Non-dominated Solutions Research Articles

Multi-Objective Evolutionary Algorithm to Optimize IoT Based Scheduling Problem Using (NSGA-II Algorithm)

Due to the continual advancements in the Internet of Things (IoT), which generate enormous volumes of data, the cloud computing infrastructure recently has received the most significance. to meet the demands made by the network of IoT devices. It is anticipated that the planned Fog computing system would constitute the next development in cloud computing. The optimal distribution of computing capacity to reduce processing times and operating costs is one of the tasks that fog computing confronts. In the IoT, fog computing is a decentralized computing approach that moves data storage and processing closer to the network's edge. This research article discusses a unique technique for lowering operating expenses and improving work scheduling in a cloud-fog environment. Non-dominated sorting genetic algorithm II (NSGA-II) is a proposal that is presented in this paper. Its purpose is to allocate service requests with the multi-objective of minimising finishing time and running cost. Determining the Pareto front that is associated with a group of perfect solutions, which are sometimes referred to as non-dominated solutions or Pareto sets, is the fundamental objective of the Pareto NSGA-II. There is a contradiction between the environmental and economic performances, which is shown by the Pareto set of sub-optimal solutions that are the consequence of the bi-objective issue.

Modelling smart energy consumption with hybrid demand management in off-grid electrical system considering techno-economic indices

This study proposes day-ahead power scheduling for electrical systems in off-grid mode, emphasizing consumer involvement. Bi-Demand Side Management (DSM) approaches like strategic conversion and demand shifting are proposed for consumer involvement. Multiple objectives are modelled to voltage profile improvement and reduce the operation energy cost. The non-dominated solutions of the voltage of buses and operation energy cost are generated by enhanced epsilon-constraint technique, simultaneously. The General Algebraic Modeling System (GAMS) software is proposed for solving optimization problems. A combination of decision-making methods like weight sum and fuzzy procedures are implemented for finding optimal solution non-dominated solutions. The proposed method’s effectiveness is confirmed through numerical simulations carried out on several case studies that utilize the 33-bus electrical system. The findings illustrate the substantial effectiveness of demand-side participation in improving power dispatch and the optimal rate of multiple objectives. By using DSM, operation cost is reduced by 21.58% and the voltage index is improved by 13.36% than the lack of implementing DSM.

A novel multi-objective stochastic model and a novel hybrid metaheuristic for designing supply chain network under disruption risks to enhance supply chain resilience

Supply chains are vulnerable to various disruption risks that can adversely affect their overall performance and objectives. This study addresses the challenge of designing a resilient and environmentally sustainable mixed open and closed-loop supply chain network that can withstand both operational and disruption risks. The research employs a bi-objective stochastic mathematical model to examine the balance between environmental sustainability and profitability within the SC. To mitigate the impact of disruptions, several resilience strategies are incorporated into the model, significantly reducing their adverse effects. Due to the inherent complexity of the problem, the study introduces a novel hybrid metaheuristic algorithm that combines ant colony optimization with teaching and learning-based optimization, named ACO-TLBO. Additionally, two other enhanced hybrid metaheuristics are proposed. The performance of these solution methods is assessed through various test problems, using specific performance metrics for comparison. Results reveal that the ACO-TLBO algorithm excels in generating high-quality, non-dominated solutions. The model's practical applicability is demonstrated through a case study in the tire industry, validating its effectiveness. The findings indicate that the proposed resilience strategies are crucial for minimizing the negative impacts of disruptions on SC objectives. Furthermore, the results underscore the importance of resilience in maintaining both sustainability and profitability within supply chains.

Surrogate-Assisted Multi-Objective Evolutionary Optimization With Pareto Front Model-Based Local Search Method.

Some local search methods have been incorporated into surrogate-assisted multi-objective evolutionary algorithms to accelerate the search toward the real Pareto front (PF). In this article, a PF model-based local search method is proposed to accelerate the exploration and exploitation of the PF. It first builds a predicted PF model with current nondominated solutions. Then, some sparse points in the predicted PF are selected to guide the search directions of the local search in order to promote the search of promising sparse areas. The approximation degree of the predicted and real PFs will influence the speed of the local search, while extreme points can significantly influence the shape of the PF. To accelerate the search progress, the optima of surrogate models are utilized to promote the progress of finding extreme points. The proposed local search method is incorporated into a surrogate-assisted multi-objective evolutionary algorithm. The proposed surrogate-assisted multi-objective evolutionary algorithm with the proposed local search method is tested with Zitzler-Deb-Thiele (ZDT), Deb-Thiele-Laummans-Zitzler (DTLZ), and MAF instances. The experimental results demonstrated the efficiency of the proposed local search method and the superiority of the proposed algorithm.

