Skip to main content
SpringerLink
Log in

Nonlinear cost network models in transportation analysis

  • Chapter
  • First Online:
Netflow at Pisa

Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 26))

Abstract

The analysis of transportation phenomena by quantitative approaches naturally gives rise to network models that represent the spatial characteristics of the transport infrastructure. This paper surveys the nonlinear cost models that arise in transportation analysis and points out the principal methods used for their solution in practice. We discuss the network equilibrium problem, the problem of estimating an origin/destination matrix and certain concave cost network flow problems.

This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

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

  • H. Aashtiani and T. Magnanti, “A linearization and decomposition algorithm for computing urban traffic equilibrium”, Proceedings of the 1982 IEEE large scale systems symposium (October, 1982).

    Google Scholar 

  • H. Aashtiani and T. Magnanti, “Equilibria on a congested transportation network,” SIAM Journal on Algebraic and Discrete Methods 2 (1981) 213–226.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Abdulaal and L. Leblanc, “Methods for combining model split equilibrium assignment models”, Transportation Science 13 (1979) 292–314.

    Article  Google Scholar 

  • B. Ahn, Computation of market equilibria for policy analysis: The Project Independence Evaluation System (PIES) approach (Garland, New York, 1979).

    Google Scholar 

  • B. Ahn and W. Hogan, “On convergence of the PIES algorithm for computing equilibria”, Operations Research 33 (1982) 281–300.

    Article  MathSciNet  Google Scholar 

  • P.A. Andersson, “On the convergence of iterative methods for the distribution balancing problem”, Transportation Research B 15B (1981) 173–201.

    Article  MathSciNet  Google Scholar 

  • R. Asmuth, “Traffic network equilibria”, Technical Report SOL 78-2, Department of Operations Research, Stanford University (Stanford, CA, 1978).

    Google Scholar 

  • A. Auslender, Optimisation, méthodes numériques (Masson, Paris, 1976).

    MATH  Google Scholar 

  • M.J. Beckmann, C.B. McGuire, and C.B. Winsten, Studies in the economics of transportation (Yale University Press, New Haven, 1956).

    Google Scholar 

  • D.P. Bertsekas, “Optimal routing and flow control methods for communications networks”, in: A. Bensoussan and J.L. Lions, eds., Analysis and optimization of systems (Springer-Verlag, Berlin, New York, 1982) pp. 615–643.

    Google Scholar 

  • D.P. Bertsekas and E.M. Gafni, “Projection methods for variational inequalities with application to the traffic assignment problem”, Mathematical Programming Study 17 (1982) 139–159.

    MATH  MathSciNet  Google Scholar 

  • D. Boyce, “A framework for constructing network equilibrium models of urban location”, Transportation Science 14 (1980) 71–96.

    Article  MathSciNet  Google Scholar 

  • L.M. Bregman, “The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming”, USSR Computational Mathematics and Mathematical Physics 7 (1967) 200–217.

    Article  Google Scholar 

  • A. Bruynooghe, A. Gibert and M. Sakarovitch, “Une méthode d’affectation du traffic”, Proceedings of the fourth symposium on the theory of traffic flow (Karlsruhe, 1968).

    Google Scholar 

  • J.E. Burrel, “Multiple route assignment: A comparison of two methods”, in: M. Florian, ed., Traffic equilibrium methods, Lecture Notes in Economics and Mathematical Systems 118 (Springer-Verlag, Berlin, 1976) pp. 229–239.

    Google Scholar 

  • D. Cantor and M. Gerla, “Optimal routing in a packet switched computer network”, IEEE Transactions on Computers 10 (1974) 1062–1068.

    Article  MathSciNet  Google Scholar 

  • S. Dafermos, “An extended traffic assignment model with applications to two-way traffic”, Transportation Science 4 (1971) 336–389.

    Google Scholar 

  • S. Dafermos, “The traffic assignment problem for multiclass-usertransportation networks”, Transportation Science 6 (1972) 73–87.

    Article  Google Scholar 

  • S. Dafermos, “Traffic equilibrium and variational inequalities”, Transportation Science 14 (1980), 42–54.

