Fundamentals of the statistical theory of the construction of continuum models of production lines

Authors

DOI:

https://doi.org/10.15587/1729-4061.2014.26280

Keywords:

PDE-model production line, work in progress, balance equations, equation of state

Abstract

The class of models of production systems with in-line production organization, introduced by the author (2003) and widely used nowadays for constructing effective control systems of production lines is discussed in the paper. Conceptual provisions of the statistical theory of production lines, operating in the transient and steady modes are considered. New types of models that allow to combine the self-consistent object-technology at the micro-level and flow at the macro-level descriptions of the production line are proposed. To build unsteady equations of state of the production line, analytical design methods of technological trajectories of objects of labor are developed. The design methods of technological trajectories are based on the laws of conservation of the number of transferred technology resources on the object of labor at a given space-time structure of the technological process. The developed design methods of technological trajectories were used to construct continuum models of production lines that operate in transient modes. To describe the stochastic process of the transfer of technology resources on the object of labor, the distribution function of objects by states is introduced. A kinetic model of transient processes, the equation of which for the first time contains terms that take into account the normative technological trajectories of objects of labor, and the mechanism of interaction of objects of labor among themselves and the process equipment is constructed. Using the kinetic equation, multi-moment balance equations of the continuum flow model of the production line are written. It is shown that the equations of the model of object-technology description are interrelated and coordinated with the balance equations of continuum flow models through the level of the kinetic description of the production process.

Author Biography

Олег Михайлович Пигнастый, National Technical University "Kharkiv Polytechnic Institute"

Ph.D., Associate Professor

Department of Computer Monitoring and logistics

References

  1. Demutsky, V. P. (2003). Enterprise Theory: Sustainability of mass production and product promotion. Kharkiv, Ukraine: KNU Karazin, 272.
  2. He, F. L., Dong, M., Shao, X. F. (2011). Modeling and analysis of material flows in re-entrant Supply Chain Networks Using modified partial differential equations. Journal of Applied Mathematics, 14.
  3. Berg, R., Lefeber E., Rooda, J. (2008). Modelling and Control of a Manufacturing Flow Line using Partial Differential Equations. IEEE Transaction son Control Systems Technology. Boston, 130–136.
  4. Lefeber, E., Berg, R., Rooda, J. (2004). Modeling, Validation and Control of Manufacturing Systems. Proceeding of the 2004 American Control Conference. Massachusetts, 4583–4588.
  5. Armbruster, D., Ringhofer, C., Jo, T-J. (2004). Continuous models for production flows. In Proceedings of the 2004 American Control Conference. Boston, 4589–4594.
  6. Bramson, M. (2008). Stability of queueing networks, lecture notes in mathematic. Journal of Probability Surveys, 5, 169–345.
  7. Schmitz, J. P. M., van Beek, D. A., Rooda, J. E. (2002). Chaos in discrete production systems? Journal of Manufacturing Systems, 21 (3), 236–246. Available at: http://mate.tue.nl/mate/pdfs/2707.pdf. doi:10.1016/s0278-6125(02)80164-9
  8. Kempf, К., Marthaler, D., Ringhofer, C., Armbruster, D., Tae-Chang, J. A. (2006). Continuum Model for a Re-entrant Factory. Operations research, 54 (5), 933–950.
  9. Vollmann, T. E., Berry, L., Whybark, D. C., Jacobs, F. R. (2005). Manufacturing Planning and Control for Supply Chain Management. New York: McGraw-Hill, 520.
  10. Tian, F., Willems, S. P., Kempf, K. G. (2011). An iterative approach to item-level tactical production and inventory planning. International Journal of Production Economics, 133 (1), 439–450. doi:10.1016/j.ijpe.2010.07.011
  11. Krasovskii, A. A. (1974). Phase space and the statistical theory of dynamical systems. Moscow: Nauka, 232.
  12. Boltzmann, L. (1953). Lectures on the theory of gases. Moscow: GITTL, 552.
  13. Gibbs, D. V. (2002). Basic principles of statistical mechanics. - Moscow: Regular and Chaotic Dynamics, 204.
  14. Lysenko, Y. G. (2007). Modeling technological flexibility of production and economic systems. Donetsk: Dondo, 238.
  15. Gross, D. (1974). Fundamentals of Queueing Theory. New York, 490.
  16. Korobetsky, Y. P., Ramazanov, S. K. (2003). Simulation models for flexible systems. Lugansk: Univ. VNU, 280.
  17. Ramadge, P. J., Wonham, W. M. (1989). The control of discrete event systems. Proceedings of the IEEE, 77 (1), 81–98.
  18. Berg, R. (2004). Partial differential equations in modelling and control of manufacturing systems. Eindhoven: Eindhoven Univ. Technol., 157.
  19. Lefeber, E. (2012). Modeling and Control of Manufacturing Systems. Decision Policies for Production Networks, 9–30. Available at: http://www.mate.tue.nl/mate/pdfs/4779.pdf. doi:10.1007/978-0-85729-644-3_2
  20. Mehdi, J. (1991). Stochastic Models in Queuing Theory. New York, 482.
  21. Forrester, J. (1961). Fundamentals of Cybernetics enterprise. Moscow: Progress Publishers, 341.
  22. Asmundsson, J., Rardin, R. L., Turkseven, C. H., Uzsoy, R. (2009). Production planning with resources subject to congestion. Naval Research Logistics, 56 (2), 142–157. doi:10.1002/nav.20335
  23. Graves, S. C. (1986). A Tactical Planning Model for a Job Shop. Operations Research, 34 (4), 522–533. doi:10.1287/opre.34.4.522
  24. Karmarkar, U. S. (1989). Capacity Loading and Release Planning with Work-in-Progress (WIP) and Leadtimes. Journal of Manufacturing and Operations Management, 2, 105–123.
  25. Zhang, Liang (2009). System-theoretic properties of Production Lines. A dissertation submitted the degree of Doctor of Philosophy. Michigan, 289.
  26. Dabagyan, A. V. (2008). Design of technical systems. Kharkov: TD "Golden Mile", 2, 280.
  27. Van Eekelen, J. A. W. M., Lefeber, E., Rooda, J. E. (2006). Coupling event domain and time domain models of manufacturing systems. Proceedings of the 45th IEEE Conference on Decision and Control, 436–441. Available at: http://www.google.com.ua/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCwQFjAA&url=http://mn.wtb.tue.nl/~lefeber/do_download_pdf.php?id=14&ei=SHN1UuTcOISK4AS5oICwBQ&usg=AFQjCNEeN5G2eu5WqV4336nWp0ZtditfzA&sig2=tC4PTJcFQ6n4YxaGptrGrQ. doi:10.1109/cdc.2006.377701
  28. Fedyukin, V. K. (2004). Quality management processes. St. Peter, 204.

Downloads

Published

2014-07-24

How to Cite

Пигнастый, О. М. (2014). Fundamentals of the statistical theory of the construction of continuum models of production lines. Eastern-European Journal of Enterprise Technologies, 4(3(70), 38–48. https://doi.org/10.15587/1729-4061.2014.26280

Issue

Vol. 4 No. 3(70) (2014): Control processes

Section

Control processes

License

Copyright (c) 2014 Олег Михайлович Пигнастый

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.