American Institute of Mathematical Sciences

Advanced
July  2006, 2(3): 237-254. doi: 10.3934/jimo.2006.2.237

Maximum flow problem in the distribution network

R.L. Sheu 1, , M.J. Ting 2, and  I.L. Wang 3,

1. 

Department of Mathematics, National Cheng-Kung University, Tainan, Taiwan
2. 

Product Development Department, Shinkong Life Insurance Co., LTD., Taipei, Taiwan
3. 

Department of Industrial and Information Management, National Cheng-Kung University, Tainan, Taiwan

Received  September 2005 Revised  December 2005 Published  July 2006

In this paper, we are concerned with the maximum flow problem in the distribution network, a new kind of network recently introduced by Fang and Qi. It differs from the traditional network by the presence of the $D$-node through which the commodities are to be distributed proportionally. Adding $ D $-nodes complicates the network structure. Particularly, flows in the distribution network are frequently increased through multiple cycles. To this end, we develop a type of depth-first-search algorithm which counts and finds all unsaturated subgraphs. The unsaturated subgraphs, however, could be invalid either topologically or numerically. The validity are then judged by computing the flow increment with a method we call the multi-labeling method. Finally, we also provide a phase-one procedure for finding an initial flow.
Keywords: labeling method, cycles., maximum flow problem, Distribution network.
Mathematics Subject Classification: 90B10, 49M3.
Citation: R.L. Sheu, M.J. Ting, I.L. Wang. Maximum flow problem in the distribution network. Journal of Industrial and Management Optimization, 2006, 2 (3) : 237-254. doi: 10.3934/jimo.2006.2.237
[1]

Zehui Shao, Huiqin Jiang, Aleksander Vesel. L(2, 1)-labeling of the Cartesian and strong product of two directed cycles. Mathematical Foundations of Computing, 2018, 1 (1) : 49-61. doi: 10.3934/mfc.2018003
[2]

I-Lin Wang, Shiou-Jie Lin. A network simplex algorithm for solving the minimum distribution cost problem. Journal of Industrial and Management Optimization, 2009, 5 (4) : 929-950. doi: 10.3934/jimo.2009.5.929
[3]

Huai-Che Hong, Bertrand M. T. Lin. A note on network repair crew scheduling and routing for emergency relief distribution problem. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1729-1731. doi: 10.3934/jimo.2018119
[4]

Weishi Yin, Jiawei Ge, Pinchao Meng, Fuheng Qu. A neural network method for the inverse scattering problem of impenetrable cavities. Electronic Research Archive, 2020, 28 (2) : 1123-1142. doi: 10.3934/era.2020062
[5]

Jia Shu, Jie Sun. Designing the distribution network for an integrated supply chain. Journal of Industrial and Management Optimization, 2006, 2 (3) : 339-349. doi: 10.3934/jimo.2006.2.339
[6]

Haodong Chen, Hongchun Sun, Yiju Wang. A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2217-2242. doi: 10.3934/jimo.2020066
[7]

Andrey Kovtanyuk, Alexander Chebotarev, Nikolai Botkin, Varvara Turova, Irina Sidorenko, Renée Lampe. Modeling the pressure distribution in a spatially averaged cerebral capillary network. Mathematical Control and Related Fields, 2021, 11 (3) : 643-652. doi: 10.3934/mcrf.2021016
[8]

Paolo Maremonti. On the Stokes problem in exterior domains: The maximum modulus theorem. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2135-2171. doi: 10.3934/dcds.2014.34.2135
[9]

Boya Li, Hongjie Ju, Yannan Liu. A flow method for a generalization of $ L_{p} $ Christofell-Minkowski problem. Communications on Pure and Applied Analysis, 2022, 21 (3) : 785-796. doi: 10.3934/cpaa.2021198
[10]

Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks and Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255
[11]

Chun Zong, Gen Qi Xu. Observability and controllability analysis of blood flow network. Mathematical Control and Related Fields, 2014, 4 (4) : 521-554. doi: 10.3934/mcrf.2014.4.521
[12]

Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Linear model of traffic flow in an isolated network. Conference Publications, 2015, 2015 (special) : 670-677. doi: 10.3934/proc.2015.0670
[13]

Ming Huang, Cong Cheng, Yang Li, Zun Quan Xia. The space decomposition method for the sum of nonlinear convex maximum eigenvalues and its applications. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1885-1905. doi: 10.3934/jimo.2019034
[14]

Ting Hu. Kernel-based maximum correntropy criterion with gradient descent method. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4159-4177. doi: 10.3934/cpaa.2020186
[15]

Fabio Scalco Dias, Luis Fernando Mello. The center--focus problem and small amplitude limit cycles in rigid systems. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1627-1637. doi: 10.3934/dcds.2012.32.1627
[16]

Yanfei Wang, Qinghua Ma. A gradient method for regularizing retrieval of aerosol particle size distribution function. Journal of Industrial and Management Optimization, 2009, 5 (1) : 115-126. doi: 10.3934/jimo.2009.5.115
[17]

Alexander Schaub, Olivier Rioul, Jean-Luc Danger, Sylvain Guilley, Joseph Boutros. Challenge codes for physically unclonable functions with Gaussian delays: A maximum entropy problem. Advances in Mathematics of Communications, 2020, 14 (3) : 491-505. doi: 10.3934/amc.2020060
[18]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial and Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067
[19]

Chia-Chun Hsu, Hsun-Jung Cho, Shu-Cherng Fang. Solving routing and wavelength assignment problem with maximum edge-disjoint paths. Journal of Industrial and Management Optimization, 2017, 13 (2) : 1065-1084. doi: 10.3934/jimo.2016062
[20]

Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control and Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (116)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy