scholarly journals Algorithm for solving one problem of optimal partition with fuzzy parameters in the target functional

2018 ◽  
Author(s):  
O. M. Kiselova ◽  
O. M. Prytomanova ◽  
S. V. Zhuravel ◽  
V. V. Sharavara
Keyword(s):  
Optimization Problems ◽  
Target Function ◽  
Optimization Methods ◽  
Set Partitioning ◽  
Modern Theory ◽  
Single Product ◽  
Infinite Dimensional ◽  
Partitioning Problems ◽  
Optimal Set ◽  
Fuzzy Parameters

The mathematical theory of optimal set partitioning (OSP) of the n-dimensional Eu-clidean space, which has been formed for todays, is the field of the modern theory of opti-mization, namely, the new section of non-classical infinite-dimensional mathematical pro-gramming. The theory is built based on a single, theoretically defined approach that sum up initial infinitedimensional optimization problems in a certain way (with the function of Lagrange) to nonsmooth, usually, finite-dimensional optimization problems, where lat-est numerical nondifferentiated optimization methods may be used - various variants r-algorithm of N.Shor, that was developed in V. Glushkov Institute of Cybernetics of the Na-tional Academy of Sciences of Ukraine. For now, the number of directions have been formed in the theory of continuous tasks of OSP, which are defined with different types of mathematical statements of partitioning problems, as well as various spheres of its application. For example, linear and nonlinear, single-product and multiproduct, deterministic and stochastic, in the conditions of com-plete and incomplete information about the initial data, static and dynamic tasks of the OSP without limitations and with limitations, both with the given position of the centers of subsets, and with definition the optimal variant of their location. Optimal set partitioning problems in uncertainty are the least developed for today is the direction of this theory, in particular, tasks where a number of parameters are fuzzy, inaccurate, or there are insuffi-cient mathematical description of some dependencies in the model. Such models refer to the fuzzy OSP problems, and special solutions and methods are needed to solve them. In this paper, we propose an algorithm for solving a continuous linear single-product problem of optimal set partitioning of n-dimensional Euclidean spaces Еn into a subset with searching of coordinates of the centers of these subsets with restrictions in the form of equalities and inequalities where target function has fuzzy parameters. The algorithm is built based on the application of neuro-fuzzy technologies and N.Shor r-algorithm

scholarly journals Solving an infinite-dimensional problem of location-allocation with fuzzy parameters

2018 ◽  
Author(s):  
O. M. Kiselova ◽  
O. M. Prytomanova ◽  
S. V. Zhuravel ◽  
V. V. Sharavara
Keyword(s):  
Mathematical Description ◽  
Mathematical Formulation ◽  
Set Partitioning ◽  
Production Costs ◽  
Dimensional Problem ◽  
Location Allocation ◽  
Infinite Dimensional ◽  
Wide Range ◽  
Optimal Set ◽  
Fuzzy Parameters

The problem of enterprises location with the simultaneous allocation of this region, coninuously filled by consumers, into consumer areas, where each of them is served by one enterprise, in order to minimize transportation and production costs, in the mathematical definition, are illustrated as infinite-dimensional optimal set partitioning problems (OSP) in non-intersecting subsets with the placement of centers of these subsets. A wide range of methods and algorithms have been developed to solve practical tasks of location-allocation, both finite-dimensional and infinite-dimensional. However, infinite-dimensional location-allocation problems are significantly complicated in uncertainty, in particular case when a number of their parameters are fuzzy, inaccurate, or an unreliable mathematical description of some dependencies in the model is false. Such models refer to the fuzzy OSP tasks, and special solutions and methods are needed to solve them. This pa-per is devoted to the solution of an infinite-dimensional problem of location-allocation with fuzzy parameters, which in mathematical formulation are defined as continuous line-ar single-product problem of n-dimensional Euclidean space Еn optimal set partitioning into a subset with the search for the coordinates of the centers of these subsets with con-straints in the form of equalities and inequalities whose target functionality has fuzzy pa-rameters. The software to solve the illustrated problem was developed. It works on the ba-sis of neuron-fuzzy technologies with r-algorithm of Shore application. The object-oriented programming language C# and the Microsoft Visual Studio development envi-ronment were used. The results for a model-based problem of location-allocation with fuzzy parameters obtained in developed software are presented. The results comparison for the solution to solve the infinite-dimensional problem of location-allocation with de-fined parameters and for the case where some parameters of the problem are inaccurate, fuzzy or their mathematical description is false

scholarly journals APPLICATION OF THE THEORY OF OPTIMAL SET PARTITIONING BEFORE BUILDING MULTIPLICATIVELY WEIGHTED VORONOI DIAGRAM WITH FUZZY PARAMETERS

2020 ◽  
Vol 6 (2(71)) ◽  
pp. 30-35
Author(s):  
O.M. Kiseliova ◽  
O.M. Prytomanova ◽  
V.H. Padalko
Keyword(s):  
Euclidean Space ◽  
Finite Number ◽  
Nonsmooth Optimization ◽  
Voronoi Diagram ◽  
Optimization Problems ◽  
Optimal Location ◽  
Set Partitioning ◽  
Dimensional Euclidean Space ◽  
Optimal Set ◽  
Fuzzy Parameters

An algorithm for constructing a multiplicatively weighted Voronoi diagram involving fuzzy parameters with the optimal location of a finite number of generator points in a limited set of n-dimensional Euclidean space 𝐸𝑛 has been suggested in the paper. The algorithm has been developed based on the synthesis of methods of solving the problems of optimal set partitioning theory involving neurofuzzy technologies modifications of N.Z. Shor 𝑟 -algorithm for solving nonsmooth optimization problems.

scholarly journals Solving a two-step continuous-discrete optimal split-distribution problem with fuzzy parameters

2019 ◽  
Author(s):  
O. M. Kiselova ◽  
O. M. Prytomanova ◽  
S. V. Dzyuba ◽  
V. G. Padalko
Keyword(s):  
Optimization Problems ◽  
Set Partitioning ◽  
Dimensional Euclidean Space ◽  
Discrete Problem ◽  
Optimal Partition ◽  
Distribution Problem ◽  
Two Stage ◽  
Method Of Potentials ◽  
Optimal Set ◽  
Fuzzy Parameters

The theory of optimal set partitioning from an n-dimensional Euclidean space En is an important part of infinite-dimensional mathematical programming. The mostly reason of high interest in development of the theory of optimal set partitioning is that its results can be applied to solving the classes of different theoretical and applied optimization problems, which are transferred into continuous optimal set partitioning problem. This paper investigates the further development of the theory of optimal set partitioning from En in the case of a two-stage continuous-discrete problem of optimal partitioningdistribution with non-determined input data, which is frequently appear in solving practical problems. The two-stage continuous-discrete problem of optimal partition-distribution under constraints in the form of equations and determined position of centers of subsets is generalized by proposed continuous-discrete problem of optimal partition-distribution in case if some parameters are presented in incomplete, inaccurate or unreliable form. These parameters can be represented as linguistic variables and the method of neurolinguistic identification of unknown complex, nonlinear dependencies can be used in purpose to recovery them. A method for solving the two-stage continuous-discrete optimal partitioning-distribution problem with fuzzy parameters in target functional which based on usage of neurolinguistic identification of unknown dependencies for recovering precise values of fuzzy parameters, methods of the theory of optimal set partitioning and the method of potentials for solving a transportation problem is proposed.

scholarly journals Construction of a multiplicatively weighted diagram of a crow with fuzzy parameters

2019 ◽  
Author(s):  
O. M. Kiselova ◽  
O. M. Prytomanova ◽  
S. V. Dzyuba ◽  
V. G. Padalko
Keyword(s):  
Euclidean Space ◽  
Finite Number ◽  
Voronoi Diagram ◽  
Optimization Problems ◽  
Optimal Location ◽  
Set Partitioning ◽  
Dimensional Euclidean Space ◽  
Bounded Set ◽  
Optimal Set ◽  
Fuzzy Parameters

An algorithm for constructing a multiplicatively weighted Voronoi diagram in the presence of fuzzy parameters with optimal location of a finite number of generator points in a bounded set of n-dimensional Euclidean space En is proposed in the paper. The algorithm is based on the formulation of a continuous set partitioning problem from En into non-intersecting subsets with a partitioning quality criterion providing the corresponding form of Voronoi diagram. Algorithms for constructing the classical Voronoi diagram and its various generalizations, which are based on the usage of the methods of the optimal set partitioning theory, have several advantages over the other used methods: they are out of thedependence of En space dimensions, which containing a partitioned bounded set into subsets, independent of the geometry of the partitioned sets, the algorithm’s complexity is not growing under increasing of number of generator points, it can be used for constructing the Voronoi diagram with optimal location of the points and others. The ability of easily construction not only already known Voronoi diagrams but also the new ones is the result of this general-purpose approach. The proposed in the paper algorithm for constructing a multiplicatively weighted Voronoi diagram in the presence of fuzzy parameters with optimal location of a finite number of generator points in a bounded set of n-dimensional Euclidean space En is developed using a synthesis of methods for solving optimal set partitioning problems, neurofuzzy technologies and modifications of the Shor’s r-algorithm for solving non-smooth optimization problems.

Domination Measure: A New Metric for Solving Multiobjective Optimization

2020 ◽  
Vol 32 (3) ◽  
pp. 565-581 ◽  
Author(s):  
Joshua Q. Hale ◽  
Helin Zhu ◽  
Enlu Zhou
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problems ◽  
Solution Space ◽  
Optimization Methods ◽  
Pareto Optimal Set ◽  
Performance Metric ◽  
Multiobjective Optimization Problems ◽  
Global Optima ◽  
Benchmark Test Functions ◽  
Optimal Set

For general multiobjective optimization problems, the usual goal is finding the set of solutions not dominated by any other solutions, that is, a set of solutions as good as any other solution in all objectives and strictly better in at least one objective. In this paper, we propose a novel performance metric called the domination measure to measure the quality of a solution, which can be intuitively interpreted as the probability that an arbitrary solution in the solution space dominates that solution with respect to a predefined probability measure. We then reformulate the original problem as a stochastic and single-objective optimization problem. We further propose a model-based approach to solve it, which leads to an ideal version algorithm and an implementable version algorithm. We show that the ideal version algorithm converges to a set representation of the global optima of the reformulated problem; we demonstrate the numerical performance of the implementable version algorithm by comparing it with numerous existing multiobjective optimization methods on popular benchmark test functions. The numerical results show that the proposed approach is effective in generating a finite and uniformly spread approximation of the Pareto optimal set of the original multiobjective problem and is competitive with the tested existing methods. The concept of domination measure opens the door for potentially many new algorithms, and our proposed algorithm is an instance that benefits from domination measure.

Software suite for determining the tilts of tower-type structures according to terrestrial laser scanning

2020 ◽  
Vol 961 (7) ◽  
pp. 2-7
Author(s):  
A.V. Zubov ◽  
N.N. Eliseeva
Keyword(s):  
Laser Scanning ◽  
Optimization Problems ◽  
Optimization Methods ◽  
User Participation ◽  
Software Suite ◽  
Current State ◽  
Visual Basic For Applications ◽  
Operational Data ◽  
Theoretical Developments

The authors describe a software suite for determining tilt degrees of tower-type structures according to ground laser scanning indication. Defining the tilt of the pipe is carried out with a set of measured data through approximating the sections by circumferences. They are constructed using one of the simplest search engine optimization methods (evolutionary algorithm). Automatic filtering the scan of the current section from distorting data is performed by the method of assessing the quality of models constructed with that of least squares. The software was designed using Visual Basic for Applications. It contains several blocks (subprograms), with each of them performing a specific task. The developed complex enables obtaining operational data on the current state of the object with minimal user participation in the calculation process. The software suite is the result of practical implementing theoretical developments on the possibilities of using search methods at solving optimization problems in geodetic practice.

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