Skip to main content
SpringerLink
Log in

Continuum approximation to dense networks and its application to the analysis of urban road networks

  • Chapter
  • First Online:
Applications

Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 20))

Abstract

A continuum approximation is proposed for problems of flows in large and dense networks, where each point of the continuum is characterized by ‘capacity’ and/or ‘distance/cost’ which are convex sets in the tangent space. The problems of flows in continua corresponding to various problems of flows in networks are formulated in variational forms, of which mathematical properties are investigated and for which numerical algorithms are derived by means of the finite-element discretization technique. A practical procedure, based upon concepts from integral geometry, for constructing the continuum which approximates the original network is also proposed.

The effectiveness of the continuum approximation is tested against flow problems on urban road networks of moderate size, to get satisfactory results. The continuum approximation has many advantages over the ordinary network model; e.g. (i) it is easier to build from the practical standpoint of data gathering; (ii) its solution helps us to intuitively understand the global characteristics of road networks; (iii) the amount of computation does not depend on the size of the original network.

This work was supported in part by the Grant in Aid for Scientific Research of the Ministry of Education, Science and Culture of Japan.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. I. Ekeland and R. Temam, Convex analysis and variational problems (North-Holland, Amsterdam, 1976).

    MATH  Google Scholar 

  2. W.H. Fleming and L.C. Young, “A generalized notion of boundary”, Transactions of the American Mathematical Society 76 (1954) 457–484.

    Article  MATH  MathSciNet  Google Scholar 

  3. W.H. Fleming, “Functions with generalized gradients and generalized surfaces”, Annali di Matematica 44 (1957) 96–103.

    MathSciNet  Google Scholar 

  4. T.C. Hu, Integer programming and network flows (Addison-Wesley, Reading, MA, 1969).

    MATH  Google Scholar 

  5. M. Iri, Network flow, transportation and scheduling—Theory and algorithms (Academic Press, New York 1969).

    MATH  Google Scholar 

  6. M. Iri, “Theory of flows in continua as approximation to flows in networks”, in: A. Prekopa, ed., Survey of mathematical programming, Vol. 2 (North-Holland, Amsterdam, 1980) pp. 263–278.

    Google Scholar 

  7. T. Koshizuka, “Fundamental studies of urban space”, Doctoral Thesis, University of Tokyo, Tokyo (1977) [In Japanese].

    Google Scholar 

  8. E. Michael, “Continuous selections I”, Annals of Mathematics 63 (1956) 361–382.

    Article  MathSciNet  Google Scholar 

  9. J.T. Oden and J.N. Reddy, An introduction to the mathematical theory of finite elements (Wiley-Interscience, New York, 1976).

    MATH  Google Scholar 

  10. M.J.D. Powell, “An efficient method for finding the minimum of a function of several variables without calculating derivatives”, The Computer Journal 7 (1964) 155–162.

    Article  MATH  MathSciNet  Google Scholar 

  11. R.T. Rockafellar, “Integrals which are convex functionals”, Pacific Journal of Mathematics 24 (1968) 525–539.

    MATH  MathSciNet  Google Scholar 

  12. R.T. Rockafellar, Convex analysis, (Princeton University Press, Princeton, NJ, 1970)

    MATH  Google Scholar 

  13. R.T. Rockafellar, “Integrals which are convex functionals, II”, Pacific Journal of Mathematics 39 (1971) 439–469.

    MATH  MathSciNet  Google Scholar 

  14. L.A. Santalo, Integral geometry and geometric probability (Addison-Wesley, Reading, MA, 1976).

    MATH  Google Scholar 

  15. J.A. Schouten, Ricci-calculus (Springer, Berlin, 1954).

    MATH  Google Scholar 

  16. G. Strang, “A minimax theory in plasticity theory”, in: M.Z. Nashed, ed., Functional analysis method in numerical analysis (Springer, Berlin 1979) pp. 319–333.

    Chapter  Google Scholar 

  17. A. Taguchi, “A method for the analysis of urban road networks”, Communications of the Operations Research Society of Japan 23 (1978) 756–763 [in Japanese].

    Google Scholar 

  18. A. Taguchi, “Maximum flow problem in discrete continuous compound system and its numerical approach”, Journal of the Operations Research Society of Japan 21 (1978) 245–272.

    MATH  MathSciNet  Google Scholar 

  19. N. Tomizawa, “A computational experiment on a practical large-scale problem”, in: M. Iri, ed., Fundamental studies of the techniques for processing network problems in operations research by computers, Technical Report Series T-73-2 (Operations Research Society of Japan, Tokyo, 1973) pp. 115–133 [In Japanese].

    Google Scholar 

  20. P. Wolfe, “A method of conjugate subgradients for minimizing nondifferentiable functions”, Mathematical Programming Study 3 (1975) 145–173.

    MathSciNet  Google Scholar 

  21. Y. Yamamoto, “A theory of saturation”, in: Proceedings of the 11th Japan national congress for applied mechanics (Japan National Committee for Theoretical and Applied Mechanics. Science Council of Japan, 1961) pp. 355–360.

    Google Scholar 

  22. S.C. Dafermos, “Continuum modelling of transportation networks”, Transportation Research (to appear).

    Google Scholar 

  23. S. Angel and G.M. Hyman, Urban fields (Pion Limited, London, 1976).

    Google Scholar 

Download references

Author information

Author notes

  1. Azuma Taguchi

    Present address: Department of Computer Science, Faculty of Engineering, Yamanashi University, Kofu, Japan

Authors and Affiliations

  1. Department of Mathematical Engineering and Instrumentation Physics, Faculty of Engineering, University of Tokyo, Bunkyo-ku, Tokyo, Japan

    Azuma Taguchi & Masao Iri

Authors
  1. Azuma Taguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masao Iri
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Jean-Louis Goffin Jean-Marc Rousseau

Rights and permissions

Reprints and permissions

Copyright information

© 1982 The Mathematical Programming Society, Inc.

About this chapter

Cite this chapter

Taguchi, A., Iri, M. (1982). Continuum approximation to dense networks and its application to the analysis of urban road networks. In: Goffin, JL., Rousseau, JM. (eds) Applications. Mathematical Programming Studies, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121231

Download citation

  • DOI: https://doi.org/10.1007/BFb0121231

  • Received:

  • Revised:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00851-1

  • Online ISBN: 978-3-642-00852-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation