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With the drastic increase in the demand for air travel, taxiway aircraft traffic scheduling is becoming increasingly important in managing air traffic. In order to reduce traffic congestion on taxiways, this thesis proposes a tool for air traffic controllers to use in decision making: a taxiway air traffic model developed using Mixed Integer Programming (MIP) that can be applied to a rolling time horizon. The objective of this model is to minimize the total taxi time, and the output is a schedule and route for each aircraft. This MIP model assumes that only the origin and destination of each aircraft is fixed; due to some uncertain factors in the air arrival and departure process, it allows for the departure time and arrival time to vary within a certain time window. This MIP model features aircraft type, and also incorporates runway crossings and runway separations. The model is programmed using C++ and Solved in CPLEX 12.1. Runways 26R and 26L of George Bush International Airport are used to find solutions. The author presents a rolling horizon method by dividing the large scheduling issue into smaller time interval problems according to the scheduled times of departure or arrival. A bound is also proposed based on the discretized time interval problems. By using partial data from George Bush International Airport (IAH), solutions are obtained. The results are compared with the bound and show fairly high optimality. Compared with the previous research, this thesis presents a model with more flexibility by considering different operations. By using the rolling horizon method, the problem is broken into smaller units that can be solved efficiently without losing much optimality.

Various kinds of optimization problems involve nonlinear functions of binary variables that exhibit a property of diminishing marginal returns. Such a property is known as submodularity. Vast amount of work has been devoted to the problem of submodular optimization. In this thesis, we exploit structural information for several classes of submodular optimization problems. We strive for polynomial time algorithms with improved approximation ratio and strong mixed-integer linear formulations of mixed-integer non-linear programs where the epigraph and hypograph of submodular functions of a specific form appear as a substructure together with other side constraints. In Chapter 2, we develop approximation algorithms for the expected utility knapsack problem. We use the sample average approximation framework to approximate the stochastic problem as a deterministic knapsack-constrained submodular maximization problem, and then use an approximation algorithm to solve the deterministic counterpart. We show that a polynomial number of samples are enough for a deterministic approximation that is close in relative error. Then, exploiting the strict monotonicity of typical utility functions, we present an algorithm that maximizes an increasing submodular function over a knapsack constraint with approximation ratio better than the classical $(1-1/e)$ ratio. In Chapter 3, we present polyhedral results for the expected utility knapsack problem. We study a mixed-integer nonlinear set that is the hypograph of $f(a'x)$ together together with a knapsack constraint. We propose a family of inequalities for the convex hull of the nonlinear set by exploiting both the structure of the submodular function $f(a'x)$ and the knapsack constraint. Effectiveness of the proposed inequalities is shown by computational experiments on expected utility maximization problem with budget constraint using a branch-and-cut framework. In Chapter 4, we study a mixed-integer nonlinear set that is the epigraph of $f(a'x)$ together with a cardinality constraint. This mixed-integer nonlinear set arises as a substructure in various constrained submodular minimization problems. We develop a strong linear formulation of the convex hull of the nonlinear set by exploiting both the submodularity of $f(a'x)$ and the cardinality constraint. We provide a full description of the convex hull of the nonlinear set when the vector a has identical components. We also develop a family of facet-defining inequalities when the vector a has nonidentical components. We demonstrate the effectiveness of the proposed inequalities by solving mean-risk knapsack problems using a branch-and-cut framework.

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We introduce a new pure integer rounding heuristic, ZI Round, and compare this heuristic to recent extremely fast pure integer rounding heuristic Simple Rounding. Simple Rounding was introduced in the non-commercial code SCIP. ZI Round attempts to round each fractional variable while using row slacks to maintain primal feasibility. We use the MIPLIB 2003 library for the test set. The average time in our run per instance for both Simple Rounding and ZI Round was 0.8 milliseconds, but ZI Round found more feasible solutions with a the same or better objective value. Also the average time to solve the lp relaxation per instance was 2.2 seconds, so these two rounding heuristics are several magnitudes faster than other heuristics which must use the lp solver, including diving heuristics.We also show that ZI Round performs well on a set covering class and a railway crew scheduling class.

The design of a chilled water plant involves selecting the size and type of chillers to be employed and determining the operating strategy. The types may include both gas engine and electric motor driven chillers. The issues that have to be considered in the selection problem are to incorporate external and internal factors into the decision making. External factors may include the utility rate schedules, the cooling load profile, and the outdoor temperature profile. Internal factors may include the chiller performance characteristics, initial and maintenance costs, and the chiller(s) operating strategy. A mathematical model representing the chilled water plant design problem is developed. The problem is approached as a mixed integer linear programming problem where non-linear chiller performance curves are transformed into linear constraints through the use of integer variables. The optimization task is to select the best cooling plant configuration and operating strategy to minimize life cycle cost. A solution procedure is developed which decomposes the optimization problem to reduce extensive computation time. Two case studies are provided to investigate the implementation of the mathematical model. / Master of Science

This dissertation explores mathematical programming optimization models and algorithms for routing and scheduling ships in a maritime transportation system. Literature surveyed on seaborne transportation systems indicates that there is a scarcity of research on ship routing and scheduling problems. The complexity and the overwhelming size of a typical ship routing and scheduling problem are the primary reasons that have resulted in the scarcity of research in this area. The principal thrust of this research effort is focused at the Kuwait Petroleum Corporation (KPC) Problem. This problem is of great economic significance to the State of Kuwait, whose economy has been traditionally dominated to a large extent by the oil sector. Any enhancement in the existing ad-hoc scheduling procedure has the potential for significant savings. A mixed-integer programming model for the KPC problem is constructed in this dissertation. The resulting mathematical formulation is rather complex to solve due to (1) the overwhelming problem size for a typical demand contract scenario, (2) the integrality conditions, and (3) the structural diversity in the constraints. Accordingly, attempting to solve this formulation for a typical demand contract scenario without resorting to any aggregation or partitioning schemes is theoretically complex and computationally intractable. Motivated by the complexity of the above model, an aggregate model that retains the principal features of the KPC problem is formulated. This model is computationally far more tractable than the initial model, and consequently, it is utilized to construct a good quality heuristic solution for the KPC problem. The initial formulation is solved using CPLEX 4.0 mixed integer programming capabilities for a number of relatively small-sized test cases, and pertinent results and computational difficulties are reported. The aggregate formulation is solved using CPLEX 4.0 MIP in concert with specialized rolling horizon solution algorithms and related results are reported. The rolling horizon solution algorithms enabled us to handle practical sized problems that could not be handled by directly solving the aggregate problem. The performance of the rolling horizon algorithms may be enhanced by increasing the physical memory, and consequently, better solutions can be extracted. The potential saving and usefulness of this model in negotiation and planning purposes strongly justifies the acquisition of more computing power to tackle practical sized test problems. An ad-hoc scheduling procedure that is intended to simulate the current KPC scheduling practice is presented in this dissertation. It is shown that results obtained via the proposed rolling horizon algorithms are at least as good, and often substantially better than, results obtained via this ad-hoc procedure. / Ph. D.

In this thesis I focus on cutting planes for large Mixed Integer Programming (MIP) problems. More specifically, I focus on two independent cutting planes studies. The first of these deals with cutting planes for the Traveling Salesman Problem (TSP), and the second with cutting planes for general MIPs. In the first study I introduce a new class of cutting planes which I call the Generalized Domino Parity (GDP) inequalities. My main achievements with regard to these are: (1) I show that these are valid for the TSP and for the graphical TSP. (2) I show that they generalize most well-known TSP inequalities (including combs, domino-parity constraints, clique-trees, bipartitions, paths and stars). (3) I show that a sub-class of these (which contains all clique-tree inequalities w/ a fixed number of handles) can be separated in polynomial time, on planar graphs. My second study can be subdivided in two parts. In the first of these I study the Mixed Integer Knapsack Problem (MIKP) and develop a branch-and-bound based algorithm for solving it. The novelty of the approach is that it exploits the notion of "dominance" in order to effectively prune solutions in the branch-and-bound tree. In the second part, I develop a Mixed Integer Rounding (MIR) cut separation heuristic, and embed the MIKP solver in a column generation algorithm in order to assess the performance of said heuristic. The goal of this study is to understand why no other class of inequalities derived from single-row systems has been able to outperform the MIR. Computational results are presented.

Single-row mixed-integer programming (MIP) problems have been studied thoroughly under many different perspectives over the years. While not many practical applications can be modeled as a single-row MIP, their importance lies in the fact that they are simple, natural and very useful relaxations of generic MIPs. This thesis will focus on such MIPs and present theoretical and computational advances in their study. Chapter 1 presents an introduction to single-row MIPs, a historical overview of results and a motivation of why we will be studying them. It will also contain a brief review of the topics studied in this thesis as well as our contribution to them. In Chapter 2, we introduce a generalization of a very important structured single-row MIP: Gomory's master cyclic group polyhedron (MCGP). We will show a structural result for the generalization, characterizing all facet-defining inequalities for it. This structural result allows us to develop relationships with MCGP, extend it to the mixed-integer case and show how it can be used to generate new valid inequalities for MIPs. Chapter 3 presents research on an algorithmic view on how to maximally lift continuous and integer variables. Connections to tilting and fractional programming will also be presented. Even though lifting is not particular to single-row MIPs, we envision that the general use of the techniques presented should be on easily solvable MIP relaxations such as single-row MIPs. In fact, Chapter 4 uses the lifting algorithm presented. Chapter 4 presents an extension to the work of Goycoolea (2006) which attempts to evaluate the effectiveness of Mixed Integer Rounding (MIR) and Gomory mixed-integer (GMI) inequalities. By extending his work, natural benchmarks arise, against which any class of cuts derived from single-row MIPs can be compared. Finally, Chapter 5 is dedicated to dealing with an important computational problem when developing any computer code for solving MIPs, namely the problem of numerical accuracy. This problem arises due to the intrinsic arithmetic errors in computer floating-point arithmetic. We propose a simple approach to deal with this issue in the context of generating MIR/GMI inequalities.

The factors limiting the performance of computer software periodically undergo sudden shifts, resulting from technological progress, and these shifts can have profound implications for the design of high performance codes. At the present time, the speed with which hardware can execute a single stream of instructions has reached a plateau. It is now the number of instruction streams that may be executed concurrently which underpins estimates of compute power, and with this change, a critical limitation on the performance of software has come to be the degree to which it can be parallelised. The research in this thesis is concerned with the means by which codes for linear programming may be adapted to this new hardware. For the most part, it is codes implementing the simplex method which will be discussed, though these have typically lower performance for single solves than those implementing interior point methods. However, the ability of the simplex method to rapidly re-solve a problem makes it at present indispensable as a subroutine for mixed integer programming. The long history of the simplex method as a practical technique, with applications in many industries and government, has led to such codes reaching a great level of sophistication. It would be unexpected in a research project such as this one to match the performance of top commercial codes with many years of development behind them. The simplex codes described in this thesis are, however, able to solve real problems of small to moderate size, rather than being confined to random or otherwise artificially generated instances. The remainder of this thesis is structured as follows. The rest of this chapter gives a brief overview of the essential elements of modern parallel hardware and of the linear programming problem. Both the simplex method and interior point methods are discussed, along with some of the key algorithmic enhancements required for such systems to solve real-world problems. Some background on the parallelisation of both types of code is given. The next chapter describes two standard simplex codes designed to exploit the current generation of hardware. i6 is a parallel standard simplex solver capable of being applied to a range of real problems, and showing exceptional performance for dense, square programs. i8 is also a parallel, standard simplex solver, but now implemented for graphics processing units (GPUs).