Solution Of Multi-objective Optimization Problem Research Articles

Multi-objective optimization for low hydrogen consumption and long useful life in fuel cell emergency power supply systems

The power system using hydrogen fuel cells (FCs) as the primary energy source holds significant potential for applications in emergency fields such as post-disaster relief and outdoor power supply. The operational environment of the FC emergency power supply system (FCEPSS) is challenging, with long continuous operation time, difficulties in hydrogen supply, high maintenance costs, and, accordingly, requirements for hydrogen consumption and useful life of FCs are heightened. Hence, this paper proposes a multi-objective energy management strategy (EMS) for FCEPSS. First, the structure of FCEPSS is designed based on the demand analysis in emergency scenarios. Then, the hydrogen consumption and life degradation models of FCEPSS are established. Aiming at the multi-objective optimization problem of minimizing hydrogen consumption and life degradation value in the FCEPSS, an improved multi-objective optimization algorithm is proposed. This paper converts complex constraints into the penalty term and incorporates it as one of the objectives, constructing a three-objective optimization model. To enhance the solution of multi-objective optimization problems, an improved strategy based on relaxation-tightening and three-objective sorting is proposed. Combining the fast non-dominated sorting genetic algorithm-II and the above improved strategy, an improved multi-objective optimization algorithm (IMOOA) is constructed. In the simulation, the advantages of the proposed IMOOA algorithm are verified. In the experimental part, compared with the logical threshold strategy, the proposed strategy saves 0.60% of hydrogen consumption and reduces 5.65% of life degradation within 24 h, respectively. Finally, the superiority of the multi-objective EMS utilizing the IMOOA is demonstrated.

Approximate Optimal Solutions for Multiobjective Optimization Problems with Infinite Constraints

Approximate Optimal Solutions for Multiobjective Optimization Problems with Infinite Constraints

Koopman-based surrogate models for multi-objective optimization of agent-based systems

Agent-based models (ABMs) provide an intuitive and powerful framework for studying social dynamics by modeling the interactions of individuals from the perspective of each individual. In addition to simulating and forecasting the dynamics of ABMs, the demand to solve optimization problems to support, for example, decision-making processes naturally arises. Most ABMs, however, are non-deterministic, high-dimensional dynamical systems, so objectives defined in terms of their behavior are computationally expensive. In particular, if the number of agents is large, evaluating the objective functions often becomes prohibitively time-consuming. We consider data-driven reduced models based on the Koopman generator to enable the efficient solution of multi-objective optimization problems involving ABMs. In a first step, we show how to obtain data-driven reduced models of non-deterministic dynamical systems (such as ABMs) that depend potentially nonlinearly on control inputs. We then use them in the second step as surrogate models to solve multi-objective optimal control problems. We first illustrate our approach using the example of a voter model, where we compute optimal controls to steer the agents to a predetermined majority, and then using the example of an epidemic ABM, where we compute optimal containment strategies in a prototypical situation. We demonstrate that the surrogate models effectively approximate the Pareto-optimal points of the ABM dynamics by comparing the surrogate-based results with test points, where the objectives are evaluated using the ABM. Our results show that when objectives are defined by the dynamic behavior of ABMs, data-driven surrogate models support or even enable the solution of multi-objective optimization problems.

Multi-objective optimization of permanent magnet motors using deep learning and CMA-ES

This paper proposes a multi-objective optimization method for permanent magnet motors using a fast optimization algorithm, Covariance Matrix Adaptation Evolution Strategy (CMA-ES), and deep learning. Multi-objective optimization with topology optimization is effective in the design of permanent magnet motors. Although CMA-ES needs fewer population size than genetic algorithm for single objective problems, this is not evident for multi-objective problems. For this reason, the proposed method generates training data by solving the single-objective optimization multiple times using CMA-ES, and constructs a deep neural network (NN) based on the data to predict performance from motor images at high speed. The deep NN is then used for fast solution of multi-objective optimization problems. Numerical examples demonstrate the effectiveness of the proposed method.

An Advanced High Dimensional Power System of Optimization Problems Algorithm

New Evolutionary Algorithm-based for High-Dimensional Power System Optimization Problems and Modern power system optimization problems face growing challenges as there are numerous decision variables and unbiased functions. Multi-objective evolutionary algorithms (MOEAs) basically are popularly applied for solving and dealing with such complicated problems in substantial power systems that have several objective functions. However, customary MOEAs works appropriately when the number of objective features comes to be less than three. Generally, the effectiveness of MOEAs begins to decrease apparently with an increase in the number of unbiased functions. This study is devoted to applying an authentic gradable evolutionary algorithm for the purpose of solving power system optimization issues which manifest numerous objective functions. In particular, this seeks an algorithm based upon NSGA-II algorithm dedicated for an efficient solution of multi-objective optimization problems. Here, we attempt to modify NSGA-II algorithm to effectively solve many target optimization problems. A novel effective grouping method is applied in the proposed algorithm in similar to the non-dominant grouping method of NSGA-II algorithm to accelerate decision convergence action in POF. The recommended algorithm is compared with modern numerous objective optimization algorithms following three test problems. The findings reveal that as the number of unbiased functions is noticeably large, the algorithm that we propose is significantly superior to other algorithms.

Optimality Conditions for Multiobjective Optimization Problems via Image Space Analysis

In this article, optimality conditions on (weak) efficient solutions in multiobjective optimization problems are investigated by using the image space analysis. A class of strong separation functions is constructed by oriented distance functions. Simultaneously, a generalized Lagrange function is introduced by the class of strong separation functions. Then, generalized Karush-Kuhn-Tucker (KKT for short) necessary optimality conditions are established without constraint qualifications or regularity conditions. Under the suitable assumptions, Lagrangian-type sufficient optimality conditions are also characterized. Moreover, the difference between strong separation and weak separation methods is explained.

Multiobjective System-by-Design for mm-Wave Automotive Radar Antennas

The computationally-efficient solution of multi-objective optimization problems (MOPs) arising in the design of modern electromagnetic (EM) microwave devices is addressed. Towards this end, a novel System-by-Design (SbD) method is developed to effectively explore the solution space and to provide the decision maker with a set of optimal trade-off solutions minimizing multiple and (generally) contrasting objectives. The proposed MO-SbD method proves a high computational efficiency, with a remarkable time saving with respect to a competitive state-of-the-art MOP solution strategy, thanks to the "smart" integration of evolutionary-inspired concepts and operators with artificial intelligence (AI) and machine learning (ML) techniques. Representative numerical results are reported to provide the interested users with useful insights and guidelines on the proposed optimization method as well as to assess its effectiveness in designing mm-wave automotive radar antennas.

A novel graph-theoretical clustering approach to find a reduced set with extreme solutions of Pareto optimal solutions for multi-objective optimization problems

Multi-objective optimization problems and their solution algorithms are of great importance as single-objective optimization problems are not usually a true representation of many real-world problems. In general, multi-objective optimization problems result in a large set of Pareto optimal solutions. Each solution in this set is optimal with some trade-offs. Therefore, it is difficult for the decision-maker to select a solution, especially in the absence of subjective or judgmental information. Moreover, an analysis of all the solutions is computationally expensive and, hence, not practical. Thus, researchers have proposed several techniques such as clustering and ranking of Pareto optimal solutions to reduce the number of solutions. The ranking methods are often used to obtain a single solution, which is not a good representation of the entire Pareto set. This paper deviates from the common approach and proposes a novel graph-theoretical clustering method. The quality of the clustering based on the Silhouette score is used to determine the number of clusters. The connectivity in the objective space is used to find representative solutions for clusters. One step forward, we identify ‘extreme solutions’. Hence, the reduced set contains both extreme solutions and representative solutions. We demonstrate the performance of the proposed method by using different 3D and 8D benchmark Pareto fronts as well as Pareto fronts from a case study in Royal Australian Navy. Results revealed that the reduced set obtained from the proposed method outperforms that from the K-means clustering, which is the most popular traditional clustering approach in Pareto pruning.

An evolutionary algorithm for the solution of multi-objective optimization problem

<span>Worldwide, the COVID-19 widespread has significant impact on a great number of people. The hospital admittance issue for patients with COVID-19 has been optimized by previous research. Identifying the symptoms that can be used to determine a patient's health status, whether they are dead or alive it is difficult task for medical professionals. To solve this issue, multi-objective group counselling optimization (MOGCO) algorithm used to control this problem. First, the zitzler-deb-thiele (ZDT)-2 benchmark function is used to evaluate the MOGCO, multi-objective particle swarm optimization (MOPSO), and non dominated sorting genetic algorithm (NSGA) II. In comparison to MOPSO and NSGA-II, MOGCO is closest to the Pareto front line according to graphic statistics on different fitness evolution values such as 4000, 6000, 8000, and 10000. As a result, MOGCO is used for the COVID-19 data optimization. Moreover, six symptoms (heart rate, oxygen saturation, fever, body pain, flue, and breath) were optimized to see if the COVID-19 patients were still alive. The information was gathered from GitHub. Based on the minimum and maximum values of these symptoms as obtained by the suggested methodology, the optimum study shows that COVID-19 patients can remain alive.</span>

