Skip to main content
SpringerLink
Log in

Uncertain minimum cost flow problem

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

The aim of this paper is to give an uncertainty distribution of the least cost of shipment of a commodity through a network with uncertain capacities. Uncertainty theory is used to deal with uncertain capacities, and an \(\alpha \)-minimum cost flow problem model is proposed. After defining the \(\alpha \)-minimum cost flow, the properties of the model are analyzed, and then an algorithm for uncertain minimum cost flow problem is developed. The algorithm can be considered as a general solution to the uncertain minimum cost flow problem. To demonstrate the efficiency of the proposed algorithm, a numerical example is illustrated.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: theory, algorithms, and applications. Prentice-Hall, New Jersey

    MATH  Google Scholar 

  • Bertsekas DP (1986) Distributed asynchronous relaxation methods for linear network flow problems. Report LIDS-P-1606, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology

  • Bertsekas DP, Castañon DA (1993) A generic auction algorithm for the minimum cost network flow problem. Comput Optim Appl 2(3):229–259

    Article  MathSciNet  MATH  Google Scholar 

  • Boyles SD, Waller ST (2010) A mean-variance model for the minimum cost flow problem with stochastic arc costs. Networks 56(3):215–227

    Article  MathSciNet  MATH  Google Scholar 

  • Busacker RG, Gowen PJ (1960) A procedure for determining a family of minimum cost network flow patterns. Technical Report ORO-TP-15, Operational Research Office, The Johns Hopkins University

  • Chen XW, Gao JW (2013) Uncertain term structure model of interest rate. Soft Comput 17(4):597–604

    Article  MathSciNet  MATH  Google Scholar 

  • Ciurea E, Ciupalâ L (2004) Sequential and parallel algorithms for minimum flows. J Appl Math Comput 15(1–2):53–75

    Article  MathSciNet  MATH  Google Scholar 

  • Connors MM, Zangwill WI (1971) Cost minimization in networks with discrete stochastic requirements. Oper Res 19(3):794–821

    Article  MathSciNet  MATH  Google Scholar 

  • Edmonds J, Karp RM (1972) Theoretical improvements in algorithmic efficiency for network flow problems. J ACM 19(2):248–264

    Article  MATH  Google Scholar 

  • Ford LR, Fulkerson DR (1962) Flows in networks. Princeton University Press, New Jersey

    MATH  Google Scholar 

  • Fulkerson DR (1961) An out-of-kilter method for minimal cost flow problems. J Soc Ind Appl Math 9(1):18–27

    Article  MATH  Google Scholar 

  • Gao Y (2011) Shortest path problem with uncertain arc lengths. Comput Math Appl 62(6):2591–2600

    Article  MathSciNet  MATH  Google Scholar 

  • Gao Y (2012) Uncertain models for single facility location problems on networks. Appl Math Model 36(6):2592–2599

    Article  MathSciNet  MATH  Google Scholar 

  • Gao XL (2013) Cycle index of uncertain graph. Information 16(2A):1131–1138

    Google Scholar 

  • Gao XL, Gao Y (2013) Connectedness index of uncertain graphs. Int J Uncertain Fuzziness Knowl Based Syst 21(1):127–137

    Article  Google Scholar 

  • Han SW, Peng ZX (2010) The maximum flow problem of uncertain network. http://orsc.edu.cn/online/101228.pdf

  • Iri M (1960) A new method of solving transportation-network problems. J Oper Res Soc Jpn 3:27–87

    Google Scholar 

  • Jewell WS (1958) Optimal flow through networks. Technical Report 8, Operations Research Center, Massachusetts Institute of Technology

  • Klein M (1967) A primal method for minimal cost flows with applications to the assignment and transportation problems. Manage Sci 14(3):205–220

    Article  MATH  Google Scholar 

  • Liu B (2007) Uncertainty theory, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  • Liu B (2009a) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10

    Google Scholar 

  • Liu B (2009b) Theory and practice of uncertain programming, 2nd edn. Springer, Berlin

    Book  MATH  Google Scholar 

  • Liu B (2010) Uncertainty theory: a branch of mathematics for modeling human uncertainty, 2nd edn. Springer, Berlin

    Book  Google Scholar 

  • Liu B (2012) Why is there a need for uncertainty theory? J Uncertain Syst 6(1):3–10

    Google Scholar 

  • Liu YH, Ha MH (2010) Expected value of function of uncertain variables. J Uncertain Syst 4(3):181–186

    Google Scholar 

  • Liu B, Yao K (2012) Uncertain multilevel programming: algorithm and applications. http://orsc.edu.cn/online/120114.pdf

  • Liu B, Chen XW (2013) Uncertain multiobjective programming and uncertain goal programming. Uncertainty Theory Laboratory, Tsinghua University Technical Report

  • Minty GJ (1960) Monotone networks. Proc R Soc A 257(1289):194–212

  • Paparrizos K, Samaras N, Sifaleras A (2009) An exterior simplex type algorithm for the minimum cost network flow problem. Comput Oper Res 36(4):1176–1190

    Google Scholar 

  • Peng ZX, Iwamura K (2010) A sufficient and necessary condition of uncertainty distribution. Information 13(3):277–285

    Google Scholar 

  • Prékopa A, Boros E (1991) On the existence of a feasible flow in a stochastic transportation network. Oper Res 39(1):119–129

    Article  MathSciNet  MATH  Google Scholar 

  • Röck H (1980) Scaling techniques for minimal cost network flows. In: Pape U (ed) Discrete structures and algorithms. Carl Hanser-Verlag, Munich, pp 181–191

    Google Scholar 

  • Wang GL, Tang WS, Zhao RQ (2013) An uncertain price discrimination model in labor market. Soft Comput 17(4):579–585

    Article  MATH  Google Scholar 

  • Williams AC (1963) A stochastic transportation problem. Oper Res 11(5):759–770

    Google Scholar 

  • Wollmer RD (1980) Investment in stochastic minimum cost generalized multicommodity networks with application to coal transport. Networks 10(4):351–362

    Article  MathSciNet  Google Scholar 

  • Yakovleva MA (1959) Problem of minimum transportation expense. In: Nemchinov VS (ed) Applications of mathematics to economics research, Moscow, pp 390–399

  • Zadeh LA (1965) Fuzzy sets. Info Control 8:338–353

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1:3–28

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by National Natural Science Foundation of China Grant No.61273044, National Soft Science Research Program No.2013GXS2B014 and Education Research Project No.YPGC2011-W03.

Author information

Authors and Affiliations

  1. School of Management, Henan University of Technology, Zhengzhou, 450001, China

    Sibo Ding

Authors
  1. Sibo Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sibo Ding.

Additional information

Communicated by F. Chiclana.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ding, S. Uncertain minimum cost flow problem. Soft Comput 18, 2201–2207 (2014). https://doi.org/10.1007/s00500-013-1194-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-013-1194-4

Keywords

Search

Navigation