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The maximum flow problem of uncertain random network

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Abstract

The maximum flow problem is an important problem of network optimization and it covers a wide range of engineering and management applications. The goal of the problem is to find the maximum amount of flow from the source to the sink in a network. This paper investigates two models of the maximum flow of an uncertain random network under the framework of chance theory. The expected value constrained maximum flow and chance constrained maximum flow models with uncertain random arc capacities are proposed. We prove that there exists an equivalence relationship between the uncertain random maximum flow models and the deterministic maximum flow models. Furthermore, some important properties of the models are analyzed, and two algorithms are proposed. Finally, as an illustration, a numerical example is presented to show the effectiveness of the models and algorithms.

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References

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

    Article  MATH  Google Scholar 

  • Carey M, Henrickson C (1984) Bounds on expected performance of networks with links subject to failure. Networks 14(3):439–456

    Article  MATH  Google Scholar 

  • Dinic EA (1970) Algorithm for solution of a problem of maximum flow in networks with power estimation. Soviet Math Dokl 11(8):1277–1280

    Google Scholar 

  • Ding S (2015) The α-maximum flow model with uncertain capacities. Appl Math Model 39(7):2056–2063

    Article  MathSciNet  Google Scholar 

  • Doulliez P (1971) Probability distribution function for the capacity of a multiterminal network. Revue Francaise d’Informatique et de Recherche Operationnelle 1(5):39–49

    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 

  • Fishman GS (1987) The distribution of maximum flow with applications to multistate reliability systems. Oper Res 35(4):607–618

    Article  MATH  Google Scholar 

  • Ford LR, Fulkerson DR (1956) Maximal flow through a network. Canad J Math 8(3):399–404

    Article  MathSciNet  MATH  Google Scholar 

  • Frank H, Frisch IT (1971) Communication, transmission, and transportation networks. Addison-Wesley, Reading

    MATH  Google Scholar 

  • Frank H, Hakimi SL (1965) Probabilistic flows through a communication network. IEEE Trans Circuit Theory 12:413–414

    Article  Google Scholar 

  • Fulkerson DR, Dantzig GB (1955) Computations of maximum flow in networks. Naval Res Logist Q 2(4):277–283

    Article  Google Scholar 

  • Gao J (2013) Uncertain bimatrix game with applications. Fuzzy Optim Decis Mak 12(1):65–78

    Article  MathSciNet  Google Scholar 

  • Gao X (2009) Some properties of continuous uncertain measure. Int J Uncertain Fuzz Knowl Syst 17(3):419–426

    Article  MathSciNet  MATH  Google Scholar 

  • Gao J, Yang X, Liu D (2017) Uncertain Shapley value of coalitional game with application to supply chain alliance. Appl Soft Comput. doi:10.1016/j.asoc.2016.06.018

  • Gao J, Yao K (2015) Some concepts and theorems of uncertain random process. Int J Intell Syst 30(1):52–65

    Article  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, Yang L, Li S (2016) Uncertain models on railway transportation planning problem. Appl Math Model 40:4921–4934

    Article  MathSciNet  Google Scholar 

  • Gao Y, Yang L, Li S, Kar S (2015) On distribution function of the diameter in uncertain graph. Inf Sci 296:61–74

    Article  MathSciNet  MATH  Google Scholar 

  • Goldberg AV, Tarjan RE (1988) A new approach to the maximum flow problem. J Assoc Comput Mach 35(4):921–940

    Article  MathSciNet  MATH  Google Scholar 

  • Guo H, Wang X (2014) Variance of uncertain random variables. J Uncertain Anal Appl. doi:10.1186/2195-5468-2-6

  • Han S, Peng Z, Wang S (2014) The maximum flow problem of uncertain network. Inf Sci 265:167–175

    Article  MathSciNet  MATH  Google Scholar 

  • Karzanov AV (1974) Determining the maximum flow in a network by the method of preflows. Soviet Math Dokl 15(3):43–47

