Maximum and Minimum Cost Flow Finding in Networks in Fuzzy Conditions

  • Alexander Vitalievich BozhenyukEmail author
  • Evgeniya Michailovna Gerasimenko
  • Janusz Kacprzyk
  • Igor Naymovich Rozenberg
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 346)


The problems of the maximum and the minimum cost flow finding with zero and nonzero lower flow bounds are relevant, since they allow solving the problems of economic planning, logistics, transportation management, etc. In the area of transportation networks flow tasks enable to find the cargo transportation of the maximum volume between given points taking into account restrictions on the arc capacities of the cargo transmission paths, choose the routes of the optimal cost with the set lower flow bounds, which can be found after the profitability analysis of the cargo transportation along the particular road section. In considering these tasks it is necessary to take into account the inherent uncertainty of the network parameters, since environmental factors, measurement errors, repair work on the roads, the specifics of the constantly changing structure of the network influence the upper and lower flow bounds and transportation costs.


Short Path Fuzzy Number Maximum Flow Fuzzy Triangular Number Flow Unit 
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.


  1. 1.
    Bozhenyuk А, Rozenberg I, Starostina T (2006) Analiz i issledovaniye potokov i zhivuchesti v transportnykx setyakh pri nechetkikh dannykh. Nauchnyy Мir, MoskvaGoogle Scholar
  2. 2.
    Bershtein L, Bozhenuk A (2008) Fuzzy graphs and fuzzy hypergraphs. In: Dopico J, de la Calle J, Sierra A (eds) Encyclopedia of artificial intelligence information SCI. Hershey, New York, pp 704–709Google Scholar
  3. 3.
    Christofides N (1975) Graph theory: an algorithmic approach. Academic Press, New YorkzbMATHGoogle Scholar
  4. 4.
    Bozhenyuk А, Gerasimenko E, Rozenberg I (2012) The methods of maximum flow and minimum cost flow finding in fuzzy network. In: Proceedings of the concept discovery in unstructured data (CDUD 2012): workshop co-located with the 10th international conference on formal concept analysis (ICFCA 2012), May 2012. Belgium, Leuven, pp 1–12Google Scholar
  5. 5.
    Cormen TH, Leiserson CE, Rivest RL, Stein C (2001) Introduction to algorithms, 3rd edn. MIT Press, CambridgeGoogle Scholar
  6. 6.
    Bershtein LS, Dzuba TA, (1998) Construction of a spanning subgraph in the fuzzy bipartite graph. In: Proceedings of EUFIT’98, Aachen, pp 47–51Google Scholar
  7. 7.
    Bozhenyuk AV, Rozenberg IN, Rogushina EM (2011) Approach of maximum flow determining for fuzzy transportation network. Izvestiya SFedU. Eng Sci 5(118):83–88Google Scholar
  8. 8.
    Kovács P (2015) Minimum-cost flow algorithms: an experimental evaluation. Optim Methods Softw 30(1):94–127MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bozhenyuk A, Gerasimenko E (2014) Flows finding in networks in fuzzy conditions. In: Kahraman C, Öztaysi B (eds) Supply chain management under fuzzines, studies in fuzziness and soft computing. Springer, Berlin, vol 313, Part III, pp 269–291. doi: 10.1007/978-3-642-53939-8_12
  10. 10.
    Murty KG (1992) Network programming. Prentice Hall, Upper Saddle RiverGoogle Scholar
  11. 11.
    Gerasimenko EM (2012) Minimum cost flow finding in the network by the method of expectation ranking of fuzzy cost functions. Izvestiya SFedU. Eng Sci 4(129):247–251Google Scholar
  12. 12.
    Busacker RG, Gowen P (1961) A procedure for determining a family of minimum-cost network flow patterns. In: Technical report 15, Operations Research Office, John Hopkins UniversityGoogle Scholar
  13. 13.
    Floyd RW (1962) Algorithm 97: shortest path. Commun ACM 5(6):345CrossRefGoogle Scholar
  14. 14.
    Dijkstra EW (1959) A note on two problems in connextion with graphs. Numer Math 1:269–271MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bellman R (1958) On a routing problem. Q Appl Math 16(1):87–90zbMATHGoogle Scholar
  16. 16.
    Ford LR Jr (1956) Network flow theory. In: Paper P-923. RAND Corporation, Santa Monica, CaliforniaGoogle Scholar
  17. 17.
    Levit BY (1971) Algoritmi poiska kratchajshikh putej na grahe. Trudi institute gidrodinamiki SO AN SSSR, Sb. “Modelirovaniye proczessov upravleniya, Novosibirsk, vip. 4, 1117–1148Google Scholar
  18. 18.
    Johnson DB (1977) Efficient algorithms for shortest paths in sparse networks. J ACM 24(1):1–13MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Levitin A (2006) Introduction to the design and analysis of algorithms, 2nd edn. Addison-Wesley, BostonGoogle Scholar
  20. 20.
    Edmonds J, Karp RM (1972) Theoretical improvements in algorithmic efficiency for network flow problems. J Assoc Comput Mach 19(2):248–264CrossRefzbMATHGoogle Scholar
  21. 21.
    Tomizawa N (1971) On some techniques useful for solution of transportation network problems. Networks 1:173–194MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pouly A (2010) Minimum cost flows, 23 Nov 2010,
  23. 23.
    Bozhenyuk А, Gerasimenko E, Rozenberg I (2012) The task of minimum cost flow finding in transportation networks in fuzzy conditions. In: Proceedings of the 10th international FLINS conference on uncertainty modeling in knowledge engineering and decision making word scientific, Istanbul, Turkey, 26–29 Aug 2012, pp 354–359Google Scholar
  24. 24.
    Bozhenyuk AV, Gerasimenko EM, Rozenberg IN (2012) Algorithm of minimum cost fuzzy flow finding in a network with fuzzy arc capacities and transmission costs 2012. Izvestiya SFedU. Eng Sci 5(130):118–122Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Alexander Vitalievich Bozhenyuk
    • 1
    Email author
  • Evgeniya Michailovna Gerasimenko
    • 1
  • Janusz Kacprzyk
    • 2
  • Igor Naymovich Rozenberg
    • 3
  1. 1.Southern Federal UniversityTaganrogRussia
  2. 2.Systems Research InstitutePolish Academy of SciencesWarsawPoland
  3. 3.Public Corporation “Research and Development Institute of Railway Engineers”MoscowRussia

Personalised recommendations