Abstract
The multicommodity flow problem arises when several different commodities are transshipped from specific supply nodes to the corresponding demand nodes through the arcs of an underlying capacity network. The maximum flow over time problem concerns to maximize the sum of commodity flows in a given time horizon. It becomes the earliest arrival flow problem if it maximizes the flow at each time step. The earliest arrival transshipment problem is the one that satisfies specified supplies and demands. These flow over time problems are computationally hard. By reverting the orientation of lanes towards the demand nodes, the outbound lane capacities can be increased. We introduce a partial lane reversal approach in the class of multicommodity flow problems. Moreover, a polynomial-time algorithm for the maximum static flow problem and pseudopolynomial algorithms for the earliest arrival transshipment and maximum dynamic flow problems are presented. Also, an approximation solution to the latter problem is obtained in polynomial-time.
Highlights
E transportation network is considered as a network associated with the transshipment of distinct commodities where the supply points, the demand points, and the junction of road segments constitute the nodes. e connections between the two nodes signify the arcs
If we do not distinguish the flow in the multicommodity flow problem, it becomes a single-commodity flow problem. e dynamic flow problem is introduced by Ford and Fulkerson [7]
By reversing the directions of arcs whenever necessary, a polynomial-time algorithm is presented by Pyakurel and Dhamala [18] for multisource single-sink earliest arrival transshipment. e major concern of partial lane reversals is to make use of the capacities of unused arcs in a network for other purposes
Summary
It does always exists for multiple sources and a single-sink [9]. e multicommodity flow problem is more complex than their single-commodity part. We introduce the partial lane reversals on the static multicommodity flow problem that makes best utilization of arc capacities to optimize the solution. E MSMCF problem with partial lane reversals sends the maximum flow from the sources si to the corresponding sinks ti in the unique pair (si, ti) for each commodity i 1, 2, . In the static single-commodity flow, the maximum static flow on the transformed network is equivalent to the maximum static flow with partial lane reversals on the original network as in [19], and capacity of unused arcs is saved. E maximum multicommodity flow problem is a linear programming problem, so the general linear programming technique (ellipsoid method or interior-point methods) solves the static multicommodity problem on the auxiliary network in polynomial-time in Step 2.
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