The maximum flow problem of uncertain network

The maximum flow problem is one of the classic combinatorial optimization problems with many applications in electrical power systems, communication networks, computer networks and logistic networks. The goal of the problem is to find the maximum amount of flow from the source to the sink in a network. A network is called uncertain if the arc capacities of the network are uncertain variables. The main purpose of this paper is to solve the maximum flow in an uncertain network by under the framework of uncertainty theory.

[1]  S. Pallottino,et al.  The maximum flow problem: A max-preflow approach , 1991 .

[2]  A. V. Karzanov,et al.  Determining the maximal flow in a network by the method of preflows , 1974 .

[3]  Louis J. Billera,et al.  Delbert Ray Fulkerson (August 14, 1924-January 10, 1976) , 1976, Math. Oper. Res..

[4]  Baoding Liu,et al.  Uncertainty Theory - A Branch of Mathematics for Modeling Human Uncertainty , 2011, Studies in Computational Intelligence.

[5]  S. Chanas,et al.  Real-valued flows in a network with fuzzy arc capacities , 1984 .

[6]  Yuhan Liu,et al.  Expected Value of Function of Uncertain Variables , 2010 .

[7]  George S. Fishman,et al.  The Distribution of Maximum Flow with Applications to Multistate Reliability Systems , 1987, Oper. Res..

[8]  Kakuzo Iwamura,et al.  A sufficient and necessary condition of uncertainty distribution , 2010 .

[9]  Michel Minoux,et al.  On robust maximum flow with polyhedral uncertainty sets , 2009, Optim. Lett..

[10]  Yuan Gao.  Uncertain models for single facility location problems on networks , 2012 .

[11]  Baoding Liu.  Some Research Problems in Uncertainty Theory , 2009 .

[12]  Fabio Somenzi,et al.  A Symbolic Algorithms for Maximum Flow in 0-1 Networks , 1997, Formal Methods Syst. Des..

[13]  Fabio Somenzi,et al.  A Symbolic Algorithms for Maximum Flow in 0-1 Networks , 1993, Proceedings of 1993 International Conference on Computer Aided Design (ICCAD).

[14]  Tianlong Gu,et al.  The symbolic algorithms for maximum flow in networks , 2007, Comput. Oper. Res..

[15]  Lixing Yang,et al.  Chance Constrained Maximum Flow Problem with Fuzzy Arc Capacities , 2006, ICIC.

[16]  B. V. Rao,et al.  Maximum flow in probabilistic communication networks , 1980 .

[17]  Yuan Gao,et al.  Shortest path problem with uncertain arc lengths , 2011, Comput. Math. Appl..

[18]  Stefan Chanas,et al.  Maximum flow in a network with fuzzy arc capacities , 1982 .

[19]  Baoding Liu.  Uncertain Risk Analysis and Uncertain Reliability Analysis , 2010 .

[20]  D. R. Fulkerson,et al.  Flows in Networks. , 1964 .

[21]  Andrew V. Goldberg,et al.  A new approach to the maximum flow problem , 1986, STOC '86.

[22]  D. R. Fulkerson,et al.  Maximal Flow Through a Network , 1956 .

[23]  Baoding Liu,et al.  Uncertainty Theory - A Branch of Mathematics for Modeling Human Uncertainty , 2011, Studies in Computational Intelligence.

[24]  Jinwu Gao,et al.  Some stability theorems of uncertain differential equation , 2012, Fuzzy Optimization and Decision Making.

[25]  E. A. Dinic.  Algorithm for solution of a problem of maximal flow in a network with power estimation , 1970 .

[26]  Baoding Liu.  Uncertain Logic for Modeling Human Language , 2011 .

[27]  Baoding Liu.  Fuzzy Process, Hybrid Process and Uncertain Process , 2008 .

[28]  Baoding Liu,et al.  Theory and Practice of Uncertain Programming , 2003, Studies in Fuzziness and Soft Computing.

[29]  Takao Asano,et al.  RECENT DEVELOPMENTS IN MAXIMUM FLOW ALGORITHMS , 2000 .

[30]  Shimon Even,et al.  Graph Algorithms , 1979 .

[31]  S. N. Maheshwari,et al.  An O(|V|³) Algorithm for Finding Maximum Flows in Networks , 1978, Inf. Process. Lett..

[32]  Fernando Ordóñez,et al.  Robust capacity expansion of network flows , 2007, Networks.

[33]  B. C. Brookes,et al.  Information Sciences , 2020, Cognitive Skills You Need for the 21st Century.

[34]  Samarjit Kar,et al.  Cross-entropy measure of uncertain variables , 2012, Inf. Sci..

[35]  Tang Bin-bing.  The Maximum Flow Problem with Fuzzy Constraints , 2001 .

[36]  X. Chen,et al.  Existence and uniqueness theorem for uncertain differential equations , 2010, Fuzzy Optim. Decis. Mak..

[37]  Stefan Chanas,et al.  Integer flows in network with fuzzy capacity constraints , 1986, Networks.

[38]  Xiang Li,et al.  Uncertain Alternating Renewal Process and Its Application , 2012, IEEE Transactions on Fuzzy Systems.

[39]  Robert E. Tarjan,et al.  Algorithms for maximum network flow , 1986 .

[40]  James J. Buckley,et al.  Fuzzy Max-Flow Problem , 2007 .

[41]  Andrew V. Goldberg,et al.  Recent Developments in Maximum Flow Algorithms (Invited Lecture) , 1998, SWAT.

[42]  Michel Minoux,et al.  Robust network optimization under polyhedral demand uncertainty is NP-hard , 2010, Discret. Appl. Math..

[43]  A. Ben-Tal,et al.  Adjustable robust solutions of uncertain linear programs , 2004, Math. Program..

[44]  Kai Yao.  No-arbitrage determinant theorems on mean-reverting stock model in uncertain market , 2012, Knowl. Based Syst..