Solution Of Multi-objective Problem Research Articles

OPSBC: A method to sort Pareto-optimal sets of solutions in multi-objective problems

In recent decades, the significance of multi-objective problems has grown substantially. One of the most popular methods for solving these problems involves the construction of Pareto sets. Pareto sets are exponentially sized relative to the input size of the problem, and so the need to reduce, or at least order, them arises. This work proposes the order of Pareto solutions by Borda count, an approach that makes use of ranking methods to establish preferences among the optimal solutions. To evaluate the proposed approach, a comparative study is conducted, evaluating its performance in comparison to other widely used and well-established methods within this domain. Finally, a case study with real-world data is used to show how the methodology works.

Recent advances in boron removal in aqueous media. An approach to the adsorption process and process optimization.

The numerous oxidation states of the element boron bring great challenges in containing its contamination in receptor bodies. This scenario increases significantly due to the widespread use of boron compounds in various industries in recent years. For this reason, the removal of this contaminant is receiving worldwide attention. Although adsorption is a promising method in boron removal, finding suitable adsorbents, that is, those with high efficiency, and feasible remains a constant challenge. Hence, this review presents the boron removal methods in comparison to costs of adsorbents, reaction mechanisms, economic viability, continuous bed application, and regeneration capacity. In addition, the approach of multivariate algorithms in the solution of multiobjective problems can enable the optimized conditions of dosage of adsorbents and coagulants, pH, and initial concentration of boron. Therefore, this review sought to comprehensively and critically demonstrate strategic issues that may guide the choice of method and adsorbent or coagulant material in future research for bench and industrial scale boron removal.

Vessel voyage schedule planning for maritime ore transportation

This paper introduces a vessel voyage schedule planning problem for maritime ore transportation arising from a real-world project in Boffa, Guinea. In this problem, delay of vessel arrivals at the loading port often occurs which interrupts the ore loading operation and further impacts the yearly throughput of ore transportation, which dues to variable weather conditions at sea and changing loading/discharging time at ports. Thus, factors regarding the uncertainty of vessel voyages must be considered in order to ensure continuous vessel arrivals and uninterrupted loading operations at the original port, leading to an increase in the overall productivity of the project. And at the same time, since every voyage entails an increase in costs, cost control is necessary to be accounted for, so there is also a need to consider minimizing operating costs. In this study, to address the uncertainty in voyage duration, a chance-constrained model is introduced to obtain sailing schedules as well as the associated costs within a given confidence level. And then, for the multi-objective requirements, we develop a multi-objective planning model and the actual system operating capacity limitation constraints are considered. After that, considering that the multi-objective model cannot be solved by the solver directly, the weighting method is introduced. Afterwards, since the direct solving of the model is rather time-consuming, valid cuts are designed to accelerate it. Finally, considering that the Pareto solutions of multi-objective problems are numerous, a nondominated sorting genetic algorithm II (NSGA-II) is launched with the aim of providing more solutions to the manager in a shorter time. In terms of experiments, each of the aforementioned issues and proposed remedies are verified. The results confirm the effectiveness of the valid cuts and NSGA-II, and demonstrate that the obtained vessel voyage schedule is more robust and outperforms manual decisions.

The Best Compromise Solution for Multi-objective Programming Problems

This paper focuses on multi-objective optimization problems that are an important part of operations research. This part is concerned with mathematical optimization problems involving two or more interdependent objectives to be optimized simultaneously. Thus, there is not a single optimal solution for multi-objective problems, but a set of solutions that represents the compromise (Pareto-optimal, efficient, non-dominated, trade-off, or non-inferior) solutions and can be visualized as Pareto front in the objective space. The best solution of this set has the shortest distance to the ideal (utopian) solution, whereas the ideal solution optimizes all objective functions, which often cannot be found. The main contribution of this paper is to introduce some methods to find the best compromise solution. These methods depend on new calculations for the normal of objectives. They can help to reduce the overall computational distance of the searching process. Therefore, they are flexible and stable. Besides, some numerical examples are presented to demonstrate the effectiveness of proposed methods with discussing their similarities and differences. The experimental results show that the proposed methods are effective and efficient for many different multi-objective (convex and non-convex) problems.

