Optimal Set Partitioning Research Articles

МАТЕМАТИЧНІ МОДЕЛІ ТА МЕТОДИ РОЗМІЩЕННЯ ОБ’ЄКТІВ І ЗОНУВАННЯ ТЕРИТОРІЙ В СИСТЕМАХ ЕКСТРЕНОЇ ЛОГІСТИКИ

The mathematical models for distribution processes related to organizing precautionary measures in the event of threats or occurrences of man-made emergencies are presented. The tasks include optimal zoning of territories with the fixing of zones by objects of social purpose for service provision. Provision is made for: the possibility of overlapping zones in case the nearest center cannot provide the service; optimal placement of a certain number of new cen-ters of emergency logistics systems with simultaneous redistribution of the load on all their structural elements; the selection of locations of structural subdivisions based on existing fa-cilities. The optimality criteria involve minimizing either the time to provide the service even to the most remote object in the given territory, or the total distance to the nearest centers from consumers that are densely distributed in the given territory, and/or the organizational costs associated with the arrangement of new centers. Mathematical models are proposed in the form of continuous problems of optimal multiplex partitioning of sets with a linear or minimax functional of quality. The latter provides such placement of centers that provides op-timal multiple coverage of the territory (with a minimum radius of multiple coverage). Meth-ods for solving the formulated problems were developed using LP-relaxation of linear prob-lems with Boolean variables, duality theory to reduce the initial problems of infinite-dimensional programming to problems of conditional optimization of a non-smooth function of several variables, and modern methods of non-differentiated optimization.

Two-stage problems of optimal location and distribution of the humanitarian logistics system’s structural subdivisions

Purpose. To ensure the rational organization of the evacuation of people from a region affected by an emergency by developing a mathematical and algorithmic toolkit that will allow for the early distribution of transport and material resources, maximizing coverage of the affected areas while minimizing evacuation time. Methodology. System analysis of evacuation processes; mathematical modeling, the theory of continuous problems of optimal partitioning of sets, non-differentiable optimization. Findings. The object of the study is the two-stage evacuation logistic processes that occur when serving the population of areas affected by emergencies of a natural or technogenic nature. The research considers the possibility of optimally distributing human flows within the transportation system, the structural subdivisions of which are first-stage centers (first aid stations that carry out the reception of citizens from areas affected by the disaster) and second-stage centers (specialized units of the emergency aid system that provide further services to the evacuated population). The proposed mathematical model deals with the problem of optimally partitioning a continuous set with the placement of subset centers and additional connections. Methods for its solution have been described. We demonstrate the versatility of these models, as they can be used to describe logistic evacuation processes, organize assembly points, intermediate locations, evacuation reception points, and those providing primary assistance to the affected population. We calculate the appropriate number of essential products and deliver them from existing warehouses through distribution centers to the affected areas. Originality. As preventive measures to increase the level of population safety during an emergency, we consider the optimal placement of rescue facilities and the zoning of the territory to distribute evacuation traffic. We also address the problem of the optimal distribution of human flows in the transport and logistics system. Practical value. The presented models, methods, and algorithms enable the solution of many practical problems related to the development of preventive measures and the planning of rescue operations to ensure the population’s safety in case of emergencies. The theoretical results obtained are translated into specific recommendations that can be utilized when addressing logistical problems related to the organization of primary evacuation of the population from affected areas and their subsequent transportation to safer locations for further assistance.

Optimal unlabeled set partitioning with application to risk-based quarantine policies

We consider the problem of partitioning a set of items into unlabeled subsets so as to optimize an additive objective, i.e., the objective function value of a partition is equal to the sum of the contribution of each subset. Under an arbitrary objective function, this family of problems is known to be an -complete combinatorial optimization problem. We study this problem under a broad family of objective functions characterized by elementary symmetric polynomials, which are “building blocks” to symmetric functions. By analyzing a continuous relaxation of the problem, we identify conditions that enable the use of a reformulation technique in which the set partitioning problem is cast as a more tractable network flow problem solvable in polynomial-time. We show that a number of results from the literature arise as special cases of our proposed framework, highlighting its generality. We demonstrate the usefulness of the developed methodology through a novel and timely application of quarantining heterogeneous populations in an optimal manner. Our case study on real COVID-19 data reveals significant benefits over conventional measures in terms of both spread mitigation and economic impact, underscoring the importance of data-driven policies.

