Integer Multicommodity Flow Problem Research Articles

Branch-and-Price Based Heuristic Algorithm for Fuzzy Multi-Depot Bus Scheduling Problem

This paper deals with fuzzy multi-depot bus scheduling (FMDBS) problem in which the objective function and constraints are defined with fuzzy attributes. Credibility relation is used to formulate the problem as an integer multicommodity flow problem. A novel combination of branch-and-price and heuristic algorithms, is proposed to efficiently solve FMDBS problem. In the proposed algorithm, the heuristic algorithm is applied to generate initial columns for the column generation method. Also, a heuristic algorithm is used to improve the generated solutions in each node of the branch-and-price tree. Two sets of benchmark examples are applied to demonstrate the efficiency of the proposed algorithm for large-scale instances. Also, the algorithm is applied to solve the classical multi-depot bus scheduling problem. The results show that the proposed algorithm decreases integrality gap and computational time in comparison with the state-of-the-art algorithms and normal branch-and-price algorithm. Finally, as a case study, the bus schedules in Tehran BRT network are generated.

Aircraft maintenance facility location planning

This investigation combines the aircraft maintenance routing problem (AMRP) with the aircraft maintenance facility location problem as they are interdependent. Unlike the classical fixed charge location model, aircraft maintenance facility location is not based on static customer demands but on how aircraft move around the airline network while undergoing periodic maintenance checks. Two important factors relevant to this analysis are: (a) aircraft overnight stays at maintenance facilities and airports away from their home bases and (b) the location and cost of maintenance facility construction. There are two possible scenarios concerning the status of maintenance facilities. The first is a tabula rasa case – for a start-up airline with no existing maintenance facilities (AMFLP). The second is an operating airline case with prepositioned maintenance facilities, which we denote as a maintenance facility location update problem (AMFUP). We formulate both as binary integer multicommodity flow problems whose aim is to find the cost-minimizing number, size, and location of the maintenance facilities. Experiments with flight schedules from two regional airlines indicate that total annualized costs are convex in the number of maintenance airports. Though new facility costs tend to dominate, costs associated with noncyclic routes become increasingly important as the number of maintenance airports decreases. Key contributions of this paper are that it (a) demonstrates the superiority of the integration of the aircraft maintenance routing and aircraft maintenance facility location problems, (b) provides a formulation and solution methodology dealing with this significant but under-researched problem, and (c) presents extensive computational experiments, cost and sensitivity analyses for two real life aircraft flight schedules.

Resolution of coupling order and station level constraints in train unit scheduling

Train unit scheduling assigns vehicles to cover all trips of a fixed timetable satisfying constraints such as seat demands. With a two-phase approach, this problem is first solved in Phase I as an integer multi-commodity flow problem. Train stations are simplified as single points and coupling orders of train units are left undetermined. In this paper, platforms and their layouts at the stations are restored to complete a fully operable schedule, defined as Phase II. An adaptive approach expanding Phase I to Phase II is proposed. The logistics of (de-)coupling operations, coupling orders and re-platforming are determined in detail to prevent unit blockage where possible, particularly focusing on developing a schedule with conflict-free coupling orders. If unresolvable station level conflicts still exist, the process loops back to Phase I with addressed Phase II constraints to avoid identified conflicts. Thus, the schedule is iteratively improved until it is fully operable.

The invisible‐hand heuristic for origin‐destination integer multicommodity network flows

AbstractOrigin‐destination integer multicommodity flow problems differ from classic multicommodity models in that each commodity has one source and one sink, and each commodity must be routed along a single path. A new invisible‐hand heuristic that mimics economic markets' behavior is presented and tested on large‐scale telecommunications networks, with solution times two orders of magnitude faster than CPLEX's LP relaxation, more dramatic MIP ratios, and small solution value differences.

Local convex hulls for a special class of integer multicommodity flow problems.