Assessing the resilience of optimal solutions in multiobjective problems

Processes and products are multidimensional so researchers and practitioners have to solve problems with multiple objectives frequently. These problems have, in general, responses in conflict so they do not have a unique solution. Different approaches have been proposed in the literature to solve these problems, but many of them, including the popular desirability function approach, are not employed with the focus on the generation of Pareto frontiers. In addition, it is important to stress that some Pareto solutions may not yield the expected outcome(s) when implemented in practice. Thus, to avoid wasting resources and time in implementing a theoretical solution which does not produce the expected outcome(s), in this paper is proposed a novel metric to assess the resilience of Pareto solutions. This way, the decision-maker may identify a solution less sensitive to changes in the variables setting when their values are implemented in production process (equipments) or during its operation. Metric usefulness is illustrated using a case study, and results analysis is complemented with plots that facilitate the decision-making process.

Passive Power Filter Optimization Problem Based on Adaptive Multipopulation NSGA-II and CRITIC-TOPSIS

The best configuration of passive filters in extra-high voltage systems is investigated in this study using a two-stage multiobjective decision-making (MODM) framework. A collection of Pareto solutions is found using an adaptive multipopulation-modified nondominated ranking genetic algorithm (AMP NSGA-II) in the first stage. The goals taken into account include the least fundamental reactive power compensation losses, overall cost and maintenance cost, and simultaneous minimum harmonic current distortion rate. The objective weights are determined and the optimum solution is chosen in the second stage using the criteria importance interrelationship (CRITIC) and similarity ranking preference technique (TOPSIS). A number of examples show how the suggested method performs more effectively than the conventional and well-known algorithm and can identify the relationship between the objective functions, indicating that the suggested scheme has better superiority in filters and has a promising future in the solution of multiobjective problems.

Investigating a citrus fruit supply chain network considering CO2 emissions using meta-heuristic algorithms

AbstractAccording to the increasing carbon dioxide released through vehicles and the shortage of water resources, decision-makers decided to combine the environmental and economic effects in the Agri-Food Supply Chain Network (AFSCN) in developing countries. This paper focuses on the citrus fruit supply chain network. The novelty of this study is the proposal of a mathematical model for a three-echelon AFSCN considering simultaneously CO2 emissions, coefficient water, and time window. Additionally, a bi-objective mixed-integer non-linear programming is formulated for production–distribution-inventory-allocation problem. The model seeks to minimise the total cost and CO+ emission simultaneously. To solve the multi-objective model in this paper, the Augmented Epsilon-constraint method is utilised for small- and medium-sized problems. The Augmented Epsilon-constraint method is not able to solve large-scale problems due to its high computational time. This method is a well-known approach to dealing with multi-objective problems. It allows for producing a set of Pareto solutions for multi-objective problems. Multi-Objective Ant Colony Optimisation, fast Pareto genetic algorithm, non-dominated sorting genetic algorithm II, and multi-objective simulated annealing are used to solve the model. Then, a hybrid meta-heuristic algorithm called Hybrid multi-objective Ant Colony Optimisation with multi-objective Simulated Annealing (HACO-SA) is developed to solve the model. In the HACO-SA algorithm, an initial temperature and temperature reduction rate is utilised to ensure a faster convergence rate and to optimise the ability of exploitation and exploration as input data of the SA algorithm. The computational results show the superiority of the Augmented Epsilon-constraint method in small-sized problems, while HACO-SA indicates that is better than the suggested original algorithms in the medium- and large-sized problems.

Solving Economic Emission Load Dispatch Using Oppositional Slime Mould Algorithm With Wavelet Mutation

This paper introduces a modified slime mould algorithm to enhance the exploitation and exploration of the algorithm. As stated in no free lunch theorem no single algorithm can proficiently solve the optimization problems at hand. Hence, hybridization of more than one heuristic algorithm is employed to tackle the situation. The proposed strategy employs opposition-based learning and wavelet mutation-based strategy to the basic slime mould algorithm for exploring the search space while avoiding local optima. These methods are tested on standard benchmark functions, CEC-14 functions and different single and multiobjective thermal load dispatch problems including small, medium and large-scale problems while satisfying non-differentiable constraints namely valve point loading, prohibited operating zone and ramp rate limit which make objective function non-convex and discontinuous. The pareto dominance technique is employed to achieve the optimal solution in multiobjective problems. This paper also compares the performance of the proposed technique with other methods available in the literature for different economic load dispatch problems and emission dispatch problems. The Wilcoxon signed-rank test performed to confirm the competitiveness of the proposed method for independent samples.

