Exact solution approaches for the discrete α\alpha-neighbor pp-center problem

The discrete $\alpha$-neighbor $p$-center problem (d-$\alpha$-$p$CP) is an emerging variant of the classical $p$-center problem which recently got attention in literature. In this problem, we are given a discrete set of points and we need to locate $p$ facilities on these points in such a way that the maximum distance between each point where no facility is located and its $\alpha$-closest facility is minimized. The only existing algorithms in literature for solving the d-$\alpha$-$p$CP are approximation algorithms and two recently proposed heuristics. In this work, we present two integer programming formulations for the d-$\alpha$-$p$CP, together with lifting of inequalities, valid inequalities, inequalities that do not change the optimal objective function value and variable fixing procedures. We provide theoretical results on the strength of the formulations and convergence results for the lower bounds obtained after applying the lifting procedures or the variable fixing procedures in an iterative fashion. Based on our formulations and theoretical results, we develop branch-and-cut (B&C) algorithms, which are further enhanced with a starting heuristic and a primal heuristic. We evaluate the effectiveness of our B&C algorithms using instances from literature. Our algorithms are able to solve 116 out of 194 instances from literature to proven optimality, with a runtime of under a minute for most of them. By doing so, we also provide improved solution values for 116 instances

Improving problem reduction for 0-1 Multidimensional Knapsack Problems with valid inequalities

© 2016 Elsevier Ltd. All rights reserved. This paper investigates the problem reduction heuristic for the Multidimensional Knapsack Problem (MKP). The MKP formulation is first strengthened by the Global Lifted Cover Inequalities (GLCI) using the cutting plane approach. The dynamic core problem heuristic is then applied to find good solutions. The GLCI is described in the general lifting framework and several variants are introduced. A Two-level Core problem Heuristic is also proposed to tackle large instances. Computational experiments were carried out on classic benchmark problems to demonstrate the effectiveness of this new method

Approximation Algorithms for Distributionally Robust Stochastic Optimization with Black-Box Distributions

Two-stage stochastic optimization is a framework for modeling uncertainty, where we have a probability distribution over possible realizations of the data, called scenarios, and decisions are taken in two stages: we make first-stage decisions knowing only the underlying distribution and before a scenario is realized, and may take additional second-stage recourse actions after a scenario is realized. The goal is typically to minimize the total expected cost. A criticism of this model is that the underlying probability distribution is itself often imprecise! To address this, a versatile approach that has been proposed is the {\em distributionally robust 2-stage model}: given a collection of probability distributions, our goal now is to minimize the maximum expected total cost with respect to a distribution in this collection. We provide a framework for designing approximation algorithms in such settings when the collection is a ball around a central distribution and the central distribution is accessed {\em only via a sampling black box}. We first show that one can utilize the {\em sample average approximation} (SAA) method to reduce the problem to the case where the central distribution has {\em polynomial-size} support. We then show how to approximately solve a fractional relaxation of the SAA (i.e., polynomial-scenario central-distribution) problem. By complementing this via LP-rounding algorithms that provide {\em local} (i.e., per-scenario) approximation guarantees, we obtain the {\em first} approximation algorithms for the distributionally robust versions of a variety of discrete-optimization problems including set cover, vertex cover, edge cover, facility location, and Steiner tree, with guarantees that are, except for set cover, within $O(1)$-factors of the guarantees known for the deterministic version of the problem

Large-scale mixed integer optimization approaches for scheduling airline operations under irregularity

Perhaps no single industry has benefited more from advancements in computation, analytics, and optimization than the airline industry. Operations Research (OR) is now ubiquitous in the way airlines develop their schedules, price their itineraries, manage their fleet, route their aircraft, and schedule their crew. These problems, among others, are well-known to industry practitioners and academics alike and arise within the context of the planning environment which takes place well in advance of the date of departure. One salient feature of the planning environment is that decisions are made in a frictionless environment that do not consider perturbations to an existing schedule. Airline operations are rife with disruptions caused by factors such as convective weather, aircraft failure, air traffic control restrictions, network effects, among other irregularities. Substantially less work in the OR community has been examined within the context of the real-time operational environment. While problems in the planning and operational environments are similar from a mathematical perspective, the complexity of the operational environment is exacerbated by two factors. First, decisions need to be made in as close to real-time as possible. Unlike the planning phase, decision-makers do not have hours of time to return a decision. Secondly, there are a host of operational considerations in which complex rules mandated by regulatory agencies like the Federal Administration Association (FAA), airline requirements, or union rules. Such restrictions often make finding even a feasible set of re-scheduling decisions an arduous task, let alone the global optimum. The goals and objectives of this thesis are found in Chapter 1. Chapter 2 provides an overview airline operations and the current practices of disruption management employed at most airlines. Both the causes and the costs associated with irregular operations are surveyed. The role of airline Operations Control Center (OCC) is discussed in which serves as the real-time decision making environment that is important to understand for the body of this work. Chapter 3 introduces an optimization-based approach to solve the Airline Integrated Recovery (AIR) problem that simultaneously solves re-scheduling decisions for the operating schedule, aircraft routings, crew assignments, and passenger itineraries. The methodology is validated by using real-world industrial data from a U.S. hub-and-spoke regional carrier and we show how the incumbent approach can dominate the incumbent sequential approach in way that is amenable to the operational constraints imposed by a decision-making environment. Computational effort is central to the efficacy of any algorithm present in a real-time decision making environment such as an OCC. The latter two chapters illustrate various methods that are shown to expedite more traditional large-scale optimization methods that are applicable a wide family of optimization problems, including the AIR problem. Chapter 4 shows how delayed constraint generation and column generation may be used simultaneously through use of alternate polyhedra that verify whether or not a given cut that has been generated from a subset of variables remains globally valid. While Benders' decomposition is a well-known algorithm to solve problems exhibiting a block structure, one possible drawback is slow convergence. Expediting Benders' decomposition has been explored in the literature through model reformulation, improving bounds, and cut selection strategies, but little has been studied how to strengthen a standard cut. Chapter 5 examines four methods for the convergence may be accelerated through an affine transformation into the interior of the feasible set, generating a split cut induced by a standard Benders' inequality, sequential lifting, and superadditive lifting over a relaxation of a multi-row system. It is shown that the first two methods yield the most promising results within the context of an AIR model.PhDCommittee Co-Chair: Clarke, John-Paul; Committee Co-Chair: Johnson, Ellis; Committee Member: Ahmed, Shabbir; Committee Member: Clarke, Michael; Committee Member: Nemhauser, Georg