Multi-objective Optimization Problem Research Articles

Strategy for obtaining robust solutions in multi-objective design with uncertainties

This paper proposes a novel definition of robustness for multi-objective optimization problems. This definition underpins an innovative strategy for obtaining robust solutions in the presence of uncertainty; it involves formulating the cost function under uncertainties as conflicting objectives during optimization. This approach aims to define the decision vectors that are not dominated in all scenarios simultaneously by any other. These solutions exhibit both optimality and robustness properties, aligning with conventional and unconventional multi-objective methods. This approach enables the implicit definition of the Pareto-optimal solutions for each scenario and robust solutions that optimize performance in worst-case scenarios. Additionally, the set of robust solutions that optimize the global performance concerning the utopian points of all uncertainty scenarios is also defined.To demonstrate the effectiveness of this method, this paper addresses two control design problems. The first example, a first-order process, illustrates the advantages and aspects of the optimization strategy. The second problem, multi-loop temperature control design of a proton exchange membrane fuel cell stack, is a more complex engineering problem involving results from previous research.

Learnable self-supervised support vector machine based individual selection strategy for multimodal multi-objective optimization

Multimodal multi-objective optimization problem (MMOP) is a frontier research problem, which can provide decision makers with more choices without making trade-offs. Many multimodal multi-objective evolutionary algorithms (MMOEAs) have been proposed to solve MMOP. However, most MMOEAs tend to prioritize the objective dominance of individuals in the process of individual selection, and only individuals with the same objective dominance will be considered the diversity, which leads to the loss of many promising solutions. To solve the above problem, this paper proposes a learnable self-supervised support vector machine (SVM) based multimodal multi-objective optimization algorithm (SVMEA). Support vector machine can learn the knowledge about distinguishing the advantages and disadvantages of individuals from the data in the existing training set and select individuals, in which the objective dominance of individuals is as important as diversity. Moreover, a crowding distance calculation method based on Manhattan distance is designed. Compared with the traditional method using Euclidean distance to calculate crowding distance, it can better evaluate the diversity of individuals in the decision space and assist the selection of elite solutions. Experimental results show that the proposed SVMEA is competitive with seven other advanced MMOEAs on 34 benchmark problems and a practical application problem.

Multi-faceted sustainability improvement in low voltage power distribution network employing DG and capacitor bank

Efficient distribution of active and reactive power in distribution networks is crucial for ensuring that consumer demand is met and that electrical power quality is maintained. This requires careful planning, particularly in terms of optimally placing and sizing distributed generator (DG) and capacitor bank (CB) units. These units play a key role in minimizing power losses and voltage fluctuations within the network. However, implementing DG and CB units in unbalanced distribution networks presents unique challenges due to inherent system unbalances. To address these challenges, this article proposes a method for allocating and sizing DG and CB units in unbalanced distribution networks. The approach accounts for the network's unbalance and targets multiple objectives, such as reducing power losses, improving multi-phase voltage stability, and minimizing phase-to-phase voltage unbalance. The fast and flexible radial power flow (FFRPF) technique is employed to model complex interactions and constraints within the network, leading to a multi-objective optimization problem. This optimization problem is solved using the weight aggregated particle swarm optimization (WA-PSO) method, a variant of particle swarm optimization tailored for multi-objective functions. WA-PSO simplifies the process by combining multiple objectives into a single function using weighted aggregation. The efficacy of this approach is validated on 19, 34, and 123-node unbalanced radial distribution networks (URDNs). The results show significant improvements in power delivery efficiency across all tested networks, especially when DG and CB units are operated simultaneously. Specifically, DG and CB integration led to a reduction in system losses by 89.90 %, 88.42 %, and 86.87 %, and a decrease in three-phase voltage unbalance indices by 88.75 %, 81.68 %, and 39.05 %, for the 19, 34, and 123-node systems, respectively, while maintaining voltage stability within acceptable limits. Additionally, CO2 emissions were reduced by 52 %, 62 %, and 53 % when microturbines were utilized instead of coal-based thermal power plants, further highlighting the environmental benefits of the proposed approach.