    Article  MathSciNet  Google Scholar 

  • S. Dafermos, “Relaxation algorithms for the general asymmetric traffic equilibrium problem”, Transportation Science 16 (1982a) 231–240.

    Article  MathSciNet  Google Scholar 

  • S. Dafermos, “The general multimodal network equilibrium problem with elastic demand”, Networks 12 (1982b) 57–72.

    Article  MATH  MathSciNet  Google Scholar 

  • S. Dafermos, “An iterative scheme for variational inequalities”, Mathematical Programming 26 (1983). 40–47.

    Article  MATH  MathSciNet  Google Scholar 

  • S. Dafermos and A. Nagurney, “Sensitivity analysis for the general asymmetric network equilibrium problem”, Technical Report LCDS M-82-7, Lefschetz Center for Dynamical Systems, Brown University (Providence, RI, 1982).

    Google Scholar 

  • S. Dafermos and T. Sparrow, “The traffic assignment problem for a general network”, Journal of Research of the National Bureau of Standards B 73B (1969) 91–117.

    MathSciNet  Google Scholar 

  • C.F. Daganzo, “Unconstrained extremal formulations of some transportation equilibrium problems”, Transportation Science 16 (1982) 332–360.

    Article  MathSciNet  Google Scholar 

  • R.S. Dembo, “Progress in large scale nonlinear optimization”, presented at XI. International Symposium on Mathematical Programming (Bonn, West Germany, 1982).

    Google Scholar 

  • R.S. Dembo and J.G. Klincewicz, “A scale reduced, gradient algorithm for network flow problems with convex separable costs”, Mathematical Programming Study 15 (1981) 125–147.

    MATH  MathSciNet  Google Scholar 

  • R.S. Dembo and U. Tulowitzki, “Successive inexact quadratic programming and its application to traffic problems”, Working Paper Series B, No. 65, School of Organization and Management, Yale University (New Haven, 1983).

    Google Scholar 

  • W.E. Deming and F.F. Stephan, “On a least squares adjustment of a sampled frequency table when the expected marginal totals are known”, Annals of Mathematical Statistics 11 (1940) 427–444.

    Article  MATH  MathSciNet  Google Scholar 

  • R.B. Dial, “A probabilistic multipath traffic assignment model which obviates path enumeration”, Transportation Research 5 (1971) 83–111.

    Article  Google Scholar 

  • P. Dow and D. Van Vliet, “Capacity restrained road assignment”, Traffic Engineering and Control 20 (1979) 296–305.

    Google Scholar 

  • R.E. Erickson, C.L. Monma and A.F. Veinott, Jr., “Minimum concave cost network flows” Technical Report (1981). Forthcoming in Mathematics of Operations Research.

    Google Scholar 

  • S. Erlander, S. Nguyen and N. Stewart, “On the calibration of the combined distribution-assignment model”, Transportation Research B 13B (1979) 259–267.

    Article  MathSciNet  Google Scholar 

  • S.P. Evans, “Derivation and analysis of some models for combining trip distribution and assignment”, Transportation Research 10 (1976) 37–57.

    Article  Google Scholar 

  • S.P. Evans and H.R. Kirby, “A three-dimensional furness procedure for calibrating gravity models”, Transportation Research 10 (1976) 37–57.

    Article  Google Scholar 

  • S.P. Evans and H.R. Kirby, “A three-dimensional Furness procedure for calibrating gravity models”, Transportation Research 8 (1974) 105–122.

    Article  Google Scholar 

  • J.A. Ferland, “Minimum cost multicommodity circulation problems with convex arc-costs”, Transportation Science 8 (1974) 355–360.

    Article  MathSciNet  Google Scholar 

  • C. Fisk, “Some developments in equilibrium traffic assignment”, Transportation Research 14B (1980) 243–255.

    MathSciNet  Google Scholar 

  • C. Fisk and D. Boyce, “Alternative variational inequality formulations of the network equilibrium—Travel Choice Problem”, Transportation Science 17 (1983) 454–463.

    Article  Google Scholar 

  • C. Fisk and D. Boyce, “A note on trip matrix estimation from link traffic count data”, Transportation Research B 17B (1983) 245–250.

