Non-dominated Solutions Research Articles

Optimization of truss structures using multi-objective cheetah optimizer

In this study, a multi-objective version of the recently proposed cheetah optimizer called multi-objective cheetah optimizer (MOCO) has been proposed. MOCO draws inspiration from the targeted hunting strategy employed by cheetahs, which involves a sequence of actions: searching for prey, patiently waiting for the right moment to attack, swiftly launching the attack, and then retreating from the prey and returning to their habitat. MOCO is the result of modification and enhancement from its single-objective counterpart, utilizing a Pareto dominance-based approach. This adaptation allows MOCO to efficiently handle multiple objectives, explores and exploits promising areas in the optimization landscape, and identifies non-dominated solutions, offering valuable tradeoff choices for decision-makers. To demonstrate its practical applications, the MOCO method has been employed to address five intricate structural design problems. These problems involve a pair of competing objectives: the minimization of structural weight and the reduction of maximum nodal displacement. To gauge the efficacy and efficiency of the proposed algorithm, a comparative analysis is conducted against three alternative state-of-the-art multi-objective algorithms. Furthermore, a rigorous evaluation is carried out utilizing hypervolume testing. The findings reveal that the MOCO algorithm surpasses the performance of the other algorithms, underscored by its capacity to uncover a diverse array of non-dominated solutions. To delve deeper into the experimental results, statistical analysis employing Friedman’s rank test is employed. The solutions generated and the convergence patterns exhibited by the MOCO approach underscore its exceptional proficiency in resolving intricate design problems.

Multimodal multiobjective differential evolution algorithm based on enhanced decision space search

Multimodal multiobjective optimization problems (MMOPs) have attracted extensive research interest. These problems are characterized by the presence of multiple equivalent optimal solutions in the decision space, all corresponding to the same optimal values in the objective space. However, effectively finding a high-quality and evenly distributed Pareto sets (PSs) remains a challenge for researchers. This paper introduces a multimodal multiobjective differential evolution algorithm based on enhanced decision space search (MMODE_EDSS). By adopting two types of strategies to enhance the decision space search capability, the algorithm generates multiple high-quality non-dominated solutions. In the early stages of evolution, neighborhood information is used to enhance search capabilities, while in the later stages, data interpolation methods following clustering are employed for searching. Moreover, to improve the overall population distribution, an environmental selection mechanism based on dual-space crowding distance is adopted. The effectiveness of the proposed algorithm, MMODE_EDSS, is evaluated by comparing it with eight state-of-the-art multimodal multiobjective evolutionary algorithms (MMOEAs). Experimental results confirm the significant advantages of MMODE_EDSS.

Structural Optimization of a Giant Magnetostrictive Actuator Based on BP-NSGA-II Algorithm

This study introduces an integrated structural optimization design method based on a BP neural network and NSGA-II multi-objective genetic algorithm. Initially, a two-dimensional axisymmetric finite element model of the Giant Magnetostrictive Actuator (GMA) was established, and the coupling simulation of the electromagnetic field, structural field, and temperature field was conducted to obtain the GMA’s performance parameters. Subsequently, the structural parameters of the GMA magnetic circuit, including the magnetic conducting ring, magnetic conducting sidewall, magnetic conducting body, and coil, were used as inputs, and the axial magnetic induction intensity, uniformity of axial magnetic induction intensity, and coil loss on the Giant Magnetostrictive Material (GMM) rod were used as outputs to establish a back propagation (BP) neural network model. This model delineated the nonlinear relationship between structural parameters and performance parameters. Then, the BP-NSGA-II algorithm was applied to perform multi-objective optimization on the actuator’s structural parameters, resulting in a set of Pareto optimal non-dominated solutions, from which a set of optimal solutions was obtained using the entropy weight method. Finally, simulation analysis of this optimal solution was conducted, indicating that under a 5 A power supply excitation, the maximum axial magnetic induction intensity on the optimized GMM rod increased from 0.87 T to 1.12 T; the uniformity of axial magnetic induction intensity improved from 93.1% to 96.5%; and the coil loss decreased from 7.79 × 104 W/m3 to 4.97 × 104 W/m3. Based on the optimization results, a prototype actuator was produced, and the test results of the prototype’s output characteristics proved the feasibility of this optimization design method.

