A FLEXIBLE OBJECTIVE-CONSTRAINT APPROACH AND A NEW ALGORITHM FOR CONSTRUCTING THE PARETO FRONT OF MULTIOBJECTIVE OPTIMIZATION PROBLEMS

In this article, a novel scalarization technique, called the improved objective-constraint approach, is introduced to find efficient solutions of a given multiobjective programming problem. The presented scalarized problem extends the objective-constraint problem. It is demonstrated that how adding variables to the scalarized problem, can lead to find conditions for (weakly, properly) Pareto optimal solutions. Applying the obtained necessary and sufficient conditions, two algorithms for generating the Pareto front approximation of bi-objective and three-objective programming problems are designed. These algorithms are easy to implement and can achieve an even approximation of (weakly, properly) Pareto optimal solutions. These algorithms can be generalized for optimization problems with more than three criterion functions, too. The effectiveness and capability of the algorithms are demonstrated in test problems.

Multiobjective Differential Evolution for Higher-Dimensional Multimodal Multiobjective Optimization

In multimodal multiobjective optimization problems(MMOPs),there are several Pareto optimal solutions corre-sponding to the identical objective vector.This paper proposes a new differential evolution algorithm to solve MMOPs with higher-dimensional decision variables.Due to the increase in the dimensions of decision variables in real-world MMOPs,it is diffi-cult for current multimodal multiobjective optimization evolu-tionary algorithms(MMOEAs)to find multiple Pareto optimal solutions.The proposed algorithm adopts a dual-population framework and an improved environmental selection method.It utilizes a convergence archive to help the first population improve the quality of solutions.The improved environmental selection method enables the other population to search the remaining decision space and reserve more Pareto optimal solutions through the information of the first population.The combination of these two strategies helps to effectively balance and enhance conver-gence and diversity performance.In addition,to study the per-formance of the proposed algorithm,a novel set of multimodal multiobjective optimization test functions with extensible decision variables is designed.The proposed MMOEA is certified to be effective through comparison with six state-of-the-art MMOEAs on the test functions.

Analysis and Research on Mechanical Stress and Multiobjective Optimization of Synchronous Reluctance Motor

The mechanical strength of the synchronous reluctance motor(SynRM)has always been a great challenge.This paper presents an analysis method for assessing stress equivalence and magnetic bridge stress interaction,along with a multiobjective optimization approach.Considering the complex flux barrier structure and inevitable stress concentration at the bridge,the finite element model suitable for SynRM is established.Initially,a neural network structure with two inputs,one output,and three layers is established.Continuous functions are constructed to enhance accuracy.Additionally,the equivalent stress can be converted into a contour distribution of a three-dimensional stress graph.The contour line distribution illustrates the matching scheme for magnetic bridge lengths under equivalent stress.Moreover,the paper explores the analysis of magnetic bridge interaction stress.The optimization levels corresponding to the length of each magnetic bridge are defined,and each level is analyzed by the finite element method.The Taguchi method is used to determine the specific gravity of the stress source on each magnetic bridge.Based on this,a multiobjective optimization employing the Multiobjective Particle Swarm Optimization(MOPSO)technique is introduced.By taking the rotor magnetic bridge as the design parameter,ten optimization objectives including air-gap flux density,sinusoidal property,average torque,torque ripple,and mechanical stress are optimized.The relationship between the optimization objectives and the design parameters can be obtained based on the response surface method(RSM)to avoid too many experimental samples.The optimized model is compared with the initial model,and the optimized effect is verified.Finally,the temperature distribution of under rated working conditions is analyzed,providing support for addressing thermal stress as mentioned earlier.

A New Multiobjective Particle Swarm Optimization Using Local Displacement and Local Guides

This paper introduces a novel variant of particle swarm optimization that leverages local displacements through attractors for addressing multiobjective optimization problems. The method incorporates a square root distance mechanism into the external archives to enhance the diversity. We evaluate the performance of the proposed approach on a set of constrained and unconstrained multiobjective test functions, establishing a benchmark for comparison. In order to gauge its effectiveness relative to established techniques, we conduct a comprehensive comparison with well-known approaches such as SMPSO, NSGA2 and SPEA2. The numerical results demonstrate that our method not only achieves efficiency but also exhibits competitiveness when compared to evolutionary algorithms. Particularly noteworthy is its superior performance in terms of convergence and diversification, surpassing the capabilities of its predecessors.