Validation of a Model Predictive Control Strategy on a High Fidelity Building Emulator

In recent years, advanced controllers, including Model Predictive Control (MPC), have emerged as promising solutions to improve the efficiency of building energy systems. This paper explores the capabilities of MPC in handling multiple control objectives and constraints. A first MPC controller focuses on the task of ensuring thermal comfort in a residential house served by a heat pump while minimizing the operating costs when subject to different pricing schedules. A second MPC controller working on the same system tests the ability of MPC to deal with demand response events by enforcing a time-varying maximum power usage limitation signal from the electric grid. Furthermore, multiple combinations of the control parameters are tested in order to assess their influence on the controller performance. The controllers are tested on the BOPTEST framework, which offers standardized test cases in high-fidelity emulation models, and pre-defined baseline control strategies to allow fair comparisons also across different studies. Results show that MPC is able to handle multi-objective optimal control problems, reducing thermal comfort violations by between 66.9% and 82% and operational costs between 15.8% up to 20.1%, depending on the specific scenario analyzed. Moreover, MPC proves its capability to exploit the building thermal mass to shift heating power consumption, allowing the latter to adapt its time profile to time-varying constraints. The proposed methodology is based on technologically feasible steps that are intended to be easily transferred to large scale, in-field applications.

An archive-assisted multi-modal multi-objective evolutionary algorithm

The multi-modal multi-objective optimization problems (MMOPs) pertain to characteristic of the decision space that exhibit multiple sets of Pareto optimal solutions that are either identical or similar. The resolution of these problems necessitates the utilization of optimization algorithms to locate multiple Pareto sets (PSs). However, existing multi-modal multi-objective evolutionary algorithms (MMOEAs) encounter difficulties in concurrently enhancing solution quality in both decision space and objective space. In order to deal with this predicament, this paper presents an Archive-assisted Multi-modal Multi-objective Evolutionary Algorithm, called A-MMOEA. This algorithm maintains a main population and an external archive, which is leveraged to improve the fault tolerance of individual screening. To augment the quality of solutions in the archive, an archive evolution mechanism (AEM) is formulated for updating the archive and an archive output mechanism (AOM) is used to output the final solutions. Both mechanisms incorporate a comprehensive crowding distance metric that employs objective space crowding distance to facilitate the calculation of decision space crowding distance. Besides, a data screening method is employed in the AOM to alleviate the negative impact on the final results arising from undesirable individuals resulting from diversity search. Finally, in order to enable individuals to effectively escape the limitation of niches and further enhance diversity of population, a diversity search method with level-based evolution mechanism (DSMLBEM) is proposed. The proposed algorithm’s performance is evaluated through extensive experiments conducted on two distinct test sets. Final results indicate that, in comparison to other commonly used algorithms, this approach exhibits favorable performance.

A method for optimizing different geometric shields of D-T neutron generators by combining BP neural network and Analytic Hierarchy Process

In this paper, an optimization of shielding structures with different geometries is established for the D-T neutron generator system by combining Back Propagation (BP) neural network and Analytic Hierarchy Process (AHP). The D-T neutron generator (Model NG-9) used in the system was developed independently by Northeast Normal University. After investigating the rule of shielding performance among spherical, cylindrical and cubic geometries, the spherical shield is selected for BP neural network prediction to determine the total dose rate through it. Information about spherical multilayer-shielding structures and properties calculated by MCNP code is used to train the neural network. The predicted result serves as a parameter of the evaluation function, which provides a comprehensive assessment of the dose rate penetrated the shield, the shielding mass, and the shielding volume. Together with AHP, the weight factors are determined for all the optimization objectives to construct the evaluation function. By comparing its values, the optimal shielding structures for spherical, cylindrical and cubic materials are found. Against MCNP simulated values, the total dose rates’ errors of the optimal shielding structures for the sphere, cylinder, and cube are 1.72 %, -4.94 %, and -5.17 %, respectively. This result demonstrates that the combination of BP neural network and AHP is more effective in addressing multi-objective optimization problems related to the design of radiation shielding for various geometries.