Multidisciplinary design analysis and optimization of the aerodynamic shape of the SpaceLiner passenger stage

The SpaceLiner is a futuristic concept of a suborbital space plane intended to provide ultra-fast intercontinental passenger transport, currently in pre-development at DLR-SART. Vehicles traveling at hypersonic speeds inevitably produce shock waves that are perceived at ground level as sonic booms, with associated disturbance of the overflown populations. The reduction of this disturbance, together with the decrease of the noise associated to the launch and ascent phase, is critical for a viable future operation of the SpaceLiner. The objective of this work is the redesign of the passenger stage aerodynamic shape in order to improve its atmospheric re-entry performance, in terms of a reduction of both the heat flux and the disturbance of the overflown populations. For this purpose, a MDAO methodology was developed with the tasks of (1) computing vehicle performance from a wing shape parametrization using fast estimation methods, (2) exploring the design space and (3) finding a new promising aerodynamic shape. First, a Python tool-chain was developed to compute vehicles performances from a wing shape parametrization using fast estimation methods. The tool-chain was then systematically exploited to explore the design space by means of parametric studies, whose results informed the implementation of the forthcoming optimization. Afterwards, the wing shape was optimized using a three-objective evolutionary algorithm that minimized the vehicle mass and maximized its lift-to-drag ratio and lift coefficient. Simplified trajectory optimizations were then run on the set of non-dominated solutions in order to identify the most performing configurations. Finally, multi-objective evolutionary trajectory optimizations were run for the most promising candidate along intercontinental point-to-point routes of interest. Comparisons with the previous design iteration showed a significant improvement in terms of a reduction of both reentry heat flux and population disturbance.

A Knee-Guided Evolutionary Algorithm for Multi-Objective Air Traffic Flow Management

Air traffic flow management plays a crucial role in efficient aviation. Most existing studies assume the flight speed as constant throughout the trip, leading to ineffective fixed-speed schedules. To address this issue, we propose a new problem model, which allows variable speed control to improve the flexibility and maneuverability of the management. In addition, we consider two conflicting objectives, which are minimizing the total flight delays and conflicts between flights, where the conflicts depend on the flight 4D trajectories (3D position plus time). To solve this new challenging problem, we propose a novel multi-objective evolutionary algorithm with new problem-specific individual representation and search operators. Specifically, the multi-chromosomes encoding scheme is designed to adapt to different types of operations. Then, to search the huge search space effectively, we develop a hybrid crossover operator that recombines the parents based on their flight routes. Furthermore, to balance the exploration and exploitation, we develop a new mutation strategy to utilize the heterogeneous search potential of different individuals. For exploitation, the knee individual in the Pareto front is improved by a new time shift operator for exploitation, and other non-dominated solutions are mutated by fixed-route mutation. For exploration, the dominated solutions are mutated randomly. To verify the effectiveness, we compare it with the real air traffic flow management schedules and the state-of-the-art algorithms on a range of real-world air traffic datasets. Extensive results show that the proposed algorithm can significantly outperform the baselines in generating safe and efficient 4D trajectories.

Solving Multiobjective Feature Selection Problems in Classification via Problem Reformulation and Duplication Handling

Reducing the number of selected features and improving the classification performance are two major objectives in feature selection, which can be viewed as a multi-objective optimization problem. Multi-objective feature selection in classification has its unique characteristics, such as it has a strong preference for the classification performance over the number of selected features. Besides, solution duplication often appears in both the search and the objective spaces, which degenerates the diversity and results in the premature convergence of the population. To deal with the above issues, in this paper, during the evolutionary training process, a multi-objective feature selection problem is reformulated and solved as a constrained multi-objective optimization problem, which adds a constraint on the classification performance for each solution (e.g., feature subset) according to the distribution of nondominated solutions, with the aim of selecting promising feature subsets that contain more informative and strongly relevant features, which are beneficial to improve the classification performance. Furthermore, based on the distribution of feature subsets in the objective space and their similarity in the search space, a duplication analysis and handling method is proposed to enhance the diversity of the population. Experimental results demonstrate that the proposed method outperforms six state-of-the-art algorithms and is computationally efficient on 18 classification datasets.

