Abstract
An algorithm is proposed to construct additive and multiplicative Voronoi diagrams under uncertainty in the initial data of a problem, which may be inaccurate, unreliable or fuzzy. To remove the uncertainty in the initial data having a fuzzy-multiple nature, a method of neurolinguistic identification of complex nonlinear dependences is used. The developed algorithm is based on a synthesis of the theory of optimal set partitioning and neuro-fuzzy technologies. Methods of optimal set partitioning are a universal mathematical apparatus for constructing various types of Voronoi diagrams. This apparatus uses an approach based on the statement of continuous problems of optimal partitioning of sets from n-dimensional Euclidian space into subsets with a partition quality criterion that provides the corresponding types of Voronoi diagrams. The universality of such approach to the construction of Voronoi diagrams makes it possible to generalize the methods of solving the optimal set partitioning problems for the case of fuzzy setting of the initial problem parameters. Along with the problem of constructing a Voronoi diagram with fuzzy parameters, this approach also allows us to state and solve a problem of finding optimal, in a sense, coordinates of the diagram’s generator points. The developed algorithm is software implemented. The work of the algorithm is demonstrated through examples of constructing additively weighted and multiplicatively weighted Voronoi diagrams with fuzzy parameters and optimal placement of generator points in a bounded set of n-dimensional Euclidian space.
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References
Atamturk, A., Nemhauser, G., Savelsbergh, M.: A combined lagrangian, linear programming and implication heuristic for large-scale set partitioning problems. J. Heuristics 1, 247–259 (1996). https://doi.org/10.1007/BF00127080
Aurenhammer, F., Klein, R., Lee, D.T.: Voronoi Diagrams and Delaunay Triangulations. Springer, New York (2013). https://doi.org/10.1007/978-1-4939-2864-4507
Bezdek, J., Dubois, D., Prade, H. (eds.): Fuzzy Sets in Approximate Reasoning and Information Systems. Springer, Boston (1999). https://doi.org/10.1007/978-1-4615-5243-7
Borisov, V., Kruglov, V., Fedulov, A.: Fuzzy models and networks. Hotline-Telecom, Moscow (2016)
Kacprzyk, J., Pedrycz, W. (eds.): Springer Handbook of Computational Intelligence. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-43505-2
Kiseleva, E.: The emergence and formation of the theory of optimal set partitioning for sets of the n-dimensional euclidean space. Theory and application. J. Autom. Inf. Sci. 50(9), 1–24 (2018). https://doi.org/10.1615/JAutomatInfScien.v50.i9.10
Kiseleva, E., Koriashkina, L.: Theory of continuous optimal set partitioning problems as a universal mathematical formalism for constructing voronoi diagrams and their generalizations. Algorithms for constructing Voronoi diagrams based on the theory of optimal set partitioning. Cybern. Syst. Anal. 51(4), 489–499 (2015). https://doi.org/10.1007/s10559-015-9740-y
Kiseleva, E., Koriashkina, L.: Theory of continuous optimal set partitioning problems as a universal mathematical formalism for constructing Voronoi diagrams and their generalizations. Theoretical foundations. Cybern. Syst. Anal. 51(3), 325–335 (2015). https://doi.org/10.1007/s10559-015-9725-x
Kiseleva, E., Prytomanova, O., Zhuravel, S.: Valuation of startups investment attractiveness based on neuro-fuzzy technologies. J. Autom. Inf. Sci. 48(9), 1–22 (2016). https://doi.org/10.1615/JAutomatInfScien.v48.i9.10
Okabe, A., Boots, B., et al.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, West Sussex (2000)
Preparata, F., Sheimos, M.: Computational Geometry: An Introduction. Springer, New York (1985). https://doi.org/10.1007/978-1-4612-1098-6
Shor, N.: Nondifferentiable Optimization and Polynomial Problems. Springer, Boston (1998). https://doi.org/10.1007/978-1-4757-6015-6
Siddique, N.: Fuzzy Sets Theory and Its Applications. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-02135-5
Voskoglou, M. (ed.): Fuzzy Sets, Fuzzy Logic and Their Applications. MDPI, Basel (2020). http://www.mdpi.com/journal/mathematics/special_issues/
Zadeh, L.: The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci. 8(3), 199–249 (1975). https://doi.org/10.1016/0020-0255(75)90036-5
Zadeh, L., Kacprzyk, J. (eds.): Computing with Words in Information/Intelligent Systems 2. Applications. Physica, Heidelberg (1999). https://doi.org/10.1007/978-3-7908-1872-7
Zimmerman, H.J.: Fuzzy Sets Theory and Its Applications. Springer, Dordrecht (2001). https://doi.org/10.1007/978-94-010-0646-0
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Kiseleva, E., Prytomanova, O., Padalko, V. (2021). An Algorithm for Constructing Additive and Multiplicative Voronoi Diagrams Under Uncertainty. In: Babichev, S., Lytvynenko, V., Wójcik, W., Vyshemyrskaya, S. (eds) Lecture Notes in Computational Intelligence and Decision Making. ISDMCI 2020. Advances in Intelligent Systems and Computing, vol 1246. Springer, Cham. https://doi.org/10.1007/978-3-030-54215-3_46
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