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Mixture problems are basic but important problems in Operations Research. In this paper we consider variants of the basic linear mixture problem and indicate mathematical links between them.

Virtual machine placement is the concept of hosting the virtual machines to appropriate physical servers so as to meet user computation requirements. An optimal placement is one of the key concerns in green cloud computing. Virtual machine placement in distributed cloud environment also imposes propagation time as a key for effective hosting of VM along with CPU and memory resource constraints. In this paper, MOEA/D a multi-objective evolutionary algorithm is used to find a non-dominated solution w.r.t. minimal wastage, minimal power consumption and less propagation delay. The proposed algorithm has been implemented, tested and compared with the existing multi-objective approaches. The statistical analysis of the simulation results proves that MOEA/D outperforms against the existing algorithms in distributed cloud VM placement. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a novel trade-off chart to support the design of multilayer acoustic packages. In this multi-objective problem, a designer has to specify a combination of layers from a set of available acoustic materials and thicknesses. Material types may include porous, mass-weighted, facing, among others. The combination must meet requirements in terms of sound absorption, sound transmission loss, cluttering, mass, etc. While predictions and analyses can be made on predetermined multilayer acoustic packages using the transfer matrix method, statistical energy analysis, finite elements methods or modal analysis, comparing a large number of possible combinations is cumbersome. On the other hand, optimization methods can be used to identify optimal thicknesses or material properties for a given layer combination, but the obtained solution may not be industrially relevant since, in general, only a limited set of acoustic materials and layer thicknesses exist commercially. In this paper, a new design methodology is proposed, which takes into account only the feasible combinations and provides guidelines for compromises between different performance parameters. The three-step methodology is demonstrated through a case study inspired by the automotive industry. First, relevant categories of layer configurations are defined, and following these patterns, all possible combinations of materials from a given inventory are calculated and stored in a database. Then, for selected performance parameters, the Pareto set of "better combinations" is identified. Finally, the "better solutions" are displayed on a trade-off chart through utility functions that allow weighting the different performance parameters. The tool developed for doing so is applied to the case study, and two example situations are presented. For each situation, the trade-off chart provides several suitable solutions, which are discussed. The use of this new tool effectively induces gains of time at the early stage of design, when it is most crucial. [ABSTRACT FROM AUTHOR]

The goal of QoS aware web service composition (QoS-WSC) is to provide new functionalities and find a best combination of services to meet complex needs of users. QoS of the resulting composite service should be optimized. QoS-WSC is a global multi-objective optimization problem belonging to NP-hard class given the number of available services. Most of existing approaches reduce this problem to a single-objective problem by aggregating different objectives, which leads to a loss of information. An alternative issue is to use Pareto-based approaches. The Pareto-optimal set contains solutions that ensure the best trade-off between conflicting objectives. In this paper, a new multi-objective meta-heuristic bio-inspired Pareto-based approach is presented to address the QoS-WSC, it is based on Elephants Herding Optimization (EHO) algorithm. EHO is characterised by a strategy of dividing and combining the population to sub population (clan) which allows exchange of information between local searches to get a global optimum. However, the application of others evolutionary algorithms to this problem cannot avoids the early stagnancy in a local optimum. In this paper a discrete and multi-objective version of EHO will be presented based on a crossover operator. Compared with SPEA2 (Strength Pareto Evolutionary Algorithm 2) and MOPSO (Multi-Objective Particle Swarm Optimization algorithm), the results of experimental evaluation show that our improvements significantly outperform the existing algorithms in term of Hypervolume, Set Coverage and Spacing metrics. [ABSTRACT FROM AUTHOR]

In this article we show that the boundary of the Pareto critical set of an unconstrained multiobjective optimization problem (MOP) consists of Pareto critical points of subproblems where only a subset of the set of objective functions is taken into account. If the Pareto critical set is completely described by its boundary (e.g., if we have more objective functions than dimensions in decision space), then this can be used to efficiently solve the MOP by solving a number of MOPs with fewer objective functions. If this is not the case, the results can still give insight into the structure of the Pareto critical set. [ABSTRACT FROM AUTHOR]

