We generalize the -relaxation method of [BPT97a] for the single commodity, linear or separable convex cost network flow problem to network flow problems with positive gains. The method maintains -complementary slackness at all iterations and adjusts the arc flows and the node prices so as to satisfy flow conservation upon termination. Each iteration of the method involves either a price change ...

In traditional multi-commodity flow theory, the task is to send a certain amount of each commodity from its start to its target node, subject to capacity constraints on the edges. However, no restriction is imposed on the number of paths used for delivering each commodity; it is thus feasible to spread the flow over a large number of different paths. Motivated by routing problems arising in rea...

This contribution addresses the consideration of time windows in the optimization of multi-commodity network flows. For each node, one interval is specified in which the visitation is allowed. Applications in freight flow consolidation let this problem become interesting. An optimization model is proposed and a construction heuristic is presented. For improving the generated solutions, a geneti...

Markovian network equilibrium generalizes the classical Wardrop in games. At a equilibrium, each player of game solves Markov decision process instead shortest path problem. We propose two novel extensions by considering (1) variable demand, which offers players quitting option, and (2) multi-commodity flow, allows to have heterogeneous ending time. further develop dynamic-programming-based ite...

We generalize the -relaxation method of [14] for the single commodity, linear or separable convex cost network flow problem to network flow problems with positive gains. The method maintains -complementary slackness at all iterations and adjusts the arc flows and the node prices so as to satisfy flow conservation upon termination. Each iteration of the method involves either a price change on a...

This tutorial is devoted to the notion of Max-Min Fairness (MMF), associated optimization problems, and their applications to multi-commodity flow networks. We first introduce a theoretical background for the MMF problem and discuss its relation to lexicographic optimization. We next present resolution algorithms for the MMF optimization, and then give some applications to telecommunication net...

In the surface mining industry, the Equipment Selection Problem involves choosing an appropriate fleet of trucks and loaders such that the long-term mine plan can be satisfied. An important characteristic for multi-location (multi-location and multi-dumpsite) mines is that the underlying problem is a multi-commodity flow problem. The problem is therefore at least as difficult as the fixed-charg...

The work at hand yields two different ways to address the assignment of inbound and outbound doors in less-than-truckload terminals. The considered optimization methods stem from two different scientific fields, which makes the comparison of the techniques a very interesting topic. The first solution approach origins from the field of discrete mathematics. For this purpose, the logistical optim...

We initiate the theoretical study a fundamental algorithmic problem: How to schedule the rerouting of k unsplittable network flows (of a certain demand) from their current paths to their respective new paths, in a congestion-free manner? This problem finds immediate applications, e.g., in traffic engineering in computer networks. We show that the problem is generally NP-hard already for k = 2 f...

In this paper we address branch-and-price algorithms for the unsplittable multicommodity flow problem. This problem is defined over a capacitated network in which we intend to route a set of commodities, each one with a given origin, destination and demand, at minimal cost, without exceeding the arc capacities. Furthermore, the flow of each commodity must be routed using a single path. Formulat...