A sustainable bi-objective inventory model with source-based emissions and plan-based green investments under inflation and the present value of money

The paper develops a finite-horizon inventory model with source-based emissions, plan-based green investments under inflation, and the present value of money. The cap-and-trade policy is used as the carbon policy. The model is solved in a bi-objective scenario where the two objectives are maximization of the present value of net profit and minimization of the total emission. We find the Pareto optimal solutions represented by a Pareto front using the ε-constraint method. A flowchart is provided to find the non-dominated solutions. Pareto solutions for three special cases (no inflation, carbon tax, and no green investments) are also derived. In our sensitivity analyses, we observe that the carbon quota does not affect the optimal policy. It only affects the optimum profit. Our model shows that green investment is beneficial for the polluting firm and also for the environment.

Research on location-routing optimization of distribution center for emergency supplies based on IMOCS-LNS hybrid algorithm

This paper establishes a location-routing optimization model of the distribution center for emergency supplies with the goals of system reaction time, total cost of consumption, psychological fear of the populace in disaster-affected locations, and material usage rate. Where the excess time, demand, and penalty coefficient are the components of the penalty cost in the total consumption cost, and where the psychological panic of those in the affected area is represented by the psychological perception function of panic developed in accordance with the prospect theory. An improved hybrid multi-objective cuckoo-large-neighborhood search algorithm was then designed to introduce tent mapping, nonlinear inertia weights, elite strategies, congestion operators, and dynamically adjusted discovery probabilities into the standard multi-objective cuckoo optimization algorithm, which generates a new solution using a large-neighborhood search algorithm after discarding part of the solution with the discovery probability, and then accepts the current nondominated solution with dynamic probabilities. The paper uses the improved algorithm to solve Christofides69, an arithmetic example from the standard dataset of the LRP problem, and the results show that the solution provided by the improved algorithm outperforms the solutions provided by the standard multi-objective cuckoo search algorithm and the NSGA-II algorithm in terms of the total cost of dissipation, the level of psychological panic of the people in the affected area, the rate of utilization of the supplies, and the number of distribution centers open. Finally, the improved algorithm was used to analyze cases of different sizes separately, and it was found that the algorithm yielded better results and was therefore able to demonstrate its effectiveness.

An improved vibrating particles system method for many-criteria engineering design applications

Optimization is getting more and more important due to its application in real engineering problems. Recently, the vibrating particles system algorithm has been developed as an efficient method for mono-objective optimization. However, in multi- and many-objective design problems, the vibrating particles system method is unable to handle simultaneously the conflicting objectives. The second drawback of the vibrating particles system algorithm is the~variability of the obtained results at each independent test, due to its inability to balance exploitation and exploration capabilities. To address these issues, this paper proposes an enhanced vibrating particles system algorithm called the many-objective vibrating particles system algorithm. The proposed many-objective vibrating particles system algorithm uses the Pareto principle to store the non-dominated solutions of multiple conflicting functions. Moreover, the implementation of the particle position enhancement mechanism to boost this algorithm's exploitation and exploration capabilities is another distinctive aspect of the suggested method. A variety of high-dimensional test functions and engineering design problems are used to evaluate the efficiency of the many-objective vibrating particles system algorithm. The obtained results show that the proposed algorithm outperforms other popular methods in terms of convergence characteristics and global search ability.

