Analysis of large scale linear programming problems with embedded network structures: Detection and solution algorithms

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Linear programming (LP) models that contain a (substantial) network structure frequently arise in many real life applications. In this thesis, we investigate two main questions; i) how an embedded network structure can be detected, ii) how the network structure can be exploited to create improved sparse simplex solution algorithms. In order to extract an embedded pure network structure from a general LP problem we develop two new heuristics. The first heuristic is an alternative multi-stage generalised upper bounds (GUB) based approach which finds as many GUB subsets as possible. In order to identify a GUB subset two different approaches are introduced; the first is based on the notion of Markowitz merit count and the second exploits an independent set in the corresponding graph. The second heuristic is based on the generalised signed graph of the coefficient matrix. This heuristic determines whether the given LP problem is an entirely pure network; this is in contrast to all previously known heuristics. Using generalised signed graphs, we prove that the problem of detecting the maximum size embedded network structure within an LP problem is NP-hard. The two detection algorithms perform very well computationally and make positive contributions to the known body of results for the embedded network detection. For computational solution a decomposition based approach is presented which solves a network problem with side constraints. In this approach, the original coefficient matrix is partitioned into the network and the non-network parts. For the partitioned problem, we investigate two alternative decomposition techniques namely, Lagrangean relaxation and Benders decomposition. Active variables identified by these procedures are then used to create an advanced basis for the original problem. The computational results of applying these techniques to a selection of Netlib models are encouraging. The development and computational investigation of this solution algorithm constitute further contribution made by the research reported in this thesis.This study is funded by the Turkish Educational Council and Mugla University

Dynamic Factorization in Large-Scale Optimization

Mathematical Programming, 64, pp. 17-51.Factorization of linear programming (LP) models enables a large portion of the LP tableau to be represented implicitly and generated from the remaining explicit part. Dynamic factorization admits algebraic elements which change in dimension during the course of solution. A unifying mathematical framework for dynamic row factorization is presented with three algorithms which derive from different LP model row structures: generalized upper bound rows, pure network rows,and generalized network TOWS. Each of these structures is a generalization of its predecessors, and each corresponding algorithm exhibits just enough additional richness to accommodate the structure at hand within the unified framework. Implementation and computational results are presented for a variety of real-world models. These results suggest that each of these algorithms is superior to the traditional, non-factorized approach, with the degree of improvement depending upon the size and quality of the row factorization identified

Fixed-Parameter Algorithms in Analysis of Heuristics for Extracting Networks in Linear Programs

We consider the problem of extracting a maximum-size reflected network in a linear program. This problem has been studied before and a state-of-the-art SGA heuristic with two variations have been proposed. In this paper we apply a new approach to evaluate the quality of SGA\@. In particular, we solve majority of the instances in the testbed to optimality using a new fixed-parameter algorithm, i.e., an algorithm whose runtime is polynomial in the input size but exponential in terms of an additional parameter associated with the given problem. This analysis allows us to conclude that the the existing SGA heuristic, in fact, produces solutions of a very high quality and often reaches the optimal objective values. However, SGA contain two components which leave some space for improvement: building of a spanning tree and searching for an independent set in a graph. In the hope of obtaining even better heuristic, we tried to replace both of these components with some equivalent algorithms. We tried to use a fixed-parameter algorithm instead of a greedy one for searching of an independent set. But even the exact solution of this subproblem improved the whole heuristic insignificantly. Hence, the crucial part of SGA is building of a spanning tree. We tried three different algorithms, and it appears that the Depth-First search is clearly superior to the other ones in building of the spanning tree for SGA. Thereby, by application of fixed-parameter algorithms, we managed to check that the existing SGA heuristic is of a high quality and selected the component which required an improvement. This allowed us to intensify the research in a proper direction which yielded a superior variation of SGA

Scheduling language and algorithm development study. Volume 3, phase 2: As-built specifications for the prototype language and module library

Detailed specifications of the prototype language and module library are presented. The user guide to the translator writing system is included

Polyhedral techniques in combinatorial optimization II: applications and computations