The main purpose of the study is to introduce the multi-objective optimization using Pareto approximations to problems of chemical kinetics. We report the setting up and solution of the multi-objective optimization problem for conditions of a chemical reaction on the basis of a kinetic model. The study addresses the reaction of alcohols with dimethyl carbonate catalyzed by cobalt or tungsten carbonyl. The objective functions for optimization of chemical reaction conditions based on a kinetic model are presented. The NSGA-II algorithm was applied to determine the Pareto set and front for the multi-objective optimization problem applied to the reaction of alcohols with dimethyl carbonate for two catalysts, which make it possible to find the compromise values of variable parameters providing extrema of the objective functions. [ABSTRACT FROM AUTHOR]

A fault on a power distribution system may cause electricity interruption for several consumers, so a good restoration plan is required to decrease such interruptions duration and, consequently, assure the quality of service. Among the measures for service restoration, there is the dispatch of inspection and maintenance crews. The routing of these teams can be classified as a case of the multiple traveling salesman problem. Although involved in series of decision problems, the power distribution system maintenance crews routing is addressed, in the most part of the literature, as a single-objective problem, an instance of a multi-objective one, or as a multi-objective aggregating approach, which generates a single solution in an optimization run, in contrast with the set of equally good solutions, known as Pareto set, the result of a multi-objective problem. In this paper, a Pareto based multi-objective discrete particle swarm optimization approach was applied with the aim of reducing the patrol duration and also the total crews displacement. Wherein the concept of epsilon-dominance was used to update the set of non-dominated solutions, resulting in a good spreading and convergence of them. To promote an uniform exploration of the Pareto set, the selection of the local leaders of the archive was based on square root distance metrics. The Dijkstra algorithm was employed to find the shortest path between two consecutive points of the route of each team. As a result, a set of solutions were obtained for the routing of maintenance crews for power distribution system restoration. [ABSTRACT FROM AUTHOR]

In this paper we address the problem of computing suitable representations of the set of approximate solutions of a given multi-objective optimization problem via stochastic search algorithms. For this, we will propose different archiving strategies for the selection of the candidate solutions maintained by the generation process of the stochastic search process, and investigate them further on analytically and empirically. For all archivers we will provide upper bounds on the approximation quality as well as on the cardinality of the limit solution set. We conclude this work by a comparative study on some test problems in order to visualize the effect of all novel archiving strategies. [ABSTRACT FROM AUTHOR]

In PID controller design, an optimization algorithm is commonly employed to search for the optimal controller parameters. The optimization algorithm is based on a specific performance criterion which is defined by an objective or cost function. To this end, different objective functions have been proposed in the literature to optimize the response of the controlled system. These functions include numerous weighted time and frequency domain variables. However, for an optimum desired response it is difficult to select the appropriate objective function or identify the best weight values required to optimize the PID controller design. This paper presents a new time domain performance criterion based on the multiobjective Pareto front solutions. The proposed objective function is tested in the PID controller design for an automatic voltage regulator system (AVR) application using particle swarm optimization algorithm. Simulation results show that the proposed performance criterion can highly improve the PID tuning optimization in comparison with traditional objective functions.

The robust design optimization (RDO) problem is generally formulated as a weighted sum of the nominal objective function and the robust term. In the RDO problem, a deterministic optimum design is regarded as one of the local optima. However, this property is not well understood. Even though robust optimum designs are known to be significantly different from deterministic designs in certain cases, they are nearly identical in other cases, for reasons that are not intuitively understandable. This is due to the fact that the trade-off relationship between deterministic and robust optimum designs and the effects of uncertainty on the latter are not evaluated by the weighted sum approach. In this study, the properties of robust optimum designs are investigated by formulating the RDO problem as a multiobjective optimization problem, where the nominal value of the performance function and the worst value in the uncertainty region are adopted as the objective functions. The problem considered in this study is limited in that for simplicity, only the design variable is assumed to have uncertainty. That is, the mean value of the random variable is regarded as the design variable. The Pareto solutions are obtained by an evolutionary algorithm whereby the worst design in each individual during the evolutionary process is selected by a sampling method so that the approximation error may be avoided. Through simple numerical examples under several distribution types for random variables, the trade-off relationship between deterministic and robust optimum designs and the effects of uncertainty on the latter are investigated.

This paper analyzes more than 40 papers with a restricted area of application of Multi-Objective Genetic Algorithm, Non-Dominated Sorting Genetic Algorithm-II and Multi-Objective Differential Evolution (MODE) to solve the multi-objective problem in agricultural water management. The paper focused on different application aspects which include water allocation, irrigation planning, crop pattern and allocation of available land. The performance and results of these techniques are discussed. The review finds that there is a potential to use MODE to analyzed the multi-objective problem, the application is more significance due to its advantage of being simple and powerful technique than any Evolutionary Algorithm. The paper concludes with the hopeful new trend of research that demand effective use of MODE; inclusion of benefits derived from farm byproducts and production costs into the model.