    Article  Google Scholar 

  • C. Fisk and S. Nguyen, “Existence and uniqueness properties of an asymetric two-mode equilibrium model”, Transportation Science 16 (1982) 318–328.

    Article  MathSciNet  Google Scholar 

  • M. Florian, “A traffic equilibrium model of travel by car and public transit modes”, Transportation Science 8 (1977a) 166–179.

    Article  MathSciNet  Google Scholar 

  • M. Florian, “An improved linear approximation algorithm for the network equilibrium (packet switching) problem”, IEEE Proceedings, Decision and Control (1977b) 812–818.

    Google Scholar 

  • M. Florian, “An introduction to network models used in transportation planning”, in: M. Florian, ed., Transportation planning models (North-Holland, Amsterdam, 1984) pp. 137–152.

    Google Scholar 

  • M. Florian, R. Chapleau, S. Nguyen, C. Achim, L. James-Lefebvre and C. Fisk, “Validation and application of EMME: An equilibrium based two-mode urban transportation planning method”, Transportation Research Record 728 (1979) 14–22.

    Google Scholar 

  • M. Florian and S. Nguyen, “A method for computing network equilibrium with elastic demand”, Transportation Science 8 (1974) 321–332.

    Article  MathSciNet  Google Scholar 

  • M. Florian and S. Nguyen, “An application and validation of equilibrium trip assignment methods”, Transportation Science 10 (1976) 374–389.

    Article  Google Scholar 

  • M. Florian and S. Nguyen, “A combined trip distribution modal split and trip assignment model”, Transportation Research 12 (1978) 241–246.

    Article  Google Scholar 

  • M. Florian, J. Ferland and S. Nguyen, “On the combined distribution-assignment of traffic”, Transportation Science 9 (1975) 43–53.

    Article  MathSciNet  Google Scholar 

  • M. Florian, J. Guelat and H. Spiess, “An efficient implementation of the PARTAN variant of the linear approximation method for the network equilibrium problem”, Publication #395, Centre de Recherche sur les Transports, Université de Montréal (1983).

    Google Scholar 

  • M. Florian, M. Rossin-Arthiat and D. de Werra, “A property of minimum concave cost flows in capacitated networks”, Canadian Journal of Operational Research 9 (1971) 293–304.

    Google Scholar 

  • M. Florian and H. Spiess, “The convergence of diagonalization algorithms for asymmetric equilibrium problems”, Transportation Research B 16B (1982) 477–483.

    Article  MathSciNet  Google Scholar 

  • M. Florian and H. Spiess “On binary mode choice/assignment models”, Transportation Science 17 (1983) 32–47.

    Article  Google Scholar 

  • C. Frank, “A study of alternative approaches to combined trip distribution-assignment modelling”, Ph.D. Thesis Department of Regional Science, University of Pennsylvania (Philadelphia, 1978).

    Google Scholar 

  • M. Frank and P. Wolfe, “An algorithm for quadratic programming”, Naval Research Logistics Quarterly 3 (1956) 95–110.

    Article  MathSciNet  Google Scholar 

  • L. Fratta, M. Gerla and L. Kleinrock, “The flow deviation method: An approach to store and forward communication network design”, Networks 3 (1973) 97–133.

    Article  MATH  MathSciNet  Google Scholar 

  • G. Gallo and C. Sodini, “Adjacent extreme flows and application to min concave cost flow problems”, Networks 9 (1979) 95–122.

    Article  MATH  MathSciNet  Google Scholar 

  • G. Gallo, C. Sandi and C. Sodini, “An algorithm for the min concave cost flow problem”, European Journal of Operations Research 4 (1980) 249–255.

    Article  MathSciNet  Google Scholar 

  • G. Gallo and S. Pallotino, “Shortest path methods in transportation models”, in: M. Florian, ed., Transportation planning models (North-Holland, Amsterdam, 1984) pp. 227–256.

    Google Scholar 

  • A. Gibert, “A method for the traffic assignment problem when demand is elastic”, unpublished report LBS TNT-B5 (1968).

    Google Scholar 

  • B. Golden, “A minimum cost multi-commodity, network flow problem concerning imports and exports”, Networks 5 (1975) 331–356.