Multiobjective particle swarm inversion algorithm for two-dimensional magnetic data

Regularization inversion uses constraints and a regularization factor to solve ill- posed inversion problems in geophysics. The choice of the regularization factor and of the initial model is critical in regularization inversion. To deal with these problems, we propose a multiobjective particle swarm inversion （MOPSOI） algorithm to simultaneously minimize the data misfit and model constraints, and obtain a multiobjective inversion solution set without the gradient information of the objective function and the regularization factor. We then choose the optimum solution from the solution set based on the trade-off between data misfit and constraints that substitute for the regularization factor. The inversion of synthetic two-dimensional magnetic data suggests that the MOPSOI algorithm can obtain as many feasible solutions as possible; thus, deeper insights of the inversion process can be gained and more reasonable solutions can be obtained by balancing the data misfit and constraints. The proposed MOPSOI algorithm can deal with the problems of choosing the right regularization factor and the initial model.

New immune multiobjective optimization algorithm and its application in boiler combustion optimization

In order to meet the requirements of combustion optimization for saving energy and reducing pollutant emission simultaneously,an immune cell subsets based multiobjective optimization algorithm（ICSMOA）is proposed.In the ICSMOA,the subset division operator and the immunological tolerance operation are defined.Preference can be easily addressed by using the subset division operator,and the distribution of the solutions can be guaranteed by the immunological tolerance operation.Using the ICSMOA,a group of Pareto optimal solutions can be obtained.However,by the traditional weighting method（WM）,only one solution can be obtained and it cannot be judged as Pareto optimal or not.In contrast to the solutions obtained by the repeatedly performed WM,the simulation results show that most solutions obtained by the ICSMOA are better than the solutions obtained by the WM.In addition,the Pareto front obtained by the ICSMOA is not as uniform as most classical multiobjective optimization algorithms.More optimal solutions which meet the preference set by the decision-maker can be obtained and they are very useful for industrial application.

MULTIOBJECTIVE OPTIMIZATION OF EIGHT-DOF VEHICLE SUSPENSION BASED ON GAME THEORY

A systematic and effective optimization is proposed for the design of a three-dimensional （3-D） vehicle suspension model with eight degrees of freedom （DOF）, including vertical seat motion, vehicle suspension, pitching and rolling motions, and vertical wheel motions using the evolutionary game theory. A new design of the passive suspension is aided by game theory to attain the best compromise between ride quality and suspension deflections. Extensive simulations are performed on three type road surface models A, B, C pavement grades based on the guidelines provided by ISO-2631 with the Matlab/Simulink environment. The preliminary results show that, when the passive suspension is optimized via the proposed approach, a substantial improvement in the vertical ride quality is obtained while keeping the suspension deflections within their allowable clearance when the vehicle moves at a constant velocity v=20 m/s, and the comfort performance of a suspension seat can be enhanced by 20%-30%.

Preference-based multiobjective artificial bee colony algorithm for optimization of superheated steam temperature control

In order to incorporate the decision maker＇s preference into multiobjective optimization a preference-based multiobjective artificial bee colony algorithm PMABCA is proposed.In the proposed algorithm a novel reference point based preference expression method is addressed.The fitness assignment function is defined based on the nondominated rank and the newly defined preference distance.An archive set is introduced for saving the nondominated solutions and an improved crowding-distance operator is addressed to remove the extra solutions in the archive.The experimental results of two benchmark test functions show that a preferred set of solutions and some other non-preference solutions are achieved simultaneously.The simulation results of the proportional-integral-derivative PID parameter optimization for superheated steam temperature verify that the PMABCA is efficient in aiding to making a reasonable decision.

Multiobjective Decision Making: the Combined Approach

In this paper, a new approach for generating all or partly efficient solutions called the Combined Approach is developed. The property of efficient solutions generated by the combined approach and its relationships with other four approaches: weighting approach, sequential approach, ε-constraint approach and hybrid approach, are discussed. Based on this combined approach, a decision-making support method called the Combined Decision-Making Method (CDMM) for multiobjective problems is developed, which is an interactive process with the decision maker. Only the aspiration levels, which reflect the decision maker's satisfying degrees for corresponding objectives, are needed to be supplied by the decision maker step by step as he will. This interactive way for objectives can easily be accepted. Finally, the application of the proposed decision making method in the resource allocation problem is discussed, and an example for the production decision analysis of the solar energy cells given.

Evolutionary Algorithm with Ensemble Classifier Surrogate Model for Expensive Multiobjective Optimization

For many real-world multiobjective optimization problems,the evaluations of the objective functions are computationally expensive.Such problems are usually called expensive multiobjective optimization problems(EMOPs).One type of feasible approaches for EMOPs is to introduce the computationally efficient surrogates for reducing the number of function evaluations.Inspired from ensemble learning,this paper proposes a multiobjective evolutionary algorithm with an ensemble classifier(MOEA-EC)for EMOPs.More specifically,multiple decision tree models are used as an ensemble classifier for the pre-selection,which is be more helpful for further reducing the function evaluations of the solutions than using single inaccurate model.The extensive experimental studies have been conducted to verify the efficiency of MOEA-EC by comparing it with several advanced multiobjective expensive optimization algorithms.The experimental results show that MOEA-EC outperforms the compared algorithms.

Multiobjective Optimization of Simulated Moving Bed by Tissue P System

The binaphthol enantiomers separation process using simulation moving bed technology is simulated with the true moving bed approach （TMB）. In order to systematically optimize the process with multiple productive objectives, this article develops a variant of tissue P system （TPS）. Inspired by general tissue P systems, the special TPS has a tissue-like structure with several membranes. The key rules of each membrane are the communication rule and mutation rule. These characteristics contribute to the diversity of the population, the conquest of the multimodal of objective function, and the convergence of algorithm. The results of comparison with a popular algorithm——the non-dominated sorting genetic algorithm 2（NSGA-2） illustrate that the new algorithm has satisfactory performance. Using the algorithm, this study maximizes synchronously several conflicting objectives, purities of different products, and productivity.

MAJOR－EFFICIENT SOLUTIONS AND WEAKLY MAJOR－EFFICIENT SOLUTIONS OF MULTIOBJECTIVE PROGRAMMING

In this papert the theory of major efficiency for multiobjective programmingis established.The major-efficient solutions and weakly major-efficient solutions of multiobjective programming given here are Pareto efficient solutions of the same multiobjectiveprogramming problem, but the converse is not true. In a ceratin sense , these solutionsare in fact better than any other Pareto efficient solutions. Some basic theorems whichcharacterize major-efficient solutions and weakly major-efficient solutions of multiobjective programming are stated and proved. Furthermore,the existence and some geometricproperties of these solutions are studied.

Multiobjective Optimization of the Industrial Naphtha Catalytic Re-forming Process

In this article, a multiobjective optimization strategy for an industrial naphtha continuous catalytic reform-ing process that aims to obtain aromatic products is proposed. The process model is based on a 20-lumped kinetics re-action network and has been proved to be quite effective in terms of industrial application. The primary objectives in-clude maximization of yield of the aromatics and minimization of the yield of heavy aromatics. Four reactor inlet tem-peratures, reaction pressure, and hydrogen-to-oil molar ratio are selected as the decision variables. A genetic algorithm, which is proposed by the authors and named as the neighborhood and archived genetic algorithm (NAGA), is applied to solve this multiobjective optimization problem. The relations between each decision variable and the two objectives are also proposed and used for choosing a suitable solution from the obtained Pareto set.

Multiobjective optimization and multivariable control of the beer fermentation process with the use of evolutionary algorithms

This paper describes empirical research on the model, optimization and supervisory control of beer fermentation.Conditions in the laboratory were made as similar as possible to brewery industry conditions. Since mathematical models that consider realistic industrial conditions were not available, a new mathematical model design involving industrial conditions was first developed. Batch fermentations are multiobjective dynamic processes that must be guided along optimal paths to obtain good results.The paper describes a direct way to apply a Pareto set approach with multiobjective evolutionary algorithms (MOEAs).Successful finding of optimal ways to drive these processes were reported.Once obtained, the mathematical fermentation model was used to optimize the fermentation process by using an intelligent control based on certain rules.

QUASI-EQUILIBRIUM PROBLEMS AND CONSTRAINED MULTIOBJECTIVE GAMES IN GENERALIZED CONVEX SPACE

A class of quasi-equilibrium problems and a class of constrained multiobjective games were introduced and studied in generalized convex spaces without linear structure. First, two existence theorems of solutions for quasi-equilibrium problems are proved in noncompact generalized convex spaces. Then, ar applications of the quasi-equilibrium existence theorem, several existence theorems of weighted Nash-equilibria and Pareto equilibria for the constrained multiobjective games are established in noncompact generalized convex spaces. These theorems improve, unify, and generalize the corresponding results of the multiobjective games in recent literatures.

