The splittable flow arc set with capacity and minimum load constraints

Cataloged from PDF version of article.We study the convex hull of the splittable flow arc set with capacity and minimum load constraints. This set arises as a relaxation of problems where clients have demand for a resource that can be installed in integer amounts and that has capacity limitations and lower bounds on utilization. We prove that the convex hull of this set is the intersection of the convex hull of the set with a capacity constraint and the convex hull of the set with a minimum load constraint

Models for the piecewise linear unsplittable multicommodity flow problems

International audienceIn this paper, we consider multicommodity flow problems, with unsplit-table flows and piecewise linear routing costs. We first focus on the case where the piecewise linear routing costs are convex. We show that this problem is N P-hard for the general case, but polynomially solvable when there is only one commodity. We then propose a strengthened mixed-integer programming formulation for the problem. We show that the linear relaxation of this formulation always gives the optimal solution of the problem for the single commodity case. We present a wide array of computational experiments, showing this formulation also produces very tight linear programming bounds for the multi-commodity case. Finally, we also adapt our formulation for the non-convex case. Our experimental results imply that the linear programming bounds for this case, are only slightly than the ones of state-of-the-art models for the splittable flow version of the problem

A comparison of different routing schemes for the robust network loading problem: polyhedral results and computation

International audienceWe consider the capacity formulation of the Robust Network Loading Problem. The aim of the paper is to study what happens from the theoretical and from the computational point of view when the routing policy (or scheme) changes. The theoretical results consider static, volume, affine and dynamic routing, along with splittable and unsplittable flows. Our polyhedral study provides evidence that some well-known valid inequalities (the robust cutset inequalities) are facets for all the considered routing/flows policies under the same assumptions. We also introduce a new class of valid inequalities, the robust 3-partition inequalities, showing that, instead, they are facets in some settings, but not in others. A branch-and-cut algorithm is also proposed and tested. The computational experiments refer to the problem with splittable flows and the budgeted uncertainty set. We report results on several instances coming from real-life networks, also including historical traffic data, as well as on randomly generated instances. Our results show that the problem with static and volume routing can be solved quite efficiently in practice and that, in many cases, volume routing is cheaper than static routing, thus possibly representing the best compromise between cost and computing time. Moreover, unlikely from what one may expect, the problem with dynamic routing is easier to solve than the one with affine routing, which is hardly tractable, even using decomposition methods

Robust network design under polyhedral traffic uncertainty

Ankara : The Department of Industrial Engineering and The Institute of Engineering and Science of Bilkent Univ., 2007.Thesis (Ph.D.) -- Bilkent University, 2007.Includes bibliographical references leaves 160-166.In this thesis, we study the design of networks robust to changes in demand estimates. We consider the case where the set of feasible demands is defined by an arbitrary polyhedron. Our motivation is to determine link capacity or routing configurations, which remain feasible for any realization in the corresponding demand polyhedron. We consider three well-known problems under polyhedral demand uncertainty all of which are posed as semi-infinite mixed integer programming problems. We develop explicit, compact formulations for all three problems as well as alternative formulations and exact solution methods. The first problem arises in the Virtual Private Network (VPN) design field. We present compact linear mixed-integer programming formulations for the problem with the classical hose traffic model and for a new, less conservative, robust variant relying on accessible traffic statistics. Although we can solve these formulations for medium-to-large instances in reasonable times using off-the-shelf MIP solvers, we develop a combined branch-and-price and cutting plane algorithm to handle larger instances. We also provide an extensive discussion of our numerical results. Next, we study the Open Shortest Path First (OSPF) routing enhanced with traffic engineering tools under general demand uncertainty with the motivation to discuss if OSPF could be made comparable to the general unconstrained routing (MPLS) when it is provided with a less restrictive operating environment. To the best of our knowledge, these two routing mechanisms are compared for the first time under such a general setting. We provide compact formulations for both routing types and show that MPLS routing for polyhedral demands can be computed in polynomial time. Moreover, we present a specialized branchand-price algorithm strengthened with the inclusion of cuts as an exact solution tool. Subsequently, we compare the new and more flexible OSPF routing with MPLS as well as the traditional OSPF on several network instances. We observe that the management tools we use in OSPF make it significantly better than the generic OSPF. Moreover, we show that OSPF performance can get closer to that of MPLS in some cases. Finally, we consider the Network Loading Problem (NLP) under a polyhedral uncertainty description of traffic demands. After giving a compact multicommodity formulation of the problem, we prove an unexpected decomposition property obtained from projecting out the flow variables, considerably simplifying the resulting polyhedral analysis and computations by doing away with metric inequalities, an attendant feature of most successful algorithms on NLP. Under the hose model of feasible demands, we study the polyhedral aspects of NLP, used as the basis of an efficient branch-and-cut algorithm supported by a simple heuristic for generating upper bounds. We provide the results of extensive computational experiments on well-known network design instances.Altın, AyşegülPh.D

Exact approaches for designing multifacility buy-at-bulk networks

We study a problem that integrates buy-at-bulk network design into the classical facility location problem. We consider a generalization of the facility location problem where multiple clients may share a capacitated network to connect to open facilities instead of requiring direct links. In this problem, we wish to open facilities, build a routing network by installing access cables of different costs and capacities, and route every client demand to an open facility. We provide a path based formulation and we compare it with the natural compact formulation for this problem. We then design an exact branch-price-and-cut algorithm for solving the path based formulation. We study the effect of two families of valid inequalities. In addition to this, we present three different types of primal heuristics and employ a hybrid approach to effectively combine these heuristics in order to improve the primal bounds. We finally report the results of our approach that were tested on a set of real world instances as well as two sets of benchmark instances and evaluate the effects of our valid inequalities and primal heuristics

Optimal Shipping Decisions in an Airfreight Forwarding Network

This thesis explores three consolidation problems derived from the daily operations of major international airfreight forwarders. First, we study the freight forwarder's unsplittable shipment planning problem in an airfreight forwarding network where a set of cargo shipments have to be transported to given destinations. We provide mixed integer programming formulations that use piecewise-linear cargo rates and account for volume and weight constraints, flight departure/arrival times, as well as shipment-ready times. After exploring the solution of such models using CPLEX, we devise two solution methodologies to handle large problem sizes. The first is based on Lagrangian relaxation, where the problems decompose into a set of knapsack problems and a set of network flow problems. The second is a local branching heuristic that combines branching ideas and local search. The two approaches show promising results in providing good quality heuristic solutions within reasonable computational times, for difficult and large shipment consolidation problems. Second, we further explore the freight forwarder's shipment planning problem with a different type of discount structure - the system-wide discount. The forwarder's cost associated with one flight depends not only on the quantity of freight assigned to that flight, but also on the total freight assigned to other flights operated by the same carrier. We propose a multi-commodity flow formulation that takes shipment volume and over-declaration into account, and solve it through a Lagrangian relaxation approach. We also model the "double-discount" scheme that incorporates both the common flight-leg discount (the one used in the unsplittable shipment problem) and the system-wide discount offered by cargo airlines. Finally, we focus on palletized loading using unit loading devices (ULDs) with pivots, which is different from what we assumed in the previous two research problems. In the international air cargo business, shipments are usually consolidated into containers; those are the ULDs. A ULD is charged depending on whether the total weight exceeds a certain threshold, called the pivot weight. Shipments are charged the under-pivot rate up to the pivot weight. Additional weight is charged at the over-pivot rate. This scheme is adopted for safety reasons to avoid the ULD overloading. We propose three solution methodologies for the air-cargo consolidation problem under the pivot-weight (ACPW), namely: an exact solution approach based on branch-and-price, a best fit decreasing loading heuristic, and an extended local branching. We found superior computational performance with a combination of the multi-level variables and a relaxation-induced neighborhood search for local branching

The robust network loading problem under hose demand uncertainty: Formulation, polyhedral analysis, and computations

We consider the network loading problem (NLP) under a polyhedral uncertainty description of traffic demands. After giving a compact multicommodity flow formulation of the problem, we state a decomposition property obtained from projecting out the flow variables. This property considerably simplifies the resulting polyhedral analysis and computations by doing away with metric inequalities. Then we focus on a specific choice of the uncertainty description, called the "hose model," which specifies aggregate traffic upper bounds for selected endpoints of the network. We study the polyhedral aspects of the NLP under hose demand uncertainty and use the results as the basis of an efficient branch-and-cut algorithm. The results of extensive computational experiments on well-known network design instances are reported. © 2011 INFORMS