Abstract
Multi-commodity flow problems concerned with the transshipment of more than one commodity from respective sources to the corresponding sinks without violating the capacity constraints on the arcs. If the objective of the problem is to send the maximum amount of flow within a given time horizon, then it becomes the maximum flow problem. In multi-commodity flow problems, the flow of different commodities departing from their sources arriving at the common intermediate node have to share the capacity through the arc. The sharing of the capacity in the common arc (bundle arc) is one of the major issues in the multi-commodity flow problems. In this paper, we introduce the maximum static and maximum dynamic multi-commodity flow problems with proportional capacity sharing and present polynomial time algorithms to solve the problems. Similarly, we investigate the maximum dynamic multi-commodity flow problems with flow-dependent capacity sharing and present a pseudo-polynomial time solution strategy.
Highlights
A topological structure with links and crossings, known as arcs and nodes, respectively, is a network in which entities are transshipped from one point to another
We introduce the maximum multi-commodity flow problem using proportional as well as flow-dependent capacity sharing on the bundle arcs
To share the capacity of the bundle arc, we propose a proportional capacity sharing technique depending on the minimum of the arc capacity of paths P[si,v], for each commodity i from their respective sources si to the tail v of bundle arc e = (v, w) as follows: Let ue be the capacity of a bundle arc e, proportional sharing of capacity ue for each commodity i ∈ K is, uie =
Summary
A topological structure with links and crossings, known as arcs and nodes, respectively, is a network in which entities are transshipped from one point to another. If the sharing of the capacity of the bundle arc is set in proportion to the bottleneck capacity of path from their respective sources to the tail node of the bundle arc, it is known as proportional capacity sharing. If the sharing of the capacity of the bundle arc is made according to the inflow rate of the flow of each commodity, it is termed as flow-dependent capacity sharing. We introduce the maximum multi-commodity flow problem using proportional as well as flow-dependent capacity sharing on the bundle arcs.
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