Show Less

# Operational Planning Problems in Intermodal Freight Transportation

## Jenny Nossak

This book addresses logistics problems (e.g., routing and partitioning problems) that are encountered in intermodal freight transportation. The reader is given an overview of the relevant literature, as well as of different mathematical formulations. Moreover, algorithmic solution approaches are suggested which are specially designed for problems that arise in the context of intermodal freight transportation, but can be applied to other areas as well.

## Prices

Show Summary Details
Restricted access

# Chapter 4 Acyclic Partitioning Problem

## Extract

This chapter addresses the problem of partitioning the vertex set of a given directed, arc- and vertex-weighted graph into disjoint subsets (i.e., clusters) as considered in Nossack and Pesch (2012a) and Briskorn et al. (2012). Clus- ters are to be determined such that the sum of the vertex weights within the clusters satisﬁes an upper bound and the sum of the arc weights within the clusters is maximized. Additionally, the digraph is enforced to partition into a directed, acyclic graph, i.e., a digraph that contains no directed cy- cle. This problem is known in the literature as acyclic partitioning problem and is proven to be NP-hard in the strong sense (refer, e.g., to Garey and Johnson (1979)). Real-life applications arise, e.g., at rail-rail transshipment yards and in VLSI (Very Large Scale Integration) design. We propose sev- eral integer programming formulations for the acyclic partitioning problem and suggest two solution approaches, a branch-and-bound framework that integrates constraint propagation and a branch-and-price algorithm. Com- putational results are reported to verify the strength of our proposals. 4.1 Problem Description Graph partitioning problems are, in general, concerned with the partition- ing of the vertex set of an undirected or directed graph into disjoint subsets such that the sum of the edge weights within the clusters is maximized (or equivalently, the sum of the edge weights between diﬀerent clusters is min- 80 Chapter 4. Acyclic Partitioning Problem (a) Digraph (b) Acyclic Partition ! ! ! ! (c) Partition Fig. 4.1: Partition and Acyclic Partition imized). Most graph...

## You are not authenticated to view the full text of this chapter or article.

This site requires a subscription or purchase to access the full text of books or journals.