A Knee Point‐Driven Many‐Objective Evolutionary Algorithm with Adaptive Switching Mechanism

The Pareto dominance‐based evolutionary algorithms can effectively address multiobjective optimization problems (MOPs). However, when dealing with many‐objective optimization problems with more than three objectives (MaOPs), the Pareto dominance relationships cannot effectively distinguish the nondominated solutions in high‐dimensional spaces. With the increase of the number of objectives, the proportion of dominance‐resistant solutions (DRSs) in the population rapidly increases, which leads to insufficient selection pressure. In this paper, to address the challenges on MaOPs, a knee point‐driven many‐objective evolutionary algorithm with adaptive switching mechanism (KPEA) is proposed. In KPEA, the knee points determined by an adaptive strategy are introduced for not only mating selection but also environmental selection, which increases the probability of generating excellent offspring. In addition, to remove dominance‐resistant solutions (DRSs) in the population, an interquartile range method is adopted, which enhances the selection pressure. Moreover, a novel adaptive switching mechanism between angle‐based selection and penalty for selecting solutions is proposed, which is aimed at achieving a balance between convergence and diversity. To validate the performance of KPEA, it is compared with five state‐of‐the‐art many‐objective evolutionary algorithms. All algorithms are evaluated on 20 benchmark problems, i.e., WFG1‐9, MaF1, and MaF4‐13 with 3, 5, 8, and 10 objectives. The experimental results demonstrate that KPEA outperforms the compared algorithms in terms of HV and IGD in most of the test instances.

NSGA-T: A novel evaluation method for renewable energy plans

In the background of exhaustion of the traditional fossil energy sources and environmental deterioration, developing renewable energy has become a strategic choice for countries to achieve energy sustainable utilization and carbon neutrality target. Different renewable energy technical plans have different characteristics under multiple criteria. Therefore, before the further exploitation of renewable energy sources, it is of great significance to evaluate the comprehensive performance of different plans and then determine the best renewable energy sources. However, the commonly used weight calculation methods in the existing renewable energy evaluation have obvious shortcomings, such as the randomness of the subjective method is strong and the gap in the weight allocation of the objective method is too large, which affect the reliability and accuracy of the sorting result. In this paper, a novel weight calculation method based on non-dominated sorting genetic algorithm (NSGA-II) is proposed, and is applied to renewable energy evaluation. The study in this paper focuses on methodology. Firstly, for the linear normalized and Zeros normalized evaluation data, the differences between renewable energy plans in the two normalized evaluation data are measured, and the ranking deviations of the two normalized data are calculated. Then, the multi-objective weight solving model is constructed in terms of plans differences minimization and ranking vectors deviations minimization. The NSGA-II algorithm is applied to the optimization of multi-objective functions to find the Pareto non-dominated solution, so as to determine the weight values of each evaluation index. Finally, the TOPSIS method is used to determine the relative closeness value of each plan, and the rankings of renewable energy technical plans are achieved. Based on the actual renewable energy development data of a province in China, experiments were carried out to investigate the effectiveness of the proposed method. Experimental results show that the proposed method performs better than some popular renewable energy evaluation methods.

Grid-based artificial bee colony algorithm for multi-objective job shop scheduling with manual loading and unloading tasks

This paper investigates a multi-objective job shop scheduling problem with manual loading and unloading tasks (MOJSSPLU) aiming to minimize makespan and total workload simultaneously. Taking human factors into consideration achieves a better balance between production efficiency and resource utilization. To address this problem, we first establish a bi-objective mixed-integer linear programming (MILP) model. Afterward, we combine the artificial bee colony algorithm with the grid technique to develop a grid-based artificial bee colony algorithm (GABC) for solving MOJSSPLU. The algorithm uses a decomposition approach with the earliest-shortest dispatching rule to reduce the complexity of MOJSSPLU. A grid coordinate system is constructed to divide the objective space into cells, facilitating individuals’ location and evaluation in the decision space. Moreover, we focus on improving the cooperation between the employed and onlooker bees and design a mathematical formula based on priority weights to generate onlooker bees to maximize information utilization and improve the exploration and exploitation capabilities of the algorithm. To ensure that the most promising individuals are retained and protected from being eliminated during the evolutionary, we propose an elitist non-dominated solutions reservation strategy for improving the diversity of the population. We further propose a dynamic adjustment of the grid-division method that focuses on improving the exploitation capacity of the algorithm in the objective space to enhance the diversity and convergence of the population. To test the performance of the GABC algorithm, we compare it to the MILP approach and other algorithms using modified benchmark instances. Furthermore, we conduct the Friedman and Wilcoxon nonparametric tests to validate the comparison results among the different algorithms. The computational results demonstrate that the algorithm GABC is efficient.