The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems. The main idea behind the technique is to consider the linear relaxation of the integer combinatorial optimization problem, and try to iteratively strengthen the linear formulation by adding violated strong valid inequalities, i.e., inequalities that are violated by the current fractional solution but satisfied by all feasible solutions, and that define high-dimensional faces, preferably facets, of the convex hull of feasible solutions. If we have the complete description of the convex hull of feasible solutions at hand all extreme points of this formulation are integral, which means that we can solve the problem as a linear programming problem. Linear programming problems are known to be computationally easy. In Part 1 of this article we discuss theoretical aspects of polyhedral techniques. Here we will mainly concentrate on the computational aspects. In particular we discuss how polyhedral results are used in cutting plane algorithms. We also consider a few theoretical issues not treated in Part 1, such as techniques for proving that a certain inequality is facet defining, and that a certain linear formulation gives a complete description of the convex hull of feasible solutions. We conclude the article by briefly mentioning some alternative techniques for solving combinatorial optimization problems

On the interplay of Mixed Integer Linear, Mixed Integer Nonlinear and Constraint Programming

In this thesis we study selected topics in the field of Mixed Integer Programming (MIP), in particular Mixed Integer Linear and Nonlinear Programming (MI(N)LP). We set a focus on the influences of Constraint Programming (CP). First, we analyze Mathematical Programming approaches to water network optimization, a set of challenging optimization problems frequently modeled as non-convex MINLPs. We give detailed descriptions of many variants and survey solution approaches from the literature. We are particularly interested in MILP approximations and present a respective computational study for water network design problems. We analyze this approach by algorithmic considerations and highlight the importance of certain convex substructures in these non-convex MINLPs. We further derive valid inequalities for water network design problems exploiting these substructures. Then, we treat Mathematical Programming problems with indicator constraints, recalling their most popular reformulation techniques in MIP, leading to either big-M constraints or disjunctive programming techniques. The latter give rise to reformulations in higher-dimensional spaces, and we review special cases from the literature that allow to describe the projection on the original space of variables explicitly. We theoretically extend the respective results in two directions and conduct computational experiments. We then present an algorithm for MILPs with indicator constraints that incorporates elements of CP into MIP techniques, including computational results for the JobShopScheduling problem. Finally, we introduce an extension of the class of MILPs so that linear expressions are allowed to have non-contiguous domains. Inspired by CP, this permits to model holes in the domains of variables as a special case. For such problems, we extend the theory of split cuts and show two ways of separating them, namely as intersection and lift-and-project cuts, and present computational results. We further experiment with an exact algorithm for such problems, applied to the Traveling Salesman Problem with multiple time windows

Optimization Approaches for Solving Large-Scale Personnel Scheduling Problems

Personnel scheduling is one of the most critical components in logistical planning for many practical areas, particularly in transportation, public services, and clinical operations. Because manpower is both an expensive and scarce resource, even a tiny improvement in utilization can provide huge expense savings for businesses. Additionally, a slightly better assignment schedule of the involved professionals can significantly increase their work satisfaction, which can in turn greatly improve the quality of the services customers or patients receive. However, practical personnel scheduling problems (PSPs) are hard to solve because modeling all of the complicated and nuanced requirements and rules is challenging. Moreover, since an iterative construction process may be necessary for handling the multiple-criteria or ill-defined objective nature of many PSPs, the model is expected to be solved in a short time while providing high-quality solutions, despite its large size and complexity. In this dissertation, we propose new models and solution approaches to address these challenges. We study in total three real-world PSPs. We first consider the crew pairing construction for a cargo airline. Each crew pairing is a sequence of flights assigned to a specific line/bid crew to operate in practice. Unlike traditional passenger aviation, due to the cargo airline's underlying network, each crew pairing will specify a complete flying schedule for the assigned crew over the entire planning horizon. Consequently, an extra and unique set of requirements must be incorporated into the construction process. We solve the problem using a delayed column generation framework. We develop a restricted shortest path model to incorporate the entire set of complicated requirements simultaneously and solve it using a labeling algorithm accelerated by a handful of proposed strategies. Computational experiments show that our approach can solve the crew pairing problem in a short time, while almost always delivering an optimal solution. Second, we consider an extension of the previous cargo crew scheduling problem, where a "break" is allowed to take place in the "middle" of each crew pairing. This break feature, working as a special type of conventional deadheading, is expected to significantly increase the flight coverage for practical deployment. However, incorporating this feature will result in an extremely dense underlying network, which introduces new computational challenges. To address this issue, we propose a bidirectional labeling based arc selection approach, which only needs to work on a tiny sub-network each time but can still guarantee the exactness of the delayed column generation process. We demonstrate through real-world instances that our proposed approach can solve this relaxed problem extension in a very short time and the resulting flight coverage will increase by over 30%. Finally, we study a medical resident annual block scheduling problem. We need to assign residents to perform services at different clinical units during each time period across the academic year so that the residents receive appropriate training while the hospital gets staffed sufficiently. We propose a two-stage partial fixing solution framework to address the long runtime issue caused by traditional approaches. A network-based model is also developed to provide a high-quality service selection to initiate this two-stage framework. Experiments using inputs from our clinical collaborator show that our approach can speed up the schedule construction at least 5 times for all instances and even over 100 times for some huge-size ones compared to a widely-used traditional approach.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169758/1/jhguo_1.pd

Integer and Constraint programming methods for mutually Orthogonal Latin Squares.

