Continuous knapsack sets with divisible capacities

We study two continuous knapsack sets (Formula presented.) and (Formula presented.) with (Formula presented.) integer, one unbounded continuous and (Formula presented.) bounded continuous variables in either (Formula presented.) or (Formula presented.) form. When the coefficients of the integer variables are integer and divisible, we show in both cases that the convex hull is the intersection of the bound constraints and (Formula presented.) polyhedra arising as the convex hulls of continuous knapsack sets with a single unbounded continuous variable. The latter convex hulls are completely described by an exponential family of partition inequalities and a polynomial size extended formulation is known in the (Formula presented.) case. We also provide an extended formulation for the (Formula presented.) case. It follows that, given a specific objective function, optimization over both (Formula presented.) and (Formula presented.) can be carried out by solving (Formula presented.) polynomial size linear programs. A further consequence of these results is that the coefficients of the continuous variables all take the values 0 or 1 (after scaling) in any non-trivial facet-defining inequality of the convex hull of such sets. © 2015, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society

Mixed n-step MIR inequalities: Facets for the n-mixing set

AbstractGünlük and Pochet [O. Günlük, Y. Pochet, Mixing mixed integer inequalities. Mathematical Programming 90 (2001) 429–457] proposed a procedure to mix mixed integer rounding (MIR) inequalities. The mixed MIR inequalities define the convex hull of the mixing set {(y1,…,ym,v)∈Zm×R+:α1yi+v≥βi,i=1,…,m} and can also be used to generate valid inequalities for general as well as several special mixed integer programs (MIPs). In another direction, Kianfar and Fathi [K. Kianfar, Y. Fathi, Generalized mixed integer rounding inequalities: facets for infinite group polyhedra. Mathematical Programming 120 (2009) 313–346] introduced the n-step MIR inequalities for the mixed integer knapsack set through a generalization of MIR. In this paper, we generalize the mixing procedure to the n-step MIR inequalities and introduce the mixed n-step MIR inequalities. We prove that these inequalities define facets for a generalization of the mixing set with n integer variables in each row (which we refer to as the n-mixing set), i.e. {(y1,…,ym,v)∈(Z×Z+n−1)m×R+:∑j=1nαjyji+v≥βi,i=1,…,m}. The mixed MIR inequalities are simply the special case of n=1. We also show that mixed n-step MIR can generate valid inequalities based on multiple constraints for general MIPs. Moreover, we introduce generalizations of the capacitated lot-sizing and facility location problems, which we refer to as the multi-module problems, and show that mixed n-step MIR can be used to generate valid inequalities for these generalizations. Our computational results on small MIPLIB instances as well as a set of multi-module lot-sizing instances justify the effectiveness of the mixed n-step MIR inequalities

Valid Inequalities and Facets for Multi-Module (Survivable) Capacitated Network Design Problem

In this dissertation, we develop new methodologies and algorithms to solve the multi-module (survivable) network design problem. Many real-world decision-making problems can be modeled as network design problems, especially on networks with capacity requirements on arcs or edges. In most cases, network design problems of this type that have been studied involve different types of capacity sizes (modules), and we call them the multi-module capacitated network design (MMND) problem. MMND problems arise in various industrial applications, such as transportation, telecommunication, power grid, data centers, and oil production, among many others. In the first part of the dissertation, we study the polyhedral structure of the MMND problem. We summarize current literature on polyhedral study of MMND, which generates the family of the so-called cutset inequalities based on the traditional mixed integer rounding (MIR). We then introduce a new family of inequalities for MMND based on the so-called n-step MIR, and show that various classes of cutset inequalities in the literature are special cases of these inequalities. We do so by studying a mixed integer set, the cutset polyhedron, which is closely related to MMND. We We also study the strength of this family of inequalities by providing some facet-defining conditions. These inequalities are then tested on MMND instances, and our computational results show that these classes of inequalities are very effective for solving MMND problems. Generalizations of these inequalities for some variants of MMND are also discussed. Network design problems have many generalizations depending on the application. In the second part of the dissertation, we study a highly applicable form of SND, referred to as multi-module SND (MM-SND), in which transmission capacities on edges can be sum of integer multiples of differently sized capacity modules. For the first time, we formulate MM-SND as a mixed integer program (MIP) using preconfigured-cycles (p-cycles) to reroute flow on failed edges. We derive several classes of valid inequalities for this MIP, and show that the valid inequalities previously developed in the literature for single-module SND are special cases of our inequalities. Furthermore, we show that our valid inequalities are facet-defining for MM-SND in many cases. Our computational results, using a heuristic separation algorithm, show that these inequalities are very effective in solving MM-SND. In particular they are more effective than compared to using single-module inequalities alone. Lastly, we generalize the inequalities for MMND for other mixed integer sets relaxed from MMND and the cutset polyhedron. These inequalities also generalize several valid inequalities in the literature. We conclude the dissertation by summarizing the work and pointing out potential directions for future research

Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression

We present an exact method, based on an arc-flow formulation with side constraints, for solving bin packing and cutting stock problems --- including multi-constraint variants --- by simply representing all the patterns in a very compact graph. Our method includes a graph compression algorithm that usually reduces the size of the underlying graph substantially without weakening the model. As opposed to our method, which provides strong models, conventional models are usually highly symmetric and provide very weak lower bounds. Our formulation is equivalent to Gilmore and Gomory's, thus providing a very strong linear relaxation. However, instead of using column-generation in an iterative process, the method constructs a graph, where paths from the source to the target node represent every valid packing pattern. The same method, without any problem-specific parameterization, was used to solve a large variety of instances from several different cutting and packing problems. In this paper, we deal with vector packing, graph coloring, bin packing, cutting stock, cardinality constrained bin packing, cutting stock with cutting knife limitation, cutting stock with binary patterns, bin packing with conflicts, and cutting stock with binary patterns and forbidden pairs. We report computational results obtained with many benchmark test data sets, all of them showing a large advantage of this formulation with respect to the traditional ones

Market-Based Allocation with Indivisible Bids

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73571/1/j.1937-5956.2007.tb00275.x.pd

Valid inequalities for the single-item capacitated lot sizing problem with step-wise costs

This paper presents a new class of valid inequalities for the single-item capacitated lotsizing problem with step-wise production costs (LS-SW). We first provide a survey of different optimization methods proposed to solve LS-SW. Then, flow cover and flow cover inequalities derived from the single node flow set are described in order to generate the new class of valid inequalities. The single node flow set can be seen as a generalization of some valid relaxations of LS-SW. A new class of valid inequalities we call mixed flow cover, is derived from the integer flow cover inequalities by a lifting procedure. The lifting coefficients are sequence independent when the batch sizes (V) and the production capacities (P) are constant and if V divides P. When the restriction of the divisibility is removed, the lifting coefficients are shown to be sequence independent. We identify some cases where the mixed flow cover inequalities are facet defining. A cutting plane algorithmis proposed for these three classes of valid inequalities. The exact separation algorithmproposed for the constant capacitated case runs in polynomial time. Finally, some computational results are given to compare the performance of the different optimization methods including the new class of valid inequalities.single-item capacitated lot sizing problem, flow cover inequalities, cutting plane algorithm

Fast approximation schemes for multi-criteria flow, knapsack, and scheduling problems

Includes bibliographical references (p. 40-42).Supported by an NSF Presidential Young Investigator grant. Supported by the Air Force Office of Scientific Research. AFOSR-88-0088 Supported by the NSF. DDM-8921835by Hershel M. Safer, James B. Orlin