Maximum flow problem in the distribution network

JOURNAL OF INDUSTRIAL AND Website: http://AIMsciences.org MANAGEMENT OPTIMIZATION

Volume 2, Number 3, August 2006 pp. 237–254

MAXIMUM FLOW PROBLEM IN THE

DISTRIBUTION NETWORK

R.L. Sheu

National Cheng-Kung University, Tainan, Taiwan

M.J. Ting

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

I.L. Wang

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

(Communicated by Shu-Cherng Fang)

Abstract. 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.

1. Introduction. Consider the network G = (N, A, d, u) with N = N

∪

N

∪

N

∪

N

∪

N

being the node set and A the arc set. The functions d : A → Q

and u : A → Q

, which take non-negative rational values, represent respectively

the minimum demand and the capacity on each arc. There are five types of nodes

in the network G: the S-node, the T -node, the O-node, the Pseudo-node, and the

D-node. The S-nodes are source nodes. They can be thought to all come from a

Pseudo

_{-S-node. The T -nodes are termination (sink) nodes which can be combined}

into a single Pseudo-T -node. The O-nodes are the intermediate nodes other than

the source and the sink nodes in the traditional network. The D-node is a very

special feature. Each D-node has only one inward arc through which the goods

must be proportioned to all its outward arcs at fixed ratios. It was first introduced

by Fang and Qi in [5] to describe some manufacturing and/or distribution processes.

Let the collection of each respective type be denoted by N

, N

, N

, N

, and

N

.

2000 Mathematics Subject Classification. 90B10, 49M35.

Key words and phrases. Distribution network, maximum flow problem, labeling method, cycles. This research work was partially supported by the National Center of Theoretic Sciences/ South of Taiwan.