Based on previous work in rolling stock scheduling problems (Alfieri et al. in Transp Sci 40:378–391, 2006; Cacchiani et al. in Math Progr B 124:207–231, 2010; Lin and Kwan in Electron Notes Discret Math 41:165–172, 2013; Schrijver in CWI Q 6:205–217, 1993; Ziarati et al. in Manag Sci 45:1156–1168, 1999), we generalize a local convex hull method for a class of integer multicommodity flow problems, and discuss its feasibility range in high dimensional cases. Suppose a local convex hull can be divided into an up hull, a main hull and a down hull if certain conditions are met, it is shown theoretically that the main hull can only have at most two nonzero facets. The numbers of points in the up and down hull are explored mainly on an empirical basis. The above properties of local convex hulls have led to a slightly modified QuickHull algorithm (the “2-facet QuickHull”) based on the original version proposed by Barber et al. (ACM Trans Math Softw 22:469–483, 1996). As for the feasibility in applying this method to rolling stock scheduling, our empirical experiments show that for the problem instances of ScotRail and Southern Railway, two major train operating companies in the UK, even in the most difficult real-world or artificial conditions (e.g. supposing a train can be served by any of 11 compatible types of self-powered unit), the standard QuickHull (Barber et al. in ACM Trans Math Softw 22:469–483, 1996) can easily compute the relevant convex hulls. For some even more difficult artificial instances that may fall outside the scope of rolling stock scheduling (e.g. a node in a graph can be covered by more than 11 kinds of compatible commodities), there is evidence showing that the “2-facet QuickHull” can be more advantageous over the standard QuickHull for our tested instances. When the number of commodity types is even higher (e.g. >19), or the number of points in a high dimensional space (e.g. 15 dimensions) is not small (e.g. >2000), the local convex hulls cannot be computed either by the standard or the 2-facet QuickHull methods within practical time.

Complexity of One Packing Optimization Problem

The authors consider packing optimization for elements of a square matrix, which are given by positive integers, into blocks of fixed size. The problem is formulated and its combinatorial intractability is discussed. The problem is proved to be NP-complete. This is done by polynomial reduction of an NP-complete minimum-cost integer multicommodity flow problem to this problem.

Integral polyhedra related to integer multicommodity flows on a cycle

The integer multicommodity flow problem on a cycle (IMFC) is to find a feasible integral routing of given demands between κ pairs of nodes on a link-capacitated undirected cycle, which is known to be polynomially solvable. Along with integral polyhedra related to IMFC, this paper shows that there exists a linear program, with a polynomial number of variables and constraints, which solves IMFC. Using the results, we also present a compact polyhedral description of the convex hull of feasible solutions to a certain class of instances of IMFC whose number of variables and constraints is O ( κ ) , which in turn means that there exists a non-trivial special case for which a minimum cost integer multicommodity flow problem can be solved in polynomial time.

A column generation heuristic for a dynamic generalized assignment problem

This paper studies the dynamic generalized assignment problem (DGAP) which extends the well-known generalized assignment problem by considering a discretized time horizon and by associating a starting time and a finishing time with each task. Additional constraints related to warehouse and yard management applications are also considered. Three linear integer programming formulations of the problem are introduced. The strongest one models the problem as an origin–destination integer multi-commodity flow problem with side constraints. This model can be solved quickly for instances of small to moderate size. However, because of its computer memory requirements, it becomes impractical for larger instances. Hence, a column generation algorithm is used to compute lower bounds by solving the linear program (LP) relaxation of the problem. This column generation algorithm is also embedded in a heuristic aimed at finding feasible integer solutions. Computational experiments on large-scale instances show the effectiveness of the proposed approach.

Multicommodity demand flow in a tree and packing integer programs

We consider requests for capacity in a given tree network T = ( V , E ) where each edge e of the tree has some integer capacity u e . Each request f is a node pair with an integer demand d f and a profit w f which is obtained if the request is satisfied. The objective is to find a set of demands that can be feasibly routed in the tree and which provides a maximum profit. This generalizes well-known problems, including the knapsack and b -matching problems. When all demands are 1, we have the integer multicommodity flow problem. Garg et al. [1997] had shown that this problem is NP-hard and gave a 2-approximation algorithm for the cardinality case (all profits are 1) via a primal-dual algorithm. Our main result establishes that the integrality gap of the natural linear programming relaxation is at most 4 for the case of arbitrary profits. Our proof is based on coloring paths on trees and this has other applications for wavelength assignment in optical network routing. We then consider the problem with arbitrary demands. When the maximum demand d max is at most the minimum edge capacity u min , we show that the integrality gap of the LP is at most 48. This result is obtained by showing that the integrality gap for the demand version of such a problem is at most 11.542 times that for the unit-demand case. We use techniques of Kolliopoulos and Stein [2004, 2001] to obtain this. We also obtain, via this method, improved algorithms for line and ring networks. Applications and connections to other combinatorial problems are discussed.

Multicommodity flows in cycle graphs

This paper considers the multicommodity flow problem and the integer multicommodity flow problem on cycle graphs. We present two linear time algorithms for solving each of the two problems.

A GRASP with path‐relinking for private virtual circuit routing

AbstractA frame relay service offers virtual private networks to customers by provisioning a set of long‐term private virtual circuits (PVCs) between customer endpoints on a large backbone network. During the provisioning of a PVC, routing decisions are made without any knowledge of future requests. Over time, these decisions can cause inefficiencies in the network and occasional offline rerouting of the PVCs is needed. In this paper, the offline PVC routing problem is formulated as an integer multicommodity flow problem with additional constraints and with an objective function that minimizes propagation delays and/or network congestion. We propose variants of a GRASP with path‐relinking heuristic for this problem. Experimental results for realistic‐size problems are reported, showing that the proposed heuristics are able to improve the solutions found with standard routing techniques. Moreover, the structure of our objective function provides a useful strategy for setting the appropriate value of its weight parameter, to achieve some quality of service (QoS) level defined by a desired balance between propagation delay and delay due to network congestion. © 2003 Wiley Periodicals, Inc.

An Integer Multicommodity Flow Model Applied to the Rerostering of Nurse Schedules

The problem of rerostering service schedules is very common in organizations that work shifts around the clock every day of the year with a set number of employees. Whenever one or more workers announce that they will not be able to attend to tasks previously assigned in their schedule, those tasks must be performed at the expense of alterations in the schedules of other workers. These changes should not conflict with the rules laid down by the administration and employment contracts and should affect the previous schedules as little as possible. This is a difficult real problem calling for a computational tool to cope with it easily. In the paper the issue is described in detail in the context of nurse scheduling and formulated as an integer multicommodity flow problem with additional constraints, in a multi-level acyclical network. A heuristic was implemented as a first approach to solving the problem. Subsequently the integer linear programming formulation of the multicommodity flow model and two linear relaxations were tested using CPLEX [2] optimizers. The computational results reported regard real instances from a Lisbon state hospital. Satisfactory rosters were obtained within acceptable computational times in all instances tested, either with the integer optimizer, or with the heuristic. This being so, refinements will be undertaken to embed these methodologies in a decision support system that may assist the head nurse in her daily rerostering activities.

Routing of Railway Carriages

In the context of organizing timetables for railway companies the following railway carriage routing problem occurs. Given a timetable containing rail links with departure and destination times/stations and the composition of the trains, find a routing of railway carriages such that the required carriages are always available when a train departs. The problem is formulated as an integer multi-commodity network flow problem with nonlinear objective function. We will present a local search approach for this NP-hard problem. The approach uses structural properties of the integer multi-commodity network flow formulation of the problem. Computational results for a real world instance are given.

A branch‐and‐price algorithm for a hierarchical crew scheduling problem

AbstractWe describe a real‐life problem arising at a crane rental company. This problem is a generalization of the basic crew scheduling problem given in Mingozzi et al. [18] and Beasley and Cao [6]. We formulate the problem as an integer programming problem and establish ties with the integer multicommodity flow problem and the hierarchical interval scheduling problem. After establishing the complexity of the problem we propose a branch‐and‐price algorithm to solve it. We test this algorithm on a limited number of real‐life instances. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 723–742, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10044

An Approximation Algorithm for Circular Arc Colouring

We consider the problem of colouring a family of n arcs of a circle. This NP-complete problem, which occurs in routing and network design problems, is modelled as a 0-1 integer multicommodity flow problem. We present an algorithm that routes the commodities in the network by augmenting the network with some extra edges which correspond to extra colours. The algorithm, which relies on probabilistic techniques such as randomized rounding and path selection, is a randomized approximation algorithm which has an asymptotic performance ratio of 1+1/e (approximately 1.37) except when the minimum number of colours required is very small (O(\ln n) ). This is an improvement over the best previously known result [7], which is a deterministic approximation algorithm with a performance ratio of 3/2. The substantial improvement is valuable, for instance in wavelength allocation strategies in communication networks where bandwidth is a precious resource.

