On continuous network flows

doi:10.1016/0167-6377(94)00035-5

Operations Research Letters

Volume 17, Issue 1, February 1995, Pages 27-36

https://doi.org/10.1016/0167-6377(94)00035-5 Get rights and content

Abstract

This work addresses two problems concerning continuous dynamic flows. A model is presented for a network that incorporates continuous time-varying flows, link capacities, node storage capacities, as well as time dependent link delays. It is an enhancement of previous results which do not incorporate time varying link delays. We present a generalized min-cut max-flow theorem for that model. A second result deals with universal flows, originally dealt with for the discrete case. We show how such flows can be constructed in a way that involves parallelism.

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Cited by (9)

Flows over time in time-varying networks: Optimality conditions and strong duality
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Citation Excerpt :
So far we reviewed the literature on network flows over time where transit times are assumed to be time-invariant. There is also a number of models that allow the transit times to vary over time in both discrete-time model (Cai, Kloks, & Wong, 1997; Cai, Sha, & Wong, 2001; Miller-Hooks & Mahmassani, 1998; Miller-Hooks & Patterson, 2004; Nasrabadi & Hashemi, 2010; Opasanon & Miller-Hooks, 2006; Wen, Çatay, & Eglese, 2013) and continuous-time model (Hashemi & Nasrabadi, 2011; Orda & Rom, 1990, 1991, 1995). In particular, Miller-Hooks and Patterson (2004) present a pseudo-polynomial time algorithm for the problem of sending a given amount of flow from a single source to a single sink at the shortest possible time in a time-varying network, where all parameters can change at discrete points in time.
There has been much research on network flows over time due to their important role in real world applications. This has led to many results, but the more challenging continuous time model still lacks some of the key concepts and techniques that are the cornerstones of static network flows. The aim of this paper is to advance the state of the art for dynamic network flows by developing the continuous time analogues of the theory for static network flows. Specifically, we make use of ideas from the static case to establish a reduced cost optimality condition, a negative cycle optimality condition, and a strong duality result for a very general class of network flows over time.
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On continuous network flows

Abstract

Linear Programming in Infinite-Dimensional Spaces

A class of continuous network flow problems

Math. Oper. Res.

On the solutions of a class of continuous linear programs

SIAM J. Control Optim.

Constructing maximal dynamic flows from static flows

Oper. Res.

Flows in Networks

Transient flows in networks

Michigan Math. J.

Graphs and Algorithms

The shortest route with time-dependent length of edges and limited delay possibilities in nodes

Z. Oper. Res.