A Multiobjective Genetic Algorithm to Evolving Local Interpretable Model-Agnostic Explanations for Deep Neural Networks in Image Classification

Deep convolutional neural networks have become a dominant solution for numerous image classification tasks. However, a main criticism is the poor explainability due to the black-box characteristic, which hurdles the extensive usage of deep convolutional neural networks. To address this issue, this paper proposes a new evolutionary multi-objective based method, which aims to explain the behaviours of deep convolutional neural networks by evolving local explanations on specific images. To the best of our knowledge, this is the first evolutionary multi-objective method to evolve local explanations. The proposed method is model-agnostic, i.e. it is applicable to explain any deep convolutional neural networks. ImageNet is used to examine the effectiveness of the proposed method. Three well-known deep convolutional neural networks -VGGNet, ResNet, and MobileNet, are chosen to demonstrate the modelagnostic characteristic. Based on the experimental results, it can be observed that the local explanations are understandable to end-users, who need to check the sensibility of the evolved explanations to decide whether to trust the predictions made by the deep convolutional neural networks. Furthermore, the local explanations evolved by the proposed method improves the confidence of deep convolutional neural networks making the predictions. Lastly, the pareto front and convergence analyses indicate that the proposed method can form a good set of nondominated solutions.

Multi-objective Approach for Dynamic Economic Emission Dispatch Problem Considering Power System Reliability and Transmission Loss Prediction Using Cascaded Forward Neural Network

This study addresses the significant problem of Dynamic Economic Emission Dispatch (DEED), a critical consideration in power systems from both economic and environmental protection viewpoints. Reliability stands as another vital facet, impacting maintenance and operation perspectives. The integration of Artificial Neural Network (ANN)-based transmission loss prediction into the DEED model is also essential to address specific limitations and enhance the overall performance of the dispatch process. Traditionally, the DEED model relies on a single B-loss coefficient to estimate transmission losses. While this approach simplifies calculations, it fails to account for the significant variations in demand that occur throughout the dispatch period and it leads to inaccuracies in loss prediction, especially in dynamic environments. Using a single coefficient, the model cannot adequately capture the complex, non-linear relationships between power generation, load, and transmission losses under different operating conditions. To overcome this limitation, this study introduces an ANN-based loss prediction method integrated into the DEED model and uses trained ANN to replace the process of finding B-loss coefficients during each dispatch period. This paper also introduces a strategy leveraging the multi-objective northern goshawk optimizer algorithm, characterized by a non-dominated sorting and crowding distance mechanism, to enhance DEED considerations incorporating reliability (DEEDR). This novel algorithm improves the solution space effectively, maintains high population diversity and enables an even distribution of individuals sharing the same rank in the objective space. The fundamental objective of this study is to balance fuel cost, emission, and system reliability in power system operations. Compared with a few existing multi-objective optimization algorithms, this study demonstrates superior performance in generating a series of non-dominated solutions. The experimental results highlight its competitive and potential as an efficient tool in the DEED and DEEDR problems, promising a synergistic coordination of economy, environmental protection, and system reliability benefits in power system management.

Leveraging innovization and transfer learning to optimize best management practices in large-scale watershed management

Recent research in evolutionary multi-objective optimization (EMO) highlights the concept of “Innovization”, which identifies essential patterns in high-quality, non-dominated solutions. This study introduces a novel method to pinpoint influential Best Management Practices (BMPs) in the Chesapeake Bay Watershed, optimizing the trade-off solution process. This approach, though innovative, demands considerable expertise and involves generating multiple solutions for expert analysis to detect commonly used BMPs. We devised three re-optimization strategies from these findings using an innovized BMP list, efficiently producing high-quality solutions. We also implemented transfer learning to adapt these strategies for new counties, demonstrating effectiveness in four West Virginia counties by reducing decision variables by 3% to 33% and achieving similar reductions in four additional counties. This showcases the potential of combining innovization with transfer learning to simplify complex optimization challenges, emphasizing its significant applicability in real-world settings.

Experimental investigation and optimization of performance, emission, and vibro-acoustic parameters of SI engine fueled with n-propanol and gasoline blends using ANN-GA coupled with NSGA3-modified TOPSIS hybrid approach