We consider the bicriteria investment Boolean problem of finding the Pareto set based on efficiency and risk criteria. The quantitative stability characteristics of the problem are investigated, and lower and upper bounds for a stability radius are obtained for the case where portfolio and financial market state spaces are endowed with the Hölder metric. Calculation of these bounds provides investors with a deeper insight into the specific problem of facilitating financial decisions more reliably in uncertain environments.

Highlights • A new formulation for multi-objective unit commitment with voltage stability and PV uncertainty is given. • The proposed method can reduce operating cost as compared with conventional methods. • This method can be an appropriate tool for planning with the stability and PV uncertainty. Abstract This paper proposes a novel multipurpose operation planning method for minimizing the prediction error of photovoltaic power generator outputs (PV); towards reducing the operating cost and improving voltage stability of power systems. The operation schedule (coordination) of demand response (DR) program and storage system are taken into account as the main parameters for achieving an improved voltage stability and reduction of PV output prediction error. In this approach, the stochastic programming algorithm is introduced for incorporating the uncertainty of PV output and the utility demand response for consumer side management. This is achieved by using the multi-objective genetic algorithm (MOGA) for multipurpose operation plan. The MATLAB optimization toolbox and neural network toolbox were applied in this research study. An IEEE-6 bus system is used to demonstrate the effectiveness of the proposed solution in power systems operation. The approach led to $ 25003.39 (= $ 99594.53 - $ 74591.14) reduction in the system operating cost, compared to the conventional approach. The simulation results also show that by using the proposed algorithm, the capacity of installed PV generators was increased and the voltage stability was improved at the same time. This accounted for the reduction in the effective operating cost and the improved operating condition of the power system. [ABSTRACT FROM AUTHOR]

Multiple optimization objectives and the Pareto set often arise from engineering structural optimization. Normalization methods (such as the weighting method) have the disadvantage that the weighted value is not set by the decision maker but the designer and is greatly influenced by the opinion of the designer. On this basis, in this paper a non-dominated sorting genetic algorithm - analytic hierarchy process (NSGA-AHP) method is proposed for decision making and analysis of the Pareto solution set of the multiple-objective optimization in a structural optimal model. In addition, illustrated by the example of a disc brake, a multiple-objective optimization model for a disc brake has been here developed. Besides, the NSGA-AHP method is adopted for the analysis optimization. The research results show that the NSGA-AHP method can be utilized to select the Pareto solution set in an effective way and that this method is effective in solving a multiple-objective problem in the structural optimization design. [ABSTRACT FROM AUTHOR]

This article optimizes an injection molding process using an efficient sequential design methodology. The goal is to set the process control variables to minimize the shrinkages of a selected collection of injection molded parts. This multiobjective optimization problem is solved by finding those process control variable settings that are Pareto minimizing values (i.e., process settings for which none of the shrinkages of the parts can be decreased by an alternative process setting without increasing the shrinkages of other parts). The sequential design uses an expected improvement criterion to guide updates. The shrinkages are estimated by a calibrated predictor of the process mean shrinkage. The calibration is based on observations of the manufacturing process supplemented by computer runs of a commercial simulator code that mimics the manufacturing process. [ABSTRACT FROM AUTHOR]

Two most important signatures of repetitive transients in the vibration signals of a faulty rotating machine are impulsiveness and cyclostationarity. In the newly proposed infogram, the time-domain and frequency-domain spectral negentropy were put forward to characterize these two aspects, respectively. However, in extension of the infogram to Bayesian inference based optimal wavelet filtering, only one spectral negentropy was employed in identifying the informative frequency band. To overcome its drawback, a novel Pareto-based Bayesian approach was proposed in this paper. The Pareto optimal solutions which can simultaneously maximize the time-domain and frequency-domain spectral negentropy were utilized in estimating the posterior wavelet parameters distributions. Moreover, the relationship between the impulsive and cyclostationary signatures was established by the domination. It can help balance the contributions due to these two aspects other than simply synthesize by the average weight in the infogram. Three instance studies including simulated and experimental signals were investigated to illustrate the effectiveness of the proposed method by challenging different noises and interferences. In addition, some comparisons with the aforementioned peer methods were also conducted to show its superiority and robustness in extracting the repetitive transients. [ABSTRACT FROM AUTHOR]