A many-objective evolutionary algorithm assisted by ideal hyperplane

In many-objective optimization problems (MaOPs), balancing convergence and diversity while rapidly converging to the Pareto front is an arduous task for evolutionary algorithms. In addition, with the increase of the number of targets, the number of non-dominant solutions increases exponentially, and the individual selection pressure is insufficient. For this problem, we propose a many-objective evolution algorithm assisted by an ideal hyperplane (MaOEA-IH). To begin, the ideal hyperplane is built from the extremums of each dimension of objective space, guiding the individual to the Pareto front of search. Second, a parallel p-norm mating selection strategy based on the ideal hyperplane is proposed to improve convergence. In addition, two other factors are taken into account: (1) different p-norms are used to measure different spatial scales; and (2) individual selection uncertainty is defined by incorporating a probabilistic perturbation mechanism. Following that, the sum of objectives is applied to shift-based density estimation, which serves as an evaluation criterion in the environmental selection operation. This method increases the chances of solutions with high convergence and diversity entering the next generation, thereby increasing selection pressure. On three benchmark problems of DTLZ, WFG, and MaF, we compare MaOEA-IH with seven excellent algorithms. The results demonstrate that the MaOEA-IH proposed is highly competitive in solving MaOPs.

Multi-objective dragonfly algorithm for optimizing a sustainable supply chain under resource sharing conditions

This research addresses optimizing production, inventory, location, and routing in a multi-level supply chain. In the studied supply chain, products are first sent to several distribution centers (DC) after production. Next, these products are delivered to several companies. Companies are divided into two independent groups considering sharing logistics resources. In this regard, vehicle routing optimization is applied for each group. For this purpose, a mixed-integer linear programming model has been developed. This model simultaneously minimizes total supply chain costs and total negative environmental impacts. A multi-objective dragonfly algorithm (MODA) has been developed to solve this model. Then, the implementation results are compared with the Non-Dominated Sorting Genetic Algorithm and Epsilon Constraint Method (EPC). Comparing the solution methods based on numerous test problems shows that the proposed MODA algorithm can be examined differently. The MODA algorithm has been able to have a significant advantage in providing solutions close to ideal points. Moreover, this algorithm has provided the most effective Pareto solutions in different test problems. However, the NSGA-II algorithm performs better regarding the spread of non-dominated solutions.

Data-driven simulation-based decision support system for resource allocation in industry 4.0 and smart manufacturing

Data-driven simulation (DDS) is fundamental to analytical and decision-support technologies in Industry 4.0 and smart manufacturing. This study investigates the potential of DDS for resource allocation (RA) in high-mix, low-volume smart manufacturing systems with mixed automation levels. A DDS-based decision support system (DDS-DSS) is developed by incorporating two RA strategies: simulation-based bottleneck analysis (SB-BA) and simulation-based multi-objective optimization (SB-MOO). To enhance the performance of SB-MOO, a unique meta-learning mechanism featuring memory, dynamic orthogonal array, and learning rate is integrated into the NSGA-II, resulting in a modified version of the NSGA-II with meta-learning (i.e., NSGA-II-ML). The proposed DSS also benefits from a post-optimality analysis that leverages a clustering algorithm to derive actionable insights. A real-life marine engine manufacturing application study is presented to demonstrate the applicability and exhibit efficacy of the proposed DSS and NSGA-II-ML. To this aim, NSGA-II-ML was tested against the original NSGA-II and differential evolution (DE) algorithm across a set of test problems. The results revealed that NSGA-II-ML surpassed the other two in terms of the number of non-dominated solutions and hypervolume, particularly in medium and large-sized problems. Furthermore, NSGA-II-ML achieved a 24% improvement in the best throughput found in the real case problem, outperforming SB-BA, NSGA-II, and DE. The post-optimality analysis led to the extraction of valuable knowledge about the key, influencing decision variables on the throughput.

Optimizing oil spill emergency logistics: a time-varying multi-resource collaborative scheduling model.