Multi-objective topological design considering functionally graded materials and coated fiber reinforcement

This study presents a multi-objective topology optimization method tailored to structures fabricated from functionally graded materials (FGMs), coated FGMs, and coated fiber-reinforced composite materials (FRCMs) with fixed fiber thickness. The design objective is the simultaneous minimization of elastic and thermal compliance. The material properties of these composite materials were derived to generate datasets using the representative volume element method under periodic boundary conditions. Subsequently, machine learning modules were developed based on the datasets to combine with the design process. The multi-objective optimization problem was addressed using the weighted sum method ensuring the generation of the Pareto front. The adaptive weighting strategy is employed to avoid biased results toward a single objective function. To define the coated boundaries within the design domain, image post-processing techniques such as convolution filters, interpolation schemes, and erosion methods were employed on the material layout information of the optimized FGM structures. Through numerical examples, optimized material layouts for coated assemblies incorporating FGMs and FRCMs are presented, with the performance verified through objective function values.

A Novel Approach for Power Quality Improvement Using D-STATCOM in Wind Integrated Distribution System

Objectives: The major goal of the proposed article is to enhance the power quality of a wind-integrated radial distribution network utilizing D-STATCOM. It is done by the appropriate allocation of wind turbine-based DGs and D-STATCOM. Methods: The multi-objective optimization problem is implemented using a novel and effective Prairie Dog Optimization (PDO) to attain the best-compromised solutions. This optimization tool is well suited for solving complex, non-linear voltage stability problems. The proposed approach is validated for the modified IEEE-13 bus highly unbalanced system using MATLAB. Findings: The findings indicate enhancements in the voltage profile, system losses, and total harmonic distortion (THD). The THD for the load current is 18%, while the THD for the source current is 2.54% for nonlinear loads using the proposed methodology. The voltage profile of 3.25% is enhanced as compared with the existing method, which is 2.15% better than the other obtainable method. Novelty: Prairie Dog Optimization (PDO) was chosen over traditional algorithms for its nature-inspired approach, simplicity, and efficacy in exploring complicated search fields. Its robust exploration and exploitation skills make it excellent for multi-modal optimization challenges. Keywords: Radial distribution system (RDS), D-STATCOM, Power quality, Prairie Dog Optimization, THD

Multi-Objective Optimization of the Forming Process Parameters of Disc Forgings Based on Grey Correlation Analysis and the Response Surface Method

This paper introduces a multi-objective optimization problem (MPO) for the forming process parameters of disc forgings using grey relational analysis (GRA) and the response surface methodology (RSM). Firstly, an experimental design based on the Box–Behnken design (BBD) principle was established, and simulations were performed in Deform to obtain response data. Secondly, GRA was used to transform the MPO into a grey relational degree (GRD) problem, and the entropic weight method was integrated to ascertain the influence weights of each variable on GRD. Then, a quadratic polynomial prediction model based on the RSM was constructed, and its accuracy was ensured through model validation. Finally, the optimal process parameter combination was determined through the particle swarm optimization algorithm, which included a friction coefficient of 0.3, an initial temperature of 1250 °C, and a downward pressing speed of 7.5 mm/s. The results of the experimental investigation indicate that optimized process parameters significantly reduce the forming load, equivalent stress, and damage value, effectively enhancing the overall quality of forged parts.

Multi-Objective Hybrid Algorithm Integrating Gradient Search and Evolutionary Mechanisms

The current multi-objective evolutionary algorithm (MOEA) has attracted much attention because of its good global exploration ability, but its local search ability near the optimal value is relatively weak, and for optimization prob lems with large-scale decision variables, the number of populations and iterations required by MOEA are very large, so the optimization efficiency is low. Gradient-based optimization algorithms can overcome these problems well, but they are difficult to be applied to multi-objective problems (MOPs). Therefore, this paper introduced random weight function on the basis of weighted average gradient, developed multi-objective gradient operator, and combined it with non-dominated genetic algorithm based on reference points (NSGA- III) proposed by Deb in 2013 to develop multi-objective optimization algorithm (MOGBA) and multi-objective Hybrid Evolutionary algorithm (HMOEA). The latter greatly enhances the local search capability while retaining the good global exploration capability of NSGA-III. Numerical experiments show that HMOEA has excellent capture capability for various Pareto formations, and the efficiency is improved by times compared with typical multi-objective algorithms. And further HMOEA is applied to the multi-objective aerodynamic optimization problem of the RAE2822 airfoil, and the ideal Pareto front is obtained, indicating that HMOEA is an efficient optimization algorithm with potential applications in aerodynamic optimization design.

A staged fuzzy evolutionary algorithm for constrained large-scale multiobjective optimization

Constrained multiobjective optimization problems (CMOPs) are prevalent in practical applications, where constraints play a significant role. Building on techniques from constrained single-objective optimization, classical methods such as the constrained dominance principle have been extended to CMOPs. However, these methods struggle with CMOPs characterized by complex infeasible regions. Furthermore, as the number of decision variables increases, the search efficiency of algorithms deteriorates dramatically. To solve those issues, we propose a staged fuzzy evolutionary algorithm (i.e., SFEA) for constrained large-scale problems. To balance exploration and exploitation, a fuzzy stage adjustment strategy based on the sigmoid function is proposed. Furthermore, this article develops an improved fuzzy operator to perform fuzzy operations on various vectors (e.g., solutions or constraint violations). Computational experiments were conducted on CMOP test suites with up to 500 decision variables and a series of real-world applications. The experimental results demonstrate that, compared to existing peer algorithms, our algorithm exhibits superior or competitive performance.

Multiobjective optimization by using cutting angle methods and hypervolume criterion

We translate a multiobjective optimization problem into a single objective Lipschitz problem by using the hypervolume criterion of the Pareto set. Deterministic global optimization methods allow one to track the whole Pareto front rather than converging to a single non-dominated solution. We augmented an efficient hypervolume computation technique with hypervolume increment strategy, and established Lipschitzianity of the resulting objective function. We used two deterministic Lipschitz optimization methods together with the hypervolume objective and benchmarked them against some state-of-the-art alternative multiobjective optimization methods, establishing the competitiveness of the proposed approach.

An adaptive co-evolutionary competitive particle swarm optimizer for constrained multi-objective optimization problems

In constrained multi-objective optimization problems, it is challenging to balance the convergence, diversity and feasibility of the population, especially encountering complex infeasible regions. In order to effectively balance the three indicators, from the aspects of the handling of infeasible solution and the quality of individuals, a multi-population co-evolutionary competitive particle swarm optimization algorithm hybridized with infeasible solution transfer and an adaptive technique (ACCPSO) is proposed. Firstly, the information of feasible and infeasible individuals is fully utilized and the individuals are classified by Hamming distance. Then, a novel constraint handling technique based on learning from the promising feasible direction is designed to make individuals cross large infeasible regions and explore more potential feasible regions. Moreover, aiming to provide robust search capability and consequently further generate high-quality solutions, the genetic operators and the particle swarm optimization operator with the competitive mechanism are introduced as operators with an adaptive mechanism. Finally, compared with the state-of-the-art methods, the performance of the proposed algorithm is verified on LIR-CMOP, MW and DTLZ, as well as two real-world problems. The results indicate that ACCPSO exhibits stronger competitiveness in terms of convergence, the solution quality, and distribution diversity on the feasible Pareto front.