Application of optimal set partitioning theory to solving problems of artificial intelligence and pattern recognition

The paper substantiates the possibility of applying the mathematical theory of continuous problems of optimal partitioning of sets of n-dimensional Euclidean space, which belong to the non-classical problems of infinite-dimensional mathematical programming, to the solution of problems of artificial intelligence and pattern recognition. The problems of pattern recognition both in conditions of certainty and in conditions of uncertainty are formulated. A particular attention is paid to the application of methods of the theory of optimal partitioning for the construction of fuzzy Voronoi diagrams. Examples of constructing fuzzy Voronoi diagrams with the optimal placement of generating points are given.

Fuzzy problem of the optimal set partition with constraints on the subsets centers location

Fuzzy problem of the optimal set partition with constraints on the subsets centers location

Construction of a generalized Voronoi diagram with optimal placement of generator points based on the theory of optimal set partitioning

The problem of construction of a generalized Voronoi diagram with optimal placement of a finite number of generator points in a bounded set of \textit{n}-dimensional Euclidean space is considered. A method is proposed for solving such a problem based on the formulation of the corresponding continuous problem of optimal partitioning of a set in \textit{n}-dimensional Euclidean space with a partition quality criterion that provides the corresponding form of the Voronoi diagram. Further, to solve such a problem, the developed mathematical and algorithmic apparatus is used, the part of which is Shor's \textit{r}-algorithm.

Algorithm for Constructing Voronoi Diagrams with Optimal Placement of Generator Points Based on Theory of Optimal Set Partitioning

An algorithm is proposed for constructing a generalized Voronoi diagram with optimal placement of a finite number of generator points in a bounded set of n-dimensional Euclidean space. The algorithm is based on the formulation of the corresponding continuous problem of optimal set partitioning with a partition quality criterion providing the corresponding form of Voronoi diagram, and on the application of the apparatus of the optimal partitioning theory to solve this problem. Herewith, the efficient method of non-differentiable optimization is used for the numerical solution of an auxiliary finite-dimensional optimization problem arising in the development of the method for solving the mentioned infinite-dimensional problem of optimal set partitioning. Namely, that is one of the variants of the generalized gradient descent method with space expansion in the direction of the difference of two successive generalized antigradients (Shor's r-algorithm). The proposed algorithm for constructing a generalized Voronoi diagram with optimal placement of a finite number of generator points in a bounded set of n-dimensional Euclidean space has some advantages compared to those known in a scientific literature. It does not depend on a dimension of Euclidean space containing the initial bounded set; it is applicable for the large-scale problems (over 300 generator points); it works not only for Euclidean metrics, but also for Chebyshev, Manhattan and other ones; the complexity of the algorithm implementation for constructing Voronoi diagram based on the proposed approach does not increase with increasing number of generator points. The results of software implementation of the developed algorithm are presented for constructing a standard Voronoi diagram with an optimal placement of generator points as well as some of its generalizations such as additive, multiplicative and additive-multiplicative Voronoi diagrams with optimal placement of generator points.

APPLICATION OF THE THEORY OF OPTIMAL SET PARTITIONING BEFORE BUILDING MULTIPLICATIVELY WEIGHTED VORONOI DIAGRAM WITH 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.

Solving a two-step continuous-discrete optimal split-distribution problem with 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.

Construction of a multiplicatively weighted diagram of a crow with 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.