Two Objective Public Service System Design Problem

The public service system serves population spread over a geographical area from a given number of service centers. One of the possible approaches to the problem with two or more simultaneously applied contradicting objectives is determination of the so-called Pareto front, i.e. set of all the feasible non-dominated solutions. The Pareto front determination represents a crucial computational deal, when a large public service system is designed using an exact method. This process complexity evoked an idea to use an evolutionary metaheuristic, which can build up a set of non-dominated solution continuously in the form of an elite set. Nevertheless, the latter approach does not assure that the resulting set of solutions represents the true Pareto front of the multi-objective problem solutions. Within this paper, authors deal with both approaches to evaluate the difference between the exact and heuristic approaches.

Multi-objective hybrid harmony search-simulated annealing for location-inventory-routing problem in supply chain network design of reverse logistics with CO2 emission

The advancement of supply chain network design in reverse logistics is gaining interest from the industries. In recent years, the multi-objective framework of the problem has been widely studied by researchers. This paper integrates three different levels of decision planning in supply chain network design: location-allocation problem for strategic planning, inventory planning management for tactical planning, and vehicle routing for operational planning. A location-inventory-routing problem based on the economic production quantity model with environmental concerns is considered. This study aims to minimise the total cost of operating facilities, inventory and distance travelled by the vehicles as the first objective while minimising the CO2 emission cost as the second objective. Due to the complexity of the problem, a non-dominated sorting and ranking procedure is applied into a Multi-Objective Hybrid Harmony Search-Simulated Annealing (MOHS-SA) algorithm to find the trade-off between these two objectives. Computational experiments on the benchmark instances indicate that the proposed MOHS-SA algorithm can produce well-distributed Pareto-optimal solutions for multi-objective problems.

Multi-objective proportional–integral–derivative optimization algorithm for parameters optimization of double-fed induction generator-based wind turbines

The meta-heuristic algorithm inspired by natural may reduce the optimization performance due to excessive imitation. This paper proposes a novel multi-objective proportional–integral–derivative optimization algorithm inspired by mathematical thought to provide a better non-dominated solution for multi-objective problems. The idea of the proportional–integral–derivative control algorithm is introduced to cooperate with multi-objective optimization problems for the first time. The proposed algorithm is employed to store and maintain non-dominated solutions. Two groups of controllers of the proposed algorithm are designed for the multi-objective optimization problems, i.e., exploitative controllers aim to obtain the local optimal solution; explorative controllers aim to obtain the global optimal solution. To verify the effectiveness of the multi-objective proportional–integral–derivative optimization algorithm, eight comparison algorithms are compared with eight benchmark functions; five comparison algorithms are compared under the multi-objective parameters optimization problem of double-fed induction generator-based wind turbines. The results of benchmark functions show that the multi-objective proportional–integral–derivative optimization algorithm has superior convergence performances and outperforms other comparison algorithms. The proposed algorithm has excellent optimization performance to obtain the minimum deviation of rotor speed and reactive power for the wind power system controller.

A new algorithm based on gray wolf optimizer and shuffled frog leaping algorithm to solve the multi-objective optimization problems

Multi-objective optimization is many important since most of the real world problems are in multi-objective category. Looking at the literature, the algorithms proposed for the solution of multi-objective problems have increased in recent years, but there is no a convenient approach for all kind of problems. Therefore, researchers aim to contribute to the literature by offering new approaches. In this study, an algorithm based on gray wolf optimizer (GWO) with memeplex structure of the shuffled frog leaping algorithm (SFLA), which is named as multi-objective shuffled GWO (MOSG), is proposed to solve the multi-objective optimization problems. Additionally, some modifications are applied on the proposed algorithm to improve the performance from different angles. The performance of the proposed algorithm is compared with the performance of six multi-objective algorithms on a benchmark set consist of 36 problems. The experimental results are presented with four different comparison metrics and statistical tests. According to the results, it can easily be said that the proposed algorithm is generally successful to solve the multi-objective problems and has better or competitive results.