In the present study, performance, emission, and vibro-acoustic studies were conducted on a spark ignition (SI) engine fueled with gasoline and an n-propanol blend at variable compression ratio (CR), speed, and propanol blend fraction (PBF). Experimental data were used to model an artificial neural network (ANN) trained with a genetic algorithm (GA). ANN predictive responses were employed to establish regression relationships between brake power (BP), brake specific fuel consumption (BSFC), brake thermal efficiency (BTE), oxides of nitrogen (NOx), carbon monoxide (CO), hydrocarbon (HC), resultant vibration acceleration (RVA), and sound pressure level (SPL) with operating parameters using response surface methodology (RSM). These models served as objective functions in the non-dominated sorting genetic algorithm-3 (NSGA3), a multi-objective optimization (MOO) technique, to optimize responses and obtain non-dominated solutions. These solutions were filtered using a modified technique for order preference by similarity to the ideal solution (TOPSIS) to obtain a compromised optimal solution. ANN-GA model outcomes showed high accuracy, with coefficient of determination (R2) and root mean square error (RMSE) values ranging from 0.979 to 0.993 and 0.0381 to 0.0643, respectively. NSGA3 coupled with modified TOPSIS identified optimal operating conditions at 1271.77 RPM, a CR of 11.96, and a PBF of 33.26 %.

A Multi-Objective Non-Dominated Sorting Gravitational Search Algorithm for Assembly Flow-Shop Scheduling of Marine Prefabricated Cabins

Prefabricated cabin modular units (PMCUs) are a widespread type of intermediate products used during ship or offshore platform construction. This paper focuses on the scheduling problem of PMCU assembly flow shops, which is summarized as a multi-objective, fuzzy-blocking hybrid flow-shop-scheduling problem based on learning and fatigue effects (FB-HFSP-LF) to minimize the maximum fuzzy makespan and maximize the average fuzzy due-date agreement index. This paper proposes a multi-objective non-dominated sorting gravitational search algorithm (MONSGSA) to solve it. In the proposed MONSGSA, the ranked-order value is used to convert continuous solutions to discrete solutions. Multi-dimensional Latin hypercube sampling is used to enhance initial population diversity. Setting up an external archive to maintain non-dominated solutions while introducing an adaptive inertia factor and a trap avoidance operator to guide individual positional updates. The results of multiple sets of experiments show that Pareto solutions of MONSGSA have better distribution and convergence compared to other competitors. Finally, the instance of PMCU manufacturer is used for validation, and the results show that MONSGSA has better applicability to practical problems.

A dynamic multi-objective evolutionary algorithm with variable stepsize and dual prediction strategies

The prediction strategy is a key method for solving dynamic multi-objective optimization problems (DMOPs), particularly the commonly used linear prediction strategy, which has an advantage in solving problems with regular changes. However, using the linear prediction strategy may have limited advantages in addressing problems with complex changes, as it may result in the loss of population diversity. To tackle this issue, this paper proposes a dynamic multi-objective optimization algorithm with variable stepsize and dual prediction strategies (VSDPS), which aims to maintain population diversity while making predictions. When an environmental change is detected, the variable stepsize is first calculated. The stepsize of the nondominated solutions is expressed by the centroid of the population, while the stepsize of the dominated solutions is determined by the centroids of the clustered subpopulations. Then, the dual prediction strategies combine an improved linear prediction strategy with a dynamic particle swarm prediction strategy to track the new Pareto-optimal front (PF) or Pareto-optimal set (PS). The improved linear prediction strategy aims to enhance the convergence of the population, while the dynamic particle swarm prediction strategy focuses on preserving the diversity of the population. There have also been some improvements made in the static optimization phase, which are advantageous for both population convergence and diversity. VSDPS is compared with six state-of-the-art dynamic multi-objective evolutionary algorithms (DMOEAs) on 28 test instances. The experimental results demonstrate that VSDPS outperforms the compared algorithms in most instances.

Numerical algorithms for generating an almost even approximation of the Pareto front in nonlinear multi-objective optimization problems

A multiobjective optimization problem (MOP) returns a set of non-dominated points, the so-called Pareto front. Since this set is usually infinite, it is impossible to generate it completely in practice. Therefore, a discrete approximation of the Pareto front is created. One of the most important features of this approximation is a uniform distribution of points on the full Pareto front in order to present a wide variety of solutions to the decision maker who chooses a final solution. While a few algorithms consider this property, two algorithms based on the Pascoletti-Serafini (PS) scalarization approach are proposed. In addition, six well-known test problems with convex and non-convex Pareto fronts are considered to show the effectiveness of the proposed algorithms. Their results are compared with some algorithms including Normal Constraint (NC), Benson type, Non-Dominated Sorting Genetic Algorithm-II (NSGA-II), S-Metric Selection Evolutionary Multiobjective Algorithm (SMS-EMOA), Differential Evolution (DE) with Binomial Crossover and MOEA/D-DE. The computational results on CPU time and reasonable distribution of points obtained on the Pareto front show that the presented algorithms perform better than other algorithms on these criteria. In addition, although the proposed algorithms compete closely with some algorithms in terms of CPU time, they have more non-dominated solutions and more appropriate distribution than they do in most problems.

Automatic grouping of wind turbine types via multi-objective formulation for nonuniform wind farm layout optimization using an analytical wake model

This study introduces a novel methodology that aims to minimize the number of different wind turbine types used in nonuniform wind farms. This approach involves a trade-off to tackle the logistical challenges associated with installing, operating, and maintaining such wind farms. To accomplish this, a Gaussian analytical wake model is used in conjunction with evolutionary algorithms (NSGA-II and SMS-EMOA) for multi-objective optimization. A multi-objective problem is formulated, regarding wind farm capacity factor (CF) and levelized cost of energy (LCOE), and addressed using two distinct approaches. In the first approach, proposed here, the number of different wind turbine types is limited and treated as an additional objective. In contrast, the second approach allows for flexibility in choosing any number of different turbine types. Comparing the results obtained, it was observed that the methodology presented here yielded a reduced number of different turbine types without compromising CF and LCOE objectives. A multi-criteria decision maker was applied to extract the final non-dominated solutions from the obtained Pareto set. Finally, economic indicators show the profitability of these solutions.

Adaptive projection plane and reference point strategy for multi-objective particle swarm optimization

Achieving a balance between convergence and diversity and their mutual enhancement is a complex task in the process of algorithm improvement. This is crucial because it is directly related to the effectiveness of the algorithm in obtaining accurate and uniformly distributed Pareto frontiers. Although significant progress has been made in particle swarm algorithms, exploring new approaches is necessary. In this paper, we construct a projection plane (projection line in 2D) based on the extreme values of the non-dominated solutions, select a set of uniform reference points on the projection plane, and then project the non-dominated solutions onto the constructed projection plane to form projection points. The reference points and projection points on the projection plane are thus utilized to guide the updating of the population as well as the maintenance of the external archive, a strategy that enhances the algorithm's global exploration and local exploitation capabilities. Secondly, we aggregate the target values of particles into a single scalar value and combine the idea of particle fusion to design a scheme for the particle selection of individual optimal particles. This paper further improves the algorithm's overall performance by using the information between populations to select individual optimal particles. Lastly, it is evaluated against a number of multi-objective algorithms that are currently in use and perform well on 22 test problems. The findings demonstrate that the algorithm this paper proposes performs better when solving multi-objective problems.

A matheuristic applied to clustering rural properties and allocating plants for biogas generation

Establishing partnerships among agro-industrial properties and selecting ideal locations for biogas plants are crucial challenges in large-scale biogas production and can influence both operational efficiency and waste management. In this context, this research proposes a new matheuristic that addresses the problems of defining a group of properties and an optimal number of groups and identifies the best allocation to the biogas plant. The group properties were defined by hierarchical and K-means cluster algorithms. The best location for the biogas plant was determined by the proposed multiobjective mathematical model. The best cluster number was decided by two strategies: (1) one that selected the closest non-dominated solutions to the ideal solution (M1) and (2) one that favored the most environmentally friendly solution (M2). The matheuristic was tested using three real databases, which yielded strategic clusters with an average daily biogas production of 544.93 m³/day (M1 and M2) for DataBase 1, 1635,156.00 m³/day (M1) and 403,497.50 m³/day (M2) for DataBase 2, and 318,662.50 m³/day (M1) and 20,479.58 m³/day (M2) for DataBase 3. This research provides an opportunity to add value to agro-industrial properties by achieving energy security and developing new business networks.

ReLeC-based clustering and multi-objective optimization for efficient energy optimization in IoT networks

In response to the escalating demand for energy-efficient wireless sensor networks (WSNs) within the expanding Internet of Things (IoT) landscape, we introduce ReLeC-MO, a novel protocol that integrates the ReLeC clustering algorithm with multi-objective optimization. Leveraging reinforcement learning-based clustering, ReLeC optimizes network topology to enhance energy efficiency. Multi-objective optimization further refines this process by identifying non-dominated solutions on the Pareto front, facilitating a balanced trade-off between network lifetime, energy consumption, and data transmission quality. Our comprehensive simulations reveal the remarkable performance improvements achieved by ReLeC-MO over existing techniques. Specifically, ReLeC-MO demonstrates a 39% reduction in delay, a 50% decrease in energy consumption, and a 25% increase in throughput, showcasing its efficacy in enhancing both energy efficiency and network performance. It also increases network lifetime by 20%, surpassing the latest existing model. Furthermore, its implementation in MATLAB ensures ease of replication and adaptation across diverse IoT applications.