We introduce a new multi-objective optimization (MOO) methodology based the splitting technique for rare-event simulation. The method generalizes the elite set selection of the traditional splitting framework, and uses both local and global sampling to sample in the decision space. In addition, an 휖-dominance method is employed to maintain good solutions. The algorithm was compared with state-of-the art MOO algorithms using a prevailing set of benchmark problems. Numerical experiments demonstrate that the new algorithm is competitive with the well-established MOO algorithms and that it can outperform the best of them in various cases. [ABSTRACT FROM AUTHOR]

Sufficient conditions are studied for the presence of the completeness or quasicompleteness property in two-criteria discrete optimization problems with the same and different weight-type criteria. Estimates are computed for the cardinalities of sets of feasible solutions, the Pareto set, and a complete set of alternatives for a number of two-criteria problems. [ABSTRACT FROM AUTHOR]

We consider a multicriteria Boolean programming problem of finding the Pareto set. Partial criteria are given as quadratic functions, and they are exposed to independent perturbations. We study quantitative characteristic of stability (stability radius) of the problem. The lower and upper bounds for the stability radius are obtained in the situation where solution space and problem parameter space are endowed with various Hölder's norms. [ABSTRACT FROM AUTHOR]

In biobjective mixed integer linear programs (BOMILPs), two linear objectives are minimized over a polyhedron while restricting some of the variables to be integer. Since many of the techniques for finding or approximating the Pareto set of a BOMILP use and update a subset of nondominated solutions, it is highly desirable to efficiently store this subset. We present a new data structure, a variant of a binary tree that takes as input points and line segments in R2 and stores the nondominated subset of this input. When used within an exact solution procedure, such as branch and bound (BB), at termination this structure contains the set of Pareto optimal solutions. We compare the efficiency of our structure in storing solutions to that of a dynamic list, which updates via pairwise comparison. Then we use our data structure in two biobjective BB techniques available in the literature and solve three classes of instances of BOMILP, one of which is generated by us. The first experiment shows that our data structure handles up to 107 points or segments much more efficiently than a dynamic list. The second experiment shows that our data structure handles points and segments much more efficiently than a list when used in a BB. [ABSTRACT FROM AUTHOR]

In the real-life multiobjective optimization, it is often required to generate a well-distributed Pareto set. Only a few methods are capable of tackling such a problem in a quite general formulation. The Directed Search Domain method (DSD) proved to be efficient and, therefore, attracted much attention. In this paper, two modifications to the rotation strategy of DSD algorithm are proposed. The first modification is meant to improve its computational efficiency. The second modification is to enhance the evenness of the Pareto set with a number of additional Pareto points. These points are obtained according to some specific rotation angles calculated in a particular way. The modified approach is verified on several test cases with three objectives, including an engineering case. The proposed algorithm is compared with both the original DSD and DSD-II algorithms. It is shown that the new approach can maintain the distribution of the Pareto set at a high level with a relatively low computational cost. [ABSTRACT FROM AUTHOR]

Consideration was given to the multicriterial approach to the problems of estimation, filtration, and control; among them under uncertainty in regard to the covariance of random factors. The Pareto-optimal estimates, filters, and controllers in the problems where the data can arrive from any of the more than one data sources with various statistical characteristics, as well as in the dual problems where it is required to minimize the rms deviations of more than one objective output, were designed on the basis of the Germeier convolution and the apparatus of linear matrix inequalities. [ABSTRACT FROM AUTHOR]

This paper analyzes more than 40 papers with a restricted area of application of Multi-Objective Genetic Algorithm, Non-Dominated Sorting Genetic Algorithm-II and Multi-Objective Differential Evolution (MODE) to solve the multi-objective problem in agricultural water management. The paper focused on different application aspects which include water allocation, irrigation planning, crop pattern and allocation of available land. The performance and results of these techniques are discussed. The review finds that there is a potential to use MODE to analyzed the multi-objective problem, the application is more significance due to its advantage of being simple and powerful technique than any Evolutionary Algorithm. The paper concludes with the hopeful new trend of research that demand effective use of MODE; inclusion of benefits derived from farm byproducts and production costs into the model. [ABSTRACT FROM AUTHOR]