Hybrid extreme gradient boosting regressor models for the multi-objective mixture design optimization of cementitious mixtures incorporating mine tailings as fine aggregates

The design of cementitious mixtures incorporating mine tailings as fine aggregates is a multi-objective optimization (MOO) problem, in which both the uniaxial compressive strength (UCS) and cost of the mixtures need to be considered simultaneously. Given that data-driven methods have shown promising results when solving similar MOO problems, this study developed an extreme gradient boosting regressor (XGBR) model on a dataset extracted from the literature to predict the UCS. Among the efforts taken to improve the models, a genetic algorithm (GA)-based XGBR model demonstrated the optimal prediction performance, with an R2 of 0.959. Next, the GA-XGBR model and a cost equation were used as objective functions in the MOO problem. The non-dominated sorting genetic algorithm with elite strategy (NSGA-II) was selected to solve the optimization problem. A case study was conducted, generating mixture designs that offered improved trade-offs between cost and UCS compared to experimental designs. Finally, a graphical user interface was developed to provide access to the prediction model and optimization method. Overall, this work can be used as a guide for optimal mixture designs as it facilitates informed decision-making before the actual applications.

A comparative study of evolutionary algorithms and particle swarm optimization approaches for constrained multi-objective optimization problems

Many real-world optimization problems contain multiple conflicting objectives as well as additional problem constraints. These problems are referred to as constrained multi-objective optimization problems (CMOPs). Many meta-heuristics for solving CMOPs, called constrained multi-objective meta-heuristics (CMOMHs) have been introduced in the literature, including those using particle swarm optimization (PSO)(Kennedy and Eberhart, 1995), genetic algorithms (GAs)(Man et al., 1996), and differential evolution (DE)(Storn and Price, 1997). CMOMHs can be grouped into four different classes: classic CMOMHs, co-evolutionary approaches, multi-stage approaches, and multi-tasking approaches. An extensive comparative study of twenty different CMOMHs on a wide variety of test problems, including real-world CMOPs in the fields of science and engineering, is conducted. A multi-swarm PSO approach called constrained multi-guide particle swarm optimization (ConMGPSO) is introduced and compared to the best-performing previous approaches according to the comparative study. The performance of each algorithm was found to be problem dependent, however the best overall approaches were ConMGPSO, paired-offspring constrained evolutionary algorithm (POCEA)(He et al., 2021), adaptive non-dominated sorting genetic algorithm III (A-NSGA-III)(Jain and Deb, 2014), and constrained multi-objective framework using Q-learning and evolutionary multi-tasking (CMOQLMT)(Ming and Gong, 2023). ConMGPSO and POCEA had the best performance on the CF benchmark set, which contains examples of bi-objective and tri-objective CMOPs with disconnected CPOFs. The CMOQLMT approach had the best performance on the DAS-CMOP benchmark set, which contain additional difficulty in terms of feasibility-, convergence-, and diversity-hardness. For the selected real-world CMOPs, A-NSGA-III had the best performance overall. ConMGPSO was shown to have the best performance on the process, design, and synthesis problems, and had competitive performance for the power system optimization problems.

A Generalization of the Concept of Proper Efficiency in the Decomposed Multi-Objective Optimization Problems

In the context of multi-objective optimization, a properly efficient solution is one that is efficient while at least one of the tradeoffs between different objectives is limited. However, in some situations, it is possible that all efficient solutions have unlimited tradeoffs, which is not always appropriate. To provide new solutions that have at least one bounded tradeoff for such problems, we first introduce a new concept of efficiency by applying a family of scalarization functions for the decomposed multi-objective optimization problem. Then, we extend the concept of proper efficiency by examining the boundedness of the tradeoffs between scalar optimization subproblems and other subproblems. Another purpose of this paper is to investigate the effect of using the scalarizing functions for some subproblems in the decomposed multi-objective optimization problem. Our findings suggest that as the number of subproblems that use scalarizing functions increases, the solution set associated with the generalized proper efficiency concept becomes smaller.