Solving Multitrip Pickup and Delivery Problem With Time Windows and Manpower Planning Using Multiobjective Algorithms

The multitrip pickup and delivery problem with time windows and manpower planning(MTPDPTW-MP)determines a set of ambulance routes and finds staff assignment for a hospital. It involves different stakeholders with diverse interests and objectives. This study firstly introduces a multiobjective MTPDPTW-MP(MO-MTPDPTWMP) with three objectives to better describe the real-world scenario. A multiobjective iterated local search algorithm with adaptive neighborhood selection(MOILS-ANS) is proposed to solve the problem. MOILS-ANS can generate a diverse set of alternative solutions for decision makers to meet their requirements. To better explore the search space, problem-specific neighborhood structures and an adaptive neighborhood selection strategy are carefully designed in MOILS-ANS. Experimental results show that the proposed MOILS-ANS significantly outperforms the other two multiobjective algorithms. Besides, the nature of objective functions and the properties of the problem are analyzed. Finally, the proposed MOILS-ANS is compared with the previous single-objective algorithm and the benefits of multiobjective optimization are discussed.

Multiobjective optimization scheme for industrial synthesis gas sweetening plant in GTL process

In industrial amine plants the optimized operating conditions are obtained from the conclusion of occurred events and challenges that are normal in the working units. For the sake of reducing the costs, time consuming, and preventing unsuitable accidents, the optimization could be performed by a computer program. In this paper, simulation and parameter analysis of amine plant is performed at first. The optimization of this unit is studied using Non-Dominated Sorting Genetic Algorithm-II in order to produce sweet gas with CO 2 mole percentage less than 2.0% and H 2 S concentration less than 10 ppm for application in Fischer-Tropsch synthesis. The simulation of the plant in HYSYS v.3.1 software has been linked with MATLAB code for real-parameter NSGA-II to simulate and optimize the amine process. Three scenarios are selected to cover the effect of （DEA/MDEA） mass composition percent ratio at amine solution on objective functions. Results show that sour gas temperature and pressure of 33.98 ？ C and 14.96 bar, DEA/CO 2 molar flow ratio of 12.58, regeneration gas temperature and pressure of 94.92 ？ C and 3.0 bar, regenerator pressure of 1.53 bar, and ratio of DEA/MDEA = 20%/10% are the best values for minimizing plant energy consumption, amine circulation rate, and carbon dioxide recovery.

Hybrid particle swarm optimization for multiobjective resource allocation

Resource allocation （RA） is the problem of allocating resources among various artifacts or business units to meet one or more expected goals, such a.s maximizing the profits, minimizing the costs, or achieving the best qualities. A complex multiobjective RA is addressed, and a multiobjective mathematical model is used to find solutions efficiently. Then, all improved particie swarm algorithm （mO_PSO） is proposed combined with a new particle diversity controller policies and dissipation operation. Meanwhile, a modified Pareto methods used in PSO to deal with multiobjectives optimization is presented. The effectiveness of the provided algorithm is validated by its application to some illustrative example dealing with multiobjective RA problems and with the comparative experiment with other algorithm.

Multiobjective optimal dispatch of microgrid based on analytic hierarchy process and quantum particle swarm optimization

Owing to the rapid development of microgrids(MGs)and growing applications of renewable energy resources,multiobjective optimal dispatch of MGs need to be studied in detail.In this study,a multiobjective optimal dispatch model is developed for a standalone MG composed of wind turbines,photovoltaics,diesel engine unit,load,and battery energy storage system.The economic cost,environmental concerns,and power supply consistency are expressed via subobjectives with varying priorities.Then,the analytic hierarchy process algorithm is employed to reasonably specify the weight coefficients of the subobjectives.The quantum particle swarm optimization algorithm is thereafter employed as a solution to achieve optimal dispatch of the MG.Finally,the validity of the proposed model and solution methodology are con firmed by case studies.This study provides refere nee for mathematical model of multiojective optimizati on of MG and can be widely used in current research field.

Multiobjective fault detection and isolation for flexible air-breathing hypersonic vehicle

An application of the multiobjective fault detection and isolation（FDI） approach to an air-breathing hypersonic vehicle（HSV） longitudinal dynamics subject to disturbances is presented.Maintaining sustainable and safe flight of HSV is a challenging task due to its strong coupling effects,variable operating conditions and possible failures of system components.A common type of system faults for aircraft including HSV is the loss of effectiveness of its actuators and sensors.To detect and isolate multiple actuator/sensor failures,a faulty linear parameter-varying（LPV） model of HSV is derived by converting actuator/system component faults into equivalent sensor faults.Then a bank of LPV FDI observers is designed to track individual fault with minimum error and suppress the effects of disturbances and other fault signals.The simulation results based on the nonlinear flexible HSV model and a nominal LPV controller demonstrate the effectiveness of the fault estimation technique for HSV.