Operating room (OR) scheduling problem is commonly recognized as a multi-objective combinatorial optimization problem with several objectives from different perspectives, e.g. minimizing waiting time from patients' perspective, reducing overtime from medical staffs' perspective, increasing resource utilization from OR management's perspective etc. Those objectives are often conflicting. A meta-heuristic approach integrating Pareto sets and Ant Colony Optimization (ACO) is proposed to solve such multi-objective OR scheduling optimization problem. The Pareto sets construction and the modified ant graph model is introduced and two types of pheromone setting and updating strategies are compared to determine a more efficient multi-objective OR scheduling algorithm. The scheduling results by four different approaches, i.e. the simulation, the ACO with single objective of makespan (ACO-SO), the ACO with multi-objective by weighted sum method (ACO-weighted-sum), and the hybrid Pareto set-ACO with multi-objectives (PSACO-MO) are compared. The test case in the literature, which is from MD Anderson Cancer Center, is also used to evaluate the performance of the proposed approach. The computational results show that the PSACO-MO achieves good results in shortening makespan, reducing nurses' overtime and balancing resources' utilization in general. [ABSTRACT FROM AUTHOR]

We consider the task to assess the quality of Pareto set (front) numerical approximation in a multi-criteria optimization (MOC) problem. We mean that Pareto-approximation is obtained by means of this or that population e.g. genetic algorithm.Eventually, the purpose of work is a comparative assessment of the efficiency of population algorithms of Pareto-approximation. The great number of characteristics (indicators) of the Pareto-approximation quality is developed. Therefore an assessment problem of the Paretoapproximation quality is also considered as multi-criteria (multi-indicator). There are a number of well-known software systems to solve an assessment problem of the Pareto-approximation quality in different degree. Common drawback of these systems is a lack of both the WEB INTERFACE and the support of a multi-indicator assessment of Pareto-approximation quality (though there is a support to calculate the values of a large number of these indicators). The PARETO RATING software system is urged to eliminate the specified shortcomings of known systems. As population algorithms of Pareto-approximation are, as a rule, stochastic, we consider statistical methods to assess the quality of two and more Pareto-approximations (and thereby the estimates of algorithms used to obtain these approximations as well) as follows: methods based on the ranging of the specified approximations; methods based on the quality indicators; methods based on the so-called empirical functions of approachability. We give formal statement of the MOC-problem and general scheme of the population algorithms of its solution, present reviews of known indicators of Pareto-approximation quality and statistical methods for assessment of Pareto-approximation quality. We describe the system architecture and main features of its software implementation and illustrate efficiency of made algorithmic and software solutions.

Water resources systems are often characterized by multiple objectives. Typically, there is no single optimal solution which can simultaneously satisfy all the objectives but rather a set of technologically efficient non-inferior or Pareto optimal solutions exists. Another point regarding multi-objective optimization is that interdependence and contradictions are common among one or more objectives. Therefore, understanding the competition mechanism of the multiple objectives plays a significant role in achieving an optimal solution. This study examines cascade reservoirs in the Heihe River Basin of China, with a focus on exploring the multi-objective competition mechanism among irrigation water shortage, ecological water shortage and the power generation of cascade hydropower stations. Our results can be summarized as follows: (1) the three-dimensional and two-dimensional spatial distributions of a Pareto set reveal that these three objectives, that is, irrigation water shortage, ecological water shortage and power generation of cascade hydropower stations cannot reach the theoretical optimal solution at the same time, implying the existence of mutual restrictions; (2) to avoid subjectivity in choosing limited representative solutions from the Pareto set, the long series of non-inferior solutions are adopted to study the competition mechanism. The premise of sufficient optimization suggests a macro-rule of ‘one falls and another rises,’ that is, when one objective value is inferior, the other two objectives show stronger and superior correlation; (3) the joint copula function of two variables is firstly employed to explore the multi-objective competition mechanism in this study. It is found that the competition between power generation and the other objectives is minimal. Furthermore, the recommended annual average water shortage are 1492 × 104 m3 for irrigation and 4951 × 104 m3 for ecological, respectively. This study is expected to provide a foundation for selective preference of a Pareto set and insights for other multi-objective research.

In this paper, the synthesis of a chitosan–montmorillonite nanocomposite material grafted with acrylic acid is presented based on its function in a case study analysis. Fuzzy optimization is used for a multi-criteria decision analysis to determine the best desirable swelling capacity (YQ) of the material synthesis at its lowest possible variable cost. For YQ, the integrating the result’s cumulative uncertainty is an essential element to investigate the feasibility of the developed model equation. The Pareto set analysis is able to set the appropriate boundary limits for YQ and the variable cost. Two case studies are presented in determining the lowest possible cost: Case 1 for maximum YQ, and Case 2 for minimum YQ. These boundary limits were used in the fuzzy optimization to determine its global optimum results that achieved the overall satisfaction ratings of 67.2% (Case 1) and 52.3% (Case 2). The synthesis of the polyacrylic acid/chitosan material for Case 1 resulted in 305 g/g YQ and 10.8 USD/kg, while Case 2 resulted in 97 g/g YQ and 12.3 USD/kg. Thus, the fuzzy optimization approach proves to be a practical method for examining the best possible compromise solution based on the desired function to adequately synthesize a material.

We consider a multiple objective combinatorial optimization problem with an arbitrary vector-criterion. The necessary and sufficient conditions for stability and quasistability are obtained for large classes of problems with partial criteria possessing certain properties of regularity. [ABSTRACT FROM AUTHOR]

We consider a novel multi-objective control problem where the criteria are generalized H -norms of transfer matrices of individual channels from the disturbance input to various objective outputs. We obtain necessary conditions for Pareto optimality. We show that synthesis of Pareto optimal controls can be done in terms of linear matrix inequalities based on optimizing Germeier's convolution, which also turns out to be the generalized H -norm of the closed-loop system with output composed of the objective outputs multiplied by scalars. As applications we consider multi-objective problems of vibration isolation and oscillation suppression with new types of criteria. [ABSTRACT FROM AUTHOR]

A method is developed for generating a well-distributed Pareto set for the upper level in bilevel multiobjective optimization. The approach is based on the Directed Search Domain (DSD) algorithm, which is a classical approach for generation of a quasi-evenly distributed Pareto set in multiobjective optimization. The approach contains a double-layer optimizer designed in a specific way under the framework of the DSD method. The double-layer optimizer is based on bilevel single-objective optimization and aims to find a unique optimal Pareto solution rather than generate the whole Pareto frontier on the lower level in order to improve the optimization efficiency. The proposed bilevel DSD approach is verified on several test cases, and a relevant comparison against another classical approach is made. It is shown that the approach can generate a quasi-evenly distributed Pareto set for the upper level with relatively low time consumption. [ABSTRACT FROM PUBLISHER]

A methodology of multiobjective design optimization of laminated composite plates with piezoelectric layers is presented in this paper. Constrained optimization is conducted for different behaviour objectives, like the maximization of buckling load or natural frequencies of specific vibration modes or prescribed displacements for example. Weight minimization can also be considered or the minimization of the electric voltages applied in the piezoelectric actuators. The optimization problems are constrained by stress based failure criteria and other structural response constraints like limits imposed on certain displacements, buckling characteristics and natural frequency constraints. The design variables considered in the present work are the fiber reinforcement orientations in the composite layers, thicknesses of individual layers and the electric potentials applied to the actuators. The optimization problems are solved with two direct search derivative-free algorithms: GLODS (Global and Local Optimization using Direct Search) and DMS (Direct MultiSearch). DMS, the multiobjective optimization solver, is started from a set of local minimizers which are initially determined by the global optimizer algorithm GLODS for each one of the objective functions. [ABSTRACT FROM AUTHOR]

Subset simulation is an efficient Monte Carlo technique originally developed for structural reliability problems, and further modified to solve single-objective optimization problems based on the idea that an extreme event (optimization problem) can be considered as a rare event (reliability problem). In this paper subset simulation is extended to solve multi-objective optimization problems by taking advantages of Markov Chain Monte Carlo and a simple evolutionary strategy. In the optimization process, a non-dominated sorting algorithm is introduced to judge the priority of each sample and handle the constraints. To improve the diversification of samples, a reordering strategy is proposed. A Pareto set can be generated after limited iterations by combining the two sorting algorithms together. Eight numerical multi-objective optimization benchmark problems are solved to demonstrate the efficiency and robustness of the proposed algorithm. A parametric study on the sample size in a simulation level and the proportion of seed samples is performed to investigate the performance of the proposed algorithm. Comparisons are made with three existing algorithms. Finally, the proposed algorithm is applied to the conceptual design optimization of a civil jet. [ABSTRACT FROM AUTHOR]