Problems of a Utopia Point Setting in Transformation of Individual Objective Functions in Multi-Objective Optimization

Abstract In an optimal processes control, where the considered goals are in general observed as concurrently conflicted, a multi-objective approach fits the best. Commonly used scalarization techniques in multi-objective optimization need a transformation of the individual single-objective functions involved into a scalar multi-criteria objective function. There are many parameters which can influence the optimization results solutions, including an unreachable utopia point value. In this study, the authors compare the multi-objective problem solutions found via two ways of the individual objectives transformation with the respect to setting the utopia point. The methods are used in the area of production control in a case study for a batch production system. To find the solutions, The Weighted Sum Method with a priori articulated preferences under specific constraints as the scalar multi-objective optimization method is applied in simulation optimization.

A Multi-Objective Optimization Problem for Optimal Site Selection of Wind Turbines for Reduce Losses and Improve Voltage Profile of Distribution Grids

In this paper, the optimal site and size selection of wind turbines (WTs) is presented considering the maximum allowable capacity constraint with the objective of loss reduction and voltage profile improvement of distribution grids based on particle swarm optimization (PSO as a multi-objective problem using weighted coefficients method. The optimal site, size, and power factor of the WTs are determined using PSO. The proposed method is implemented on 84- and 32-bus standard grids. In this study, PSO algorithm is applied to determine the size, site, and power factor of WTs considering their maximum size constraint (with constraint, variant size) and also not considering their maximum size constraint (without constraint, constant size). The simulation results showed that the PSO is effective to find the site, size, and power factor of WTs optimally in the single and multi-objective problem. The results of this method showed that the power loss is reduced more and voltage profile improved more considering WTs maximum allowable size versus not considering this constraint. Additionally, the multi-objective results showed that there is a compromise between the objectives in the multi-objective WTs site selection and the multi-objective problem solution is a more realistic and accurate approach in comparison with the single-objective problem solution.

Solving Multi-objective Optimal Control Problems of chemical processes using Hybrid Evolutionary Algorithm

Evolutionary algorithms have been recognized to be suitable for extracting approximate solutions of multi-objective problems because of their capability to evolve a set of non-dominated solutions distributed along the Pareto frontier‎. ‎This paper applies an evolutionary optimization scheme‎, ‎inspired by Multi-objective Invasive Weed Optimization (MOIWO) and Non-dominated Sorting (NS) strategies‎, ‎to find approximate solutions for multi-objective optimal control problems (MOCPs)‎. ‎The desired control function may be subjected to severe changes over a period of time‎. ‎In response to deficiency‎, ‎the process of dispersal has been modified in the MOIWO‎. ‎This modification will increase the exploration power of the weeds and reduces the search space gradually during the iteration process‎. ‎ ‎The performance of the proposed algorithm ‎is compared with conventional Non-dominated Sorting Genetic Algorithm (NSGA-II) and Non-dominated Sorting Invasive Weed Optimization (NSIWO) algorithm‎.The results show that the proposed algorithm has better performance than others in terms of computing time‎, ‎convergence rate and diversity of solutions on the Pareto ‎frontier.

Two-stage multicriteria georeferenced express analysis of new electric transmission line projects

The objective of the present research is to develop methodological and computing tools to support the analysis and prioritization of new electric transmission line projects, in the prospection stage, aimed at greater efficiency and effectiveness of the advanced preparation of concessionaires and investors for participating in the auctions. These auctions are organized by the granting authorities (usually, regulatory agencies). The process of elaborating decisions on auction lots (possible “candidates”) is divided into two stages. The first one is associated with a study on relevant criteria modeled on the basis of applying Geographic Information Systems. The purpose of the first stage is to form the most rational multiobjective estimates of new transmission lines, which can be constructed and operated on analyzed lots. The included criteria are defined with a high level of uncertainty. Considering this, the general scheme of multiobjective decision making in conditions of uncertainty is applied to construct robust multiobjective estimates. This scheme presumes the building and analysis of payoff matrices. The solution of multiobjective problems for representative combinations of initial data, states of nature or scenarios is based on the use of a generalization of the well-known Dijkstra's algorithm (related to graph optimization, permitting the construction of transmission line routes) to analyze multiobjective problems. The second stage is directed at the definition, from the set of lots analyzed at the first stage, of a portfolio of the most appropriate and favorable areas for implanting transmission lines. This stage is based on applying techniques for preference modeling in a fuzzy environment within the framework of models of multiattribute decision making. Their use permits one to adequately consider criteria of quantitative character as well as criteria of qualitative character, whose estimates are based on knowledge, experience, and intuition of involved experts. The paper results are illustrated by a case study which demonstrates the rationality of their practical application, in addition to the strategic role that these results can play for concessionaires and investors, participating in auctions due to their ability to meet a wide range of criteria.

