• Alexander S. Belenky
Part of the Applied Optimization book series (APOP, volume 20)


Mathematical methods of optimization for strategic planning and operations management in transportation systems considered in the present book provide solutions for basic problems in the field detected and formulated in [1] as optimization problems. The list of such problems is presented in the table contained in the Introduction. These problems encompass a variety of practical situations beginning with those for various types of transport and finishing with problems of analyzing transport as a part of an infrastructure of the national economy complex [2]. These methods form the main mathematical tools used today in the research and practical strategic planning and operations management in transportation systems. However, the process of increasing the activity in mathematical modeling in transportation systems and in that of detecting new problems that may be formalized and researched on the basic of mathematical models is extremely intensive. In the course of this process, new optimization methods within the framework of known classes of problems, as well as new formulations of problems necessitating the use of different existing mathematical tools or working out the new ones appear [3]. In the author’s opinion, two directions of researching transportation systems will shortly acquire a practical turn and demand using and working out new mathematical tools. These directions for which outlines of the tools are becoming more and more evident in the process of the directions development are briefly discussed below.


Variational Inequality Transportation System Optimal Trajectory USSR Academy Economic Dynamic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Belen’kii, A. S. Matematicheskie Modeli Optimal’nogo Planirovanija v Transportnykh Sistemakh. Itogi nauki i tekhniki. Seria Organizatzija Upravlenija Transportom (Mathematical Models of Optimal Planning in Transportation Systems. Frontiers of Science and Technology. Series Organization of Transportation Management). Moscow: VINITI, 7, 1988 [in Russian].Google Scholar
  2. [2]
    Belen’kii, A. S. Metody Optimal’nogo Planirovanija na Transporte (Optimum Planning Methods for Transport). Moscow: Znanie, 1988 [in Russian]Google Scholar
  3. [3]
    Belen’kii, A. S. Prikladnaja Matematika v Narodnom Khozjaistve (Applied Mathematics in National Economy). Moscow Znanie, 1985 [in Russian].Google Scholar
  4. [4]
    Tzvirkun, A. D., Akinfiev, V. K., and Solov’ev, M. M. Modelirovanie Razvitija Krupnomasschtabnykh Sistem (Modeling of Developing Large-Scale Systems). Moscow: Ekonomika, 1983 [in Russian].Google Scholar
  5. [5]
    Livshitz, V. N. Sistemnyj Analiz Ekonomicheskikh Processov na Transporte (System Analysis of Economic Processes for Transport). Moscow: Transport, 1986 [in Russian].Google Scholar
  6. [6]
    Dyukalov, A. N., and Ilyutovich, A. E. Asymptotic properties of optimal trajectories in economic dynamics. Automation and Remote Control. 1973; 34, No. 3: 423–434.Google Scholar
  7. [7]
    Tzvirkun, A. D., Karibskii, A. V., and Yakovenko, S. Yu. Matematicheskoe Modelirovanie Upravlenija Razvitiem Struktur Krupnomasschtabnykh Sistem. Preprint (Mathematical Modeling of Control of Developing Large-Scale Systems. Preprint). Moscow: Izd. Institut Problem Upravleniya ( Institute of Control Sciences, USSR Academy of Sciences ), 1985 [in Russian].Google Scholar
  8. [8]
    Rubinov, A. M. Ekonomicheskaja Dinarnika. Sovremennye Problemy (Economic Dynamics. Contemporary Problems of Mathematics). Moscow: VINITI, 1982, 19 [in Russian].Google Scholar
  9. [9]
    Cheremnykh, Yu. N. Matematicheskoe Modelirovanie Narodnokhozjaistvennoi Dinamiki (Mathematical Modeling of Dynamics of National Economy). Moscow: Nauka, 1982 [in Russian].Google Scholar
  10. [10]
    Cheremnykh, Yu. N. Analiz Povedenija 7’raektorii Dinamiki Narodnokhozjaistvennykh Modelei (Behavioral Analysis of Trajectories of National Economy Models Dynamics). Moscow: Nauka, 1982 [in Russian].Google Scholar
  11. [11]
    Teplova, T. V. “Turnpike approach in dynamic multi-sector models with nonlinear dependences.” In Matematicheskie Metody Analiza Ekonomiki (Mathematical Methods of Economy Analysis). Moscow: Izd. MGU (Moscow State University), 1987; 188–199 [in Russian].Google Scholar
  12. [12]
    Hori, H. A turnpike theorem for rolling plans. Journal of Mathematical Economics. 1987; 16, No. 3: 223–235.Google Scholar
  13. [13]
    Zaslaysky, A. Ya. “Turnpike sets in models of economic dynamics.” In Optimizatzija (Optimization). Novosibirsk: Institute of Mathematics, Novosibirsk: 1988; No. 3: 427–441 [in Russian].Google Scholar
  14. [14]
    Borisov, K. Yu. “Turnpike features of optimal trajectories in models of economic dynamics.” In Optimizatzija (Optimization). Novosibirsk: Institute of Mathematics, Siberian Division of the USSR Academy of Sciences, 1987; 41 (58), 76–87 [in Russian].Google Scholar
  15. [15]
    Marcotte, P. A new algorithm for solving variational inequalities with application to the traffic assignment problem. Mathematical Programming. 1985; 33, No. 3: 339–351.Google Scholar
  16. [16]
    Smith, M. J. The existence of a time-dependent equilibrium distribution of arrivals at a single bottleneck. Transportation Science. 1984; 18, No. 4: 385–394.Google Scholar
  17. [17]
    Iwantschew, D. Optimierungsprobleme zur Vitalitat von Netzwerken. Int. Tag.: Math. Optimierungstheor. and Anwend. Eisenach, 10–15 Nov.. Vortragsanzuge. Ilmenau, s.a., 1986; 180–181.Google Scholar
  18. [18]
    Kanemoto, Y., and Mera, K. General equilibrium analysis of the benefits of large transportation improvements. Discussion Paper. Institute of Economics Research. Queen’s University. 1984; No. 567.Google Scholar
  19. [19]
    Granja, L., and Mora-Catino, F. Modeles entropiques multimodaux de prevision de flux de transport. RAIRO Recherche Operationnelle/Operations Research. 1985; 19, No. 2: 143–158.zbMATHGoogle Scholar
  20. [20]
    Scott, C. H., McMillan, H., and Jefferson, T. R. Equilibria and convex cost networks. International Journal of Systems Science. 1986; 17, No. 7: 1007–1013.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Maugeri, A. Convex programming, variational inequalities, and applications to the traffic equilibrium problem. Applied Mathematics and Optimization. 1987; 16, No. 2: 169–185.MathSciNetzbMATHGoogle Scholar
  22. [22]
    Fukushima, M., and Itoh, T. A dual approach to asymmetric traffic equilibrium problems. Mathematica Japonica. 1987; 32, No. 5: 701–721.MathSciNetzbMATHGoogle Scholar
  23. [23]
    Itoh, T., Fukushima, M., and Ibaraki, T. An iterative method for variational inequalities with application to traffic equilibrium problems. Journal of the Operations Research Society of Japan. 1988; 31, No. 1: 82–103.MathSciNetzbMATHGoogle Scholar
  24. [24]
    Maugeri, A. Stability results for variational inequalities and applications to traffic equilibrium problem. Rendiconti. Circolo Mathematico di Palermo. 1986; 34, Suppl. No. 8: 269–280.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • Alexander S. Belenky

There are no affiliations available

Personalised recommendations