pareto optimal solutions of the multiobjective optimization problem with interval-valued coefficients.

Multiobjective sparse unmixing based hyperspectral change detection

Hyperspectral unmixing can provide the composition of ground objects, while change detection can identify the changes of the same region over time. Therefore, unmixing based hyperspectral change detection can investigate changes in a subpixel-level. It can provide not only whether the change happens or not but also how to change. This paper will gradually establish an unconstrained sparse unmixing based change detection model and design a multiobjective optimization sparse unmixing approach (termed MoSU-CD) to solve it for more subpixel-level change details. Compared with the existing sparse unmixing based change detection approaches, MoSU-CD directly unmixes the difference map of multi-temporal images rather than multi-temporal images themselves respectively. Then it explores the change details of one substance in each pixel based on the unmixing results, i.e., endmembers and abundances. In addition, in order to improve the change detection accuracy, an ensemble decision strategy is designed for the Pareto optimal solutions obtained by multiobjective optimization method. Based on this strategy, the subpixel-level changes can be aggregated to generate an aggressive pixel-level change map. The experimental results on synthetic and real data sets demonstrate that the proposed MoSU-CD not only outperforms the benchmark and state-of-art approaches in terms of pixel-level change detection but also provides additional subpixel-level change information.

Experimental Research and Improved Neural Network Optimization Based on the Ocean Thermal Energy Conversion Experimental Platform

With the progress of research on ocean thermal energy conversion, the stabI have checked and revised all. le operation of ocean thermal energy conversion experiments has become a problem that cannot be ignored. The control foundation for stable operation is the accurate prediction of operational performance. In order to achieve accurate prediction and optimization of the performance of the ocean thermal energy conversion experimental platform, this article analyzes the experimental parameters of the turbine based on the basic experimental data obtained from the 50 kW OTEC experimental platform. Through the selection and training of experimental data, a GA-BP-OTE (GBO) model that can automatically select the number of hidden layer nodes was established using seven input parameters. Bayesian optimization was used to complete the optimization of hyperparameters, greatly reducing the training time of the surrogate model. Analyzing the prediction results of the GBO model, it is concluded that the GBO model has better prediction accuracy and has a very low prediction error in the prediction of small temperature changes in ocean thermal energy, proving the progressiveness of the model proposed in this article. The dual-objective optimization problem of turbine grid-connected power and isentropic efficiency is solved. The results show that the change in isentropic efficiency of the permeable device is affected by the combined influence of the seven parameters selected in this study, with the mass flow rate of the working fluid having the greatest impact. The MAPE of the GBO model turbine grid-connected power is 0.24547%, the MAPE of the turbine isentropic efficiency is 0.04%, and the MAPE of the turbine speed is 0.33%. The Pareto-optimal solution for the turbine grid-connected power is 40.1792 kW, with an isentropic efficiency of 0.837439.

Optimizing frequencies of functionally graded material plates via multi-objective genetic algorithm: Positioning point supporters for maximum performance

In the presented study, an approach is introduced for optimizing the vibrational characteristics of aerospace structures that are composed of Functionally Graded Materials (FGM). The focus is placed on the strategic positioning of point supporters to enhance the structural performance under various operational conditions. The effective elasticity properties of the FGMs are determined using the Mori-Tanaka method for homogenization. The deformation behavior of these structures is analyzed by employing the first-order shear deformation theory. Solutions are computed through the 2D Ritz method, which utilizes Chebyshev polynomials, forming the basis for the objective function of a multi-objective genetic algorithm. This algorithm is designed to optimize the positioning and the number of point supporters aimed at targeting specific vibration frequencies. The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is utilized to efficiently generate Pareto optimal solutions, which illustrate the trade-offs between conflicting objectives. The validation of the solution method is achieved through comparisons with experimental data, confirming the accuracy and practical relevance of the approach. The application of the optimization framework is extended to various configurations of aerospace structures, including diverse compositions of FGM and different boundary conditions. This methodology is shown not only to enhance the operational performance of aerospace structures but also to contribute to the precise design and manufacturing of components in aircraft and satellites, aligning with the central interests of aerospace engineering.

Dynamic switched crowding-based multi-objective particle swarm optimization algorithm for solving multi-objective AC-DC optimal power flow problem

In this paper, the multi-objective AC-DC optimal power flow (MO/AC-DC OPF) problem in the presence of renewable energy sources (RESs), flexible AC transmission system (FACTS) devices and multi-terminal direct current (MTDC) systems is introduced for the first time. Conflicting objective functions and the high complexity of the objective and constraint spaces are the main challenges in finding optimal solutions for MO/AC-DC OPF. To overcome these challenges, twelve different versions of the dynamic switched crowding-based multi-objective particle swarm optimization (DSC-MOPSO) algorithm are introduced in this paper. Studies on multimodal optimization problems have shown that all DSC-MOPSO versions have better performance metrics than the MOPSO algorithm. Using the developed DSC-MOPSO and its strong competitors, the Pareto-optimal solution sets of the MO/AC-DC OPF problem are investigated. In these investigations, the performances of the algorithms are tested for the minimization of dual and triple objectives such as fuel cost, voltage level deviation, emission and power loss in a modified IEEE 30-bus power grid. According to the simulation results, the proposed DSC-MOPSO achieved an improvement in fuel cost between 0.02 % and 5.05 % and a reduction in active power loss between 0.44 % and 30.74 % compared to its competitors. The Hypervolume (HV) performance metric was used to evaluate the Pareto-front coverage performance of DSC-MOPSO and other optimizers. The results from nine case studies of the MO/AC-DC OPF were statistically analyzed by the Friedman test according to the 1/HV metric. According to the Friedman test results, the rankings of DSC-MOPSO and MOMA are 1.984 and 3.079, respectively, ranking first and second among all competitors. Finally, in this study, feasible solutions for MO/AC-DC OPF problem are identified for the first time and the stability of competitive algorithms in finding these solutions is analyzed for the first time. The success rates and search times of DSC-MOPSO and MOMA algorithms in finding feasible solutions for MO/AC-DC OPF are 91.01 % (30.641 s) and 82.01 % (46.038 s), respectively.