Finding top-K solutions for the decision-maker in multiobjective optimization

Multiobjective optimization problems (MOPs) are the optimization problem with multiple conflicting objectives. Generally, an optimization algorithm can find a large number of optimal solutions for MOPs, which easily overwhelm decision makers (DMs) and make it difficult for decision-making. Preference-based evolutionary multiobjective optimization (EMO) aims to find the partial optima in the regions preferred by the DM. Although it narrows the scope of the optimal solutions, it usually still returns a population of optimal solutions (typically 100 or larger in EMO) with a small distance between adjacent optima. Top-K, which is a well-established research subject in many fields to find the best K solutions, may be a direction to reduce the number of optimal solutions. In this paper, first, we introduce the top-K notion into preference-based EMO and propose the top-K model to obtain the best K individuals of multiobjective optimization problems (MOPs). Then, with the top-K model, we propose NSGA-II-TopK and SPEA2-TopK to search for the top-K preferred solutions for preference-based continuous and combinatorial MOPs, respectively. Finally, the proposed algorithms with several representative preference-based EMO algorithms are compared in different preference situations for MOPs. Experimental results show the proposed algorithms have strong performances against the compared algorithms.

Multi-objective optimization of the Atkinson cycle gasoline engine using NSGA Ⅲ coupled with support vector machine and back-propagation algorithm

This paper presents an optimization method using Non-dominated Sorting Genetic Algorithm (NSGA) Ⅲ to drive support vector machine (SVM). In the NSGA Ⅲ algorithm, brake specific fuel consumption (BSFC), NOx and CO2 are optimized by changing the engine control parameters including spark angle, VVT-I (intake), VVT-E (exhaust) and exhaust gas recirculation (EGR). The engine GT-Power physical model is used to generate training data for the SVM, and verify the accuracy of the results of NSGA Ⅲ algorithm during the optimization process. The SVM with fast calculation speed is used in the calculation of NSGA Ⅲ fitness evaluation. In addition, enhancing training is utilized to improve the accuracy of the SVM model in this research. When the optimization method is applied to the Atkinson cycle gasoline engine, its high efficiency has been presented. In the three plans obtained by GT-Power physical model with all four parameters optimized, the maximum reduction rates of BSFC, NOx, CO2 and CO (g/kW·h) reached 7.07%, 35.90%, 6.62% and 5.50% respectively. The SVM model is compared with back-propagation algorithm, and the result proves that SVM is more suitable for such problems. Finally, based on the Pareto optimal solution obtained by the optimization method, it significantly promotes the solution of multi-objective optimization problems. Theoretically, the time cost of the optimization method in this paper can reach 1/23 of that for the optimization algorithm directly driving physical model.

A decomposition-based many-objective evolutionary algorithm with adaptive weight vector strategy

Evolutionary multi-objective optimization methods have become increasingly popular in finding a representative set of Pareto optimal solutions for multi-objective optimization problems (MOPs). As an important multi-objective optimization method, MOEA/D has been successfully applied to resolve various of MOPs. However, there still exist two issues in MOEA/D. First, the distribution of result set is greatly limited in complex Pareto Front (PF) shapes. Second, the effectiveness is relatively poor for solving many-objective optimization problems (MaOPs). Although many researchers attempt to improve MOEA/D, these two issues have not been solved well. This paper proposes a modified MOEA/D method called MOEA/D-VW which involves three optimization strategies. First, an adaptive weight vector adjustment strategy is adopted to fine-tune the subproblems, thus changing the evolution direction of the subpopulation. Second, a weight vector initialization strategy with preference parameters is added into MOEA/D-VW. Third, a modified crossover operator is leveraged to generate new population with good diversity and overcome the problem of high computational cost. Experiments are carried out on the set of 23 continuous optimization problems. The experimental results show that the performance and the distribution of the Pareto Set (PS) of MOEA/D-VW are effectively improved. Meanwhile, the experimental results demonstrate that the modified crossover operator can effectively reduce the computational cost of MOEA/D-VW.

Multi-objective whale optimization approach for cost and emissions scheduling of thermal plants in energy hubs

To a great extent, most of the power generation systems are dependent on fossil fuels, and their rapid usage is not only declining the reserves but also producing the greenhouse gasses that pollutes the ecology of planet and causes significant long haul harms to the earth habitats. The continuous dependency of hydrocarbon deposit will also cause shortage of fuel, causing excessively costly generation in future. In this article, a soft computing framework named as whale optimization algorithm (WOA) is proposed for the solution of multi-objective optimization problem known as economic/environmental dispatch problem (EEDP), which is among the most vibrant problem in the power system control area. Our designed architecture provides fast convergence rate, optimal generation cost, and reduction in environmental pollutants while handling the complex system constraints such as valve-point loading effect (VPLE) and prohibited zones. To demonstrate feasibility and effectiveness of the provided scheme, three highly nonlinear case studies are computed by while considering all system contingencies. The statistical results demonstrate significant reduction in fuel emission costs, and the superiority and performance of suggested technique is compared with the recently published approaches in the literature.

Multi-Objective Quantum-Inspired Seagull Optimization Algorithm

Objective solutions of multi-objective optimization problems (MOPs) are required to balance convergence and distribution to the Pareto front. This paper proposes a multi-objective quantum-inspired seagull optimization algorithm (MOQSOA) to optimize the convergence and distribution of solutions in multi-objective optimization problems. The proposed algorithm adopts opposite-based learning, the migration and attacking behavior of seagulls, grid ranking, and the superposition principles of quantum computing. To obtain a better initialized population in the absence of a priori knowledge, an opposite-based learning mechanism is used for initialization. The proposed algorithm uses nonlinear migration and attacking operation, simulating the behavior of seagulls for exploration and exploitation. Moreover, the real-coded quantum representation of the current optimal solution and quantum rotation gate are adopted to update the seagull population. In addition, a grid mechanism including global grid ranking and grid density ranking provides a criterion for leader selection and archive control. The experimental results of the IGD and Spacing metrics performed on ZDT, DTLZ, and UF test suites demonstrate the superiority of MOQSOA over NSGA-II, MOEA/D, MOPSO, IMMOEA, RVEA, and LMEA for enhancing the distribution and convergence performance of MOPs.

AUTOMATION CHOICE OF THE SOLUTIONS WITH PRECEDENTS USING BASED ON THE ALTERNATIVES FROM PARETO SET

The paper introduces the approach to automation of Pareto set reduction and solutions selection using precedents. In multi-objective optimization process the problem to select Pareto-optimal solutions emerges very frequently. Such a choice is possible not based only on the expert’s preference scheme but with precedents using. They reflect the accumulated experience in decision making on the similar problems solving. The generating of these precedents based on the alternatives from Pareto set is suggested. The procedures for automation of the precedents base creating and precedents matching for new solutions analysis are offered also. In this approach the precedent is an aggregate of some obtained Pareto-optimal solution and their fitness rating for expert. Such fitness value may be assigned directly by him or determined by results of clusterization, so that the solutions from same cluster have been got the equal fitness values. Further the fitness value estimate of new solution of multi-objective optimization problem may be performed with machine learning methods using such as decision trees, artificial neural network without calling expert. The series of experiments for accuracy testing on fitness assessment of new solutions were conducted for a multi-objective optimization test problem. The high accuracy was achieved for a compact base of the precedents with a decision trees method using and neural networks using for relatively large base of the precedents were more effectively.

Evolutionary Computation System Solving Group Decision Making Multiobjective Problems for Human Groups

In a previous work, we proposed an evolutionary computation system designed to solve group decision making multiobjective problems for human groups, which is equivalent to obtaining consensus solutions to multiobjective optimization problems. Multi-human-agent-based evolutionary computation (Mhab-EC) is a primary component of the system, used to obtain converged solutions for multiobjective optimization problems. The other main component is a mechanism that allows owners of simulated human agents to review simulation results thus far and adjust their agents accordingly between successive simulation runs of the Mhab-EC. However, in our previous study, we simply conducted simulations to demonstrate that a single run yielded converged solutions. Consensus solutions were assumed to be obtained through iterations of the Mhab-EC run and agent adjustment. Therefore, in this study, we conducted simulations of the entire system, including the agent adjustment mechanism. For this purpose, we implemented a simple model of agent adjustment by owners to facilitate solution convergence. Simulation results showed that the system indeed yielded converged solutions, which are considered to indicate consensus.