    Article  MATH  MathSciNet  Google Scholar 

  • J. Guélat, “Algorithmes pour le probleme d’affectation du trafic avec demandes fixes—Comparaisons”, Publication #299, Centre de Recherche sur les Transports Université de Montréal (1982).

    Google Scholar 

  • M.D. Hall, D. Van Vliet and L.G. Willumsen, “SATURN—A simulation assignment model for the evaluation of traffic management schemes”, Traffic Engineering and Control (1980) 168–176.

    Google Scholar 

  • D.W. Hearn, “The gap function of a convex program”, Operations Research Letters 1 (1982) 67–71.

    Article  MATH  MathSciNet  Google Scholar 

  • D.W. Hearn and S. Nguyen, “Dual and saddle functions related to the gap function”, Research Report 82-4, Department of Industrial and Systems Engineering, University of Florida (Gainesville, 1982).

    Google Scholar 

  • W.W. Hogan, “Energy policy models for Project Independence”, Computers and Operations Research 2 (1975) 251–271.

    Article  Google Scholar 

  • C.A. Holloway, “An extension of the Frank and Wolfe method of feasible directions”, Mathematical Programming 6 (1974) 14–27.

    Article  MATH  MathSciNet  Google Scholar 

  • B. Hutchinson, “Land use transport planning models”, in: M. Florian, ed., Transportation Planning models (North-Holland, Amsterdam, 1984) pp. 499–510.

    Google Scholar 

  • E. Jaynes, “Information theory and statistical mechanics”, Physical Review 106 (1957a) 171–190.

    Article  Google Scholar 

  • E. Jaynes, “Information theory and statistical mechanics”, Physical Review 108 (1957b) 620–630.

    Article  Google Scholar 

  • T.R. Jefferson and C.H. Scott, “The analysis of entropy models with equality and inequality constraints”, Transportation Research B 13B (1979) 123–132.

    Article  Google Scholar 

  • K. Jornsten and S. Nguyen, “On the estimation of a trip matrix from network data”, Publication #153, Centre de Recherche sur les Transports, Université de Montréal (1979).

    Google Scholar 

  • N. Josephy, “Newton’s method for generalized equations”, Technical Report 1965, Mathematics Research Center, University of Wisconsin (Madison, 1979).

    Google Scholar 

  • D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and applications (Academic Press, New York, 1980).

    MATH  Google Scholar 

  • F. H. Knight, “Some fallacies in the interpretation of social cost”, Quarterly Journal of Economics 38 (1924) 582–606.

    Article  Google Scholar 

  • J. Kruithof, “Calculation of telephone”, De Ingenieur 52 (1937) E15–E25. [in Flemish.]

    Google Scholar 

  • S. Kullback, Information theory and statistics (Wiley, New York, 1959).

    MATH  Google Scholar 

  • B. Lamond and N.F. Stewart, “Bregman’s balancing method”, Transportation Research B 15B (1981) 239–248.

    Article  MathSciNet  Google Scholar 

  • S. Lawphongpanich and D.W. Hearn, “Simplicial decomposition of the asymmetric traffic assignment problem,” Research Report 82-12, Department of Industrial and Systems Engineering, University of Florida (Gainesville, 1982).

    Google Scholar 

  • L. Le Blanc, “Global solutions for a non convex, non concave rail network model”, Mathematical Science 23 (1978) 131–139.

    MathSciNet  Google Scholar 

  • L. Le Blanc and K. Farhangian, “Efficient algorithms for solving elastic demand traffic assignment problems and mode split assignment problems”, Transportation Science 15 (1981) 306–312.

    Article  MathSciNet  Google Scholar 

  • L.J. Leblanc, R.V. Helgason and D.E. Boyce, “Improved efficiency of the Frank-Wolfe algorithm”, Working Paper 381-131, Owen Graduate School of Management, Vanderbilt University (Nashville, TN, 1982).

    Google Scholar 

  • L.J. Le Blance, E.K. Morlok and W.P. Pierskalla, “An efficient approach to solving the road network equilibrium traffic assignment problem”, Transportation Research 5 (1975) 309–318.

    Article  Google Scholar 

  • T. Leventhal, G. Nemhauser and L.E. Trotter, Jr., “A column generation algorithm for optimal traffic assignment”, Transportation Science 7 (1973) 168–176.

    Article  MathSciNet  Google Scholar 

  • M. Los, “Combined residential-location and transportation models”, Environment and Planning A 11 (1979) 1241–1265.

    Article  Google Scholar 

  • D.G. Luenberger, Introduction to linear and nonlinear programming (Addison-Wesley, Reading, MA, 1973).

    MATH  Google Scholar 

  • T.L. Magnanti, “Equilibria and congested networks: Modeling and computation”, ORSA/TIMS Bulletin 10 (1980).

    Google Scholar 

  • T.L. Magnanti, “Models and algorithms for predicting urban traffic equilibrium”, in: M. Florian, ed., Transportation planning models (North-Holland, Amsterdam, 1984) pp. 153–186.

    Google Scholar 

  • J.D. Murchland, “Road network traffic distribution in equilibrium”, in: Proceedings of the conference “Mathematical methods in economic sciences”, Mathematisches Forschungsinstitut, Oberwolfach (W. Germany, 1969).

    Google Scholar 

  • B. Murtagh and M.A. Saunders, “MINOS: A large-scale nonlinear programming system—User’s guide”, Technical Report SOL 77-9, Systems Optimization Laboratory, Department of Operations Research, Stanford University (Stanford, CA, 1977).

    Google Scholar 

  • S. Nguyen, “A mathematical programming approach to equilibrium methods of traffic assignment”, Publication #138, Département d’informatique et de recherche opérationnelle, Université de Montréal (1973).

    Google Scholar 

  • S. Nguyen, “An algorithm for the traffic assignment problem”, Transportation Science 8 (1974) 203–216.

    Article  Google Scholar 

  • S. Nguyen, “A unified approach to equilibrium methods for traffic assignment”, in: M. Florian, ed., Traffic equilibrium methods, Lecture Notes in Economics and Mathematical Systems 118 (Springer-Verlag, Berlin, 1976) pp. 148–182.

    Google Scholar 

  • S. Nguyen, “Estimating an OD matrix from network dates: A network equilibrium approach”, Publication #60, Centre de Recherche sur les Transports, Université de Montréal (1977a).

    Google Scholar 

  • S. Nguyen, “Procedures for equilibrium traffic assignment with elastic demand”, Publication #339, Centre de recherche sur les Transports, Université de Montréal (1977b).

    Google Scholar 

  • S. Nguyen, “Modeles de distribution spatiale tenant compte des itineraires”, Publication #225, Centre de Recherche sur les Transports, Université de Montréal (1981). To appear in INFOR.

    Google Scholar 

  • S. Nguyen and C. Dupuis, “An efficient method for computing traffic equilibria in network with asymmetric transportation costs”, Transportation Science 18 (1984) 185–232.

    Article  Google Scholar 

  • A. Ohuchi and I. Kaji, “An algorithm for the Hitchcock transportation problems with quadratic cost functions”, Journal of the Operations Research Society of Japan 24 (1981) 170–191.

    MATH  MathSciNet  Google Scholar 

  • J.-S. Pang and D. Chan, “Gauss-Seidel methods for variational inequality problems over product sets”, Working Paper, GSIA, Carnegie-Mellon University (Pittsburgh, PA, 1982).

    Google Scholar 

  • J.-S. Pang and D. Chan, “Iterative methods for variational and complementarity problems”, Mathematical Programming 24 (1982) 284–313.

    Article  MATH  MathSciNet  Google Scholar 

  • E.R. Peterson and D.M. Tulett, “An algorithm for calculating equilibrium demand with fixed traffic routing”, Technical Report, School of Business, Queens University (Kingston, Ontario, Canada, 1982).

    Google Scholar 

  • R.B. Potts and R.M. Oliver, Flows in transportation networks (Academic Press, New York, 1972).

    MATH  Google Scholar 

  • W.B. Powell and Y. Sheffi, “The convergence of equilibrium algorithms with predetermined step sizes”, Transportation Science 16 (1982) 45–55.

    Article  MathSciNet  Google Scholar 

  • P. Rech and L.G. Barton, “A non-convex transportation algorithm,” in: E.M.L. Beale, ed., Applications of mathematical programming techniques (American Elsevier, New York, 1970) pp. 250–260.