基于微粒群-马尔科夫复合模型的生态空间预测模拟——以长株潭城市群为例

PDF HTML阅读 XML下载 导出引用 引用提醒 基于微粒群-马尔科夫复合模型的生态空间预测模拟——以长株潭城市群为例 DOI: 10.5846/stxb201612082530 作者: 作者单位: 作者简介: 通讯作者: 中图分类号: 基金项目: 国家重点研发计划项目（2016YFC0502406）；教育部人文社会科学重点研究基地重大项目（14JJD720016）；江西省社科规划项目（15SH10）；江西省高校人文社会科学项目（SH1401）；江西省教育厅科技项目（GJJ151018）；江西省教育厅教改项目（JXJG-15-14-9） Predictive simulation of ecological space based on a particle swarm optimization-Markov composite model: A case study for Chang-Zhu-Tan urban agglomerations Author: Affiliation: Fund Project: 摘要 | 图/表 | 访问统计 | 参考文献 | 相似文献 | 引证文献 | 资源附件 | 文章评论 摘要:生态空间具有重要的生态功能，对生态空间进行科学预测模拟可为保护国土空间生态安全提供决策依据。利用ArcGIS及MATLAB软件，在生态空间风险评价的基础上构建了微粒群-马尔科夫复合模型，并以长株潭城市群为研究区，基于2013年土地利用现状数据，对2020年的生态空间进行了预测模拟，最后在此基础上提出了生态空间重构的基本思路。结果表明：1）微粒群-马尔科夫复合模型（PSO-Markov）构建的基本步骤为：第一步：粒子的选择与设计，以2000 m×2000 m的正方形单元作为基本粒子。第二步：粒子的初始化设定，根据生态空间风险由低到高的原则进行选择。第三步：适应度函数的建立，用生态空间的风险值来确定生态空间的空间格局。第四步：空间位置的更新，根据自身的历史最优值及粒子群的全局最优值进行速度和位置更新。2）微粒群-马尔科夫复合模型（PSO-Markov）是一种土地利用格局预测的新途径，生态空间的数量规模可以通过改进后的马尔科夫模型进行预测，生态空间的格局可以通过微粒群模型进行预测。3）微粒群-马尔科夫复合模型具有4个特点：第一、数量预测较为合理。第二、搜索范围大、较好地考虑到局部对全局的影响。第三、受问题维数变化影响小，在求解多目标问题时具有明显优势。第四、收敛时间短、运算速度快、易于实现。4）2020年，长株潭城市群的生态空间总体数量减少，其中林地和未利用地面积变化最明显，空间变化主要集中分布在西南部地区。生态空间总面积减小的主要原因是建设用地的扩张。因此，要控制城市群的人口密度，优化城市群生产-生活-生态的数量结构及空间布局，尤其要合理规划与利用城市建设用地，充分发挥水体与未利用地的生态价值，重点保护好生态源地、廊道及关键结点，构建结构合理、功能齐全的生态网络系统，提高系统的生态服务价值功能，要在规划的指导下合理调整城市群的城乡局部空间结构，保护生态环境，提高生境质量和景观多样性。这是今后一段时期面临的主要任务。 Abstract:The distribution of ecological space has important functions, and its scientific prediction can provide a basis for making decisions regarding protecting the ecological security of a national landscape. A particle swarm optimization-Markov (PSO-Markov) composite model based on a risk assessment of ecological space was conducted using ArcGIS and MATLAB. The ecological space of the Chang-Zhu-Tan urban agglomeration in 2020 was predicted based on land-use data for 2013, and the resulting particle swarm was used to reconstruct the ecological space. The PSO-Markov composite model was established according to the following steps:First, a particle swam was selected and designed, and a 2000 m×2000 m square unit was selected as the basic particle. Second, the particle was initialized from low to high risk based on the principle of ecological space. Third, a fitness function was established, and the value of ecological space risk was used to determine the spatial pattern of ecological space. Finally, the spatial position and speed of a particle swarm were updated according to the history of optimal value and the global optimal value of the particle swarm. The PSO-Markov composite model is a relatively new method for land-use pattern prediction. The quantitative scale of ecological space is predictable with an improved Markov model, and the pattern of ecological space is predictable with a PSO model. Thus, the PSO-Markov composite model has the following features compared with other models:First, this model can yield more reasonable quantitative predictions. Second, it utilizes a large search area and thoroughly considers local and global influence. Third, it is less influenced by the problem of dimension change, and it has a significant advantage regarding the solution of multi-objective problems. Finally, with short convergence time and high arithmetic speed, it is easy to realize. The ecological space of the Chang-Zhu-Tan urban agglomeration was predicted to decrease by 2020 with woodland and unused land changing the most dramatically and spatial variation primarily concentrating in the southwest region. The decrease in total ecological space area was shown to be primarily due to the expansion of land development. Thus, we must control the population density of urban agglomerations and optimize the structure and spatial distribution of agriculture/industry, human settlements, and the ecosystem in the urban agglomeration. In particular, we must reasonably plan and utilize urban developed land, as well as make full use of the ecological value of water bodies and undeveloped land, with emphasis on the protection of ecological resources, ecological corridors, and key ecological areas so as to establish a rationally structured, functioning ecological net system and to improve the ecological services of the ecosystem. Urban and rural spatial structure must be regulated rationally under the guidance of city planning. Moreover, measures should be carried out to protect and improve the quality of the environment and to enhance landscape diversity. These are our priorities for the future. 参考文献 相似文献 引证文献

Multi-objective imperialistic competitive algorithm with multiple non-dominated sets for the solution of global optimization problems

In this paper, we propose a multi-objective imperialistic competitive algorithm (MOICA) for solving global multi-objective optimization problems. The MOICA is a modified and improved multi-objective version of the single-objective imperialistic competitive algorithm previously proposed by Atashpaz-Gargari and Lucas (IEEE Congr Evolut Comput 7:4661–4666. doi: 10.1109/CEC.2007.4425083 , 2007). The presented algorithm utilizes the metaphor of imperialism to solve optimization problems. Accordingly, the individuals in a population are referred to as countries, of which there are two types—colonies and imperialists. The MOICA incorporates competition between empires and their colonies for the solution of multi-objective problems. To this end, it employs several non-dominated solution sets, whereby each set is referred to as a local non-dominated solution (LNDS) set. All imperialists in an empire are considered non-dominated solutions, whereas all colonies are considered dominated solutions. In addition to LNDS sets, there is one global non-dominated solution (GNDS) set, which is created from the LNDS sets of all empires. There are two primary operators in the proposed algorithm, i.e., assimilation and revolution, which use the GNDS and LNDS sets, respectively. The significance of this study lies in a notable feature of the proposed algorithm, which is that no special parameter is used for diversity preservation. This enables the algorithm to prevent extra computation to maintain the spread of solutions. Simulations and experimental results on multi-objective benchmark problems show that the MOICA is more efficient compared to a few existing major multi-objective optimization algorithms because it produces better results for several test problems.

Multiobjective Problems: Enhanced Necessary Conditions and New Constraint Qualifications through Convexificators

ABSTRACTIn this paper, we study necessary optimality conditions for local Pareto and weak Pareto solutions of multiobjective problems involving inequality and equality constraints in terms of convexificators. We develop the enhanced Karush–Kuhn–Tucker conditions and introduce the associated pseudonormality and quasinormality conditions. We also introduce several other new constraint qualifications which entirely depend on the feasible set. Then a connecting link between these constraint qualifications is presented. Moreover, we provide several examples that clarify the interrelations between the different results that we have established.

ЗАСТОСУВАННЯ МЕТОДІВ ОБЧИСЛЮВАЛЬНОГО ІНТЕЛЕКТУ ДЛЯ ВИРІШЕННЯ ПРЯМИХ ЗАДАЧ РОЗРАХУНКУ КОНСТРУКТОРСЬКИХ РОЗМІРНИХ ЛАНЦЮГІВ В УМОВАХ ПАРАМЕТРИЧНОЇ АПРІОРНОЇ НЕВИЗНАЧЕНОСТІ

Statement of a nonlinear problem of calculation of design dimensional chains under conditions of parametric a priori uncertainty is considered. The methodology of synthesis of solutions of multi-objective problems of stochastic optimization with the mixed conditions (MV-tasks) is offered. The effective memetic synthesis algorithm of solutions of MV- tasks is developed. Results of the solution of a problem robust optimal designing of centrifugal impeller fitted with backward curved blades in the conditions of stochastic nature of the input data, the decision-making supports got by means of interactive computer system «Concept_Pro_St®» are presented