Optimal shape design and transition uncertainty analysis of transonic axisymmetric natural laminar flow nacelle at high Reynolds number

Laminar flow drag reduction is an important way to improve the cost effectiveness of civil aircraft. In this paper, a hybrid game based optimization method is applied to optimize an axisymmetric natural laminar flow nacelle with inflow and geometric constraints at high Reynolds number. The effectiveness of the optimization algorithm and the ability to explore the solution of multi-objective optimization problem from engineering science are confirmed. In order to maintain the stable laminar flow performance and to achieve robust aerodynamic drag reduction in changing flight conditions, the global sensitivity and uncertainty analysis of the designed axisymmetric laminar nacelle with respect to the coupling of geometric parameters and the coupling of Mach number and turbulence intensity are finally performed, and the effects of the input parameters on the laminar flow range and the total drag coefficient are analyzed. The sensitivities of each parameter and its contribution to the laminar flow region were obtained. This lays the foundation for the subsequent development of a robust optimization modeling and the specification of its effective search space for the three-dimensional natural laminar flow nacelle.

A novel approach for the solution of multiobjective optimization problem using hesitant fuzzy aggregation operator

Many decision-making problems can solve successfully by traditional optimization methods with a well-defined configuration. The formulation of such optimization problems depends on crisply objective functions and a specific system of constraints. Nevertheless, in reality, in any decision-making process, it is often observed that due to some doubt or hesitation, it is pretty tricky for decision-maker(s) to specify the precise/crisp value of any parameters and compelled to take opinions from different experts which leads towards a set of conflicting values regarding satisfaction level of decision-maker(s). Therefore the real decision-making problem cannot always be deterministic. Various types of uncertainties in parameters make it fuzzy. This paper presents a practical mathematical framework to reflect the reality involved in any decision-making process. The proposed method has taken advantage of the hesitant fuzzy aggregation operator and presents a particular way to emerge in a decision-making process. For this purpose, we have discussed a couple of different hesitant fuzzy aggregation operators and developed linear and hyperbolic membership functions under hesitant fuzziness, which contains the concept of hesitant degrees for different objectives. Finally, an example based on a multiobjective optimization problem is presented to illustrate the validity and applicability of our proposed models.

Using Shapley Values and Genetic Algorithms to Solve Multiobjective Optimization Problems

This paper proposes a new methodology to solve multiobjective optimization problems by invoking genetic algorithms and the concept of the Shapley values of cooperative games. It is well known that the Pareto-optimal solutions of multiobjective optimization problems can be obtained by solving the corresponding weighting problems that are formulated by assigning some suitable weights to the objective functions. In this paper, we formulated a cooperative game from the original multiobjective optimization problem by regarding the objective functions as the corresponding players. The payoff function of this formulated cooperative game involves the symmetric concept, which means that the payoff function only depends on the number of players in a coalition and is independent of the role of players in this coalition. In this case, we can reasonably set up the weights as the corresponding Shapley values of this formulated cooperative game. Under these settings, we can obtain the so-called Shapley–Pareto-optimal solution. In order to choose the best Shapley–Pareto-optimal solution, we used genetic algorithms by setting a reasonable fitness function.

Development of the Multi-Objective Adaptive Guided Differential Evolution and optimization of the MO-ACOPF for wind/PV/tidal energy sources

Currently, one of the most popular research topics is the development of a new meta-heuristic algorithm for solving multi-objective optimization problems. However, few of the proposed algorithms have been successful in finding sets that represent feasible solutions. One reason for this failure is that algorithms originally designed for single-objective problems do not provide sufficient exploitation and exploration capabilities for the multi-objective optimization process. Another reason is that a strong harmony is lacking between the Pareto-based archiving approach and the meta-heuristic search (MHS) method (the two basic elements of multi-objective optimization). This study presents the Multi-Objective Adaptive Guided Differential Evolution (MOAGDE) as a powerful and stable algorithm. The MOAGDE can effectively find Pareto optimal solutions for multi-objective optimization problems with different types of high-complexity decision/objective spaces. The proposed MOAGDE was developed by redesigning the adaptive guided differential evolution algorithm for multi-objective optimization. In addition, in the MOAGDE, non-dominated sorting strategy was integrated with crowding distance to archive Pareto optimal solutions of multiple conflicting functions. The CEC 2020, the most recent and advanced benchmark problem suite, was used along with the strongest competitors in the literature to test and verify the performance of the MOAGDE. Finally, using the proposed method, the best Pareto optimal solutions in the literature were applied to a real-world engineering problem: the multi-objective-alternating current optimal power flow (MO-ACOPF) problem involving wind/PV/tidal energy sources. Although the multi-objective version of this problem is rarely studied, the optimal power flow (OPF) is one of the major problems encountered in the planning and operation of modern power systems. The performance of the proposed approach was studied and tested for various objective functions on an IEEE 30-bus test system and the simulation results were compared with the results of the OMNI and MO_Ring_PSO_SCD algorithms.