Abstract
The multicommodity flow problem deals with the transshipment of more than one commodity from respective sources to corresponding sinks without violating the capacity constraints. Due to the capacity constraints, flows out from the sources may not reach their sinks, and so, the storage of excess flows at intermediate nodes plays an important role in the maximization of flow values. In this paper, we introduce the maximum static as well as maximum dynamic multicommodity flow problems with intermediate storage. We present polynomial and pseudopolynomial time algorithms for the former and latter problems, respectively. We also present the solution procedures to these problems in contraflow network having symmetric as well as asymmetric arc transit times. We transform the solutions in continuous-time settings by using natural transformation.
Highlights
Network is a topological structure with links, with its crossings being its components. e transportation network is one of the relevant examples of a network topology, in which road segments are considered as the arcs and their crossings as nodes
By using Algorithm 3, the optimal solution for maximum static and maximum dynamic multicommodity contraflow problems with intermediate storage can be obtained for a given network with asymmetric transit times, where transit time is considered as a cost in static problem
A continuous dynamic flow function ψc with intermediate storage is defined as the flow rate per unit time that leaves from the source at each moment of time by allowing the storage of excess flow at intermediate nodes without violating the capacity constraints
Summary
Network is a topological structure with links (arcs), with its crossings (nodes) being its components. e transportation network is one of the relevant examples of a network topology, in which road segments are considered as the arcs and their crossings as nodes. Pyakurel and Dempe [10] introduced the concept of maximum static and maximum dynamic flow problems with intermediate storage and presented polynomial time algorithms to solve them. In case of multisource multisink network, Pyakurel et al [11] solved the prioritized maximum flow problem with intermediate storage and presented polynomial time algorithm to solve the problem, where priority is given to the farthest element from the source. We present polynomial time algorithm for static multicommodity flow problem and pseudopolynomial time algorithm for dynamic multicommodity flow problem by allowing the storage of excess flow at intermediate nodes.
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