Multi-Objective Real-Time Planning of Evacuation Routes for Underground Mine Fires

When a fire occurs underground, pre-existing emergency escape routes may become ineffective. Such scenarios necessitate the real-time planning of escape routes that consider evolving conditions and prioritize safety. This paper proposes a multi-objective real-time search method for emergency escape routes using dynamic programming to address these challenges. Firstly, this paper discusses how the evacuation of an underground mine fire caused by external factors depends on the fire monitoring system, personnel positioning system, and emergency evacuation system, as well as the modeling and solution methods of evacuation route planning. Then, based on fire smoke, CO, and temperature sensor data from the mine ventilation network structure and monitoring system, the possible smoke spread in the event of an underground fire was calculated. The objectives are to optimize both the shortest equivalent length of the path and the shortest time to walk through the smoke flow while ensuring that temporary escape time does not exceed the rated protection time of the self-rescuer. A mathematical model for emergency escape route planning is established under these conditions. A labeling algorithm based on dynamic programming is employed to find the Pareto optimal solution set of emergency evacuation routes that meet emergency requirements. Finally, two path evaluation indicators, namely “escape target priority” and “personnel temporary escape time”, are introduced to re-rank the solutions in the Pareto optimal set, thereby obtaining disaster evacuation routes with different priorities. Example verification shows that the algorithm can quickly solve the disaster evacuation routes that meet the actual disaster evacuation needs in complex networks. Example verification shows that the algorithm can quickly solve the emergency escape routes in complex networks that meet actual emergency escape needs.

Multi-objective optimization and long-time simulation of a multi-borehole ground heat exchanger system

Multi-objective optimization and CFD simulation are conducted to optimize the design of a multi-borehole ground heat exchanger (GHE) system and assess its long-time performance. The multi-objective optimization is performed to minimize the entropy generation number (EGN) and total cost rate by using various evolutionary algorithms, including NSGA-II, GDE-3, MOEA/D, PESA-II, SPEA-II, and SMPSO. NSGA-II and GDE-3 algorithms perform best in obtaining Pareto optimal solutions. Three prominent points on the NSGA-II Pareto frontier, representing the results of single-objective thermodynamic, single-objective economic, and multi-objective optimizations, are simulated in three dimensions over three months. The trends of EGN variations extracted from the transient CFD simulation agree well with those from the steady analytical model. The EGN obtained from multi-objective optimization is 58.8% lower than the EGN obtained using single-objective economic optimization and 1.9 times higher than that calculated from single-objective thermodynamic optimization. Likewise, the total cost rate obtained from multi-objective optimization is 64.4% lower than the value obtained from single-objective thermodynamic optimization and four times higher than that calculated using single-objective economic optimization. The proposed optimization approach can be reliably applied to improve the design of multi-borehole GHE systems.

Pareto Optimization and Tuning of a Laser Wakefield Accelerator.

Optimization of accelerator performance parameters is limited by numerous trade-offs, and finding the appropriate balance between optimization goals for an unknown system is challenging to achieve. Here, we show that multiobjective Bayesian optimization can map the solution space of a laser wakefield accelerator (LWFA) in a very sample-efficient way. We observe that there exists a wide range of Pareto-optimal solutions that trade beam energy versus charge at similar laser-to-beam efficiency. Moreover, many applications such as light sources require particle beams at certain target energies. We demonstrate accurate energy tuning of the LWFA from 150 to 400MeV via the simultaneous adjustment of eight parameters. To further advance this use case, we propose an inverse model that allows a user to specify desired beam parameters. Trained on the forward Gaussian process model, the inverse model generates input parameter value ranges within which the desired setting is likely to be reached. The method reveals different strategies for accelerator tuning and is expected to drastically facilitate the operation of LWFAs in the near future.

Self-organizing surrogate-assisted non-dominated sorting differential evolution

Multi-objective optimization problems (MOPs) involve optimizing multiple conflicting objectives simultaneously, resulting in a set of Pareto optimal solutions. Due to the high computational or financial cost associated with evaluating fitness, expensive multi-objective optimization problems (EMOPs) further complicate the optimization process. Surrogate-assisted evolutionary algorithms (SAEAs) have emerged as a promising approach to address EMOPs by substituting costly evaluations with computationally efficient surrogate models. This paper introduces the self-organizing surrogate-assisted non-dominated sorting differential evolution (SSDE), which uses surrogate model based on a self-organizing map (SOM) to approximate the fitness function. SSDE offers advantages such as reduced computational cost, improved accuracy, and the speed of enhanced convergence. The SOM-based surrogate models effectively capture the underlying structure of the Pareto optimal set and Pareto optimal front, leading to superior approximations of the fitness function. Experimental results on benchmark functions and real-world problems, including Model-Free Adaptive Control (MFAC) and the Yagi-Uda Antenna design, demonstrate the competitiveness and efficiency of SSDE compared to other algorithms.