# American Institute of Mathematical Sciences

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

## Maximum flow problem in the distribution network

 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.
Citation: R.L. Sheu, M.J. Ting, I.L. Wang. Maximum flow problem in the distribution network. Journal of Industrial & Management Optimization, 2006, 2 (3) : 237-254. doi: 10.3934/jimo.2006.2.237
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