Using Branch-and-Price-and-Cut to Solve Origin-Destination Integer Multicommodity Flow Problems

We present a column-generation model and branch-and-price-and-cut algorithm for origin-destination integer multicommodity flow problems. The origin-destination integer multicommodity flow problem is a constrained version of the linear multicommodity flow problem in which flow of a commodity (defined in this case by an origin-destination pair) may use only one path from origin to destination. Branch-and-price-and-cut is a variant of branch-and-bound, with bounds provided by solving linear programs using column-and-cut generation at nodes of the branch-and-bound tree. Because our model contains one variable for each origin-destination path, for every commodity, the linear programming relaxations at nodes of the branch-and-bound tree are solved using column generation, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality. We devise a new branching rule that allows columns to be generated efficiently at each node of the branch-and-bound tree. Then, we describe cuts (cover inequalities) that can be generated at each node of the branch-and-bound tree. These cuts help to strengthen the linear programming relaxation and to mitigate the effects of problem symmetry. We detail the implementation of our combined column-and-cut generation method and present computational results for a set of test problems arising from telecommunications applications. We illustrate the value of our branching rule when used to find a heuristic solution and compare branch-and-price and branch-and-price-and-cut methods to find optimal solutions for highly capacitated problems.

A polyhedral approach to an integer multicommodity flow problem

In this paper we propose a branch-and-cut algorithm for the exact solution of an integer multicommodity flow problem. This NP -hard problem finds important applications in transportation, VLSI design, and telecommunications. We consider alternative formulations of the problem and describe several classes of valid inequalities. We describe lifting procedures to extend a given valid inequality for the problem with k commodities, to that having a larger number of commodities. We introduce a new large class of valid constraints, the multi-handle comb inequalities. The polyhedral structure of the integer multicommodity polytope is studied in the case of unit edge capacities. We prove that this polytope is full dimensional and show that some multi-handle comb inequalities are facet defining. Also, the lifting procedures are facet preserving under certain conditions. A branch-and-cut algorithm for the exact solution of the problem is then outlined, and separation algorithms for the main classes of inequalities studied in the paper are proposed. Computational results on several classes of test problems are finally reported, with a comparison between two different formulations.

On the Inapproximability of Disjoint Paths and Minimum Steiner Forest with Bandwidth Constraints

In this paper, we study the inapproximability of several well-known optimization problems in network optimization. We show-that the max directed vertex-disjoint paths problem cannot be approximated within ratio 2log1−εn unless NP⊆DTIME[2polylogn], the max directed edge-disjoint paths problem cannot be approximated within ratio 2log1−εn unless NP⊆DTIME[2polylogn], the integer multicommodity flow problem in directed graphs cannot be approximated within ratio 2log1−εn unless NP⊆DTIME[2polylogn], the max undirected vertex-disjoint paths problem does not have a polynomial time approximation scheme unless P=NP, and the minimum Steiner forest with bandwidth constraints problem cannot be approximated within ratio exp(poly(n)) unless P=NP.

SIGEST

Previous article Next article Full AccessSIGESThttps://doi.org/10.1137/SIREAD000041000004000775000001PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThis issue's SIGEST paper, "The ring loading problem,' by Alexander Schrijver, Paul Seymour, and Peter Winkler, appeared originally in volume 11 of SIAM Journal on Discrete Mathematics, February 1998. Beginning with an application of great and growing practical importance to modern communication networks, the authors show that the associated mathematical formulation, a highly special integer multicommodity flow problem, is NP-complete. They then describe their (ultimately successful) search for a fast algorithm that provably produces a solution within 5% of the optimum load. The paper is a lively, appealing mix of algorithmic theoretical computer science, operations research, and graph theory and has already inspired several other papers in the area. The exposition is outstanding, allowing even novices to understand both theoretical and practical aspects of the problem. The continuing, fruitful interactions between mathematics and numerical experiments are exceptionally interesting. For the paper's appearance in SIGEST, the authors have added a new approximation algorithm and comments about a further application area. We are grateful to them for their contribution to SIAM Review. Previous article Next article FiguresRelatedReferencesCited ByDetails Volume 41, Issue 4| 1999SIAM Review635-851 History Published online:02 August 2006 InformationCopyright © 1999 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/SIREAD000041000004000775000001Article page range:pp. 775-775ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics