Study of the mathematical models of optimal partitioning for particular cases

Authors

DOI:

https://doi.org/10.15587/1729-4061.2018.123261

Keywords:

optimal partitioning, continuous set, minimization, arc length, set center, metric, placement

Abstract

The basic problem of optimal sets partitioning (OSP) for the case, where a segment of a plane curve is a set, was stated. The problem is stated as follows: let us assume there is a segment of a plane curve, it is required to place on it a specified number of sources of a certain resource and allocate each point of the curve to a particular source. In addition, it is necessary to minimize the costs of transportation of a resource is from the sources to the corresponding points of the curve along the shortest route. The basic problem was refined by taking into account geometrical characteristics of the curve. For this, the function of the cost was changed according to such parameters as the length of the curve and its curvature.

As a result, new statements of OSP problems were obtained. It was shown that geometric characteristics of the curve correspond to a subject area. Each of the problems was solved by using the known methods and the numerical experiment was conducted. Analysis of the obtained results was carried out. Thus, the general OSP theory was supplemented with the new models that are applicable for solving optimization problems with taking into account surfaces of a relief.

Author Biography

Alexander Firsov, University of customs and finance Volodymyra Vernadskoho str., 2/4, Dnipro, Ukraine, 49000

PhD, Associate Professor

Department of transport systems and technologies

References

  1. Us, S. A. (2010). O modelyah optimal'nogo razbieniya mnozhestv v usloviyah neopredelennosti. Pytannia prykladnoi matematyky i matematychnoho modeliuvannia, 320–326.
  2. Kiseleva, E. M., Lozovskaya, L. I., Timoshenko, E. V. (2009). Reshenie nepreryvnyh zadach optimal'nogo pokrytiya sharami s ispol'zovaniem teorii optimal'nogo razbieniya mnozhestv. Kibernetika i sistemnyy analiz, 3, 98–117.
  3. Kiseleva, E. M., Koryashkina, L. S., Shevchenko, T. A. (2014). O reshenii dinamicheskoy zadachi optimal'nogo razbieniya mnozhestv s razmeshcheniem centrov podmnozhestv. Kibernetika i sistemniy analiz, 6, 29–40.
  4. Shevchenko, T., Kiseleva, E., Koriashkina, L. (2009). The Features of Solving of the set Partitioning Problems with Moving Boundaries Between Subsets. Operations Research Proceedings 2008, 533–538. doi: 10.1007/978-3-642-00142-0_86
  5. Koriashkina, L. S., Shevchenko, T. O. (2009). Novi pidkhody do rozviazannia dynamichnoi zadachi optymalnoho rozbyttia mnozhyn. Pytannia prykladnoi matematyky i matematychnoho modeliuvannia, 220–231.
  6. Kiseleva, E. M., Zhil'cova, A. A. (2009). Nepreryvnaya zadacha optimal'nogo nechetkogo razbieniya mnozhestv bez ogranicheniy s zadannym polozheniem centrov podmnozhestv. Pytannia prykladnoi matematyky ta matematychnoho modeliuvannia, 121–136.
  7. Hungerford, J. T. (2012). Research Statement: New Methods For Transforming Discrete Optimization Problems Into Continuous Problems, 4.
  8. Bakolas, E., Tsiotras, P. (2010). The Zermelo-Voronoi diagram: A dynamic partition problem. Automatica, 46 (12), 2059–2067. doi: 10.1016/j.automatica.2010.09.003
  9. Balzer, M. (2009). Capacity-Constrained Voronoi Diagrams in Continuous Spaces. 2009 Sixth International Symposium on Voronoi Diagrams. doi: 10.1109/isvd.2009.28
  10. Jooyandeh, M., Mohades, A., Mirzakhah, M. (2009). Uncertain Voronoi diagram. Information Processing Letters, 109 (13), 709–712. doi: 10.1016/j.ipl.2009.03.007
  11. Kiseleva, E. M., Koriashkina, L. S. (2015). Theory of Continuous Optimal Set Partitioning Problems as a Universal Mathematical Formalism for Constructing Voronoi Diagrams and Their Generalizations. I. Theoretical Foundations. Cybernetics and Systems Analysis, 51 (3), 325–335. doi: 10.1007/s10559-015-9725-x
  12. Guruprasad, K. R. (2012). Effectiveness-based Voronoi partition: a new tool for solving a class of location optimization problems. Optimization Letters, 7 (8), 1733–1743. doi: 10.1007/s11590-012-0519-z
  13. Zhao, X., Zhang, H., Jiang, Y., Song, S., Jiao, X., Gu, M. (2013). An Effective Heuristic-Based Approach for Partitioning. Journal of Applied Mathematics, 2013, 1–8. doi: 10.1155/2013/138037
  14. Anders, G., Siefert, F., Steghöfer, J.-P., Reif, W. (2012). A Decentralized Multi-agent Algorithm for the Set Partitioning Problem. Lecture Notes in Computer Science, 107–121. doi: 10.1007/978-3-642-32729-2_8
  15. Koriashkina, L., Saveliev, V., Zhelo, A. (2017). On Mathematical Models of Some Optimization Problems Arising in the Production of Autoclaved Aerated Concrete. Advanced Engineering Forum, 22, 173–181. doi: 10.4028/www.scientific.net/aef.22.173
  16. Lau, B., Sprunk, C., Burgard, W. (2013). Efficient grid-based spatial representations for robot navigation in dynamic environments. Robotics and Autonomous Systems, 61 (10), 1116–1130. doi: 10.1016/j.robot.2012.08.010
  17. Us, S., Stanina, O. (2017). The methods and algorithms for solving multi-stage location-allocation problem. Power Engineering and Information Technologies in Technical Objects Control. doi: 10.1201/9781315197814-21
  18. Blyuss, O., Koriashkina, L., Kiseleva, E., Molchanov, R. (2015). Optimal Placement of Irradiation Sources in the Planning of Radiotherapy: Mathematical Models and Methods of Solving. Computational and Mathematical Methods in Medicine, 2015, 1–8. doi: 10.1155/2015/142987

Downloads

Published

2018-02-09

How to Cite

Firsov, A. (2018). Study of the mathematical models of optimal partitioning for particular cases. Eastern-European Journal of Enterprise Technologies, 1(4 (91), 69–76. https://doi.org/10.15587/1729-4061.2018.123261

Issue

Vol. 1 No. 4 (91) (2018): Mathematics and Cybernetics - applied aspects

Section

Mathematics and Cybernetics - applied aspects

License

Copyright (c) 2018 Alexander Firsov

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.

A license agreement is a document in which the author warrants that he/she owns all copyright for the work (manuscript, article, etc.).
The authors, signing the License Agreement with TECHNOLOGY CENTER PC, have all rights to the further use of their work, provided that they link to our edition in which the work was published.
According to the terms of the License Agreement, the Publisher TECHNOLOGY CENTER PC does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.