A reinforcement learning-based multiobjective heuristic algorithm for multiple-truck routing problems with heterogeneous drones

As a result of new technological developments, drone usage has become common in last-mile logistics activities. Owing to the limited flight range of drones, drones may service customers by using trucks as hubs. Although the truck routing problem with drones (TRP-D) has been studied frequently, studies that have addressed this problem in a multiobjective manner are rare. In this study, multiobjective TRP-D is discussed. In this problem, multiple trucks are considered. Each truck is equipped with multiple heterogeneous drones. A multiobjective mixed-integer linear programming (MILP) model is proposed. Since the problem is NP-hard, a reinforcement learning (RL)-based multiobjective heuristic algorithm is proposed to obtain solutions in large-scale cases. A three-phase heuristic feasible-solution-generation algorithm is proposed. The variable neighbourhood descent (VND) algorithm with RL is proposed as a local search algorithm for the multiobjective optimization problem. The results of the proposed heuristic algorithm are compared with the Pareto optimal solutions for small-sized test problems via the augmented ε-constraint (AUGMECON) method and the CPLEX solver. The proposed algorithm is also compared with state-of-the-art multiobjective heuristic algorithms for large-scale problems. According to the results, the proposed algorithm is an efficient multiobjective heuristic algorithm for the addressed problem.

Pareto-critical equilibrium condition for non-linear multiobjective optimization problems with applications

The new (necessary) Pareto-critical equilibrium condition (PCEC) discussed in this study is based on the Pareto-critical condition for non-linear optimization problems. The proposed condition is evaluated by defining a multiobjective optimization problems (MOOPs) subject to a set of constraints. Then this MOOPs is converted to a single objective optimization problem by using the available optimization techniques, for instance, the weighted sum method, and solved by the non-linear Nelder-Mead simplex method. The obtained solution points satisfy the proposed PCEC of an unconstrained multiobjective optimization problem (MOOPs) under the convexity assumption. This PCEC is evaluated in a search that either results in a point inside the feasible region or a point for which the PCEC holds. The resulting Pareto-critical equilibrium condition is globally valid for convex problems. Numerical experiments are carried out in order to clarify the proposed condition's validity. Moreover, the proposed PCEC is used to place a downlink transmit beamforming system base station in an ideal position as a first application and is also employed in situations where there are multiple agents working towards a common goal as a second application.

Maximizing the spread of information through content optimization

As data-driven prediction models advance, an increasing number of people are enjoying news personalized to their interests. The primary problem such recommendation models solve is to precisely match information with users and, in so doing, ensure that news spreads with greater efficiency. However, these techniques only help the media platform; they do not help those who produce the news. Hence, we devised a propagation framework based on a human-in-the-loop simulation that helps content authors maximize the spread of their messages through social networks. The framework works by acting on feedback provided by the simulation model. Additionally, the spread of information is formulated as a multi-objective optimization problem in which propagation is data-driven and simulated with machine learning techniques that leverage data on the historical behaviors of users. We additionally describe an implementation for this framework as an example of how the framework might be used in real life. On the practical side, the implementation uses text data from a blog to simulate the message's propagation, while, from a technical point of view, the multi-objective optimization problem is divided into an information retrieval problem and an integer programming problem, the results of which are fed back into the content editor as content operation strategies. A case study with the Sina Weibo microblog site not only validates the framework but also provides practitioners with insights into how to maximize the spread of information through social networking platforms. The results show that the proposed propagation framework is capable of increasing retweets by 7.9575 %. As an interesting aside, our experiments also show that the Weibo retweet lottery is both popular and a highly effective mechanism for increasing reposts.

Solving Multi-Objective Optimal Control Problems Using a Multiresolution Approach

This paper presents an adaptive multiresolution strategy for multi-objective optimal control problems. The optimal control problem is solved using a direct approach, with individualistic grid adaptation facilitated by a local error analysis at element boundaries. Multiple objectives are considered using a dominance-based approach applying both local and global search methods to a collaborative population of unique solutions. These aspects are simultaneously incorporated via a novel application of evolutionary algorithms for adaptive optimal control problems. Together, this avoids the need for a priori specification of the quantity and temporal location of element boundaries and the set of scalarization weights defining the multi-objective descent directions. Solution fidelity can thus increase concurrently with the exploration of the design space, which leads to increased numerical efficiency while propagating and maintaining population diversity. The benefits of the proposed approach over traditional uniform-grid implementations are demonstrated. Results show that the multiresolution approach is capable of striking an effective balance between solution fidelity, population diversity, and computational cost unachievable using uniform grids.