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Due to the growing interest in approximation for multiobjective optimization problems (MOPs), a theoretical framework for defining and classifying sets representing or approximating solution sets for MOPs is developed. The concept of tolerance function is proposed as a tool for modeling representation quality. This notion leads to the extension of the traditional dominance relation to $$t\hbox {-}$$ dominance. Two types of sets representing the solution sets are defined: covers and approximations. Their properties are examined in a broader context of multiple solution sets, multiple cones, and multiple quality measures. Applications to complex MOPs are included. [ABSTRACT FROM AUTHOR]

We consider a multi-criteria problem of suppressing disturbances with linear feedback with respect to the state or output measured with noise. We assume that the system has N potentially possible inputs for disturbances from given classes, and the criteria are induced norms of operators generated by the system from the corresponding input to the common target output. We obtain necessary Pareto optimality conditions. We show that based on scalar optimization of the suppression level for the disturbances that act on all inputs we can synthesize Pareto suboptimal controllers whose relative losses compared to Pareto optimal controllers do not exceed 1 − $$\sqrt N /N$$ . Our results generalize to the case when disturbances from different classes may form coalitions. [ABSTRACT FROM AUTHOR]

We consider two-criteria control or filtering problems for linear systems, where one criterion is the level of suppression for Gaussian white noise with unknown covariance, and another is the level of suppression for a deterministic signal of bounded power. We define a new criterion, the level of suppression for stochastic and deterministic disturbances that act jointly in the general case on different inputs. This criterion is characterized in terms of solutions of Riccati equations or linear matrix inequalities. We establish that for the choice of optimal controller or filter with respect to this criterion relative losses with respect to each of the original criteria compared to Pareto optimal solutions do not exceed the value $$1 - {{\sqrt 2 } \mathord{\left/ {\vphantom {{\sqrt 2 } 2}} \right. \kern-\nulldelimiterspace} 2}$$ . We extend these results to dual control and filtering problems for systems with one input and two outputs, generalize them to the case of N criteria with loss estimate $$1 - {{\sqrt N } \mathord{\left/ {\vphantom {{\sqrt N } N}} \right. \kern-\nulldelimiterspace} N}$$ , and also apply them for systems with external and initial disturbances. We show a numerical example. [ABSTRACT FROM AUTHOR]

In order to enhance the convergence ability of multi-objective group search optimizer and improve solution distribution of non-dominated Pareto set, we put forward a novel multi-objective group search optimizer based on multiple producers and crossover operator of genetic algorithm (MCGSO) in this paper. The producer of MCGSO is extended from one to multiple ones, which explores more solutions and improves solution distribution of non-dominated Pareto set. For the purpose of preventing a local optimal solution, the metropolis rule of simulation annealing algorithm is introduced into the search pattern of producers. Rangers' search strategies and crossover operator are combined to enhance algorithm's ability to find new solutions and expand the range of non-dominated optimal set. Experimental results on DZTL serial benchmark functions demonstrate that MCGSO can effectively and efficiently solve multi-objective optimization problems compared with other similar multi-objective evolutionary algorithms. [ABSTRACT FROM AUTHOR]

This paper deals with multi-objective optimization in the case of expensive objective functions. Such a problem arises frequently in engineering applications where the main purpose is to find a set of optimal solutions in a limited global processing time. Several algorithms use linearly combined criteria to use directly mono-objective algorithms. Nevertheless, other algorithms, such as multi-objective evolutionary algorithm (MOEA) and model-based algorithms, propose a strategy based on Pareto dominance to optimize efficiently all criteria. A widely used model-based algorithm for multi-objective optimization is Pareto efficient global optimization (ParEGO). It combines linearly the objective functions with several random weights and maximizes the expected improvement (EI) criterion. However, this algorithm tends to favor parameter values suitable for the reduction of the surrogate model error, rather than finding non-dominated solutions. The contribution of this article is to propose an extension of the ParEGO algorithm for finding the Pareto Front by introducing a double Kriging strategy. Such an innovation allows to calculate a modified EI criterion that jointly accounts for the objective function approximation error and the probability to find Pareto Set solutions. The main feature of the resulting algorithm is to enhance the convergence speed and thus to reduce the total number of function evaluations. This new algorithm is compared against ParEGO and several MOEA algorithms on a standard benchmark problems. Finally, an automotive engineering problem allowing to illustrate the applicability of the proposed approach is given as an example of a real application: the parameter setting of an indirect tire pressure monitoring system. [ABSTRACT FROM AUTHOR]