    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

    MathSciNet  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. Springer, Berlin

    Book  Google Scholar 

  • Liu B (2014) Uncertain random graph and uncertain random network. J Uncertain Syst 8(1):3–12

    Google Scholar 

  • Liu Y (2013a) Uncertain random variables: a mixture of uncertainty and randomness. Soft comput 17(4):625–634

    Article  MATH  Google Scholar 

  • Liu Y (2013b) Uncertain random programming with applications. Fuzzy Optim Decis Mak 12(2):153–169

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  • Liu Y, Ralescu D (2014) Risk index in uncertain random risk analysis. Int Uncertain Fuzz Knowl Syst 22(4):491–504

    Article  MathSciNet  MATH  Google Scholar 

  • Nawathe SP, Rao BV (1980) Maximum flow in probabilistic communication networks. J Circuit Theory Appl 8:167–177

    Article  Google Scholar 

  • Ni Y (2017) Sequential seeding to optimize influence diffusion in a social network. Appl Soft Comput. doi:10.1016/j.asoc.2016.04.025

  • Onaga K (1968) Bounds on the average terminal capacity of probabilistic nets. IEEE Trans Inf Theory 14(5):766–768

    Article  MathSciNet  MATH  Google Scholar 

  • Peng Z, Iwamura K (2010) A sufficient and necessary condition of uncertainty distribution. J Interdiscip Math 13(3):277–285

    Article  MathSciNet  MATH  Google Scholar 

  • Sheng Y, Gao J (2014) Chance distribution of maximum flow of uncertain random network. J Uncertain Anal Appl. doi:10.1186/s40467-014-0015-3

    Google Scholar 

  • Sheng Y, Gao Y (2016) Shortest path problem of uncertain random network. Comput Ind Eng 99:97–105

    Article  Google Scholar 

  • Sheng Y, Yao K (2014) Some formulas of variance of uncertain random variable. J Uncertain Anal Appl. doi:10.1186/2195-5468-2-12

    Google Scholar 

  • Yang X, Gao J (2017) Bayesian equilibria for uncertain bimatrix game with asymmetric information. J Intell Manuf 28(3):515–525

    Article  Google Scholar 

  • Yang X, Gao J (2016) Linear-quadratic uncertain differential games with applicatians to resource extraction problem. IEEE Trans Fuzzy Syst 24(4):819–826

    Article  MathSciNet  Google Scholar 

  • Yao K (2015) A formula to calculate the variance of uncertain variable. Soft Comput 19(10):2947–2953

    Article  MATH  Google Scholar 

  • Yao K, Gao J (2015) Uncertain random alternating renewal process with application to interval availability. IEEE Trans Fuzzy Syst 23(5):1333–1342

    Article  Google Scholar 

  • Yao K, Zhou J (2016) Uncertain random renewal reward process with application to block replacement policy. IEEE Trans Fuzzy Syst 24(6):1637–1647

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grants No. 61462086, No. 61563050) and in part by Xinjiang University (No. BS150206). Professor Dan A. Ralescu’s work was partly supported by a Taft Travel Grant for Research.

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Authors and Affiliations

  1. Department of Computer Science and Technology, Tsinghua University, Beijing, 100084, China

    Gang Shi

  2. School of Information Science and Engineering, Xinjiang University, Urumqi, 830046, China

    Gang Shi

  3. College of Mathematical and System Sciences, Xinjiang University, Urumqi, 830046, China

    Yuhong Sheng

  4. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, 45221-0025, USA

    Dan A. Ralescu

Authors
  1. Gang Shi
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  2. Yuhong Sheng
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  3. Dan A. Ralescu
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Correspondence to Dan A. Ralescu.

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Shi, G., Sheng, Y. & Ralescu, D.A. The maximum flow problem of uncertain random network. J Ambient Intell Human Comput 8, 667–675 (2017). https://doi.org/10.1007/s12652-017-0495-3

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  • DOI: https://doi.org/10.1007/s12652-017-0495-3

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