This thesis examines the Orthogonal Latin Squares (OLS) problem from the viewpoint of Integer and Constraint programming. An Integer Programming (IP) model is proposed and the associated polytope is analysed. We identify several families of strong valid inequalities, namely inequalities arising from cliques, odd holes, antiwebs and wheels of the associated intersection graph. The dimension of the OLS polytope is established and it is proved that certain valid inequalities are facet-inducing. This analysis reveals also a new family of facet-defining inequalities for the polytope associated with the Latin square problem. Separation algorithms of the lowest complexity are presented for particular families of valid inequalities. We illustrate a method for reducing problem's symmetry, which extends previously known results. This allows us to devise an alternative proof for the non-existence of an OLS structure for n = 6, based solely on Linear Programming. Moreover, we present a more general Branch & Cut algorithm for the OLS problem. The algorithm exploits problem structure via integer preprocessing and a specialised branching mechanism. It also incorporates families of strong valid inequalities. Computational analysis is conducted in order to illustrate the significant improvements over simple Branch & Bound. Next, the Constraint Programming (CP) paradigm is examined. Important aspects of designing an efficient CP solver, such as branching strategies and constraint propagation procedures, are evaluated by comprehensive, problem-specific, experiments. The CP algorithms lead to computationally favourable results. In particular, the infeasible case of n = 6, which requires enumerating the entire solution space, is solved in a few seconds. A broader aim of our research is to successfully integrate IP and CP. Hence, we present ideas concerning the unification of IP and CP methods in the form of hybrid algorithms. Two such algorithms are presented and their behaviour is analysed via experimentation. The main finding is that hybrid algorithms are clearly more efficient, as problem size grows, and exhibit a more robust performance than traditional IP and CP algorithms. A hybrid algorithm is also designed for the problem of finding triples of Mutually Orthogonal Latin Squares (MOLS). Given that the OLS problem is a special form of an assignment problem, the last part of the thesis considers multidimensional assignment problems. It introduces a model encompassing all assignment structures, including the case of MOLS. A necessary condition for the existence of an assignment structure is revealed. Relations among assignment problems are also examined, leading to a proposed hierarchy. Further, the polyhedral analysis presented unifies and generalises previous results

Examining recombination and intra-genomic conflict dynamics in the evolution of anti-microbial resistant bacteria

The spread of antimicrobial resistance (AMR) among pathogenic bacterial species threatens to undercut much of the progress made in treating infectious diseases. AMR genes can disseminate between and within populations via horizontal gene transfer (HGT). Selfish mobile genetic elements (MGEs) can encode resistance and spread between host cells. Homologous recombination can alter the core genes of pathogens with resistant donors via HGT too. MGEs may be cured from host genomes through transformation. Hence, MGEs may be able to avoid deletion by disrupting transformation. This work aims to understand how the dynamics of these processes affect the epidemiology of AMR pathogens. To understand these dynamics, I co-developed a new version of the popular recombination detection tool Gubbins. Through simulation studies, I find this new version to be both accurate in reconstructing the relationships between isolates, and efficient in terms of its use of computational resources. I then apply Gubbins to both AMR lineages and species-wide datasets of the pathogen Streptococcus pneumoniae. I find that recombination frequently occurs around core genes involved in both drug resistance and the host immune response. Additionally, an MGE was able to successfully spread within a population by disrupting the transformation machinery, preventing its loss from the host. Finally, I investigate two recent examples of MGEs disrupting transformation in the gram-negative species Acinetobacter baumannii and Legionella pneumophila. I find that while these insertions may decrease the efficiency of transformations within cells, the observed recombination rates largely reflect the selection pressures on isolates. With MGEs only partially able to inhibit these observable transformation events. These results show how selection pressures from clinical interventions shape pathogen genomes through diverse, often interspecies, recombination events. The spread of MGEs can also be favoured by both these selection pressures, and their ability to disrupt host cell machinery.Open Acces