Resolution of the bi-objective optimization problem for the dispatch of hydroelectric plants under conditions of low inflow using the NSGA II algorithm

Among the consequences of climate change are increased temperatures and changes in rainfall patterns that bring longer periods of drought. This creates limitations in the administration of hydroelectric plant reservoirs, restricting, in some cases, the amount of electrical energy generated. The objective of this research is to solve the multi-objective optimization problem that seeks to minimize the production of electrical energy from hydroelectric plants with low inflow and, at the same time, minimize electrical rationing due to this low production. As these objectives conflict with each other, it is necessary to apply multi-objective optimization problem-solving methodologies, among which are genetic algorithms. The mathematical model is built considering the operating conditions of the reservoirs of the hydroelectric plants under study, including their minimum operating levels, which are included in the model restrictions. The non-dominated genetic classification algorithm II is used to obtain the Pareto front, which is composed of a total of 78 non-dominated solutions that are useful to manage the considered reservoirs at the time of maximum demand. It is recommended to use other multi-objective optimization algorithms for comparison purposes, selecting the ideal indicators to evaluate the performance of each algorithm used, in addition to incorporating monetary and environmental cost restrictions into the model

A multi-objective mixed integer linear programming approach for simultaneous optimization of cost and resilience of power distribution networks

In recent years, customers have experienced a significant increase in weather-related power outages. The power distribution network (PDN), a subset of the power system, in particular is more susceptible to extreme events. Therefore, ensuring the resilient and cost-effective operation of PDNs following extreme weather conditions poses a significant challenge for distribution network operators. This paper presents an approach for simultaneously optimizing the cost and load restoration for resilience enhancement of power distribution networks while determining the optimal positioning and generation levels of mobile emergency generators. The proposed method, in addition, employs a distribution network reconfiguration to improve the load restoration process, by optimally determining the status of switches. The multi-objective formulation involves the minimization of load shedding to increase the resilience of the system, while the other objective is formulated to minimize the cost of load restoration. A weighted sum method is employed to address the multi-objective mixed-integer linear programming (MILP) model. A set of non-dominated solutions determined using the proposed formulation provides opportunities to the distribution system operator in choosing a resilience improvement strategy according to the availability of the operational budget. The proposed model is implemented on 33-bus distribution system to validate the efficacy of the proposed model.

Improvement of soft clay soil with a combination of pre-gelatinized corn starch and nanoparticle agro waste: A multi-objective grey wolf optimizer approach

The present study aimed to assess the potential of a bio-inspired algorithm, multi-objective grey wolf optimization algorithm (MOGWO), to optimize the strength properties (California bearing ratio (CBR) and unconfined compressive strength (UCS)) of an expansive subgrade soil. This optimization process involves the use of two additives, namely a bio-polymer, pregelatinized corn starch (PGCS), and a nanoparticle agro waste, rice husk ash (RHA), blended with the soil in different mix ratios determined by a 32 factorial experimental design. The CBR samples were cured for 7 days, while the UCS samples were cured for 1, 7, and 28 days. To optimize the expansive subgrade soil strength, regression models were developed using PGCS and RHA as predictors for CBR and UCS, serving as fitness functions in the slightly modified MOGWO optimization technique. Next, the optimization analysis produced non-dominated solutions. The results obtained from the laboratory experiments and optimization analysis revealed that there was significant improvement in the UCS and CBR of the soil. These improvements can be attributed to the pozzolanic reaction between the soil-RHA matrix, the formation of intercalated and exfoliated nanocomposites, and the hydrophilic interaction of PGCS. By applying the slightly modified MOGWO technique, the study achieved optimal enhancements in UCS (710.3 kN/m2) and CBR (24.2%) when the expansive subgrade soil was mixed with 0.2637% PGCS and 12.2413% RHA. The results demonstrate the potential of the MOGWO technique in improving the properties of expansive subgrade soil.