Emergency resource scheduling is at the heart of the response to an oil spill, as it lays the foundation for all other emergency operations. Extant studies address the dynamicity inherent to these operations primarily by modeling a dynamic network flow with static data, which is not applicable to continuously changing conditions resulting from oil film movement. To enhance the responsiveness and cost-efficiency of the response to oil spills, this paper takes a novel approach and formulates a multi-objective location-routing model for multi-resource collaborative scheduling, namely, harnessing time-varying parameters rather than static data to model real-time changes in the demand for emergency resources and the transportation network. Additionally, the model considers various operational factors, including the transportation of multiple resources in the order of operating procedures; the coordination of split delivery with the consumption of emergency resources; and the matching of multiple resources with suitable vehicles. To solve the proposed model, a hybrid heuristic algorithm of PSO-PGSA is developed, which utilizes particle swarm optimization (PSO) to search widely for non-dominated solutions. The algorithm then makes use of the plant growth simulation algorithm (PGSA) to find the more effective vehicle routes based on the obtained solutions. Finally, a numerical analysis is used to illustrate the practical capabilities of the developed model and solution strategies. Most significantly, our work not only validates the methodology proposed here but also underlines the importance of incorporating the features of an oil spill emergency response into emergency logistics in general.

A bi-criteria approach to the truck-multidrone routing problem

In the rapidly expanding field of e-commerce logistics, the optimisation of last-mile delivery solutions is paramount. This paper introduces a novel methodology that addresses this challenge by generating approximate Pareto fronts in a hybrid truck-drone delivery system. Specifically, we examine a generalized single truck multi-drone problem that allows multi-visit flight missions and rendezvous points distinct from launch locations. Our goal is providing decision-makers with a portfolio of optimal routing solutions that balance service time and environmental impact, criteria that are increasingly shaping decision-making in this domain. To achieve this, we introduce a bivector coding scheme inspired by flow-shop scheduling problems and implement a Simulated Annealing algorithm. This algorithm features an advanced stopping mechanism, negating the need for manual adjustments by utilizing a distinctive blend of a domination rate and a Kalman filter. Importantly, our framework employs an iterated greedy search algorithm to evolve from initial solutions towards identifying non-dominated solutions sets, which are then ranked using a hypervolume coefficient. To validate our methodology, we conduct a sensitivity analysis on two different size instances using a full factorial design of experiments. Our analysis reveals crucial insights into the impact of the number of drones, their autonomy, and their flight speed settings. From it, we conclude that it is a robust and adaptable framework for its practical application for obtaining Pareto fronts solutions among which picking the ultimate routing to be implemented.

Solving Fuzzy Optimization Problems Using Shapley Values and Evolutionary Algorithms

The fusion of evolutionary algorithms and the solution concepts of cooperative game theory is proposed in this paper to solve the fuzzy optimization problems. The original fuzzy optimization problem is transformed into a scalar optimization problem by assigning some suitable coefficients. The assignment of those coefficients is frequently determined by the decision-makers via their subjectivity, which may cause some biases. In order to avoid these subjective biases, a cooperative game is formulated by considering the α-level functions of the fuzzy objective function. Using the Shapley values of this formulated cooperative game, the suitable coefficients can be reasonably set up. Under these settings, the transformed scalar optimization problem is solved to obtain the nondominated solution, which will depend on the coefficients. In other words, we shall obtain a bunch of nondominated solutions depending on the coefficients. Finally, the evolutionary algorithms are invoked to find the best nondominated solution by evolving the coefficients.

Multi-objective optimization of truss structure using multi-agent reinforcement learning and graph representation

This paper presents a novel method for multi-objective optimization of truss design utilizing multi-agent reinforcement learning and graph representation. The agents are trained to modify solutions to increase the hypervolume and spread the distribution of the non-dominated solutions. This iterative modification is modeled as a Markov game where multiple agents interact with the environment by changing the state of the environment, i.e., the solution and the current non-dominated solutions. Agents observe data representations of the state, modify each solution in the current non-dominated solutions, and receive their reward based on the improvement of the current non-dominated solutions. Each agent is modeled by the multi-agent deep deterministic gradient consisting of a policy function that predicts actions from its observation data and a value function that estimates rewards from the observation data, its action, and the actions of the other agents. The non-dominated solutions are represented as graph data and observed by the agents as the observation data through graph representation. The proposed method is applied to three multi-objective optimization problems: (1) a simple mathematical problem, (2) a 10-bar truss problem to minimize the structural volume and the displacement, and (3) trade-off designs of trusses to minimize the structural volume and bring the shape closer to the target shape. These numerical examples show the versatility of the trained agents to obtain superior solutions in practical structural optimization problems, compared to a conventional method.

Multipopulation-Based Differential Evolution for Large-Scale Many-Objective Optimization.

In recent years, numerous efficient many-objective optimization evolutionary algorithms have been proposed to find well-converged and well-distributed nondominated optimal solutions. However, their scalability performance may deteriorate drastically to solve large-scale many-objective optimization problems (LSMaOPs). Encountering high-dimensional solution space with more than 100 decision variables, some of them may lose diversity and trap into local optima, while others may achieve poor convergence performance. This article proposes a multipopulation-based differential evolution algorithm, called LSMaODE, which can solve LSMaOPs efficiently and effectively. In order to exploit and explore the exponential decision space, the proposed algorithm divides the population into two groups of subpopulations, which are optimized with different strategies. First, the randomized coordinate descent technique is applied to 10% of individuals to exploit the decision variables independently. This subpopulation maintains diversity in the decision space to avoid premature convergence into local optimum. Second, the remaining 90% of individuals are optimized with the nondominated guided random interpolation strategy, which interpolates individual among three nondominated solutions randomly. The strategy can guide the population convergent toward the nondominated solutions quickly, meanwhile, maintain good distribution in objective space. Finally, the proposed LSMaODE is evaluated on the LSMOP test suites from the scalability in both decision and objective dimensions. The performance is compared against five state-of-the-art large-scale many-objective evolutionary algorithms. The experimental results show that LSMaODE provides highly competitive performance.

Density Function-Based Trust Region Algorithm for Approximating Pareto Front of Black-Box Multiobjective Optimization Problems

In this paper, we consider a black-box multiobjective optimization problem, whose objective functions are computationally expensive. We propose a density function-based trust region algorithm for approximating the Pareto front of this problem. At every iteration, we determine a trust region and then in this trust region, select several sample points, at which are evaluated objective function values. In order to obtain non-dominated solutions in the trust region, we convert given objective functions into one function: scalarization. Then, we construct quadratic models of this function and the objective functions. In current trust region, we find optimal solutions of all single-objective optimization problems with these models as objectives. After that, we remove dominated points from the set of obtained solutions. In order to estimate the distribution of non-dominated solutions, we introduce a density function. By using this density function, we obtain the most “isolated” point among the non-dominated points. Then, we construct a new trust region around this point and repeat the algorithm. We prove convergence of proposed algorithm under the several assumptions. Numerical results show that even in case of tri-objective optimization problems, the points generated by proposed algorithm are uniformly distributed over the Pareto front.

Many-objective optimization of BEV design parameters based on gradient boosting decision tree models and the NSGA-III algorithm considering the ambient temperature

In this study, extensive efforts have been devoted to optimizing the energy distribution and performance of a battery electric vehicle (BEV). An integrated simulation model based on energy flow test data is built and validated, and a parallel framework to implement automatic batch simulations is developed. On this basis, eXtreme Gradient Boosting (XGBoost), Light Gradient Boosting Machine (LightGBM) and Categorical Boosting (CatBoost) models of the BEV are established and compared, and many-objective optimization is carried out based on the Non-dominated Sorting Genetic Algorithm-III (NSGA-III) algorithm. The results indicate that the developed parallel framework can efficiently perform automatic batch computations of the integrated simulation model for the BEV, and the CatBoost model demonstrates superior prediction performance on both the train and test sets. A uniformly distributed non-dominated set approximating the Pareto Front (PF) is obtained by the many-objective optimization, with the fourth non-dominated solution exhibiting a favorable optimization effect. The electricity consumption per 100 km, half-axis effective work, electricity recovered by battery, energy utilization and recovery efficiency are improved by 9.4 %, 12.2 %, 6.4 %, 96.3 % and 16.0 %, respectively. These findings can provide the theoretical basis, directional guidance and data support for the accurate modeling, batch simulation and many-objective optimization of BEVs.