    Google Scholar 

  • P. Robillard and N.F. Stewart, “Iterative numerical methods for trip distribution problems”, Transportation Research 8 (1974) 575–582.

    Article  Google Scholar 

  • B. Roy, Algebre moderne et théorie des graphes, Tome 2 (Dunod, Paris, 1970).

    Google Scholar 

  • R. Rust and C. Bornstein, “Um algorithmo branch-and-bound para problemas de localizqçao com custos expressos por funçoes escadas”, Technical Report, COPPE, Federal University of Rio de Janeiro (1982).

    Google Scholar 

  • B.V. Shah, R.J. Buehler and O. Kempthorne, “Some algorithms for minimizing a function of several variables”, SIAM Journal on Applied Mathematics 12 (1964) 74–92.

    Article  MATH  MathSciNet  Google Scholar 

  • M.J. Smith, “Existence, uniqueness and stability of traffic equilibria”, Transportation Research B 1B (1979) 295–304.

    Article  Google Scholar 

  • F. Snickars and J.W. Weibull, “A minimum information principle, theory and practice”, Regional Science and Urban Economics 7 (1977) 137–168.

    Article  Google Scholar 

  • R.M. Soland, “Optimal facility location with concave costs”, Operations Research 22 (1974) 373–382.

    Article  MATH  MathSciNet  Google Scholar 

  • F. Soumis, “Planification d’une flotte d’avions”, Publication #133, Centre de Recherche sur les Transports, Université de Montréal (1978).

    Google Scholar 

  • N.F. Stewart, “Notes on the mathemtical structure of equilibrium models”, Technical Report LiTH-MAT-12-79-8, Department of Mathematics, Linkøping Institute of Technology (Linkøping, Sweden, 1979).

    Google Scholar 

  • J. Wardrop, “Some theoretical aspects of road traffic research”, in: Proceedings of the Institute of Civil Engineers, Part II, Vol. 1 (1952) 325–378.

    Google Scholar 

  • A. Weintraub and J. Gonzales, “An algorithm for the traffic assignment problem”, Networks. 10 (1980) 191–210.

    Article  Google Scholar 

  • A.G. Wilson, “A statistical theory of spatial distribution models”, Transportation Research 1 (1967) 253–269.

    Article  Google Scholar 

  • A.G. Wilson, Entropy in urban and regional modelling (Pion, London, 1970).

    Google Scholar 

  • P. Wolfe, “Convergence theory in nonlinear programming,” in: J. Abadie, ed., Nonlinear and integer programming (North-Holland, Amsterdam, 1970) pp. 1–36.

    Google Scholar 

  • B. Yaged, Jr., “Minimum cost routing for static network models,” Networks 1 (1971) 139–172.

    Article  MATH  MathSciNet  Google Scholar 

  • N. Zadeh, “On building minimum cost communication networks”, Networks 3 (1973) 315–331.

    Article  MATH  MathSciNet  Google Scholar 

  • N. Zadeh, “On building minimum cost communication networkds over time”, Networks 4 (1974) 19–34.

    Article  MATH  Google Scholar 

  • W.I. Zangwill, “Minimum concave cost flows in certain networks”, Management Science 14 (1968) 429–450.

    Article  MATH  MathSciNet  Google Scholar 

  • W.I. Zangwill, Nonlinear programming: A unified approach (Prentice-Hall, Englewood, Cliffs, NJ, 1969).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département d’informatique et de recherche opérationnelle and Centre de recherche sur les transports, Université de Montréal, H3C 3J7, Quebec, Canada

    Michael Florian

Authors
  1. Michael Florian
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Giorgio Gallo Claudio Sandi

Rights and permissions

Reprints and permissions

Copyright information

© 1986 The Mathematical Programming Society, Inc.

About this chapter

Cite this chapter

Florian, M. (1986). Nonlinear cost network models in transportation analysis. In: Gallo, G., Sandi, C. (eds) Netflow at Pisa. Mathematical Programming Studies, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121092

Download citation

  • DOI: https://doi.org/10.1007/BFb0121092

  • Received:

  • Revised:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00922-8

  • Online ISBN: 978-3-642-00923-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation