J. Cole Smith

A Multiple-Criterion Model for Machine Scheduling

We consider a scheduling problem involving a single processor being utilized by two or more customers. Traditionally, such scenarios are modeled by assuming that each customer has the same criterion. In practice, this assumption may not hold. Instead of using a single criterion, we examine the implications of minimizing an aggregate scheduling objective function in which jobs belonging to different customers are evaluated based on their individual criteria. We examine three basic scheduling criteria: minimizing makespan, minimizing maximum lateness, and minimizing total weighted completion time. Although determining a minimum-cost schedule according to any one of these criteria is polynomially solvable, we demonstrate that when minimizing a mix of these criteria, the problem becomes NP-hard.

Improving Discrete Model Representations via Symmetry Considerations

In this paper, we focus on a useful modeling concept that is frequently ignored while formu lating discrete optimization problems. Very often, there exists a natural symmetry inherent in the problem itself that, if propagated to the model, can hopelessly mire a branch-and-bound solver by burdening it to explore and eliminate such alternative symmetric solutions. We discuss three applications where such a symmetry arises: a telecommunications network design problem, a noise pollution problem, and a machine procurement and operation problem. For each case, we identify the indistinguishable objects in the model that create the problem symmetry and show how imposing certain decision hierarchies within the model significantly enhances its solvability, while using a popular modern-day commercial branch-and-cut software (CPLEX 6.5).

Algorithms for discrete and continuous multicommodity flow network interdiction problems

We consider a network interdiction problem on a multicommodity flow network, in which an attacker disables a set of network arcs in order to minimize the maximum profit that can be obtained from shipping commodities across the network. The attacker is assumed to have some budget for destroying (or “interdicting”) arcs, and each arc is associated with a positive interdiction expense. In this paper, we examine problems in which interdiction must be discrete (i.e., each arc must either be left alone or completely destroyed), and in which interdiction can be continuous (the capacities of arcs may be partially reduced). For the discrete problem, we describe a linearized model for optimizing network interdiction that is similar to previous studies in the field, and compare it to a penalty model that does not require linearization constraints. For the continuous case, we prescribe an optimal partitioning algorithm along with a heuristic procedure for estimating the optimal objective function value. We demonstrate on a set of randomly generated test data that our penalty model for the discrete interdiction problem significantly reduces computational time when compared to that consumed by the linearization model.

Survivable network design under optimal and heuristic interdiction scenarios

We examine the problem of building or fortifying a network to defend against enemy attacks in various scenarios. In particular, we examine the case in which an enemy can destroy any portion of any arc that a designer constructs on the network, subject to some interdiction budget. This problem takes the form of a three-level, two-player game, in which the designer acts first to construct a network and transmit an initial set of flows through the network. The enemy acts next to destroy a set of constructed arcs in the designer’s network, and the designer acts last to transmit a final set of flows in the network. Most studies of this nature assume that the enemy will act optimally; however, in real-world scenarios one cannot necessarily assume rationality on the part of the enemy. Hence, we prescribe optimal network design algorithms for three different profiles of enemy action: an enemy destroying arcs based on capacities, based on initial flows, or acting optimally to minimize our maximum profits obtained from transmitting flows.

Exact interdiction models and algorithms for disconnecting networks via node deletions

Abstract This paper analyzes the problem of maximizing the disconnectivity of undirected graphs by deleting a subset of their nodes. We consider three metrics that measure the connectivity of a graph: the number of connected components (which we attempt to maximize), the largest component size (which we attempt to minimize), and the minimum cost required to reconnect the graph after the nodes are deleted (which we attempt to maximize). We formulate each problem as a mixed-integer program, and then study valid inequalities for the first two connectivity objectives by examining intermediate dynamic programming solutions to k -hole subgraphs. We randomly generate a set of test instances, on which we demonstrate the computational efficacy of our approaches.

Polynomial-time algorithms for solving a class of critical node problems on trees and series-parallel graphs

We examine variants of the critical node problem on specially structured graphs, which aim to identify a subset of nodes whose removal will maximally disconnect the graph. These problems lie at the intersection of network interdiction and graph theory research and are relevant to several practical optimization problems. The two different connectivity metrics that we consider regard the number of maximal connected components (which we attempt to maximize) and the largest component size (which we attempt to minimize). We develop optimal polynomial-time dynamic programming algorithms for solving these problems on tree structures and on series-parallel graphs, corresponding to each graph-connectivity metric. We also extend our discussion by considering node deletion costs, node weights, and solving the problems on generalizations of tree structures. Finally, we demonstrate the computational efficacy of our approach on randomly generated graph instances.

Optimal Multileaf Collimator Leaf Sequencing in IMRT Treatment Planning

We consider a problem dealing with the efficient delivery of intensity modulated radiation therapy (IMRT) to individual patients. IMRT treatment planning is usually performed in three phases. The first phase determines a set of beam angles through which radiation is delivered, followed by a second phase that determines an optimal radiation intensity profile (or fluence map). This intensity profile is selected to ensure that certain targets receive a required amount of dose while functional organs are spared. To deliver these intensity profiles to the patient, a third phase must decompose them into a collection of apertures and corresponding intensities. In this paper, we investigate this last problem. Formally, an intensity profile is represented as a nonnegative integer matrix; an aperture is represented as a binary matrix whose ones appear consecutively in each row. A feasible decomposition is one in which the original desired intensity profile is equal to the sum of a number of feasible binary matrices multiplied by corresponding intensity values. To most efficiently treat a patient, we wish to minimize a measure of total treatment time, which is given as a weighted sum of the number of apertures and the sum of the aperture intensities used in the decomposition. We develop the first exact algorithm capable of solving real-world problem instances to optimality within practicable computational limits, using a combination of integer programming decomposition and combinatorial search techniques. We demonstrate the efficacy of our approach on a set of 25 test instances derived from actual clinical data and on 100 randomly generated instances.

An improved linearization strategy for zero-one quadratic programming problems

We present a new linearized model for the zero-one quadratic programming problem, whose size is linear in terms of the number of variables in the original nonlinear problem. Our derivation yields three alternative reformulations, each varying in model size and tightness. We show that our models are at least as tight as the one recently proposed in [7], and examine the theoretical relationship of our models to a standard linearization of the zero-one quadratic programming problem. Finally, we demonstrate the efficacy of solving each of these models on a set of randomly generated test instances.

Algorithms for Network Interdiction and Fortification Games

This chapter explores models and algorithms applied to a class of Stackelberg games on networks. In these network interdiction games, a network exists over which an operator wishes to execute some function, such as finding a shortest path, shipping a maximum flow, or transmitting a minimum cost multicommodity flow. The role of the interdictor is to compromise certain network elements before the operator acts, by (for instance) increasing the cost of flow or reducing capacity on an arc. We begin by reviewing the field of network interdiction and its related theoretical and mathematical foundations. We then discuss recent applications of stochastic models, valid inequalities, continuous bilinear programming techniques, and asymmetric analysis to network interdiction problems. Next, note that interdiction problems can be extended to a three-stage problem in which the operator fortifies the network (by increasing capacities, reducing flow costs, or defending network elements from the interdictor) before the interdictor takes action. We devote one section to ongoing research in this area and conclude by discussing areas for future research.

An Airspace Planning Model for Selecting Flight-plans Under Workload, Safety, and Equity Considerations

In this paper, we present an airspace planning model (APM) that has been developed for use in both tactical and strategic planning contexts under various airspace scenarios. Given a set of flights for a particular time horizon, along with (possibly several) alternative flight-plans for each flight that are based on delays and diversions, due to special-use airspace (SUA) restrictions prompted by launches at spaceports or adverse weather conditions, this model prescribes a set of flight-plans to be implemented. The model formulation seeks to minimize and delay fuel-cost-based objective function, subject to the constraints that each flight is assigned one of the designated flight-plans, and that the resulting set of flight-plans satisfies certain specified workload, safety, and equity criteria. These requirements ensure that the workload for air-traffic controllers in each sector is held under a permissible limit, that any potential conflicts are routinely resolvable, and that the various airlines involved derive equitable levels of benefits from the overall implemented schedule. To solve the resulting 0--1 mixed-integer programming problem more effectively using commercial software (e.g., CPLEX-MIP), we explore the use of reformulation techniques designed to more closely approximate the convex hull of feasible solutions to the problem. We also prescribe a polynomial-time heuristic procedure that is demonstrated to provide solutions to the problem within 0.01% of optimality. Computational results are reported on several scenarios based on actual flight data obtained from the Federal Aviation Administration (FAA) to demonstrate the efficacy of the proposed approach for air-traffic management (ATM) purposes.