6 research outputs found
Solving the bifurcated and nonbifurcated robust network loading problem with k-adaptive routing
International audienceWe experiment with an alternative routing scheme for the robust network loading problem with demand uncertainty. Named kâadaptive, it is based on the fact that the decisionâmaker chooses k secondâstage solutions and then commits to one of them only after realization of the uncertainty. This routing scheme, with its corresponding kâpartition of the uncertainty set, is dynamically defined under an iterative method to sequentially improve the solution. The method has an inherent characteristic of multiplying the number of variables and constraints after each iteration, so that additional measures are introduced in the solution strategy in order to control time performance. We compare our kâadaptive results with the ones obtained through other routing schemes and also verify the effectiveness of the methods utilized using several realistic networks from SNDlib and other sources
A Systematic Review on Integer Multi-objective Adjustable Robust Counterpart Optimization Model Using Benders Decomposition
Multi-objective integer optimization model that contain uncertain parameter can be handled using the Adjustable Robust Counterpart (ARC) methodology with Polyhedral Uncertainty Set. The ARC method has two stages of completion, so completing the second stage can be assisted by the Benders Decomposition. This paper discusses the systematic review on this topic using the Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA). PRISMA presents a database mining algorithm for previous articles and related topics sourced from Scopus, Science Direct, Dimensions, and Google Scholar. Four stages of the algorithm are used, namely Identification, Screening, Eligibility, and Included. In the Eligibility stage, 16 articles obtained and called Dataset 1, used for bibliometric mapping and evolutionary analysis. Moreover, in the Included stage, six final databases obtained and called Dataset 2, which was used to analyze research gaps and novelty. The analysis was carried out on two datasets, assisted by the output visualisation using RStudio software with the " bibliometrix" package, then we use the command 'biblioshiny()' to create a link to the âshiny web interfaceâ. At the final stage of the article using six articles analysis, it is concluded that there is no research on the ARC multi-objective integer optimization model with Polyhedral Uncertainty Sets using the Benders Decomposition Method, which focuses on discussing the general model and its mathematical analysis. Moreover, this research topic is open and becomes the primary references for further research in connection
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
Singleâcommodity stochastic network design under demand and topological uncertainties with insufficient data
Stochastic network design is fundamental to transportation and logistic problems in practice, yet faces new modeling and computational challenges resulted from heterogeneous sources of uncertainties and their unknown distributions given limited data. In this article, we design arcs in a network to optimize the cost of singleâcommodity flows under random demand and arc disruptions. We minimize the network design cost plus cost associated with network performance under uncertainty evaluated by two schemes. The first scheme restricts demand and arc capacities in budgeted uncertainty sets and minimizes the worstâcase cost of supply generation and network flows for any possible realizations. The second scheme generates a finite set of samples from statistical information (e.g., moments) of data and minimizes the expected cost of supplies and flows, for which we bound the worstâcase cost using budgeted uncertainty sets. We develop cuttingâplane algorithms for solving the mixedâinteger nonlinear programming reformulations of the problem under the two schemes. We compare the computational efficacy of different approaches and analyze the results by testing diverse instances of random and realâworld networks. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 154â173, 2017Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/137236/1/nav21739_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/137236/2/nav21739.pd
Robust Design of Single-Commodity Networks
The results in the present work were obtained in a collaboration with Eduardo Ălvarez-
Miranda, Valentina Cacchiani, Tim Dorneth, Michael JĂŒnger, Frauke Liers, Andrea Lodi
and Tiziano Parriani.
The subject of this thesis is a robust network design problem, i.e., a problem of the type
âdimension a network such that it has sufficient capacity in all likely scenarios.â In our case,
we model the network with an undirected graph in which each scenario defines a supply or
demand for each node. We say that a flow in the network is feasible for a scenario if it can
balance out its supplies and demands. A scenario polytope B defines which scenarios are
relevant. The task is now to find integer capacities that minimize the total installation costs
while allowing for a feasible flow in each scenario. This problem is called Single-Commodity
Robust Network Design Problem (sRND) and was introduced by Buchheim, Liers and SanitĂ
(INOC 2011). The problem contains the Steiner Tree Problem (given an undirected graph
and a terminal set, find a minimum cost subtree that connects all terminals) and therefore
is N P-hard. The problem is also a natural extension of minimum cost flows.
The network design literature treats the case that the scenario polytope B is given as
the finite set of its extreme points (finite case) and that it is given as the feasible region
of finitely many linear inequalities (polyhedral case). Both descriptions are equivalent,
however, an efficient transformation is not possible in general.
Buchheim, Liers and SanitĂ (INOC 2011) propose a Branch-and-Cut algorithm for the
finite case. In this case, there exists a canonical problem formulation as a mixed integer
linear program (MIP). It contains a set of flow variables for every scenario. Buchheim, Liers
and SanitĂ enhance the formulation with general cutting planes that are called target cuts.
The first part of the dissertation considers the problem variant where every scenario has
exactly two terminal nodes. If the underlying network is a complete, unweighted graph,
then this problem is the Network Synthesis Problem as defined by Chien (IBM Journal of
R&D 1960). There exist polynomial time algorithms by Gomory and Hu (SIAM J. of Appl.
Math 1961) and by Kabadi, Yan, Du and Nair (SIAM J. on Discr. Math.) for this special
case. However, these algorithms are based on the fact that complete graphs are Hamiltonian.
The result of this part is a similar algorithm for hypercube graphs that assumes a special
distribution of the supplies and demands. These graphs are also Hamiltonian.
The second part of the thesis discusses the structure of the polyhedron of feasible sRND
solutions. Here, the first result is a new MIP-based capacity formulation for the sRND
problem. The size of this formulation is independent of the number of extreme points
of B and therefore, it is also suited for the polyhedral case. The formulation uses so-called
cut-set inequalities that are known in similar form from other network design problems. By
adapting a proof by Mattia (Computational Optimization and Applications 2013), we show
that cut-set inequalities induce facets of the sRND polyhedron. To obtain a better linear
programming relaxation of the capacity formulation, we interpret certain general mixed
integer cuts as 3-partition inequalities and show that these inequalities induce facets as well.
The capacity formulation has exponential size and we therefore need a separation algorithm
for cut-set inequalities. In the finite case, we reduce the cut-set separation problem to
a minimum cut problem that can be solved in polynomial time. In the polyhedral case,
however, the separation problem is N P-hard, even if we assume that the scenario polytope
is basically a cube. Such a scenario polytope is called Hose polytope. Nonetheless, we can
solve the separation problem in practice: We show a MIP based separation procedure for
the Hose scenario polytope. Additionally, the thesis presents two separation methods for
3-partition inequalities. These methods are independent of the encoding of the scenario
polytope. Additionally, we present several rounding heuristics.
The result is a Branch-and-Cut algorithm for the capacity formulation. We analyze the
algorithm in the last part of the thesis. There, we show experimentally that the algorithm
works in practice, both in the finite and in the polyhedral case. As a reference point, we
use a CPLEX implementation of the flow based formulation and the computational results by
Buchheim, Liers and SanitĂ . Our experiments show that the new Branch-and-Cut algorithm
is an improvement over the existing approach. Here, the algorithm excels on problem
instances with many scenarios. In particular, we can show that the MIP separation of the
cut-set inequalities is practical