AN OPTIMAL TWO-STAGE ALLOCATION OF MATERIAL FLOWS IN A TRANSPORT-LOGISTIC SYSTEM WITH CONTINUOUSLY DISTRIBUTED RESOURCE

Context. The object of the research is a two-stage process of material flows allocation in the transport-logistic system, the structural elements of which are enterprises that collect a resource, is been distributed in a certain territory (centers of the first stage), and the enterprises that consume or process this resource. A mathematical model of such process is a two-stage problem of the optimal partitioning of a continual set with the locating of subset centers under additional constraints presented in the paper. Objective. The goal of the work is to ensure the reduction of transport costs in the organization of multi-stage production, the raw material resource of which is distributed in some territory, through the development of appropriate mathematical apparatus and software. The urgency of the work is explained by one of the most pronounced tendencies in extracting and processing branches of industry and agriculture, namely, the creation of territorially-distributed multilevel companies that include dozens of large enterprises and carry out a full cycle of production from raw material harvesting with its integrated use and the product manufacturing to its transportation to end consumers. Method. Mathematical apparatus for two-stage problems of optimal partitioning of sets with additional couplings was developed using the basic concepts of the theory of continuous linear problems of optimal set partitioning, duality theory, and methods for solving linear programming problems of transport type. The research shows that the formulation of a multi-stage transport-logistic problem in a continuous variant (in the form of an infinite-dimensional optimization problem) is expedient when the number of resource suppliers is limited but very large. The application of the developed mathematical apparatus makes it possible to find the optimal solution of the two-stage allocation-distribution problem in an analytic form (the analytic expression includes parameters that are the optimal solution of the auxiliary finite-dimensional optimization problem with a nondifferentiable objective function). The proposed iterative algorithm for solving the formulated problem bases on modification of Shor’s r-algorithm and the method of potentials for solving the transport problem. Results. Developed mathematical models, methods and algorithms for solving continuous multi-stage problems for locating enterprises with a continuously distributed resource can be used to solve a wide class of continuous linear location-allocation problems. The presented methods, algorithms and software allow solving several practical problems connected, for example, with the strategic planning in the production, social and economic fields. The theoretical results obtained are been brought to the level of specific recommendations that can be used by state-owned and private enterprises in solving logistics tasks related to the organization of collection of a certain resource and its delivery to processing points, as well as further transportation of the product received to places of destination. Conclusions. The results of the computational experiments testify to the correctness of the developed algorithms operation for solving two-stage optimal set partitioning problems with additional couplings. Furthermore, it is confirmed the feasibility of formulating such problems when it is necessary to determine the location of new objects in a given territory, considering the multistage raw material resource distribution process. Further research is subject to the theoretical justification of the convergence of the iterative process realized in the proposed algorithm for solving continuous problems of OPS with additional couplings. In future, the development of software to solve such problems with the involvement of GIS-technologies is planned.

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An approach is proposed to solve one continuous linear single-product problem of optimal partitioning of set Ω from n-dimensional Euclidean space Ån into subsets Ω1,...,ΩN by nding the coordinates of the centers τ1,...,τN of these subsets with fuzzy parameters in constraints. Real situations, which are described by models of optimal partitioning of sets, are most often characterized by a certain degree of uncertainty. In these cases, the quality of the decisions made in the optimization partitioning set problems is directly dependent on the completeness of all the uncertain factors that are signicant for the consequences of the decisions made. The class of the optimal partitioning set problem which makes it possible to consider uncertainty factors of not probabilistic-statistical nature are the optimal partitioning set problems in which either certain parameters in the model description are fuzzy, inaccurate, underdetermined or there exists an invalid mathematical description of some dependencies in the models (for example, demand functions and the cost of transporting a unit of production in innite dimensional transport problems) or the criteria themselves and (or) constraint systems are formulated inaccurately, or the optimization model admits the inuence of external uncontrolled disturbances of dierent forms and etc. The problem-solving algorithm was developed using a synthesis of methods for solving problems of the theory of optimal partitioning of sets with neuro-fuzzy technologies and modications of N.Z.Shor's r-algorithm for solving non-smooth optimization problems. The results are presented for the model problem of optimal partitioning of a set into three subsets with fuzzy parameters on constraints obtained by the developed approach. The results of the solution of the problem with the clear parameters are compared with the results when some parameters of the solved problem are fuzzy, unclear or their mathematical description is incorrect.

An optimal two-stage distribution of material flow at the fuel and energy complex enterprises

The two-stage distributing of material flows in the transport-logistics system of fuel and energy complex is considered. The structural elements of such system are mines (centers of the first stage), extracting coal from various mineral deposits in a certain area, and enterprises that consume or process coal (centers of the second stage). A presented method for solving this problem is based on elements of the theory of continuous linear problems of optimal set partitioning, duality theory, and methods for solving linear programming problems of transport type. The optimal solution of the two-stage location-allocation problem is obtained in an analytical form, which contains parameters that are the optimal solution of the auxiliary finite-dimensional optimization problem with a non-differentiable objective function. Therefore, the part of numerical algorithm is non-differentiable optimization method – modification of Shor’s r-algorithm. The results of computational experiments solving model problems confirm the correctness of the presented method and algorithm. It is demonstrated the synergistic effect obtained from formulation of continuous problems of optimal partitioning sets with additional constraints. It is showed, how important to take into account the multi-stage distributing of raw materials when it is necessary to locate new transport-logistics system objects in a given territory.

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

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

Solving an infinite-dimensional problem of location-allocation with 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

Numerical study of models of optimal distribution of recreational resources

Numerical realization of the problems of optimal distribution of recreational resources adapted to the continuous non-linear multiproduct problem of optimal partitioning of sets, the solution of which can be interesting in the sense of the development of recreation and tourism, is carried out. The success of most management tasks depends on the best way to use resources. In general, this problem is reduced to the problems of optimal resource allocation. The tasks of this type include the tasks of optimal placement of tourist complexes on a given territory, which is attractive in the sense of the tourism industry. In this case, the models of functioning and development of tourist and recreational systems are considered that can be used for analysis and forecasting of the tourism industry, both at the regional and national and international levels. The problem of the optimal distribution of recreational resources in the conditions of the modern and perspective structure of recreational needs was investigated. A mathematical model of the problem is constructed, which is a nonlinear continuous multi-product problem of the optimal partition of a set into its disjoint subsets (among which there may be empty ones) with fixed coordinates of the centers of these subsets under constraints in the form of equalities and inequalities. An analytical solution of the problem is obtained, which includes parameters that are sought as the optimal solution of the auxiliary dual finite-dimensional optimization problem with a nonsmooth target functional. On the basis of the analytical solution of the problem studied, a theoretically grounded solution algorithm was developed. A numerical investigation of the problem is carried out. The presented research results can serve as a useful tool in the sense of effective optimal partitioning of the functioning zones of existing tourist and recreational facilities that are able to produce service complexes to meet the necessary needs of recreants from the service area in conditions of the optimal cost of recreational needs

The Emergence and Formation of the Theory of Optimal Set Partitioning for Sets of the n Dimensional Euclidean Space. Theory and Application

The Emergence and Formation of the Theory of Optimal Set Partitioning for Sets of the n Dimensional Euclidean Space. Theory and Application

Multi-view ensemble learning: an optimal feature set partitioning for high-dimensional data classification

Multi-view ensemble learning has the potential to address issues related to the high dimensionality of data. It attempts to utilize all the relevant only discarding the irrelevant features. The view of a dataset is the sub-table of the training data with respect to a subset of the feature set. The problem of discarding the irrelevant features and obtaining subsets of the relevant features is useful for dimension reduction and dealing with the problem of having fewer training examples than even the reduced set of relevant features. A feature set partitioning resulting in the blocks of relevant features may not yield multiple-view-based classifiers with good classification performance. In this work the optimal feature set partition approach has been proposed. Further, the ensemble learning from views aims to maximize the performance of the classifier. The experiments study the performance of random feature set partitioning, attribute bagging, view generation using attribute clustering, view construction using genetic algorithm and OFSP proposed method. The blocks of relevant feature subsets are used to construct the multi-view classifier ensemble using K-nearest neighbor, Naive Bayesian and support vector machine algorithm applied to sixteen high-dimensional data sets from UCI machine learning repository. The performance parameters considered for comparison are classification accuracy, disagreement among the classifiers, execution time and percentage reduction of attributes.

Recommendations Using Information from Multiple Association Rules: A Probabilistic Approach

Business analytics has evolved from being a novelty used by a select few to an accepted facet of conducting business. Recommender systems form a critical component of the business analytics toolkit and, by enabling firms to effectively target customers with products and services, are helping alter the e-commerce landscape. A variety of methods exist for providing recommendations, with collaborative filtering, matrix factorization, and association-rule-based methods being the most common. In this paper, we propose a method to improve the quality of recommendations made using association rules. This is accomplished by combining rules when possible and stands apart from existing rule-combination methods in that it is strongly grounded in probability theory. Combining rules requires the identification of the best combination of rules from the many combinations that might exist, and we use a maximum-likelihood framework to compare alternative combinations. Because it is impractical to apply the maximum likelihood framework directly in real time, we show that this problem can equivalently be represented as a set partitioning problem by translating it into an information theoretic context—the best solution corresponds to the set of rules that leads to the highest sum of mutual information associated with the rules. Through a variety of experiments that evaluate the quality of recommendations made using the proposed approach, we show that (i) a greedy heuristic used to solve the maximum likelihood estimation problem is very effective, providing results comparable to those from using the optimal set partitioning solution; (ii) the recommendations made by our approach are more accurate than those made by a variety of state-of-the-art benchmarks, including collaborative filtering and matrix factorization; and (iii) the recommendations can be made in a fraction of a second on a desktop computer, making it practical to use in real-world applications.

Theory of Continuous Optimal set Partitioning Problems as a Universal Mathematical Formalism for Constructing Voronoi Diagrams and Their Generalizations. Ii. Algorithms for Constructing Voronoi Diagrams Based on the Theory of Optimal set Partitioning1

Theory of Continuous Optimal set Partitioning Problems as a Universal Mathematical Formalism for Constructing Voronoi Diagrams and Their Generalizations. Ii. Algorithms for Constructing Voronoi Diagrams Based on the Theory of Optimal set Partitioning1